Volume 1 July–December 1995 ISSN 1083-0464

SOUTHWEST JOURNAL OF

PURE AND APPLIED MATHEMATICS

EDITOR IN CHIEF: IOANNIS ARGYROS MANAGING EDITOR: MOHAMMAD TABATABAI

TECHNICAL EDITOR: GARY HUCKABAY

EDITORIAL BOARD

G. ANASTASSIOU  I. ARGYROS
D. CHEN  P. VAN FLEET
D. JANKOVIC  T. McKELLIPS
C. RIECKE  D. RUCH
F. SZIDAROVSZKY  M. TABATABAI

CAMERON UNIVERSITY DEPARTMENT OF MATHEMATICAL SCIENCES LAWTON, OK 73505, U.S.A Southwest Journal of Pure and Applied Mathematics

Editor in Chief: Ioannis Argyros Managing Editor: Mohammad Tabatabai Technical Editor: Gary Huckabay

Editorial Board

G. Anastassiou  anastasg@hermes.msci.memst.edu
I. Argyros  ioannisa@cua.cameron.edu
D. Chen  dchen@uafsysb.uark.edu
P. Van Fleet  mth pvf@shsu.edu
D. Jankovic  draganj@cuok.cameron.edu
T. McKellips  terral@dragon.cameron.edu
C. Riecke  rick@penguin.cameron.edu
D. Ruch  mth dkr@shsu.edu
F. Szidarovszky  szidar@sie.arizona.edu
M. Tabatabai  mohammad@tiger.cameron.edu
Editorial Office  Publication Program
SWJPAM  2 issues per year
Cameron University  July, December

Department of Mathematical Sciences Lawton, OK 73505, U.S.A.

Information for Authors

Scope. SWJPAM is an electronic journal devoted to all aspects of Pure and Applied mathematics, and related topics. Authoritative expository and survey articles on subjects of special interest are also welcomed.

Since SWJPAM aims to serve not only the specialists, but also the user of mathematics (promoting, in such a way, a strong interaction between them), particular emphasis is placed on applications. Thus, SWJPAM publishes papers which, although not directly contributing to mathematics, contain, however, significant results (from any field) obtained by making an essential and well stressed use of the methods and results of mathematics.

Submission of Manuscripts. Since SWJPAM is a completely electronic journal, all manuscripts must be submitted in either AMSTEX, LaTEX, TEX, or post script form. Papers for publication should be submitted to ftp://rattler.cameron.edu/SWJPAM incoming by anonymous ftp. The articles must be placed in the ‘incoming’ directory. The submission should include the title of the paper, the author’s name and affiliation, and a short abstract which summarizes the paper; a list of keywords and a classification according to the Mathematical Reviews’ Subject Classification (1991). Authors are informed by e-mail of referee communication.

Key Words. Key words should be comprehensive and subject specific. It is not necessary to list the subject area of the journal’s coverage as a key word. Four key words should be sufficient to cover the major subjects off a given paper, although more can always be supplied if the author deems it necessary.

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University (continued on inside back cover)

SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS VOLUME 1, JULY–DECEMBER 1995

CONTENTS

I. Argyros  Perturbed Newton-Like Methods and Nondifferentiable Operator Equations
on Banach Spaces with a Convergence Structure . . . . . . . . . . 1–12
G. Anastassiou Differentiated Shift-Invariant Multivariate Integral Operators . . . . . . 13–16
D. Pantić  Stochastic Calculus and Initial Value Analysis on the One Dimensional
Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–26
R. Forslund  A Logical Alternative to the Existing Positional Number System …. 27–29
I. Argyros  Results on Controlling the Residuals of Perturbed Newton-Like Methods
on Banach Spaces with a Convergence Structure . . . . . . . . . . 30–36
M. Tabatabai A Probabilistic Approach to the Estimation of the Distribution of the
Target Location Point . . . . . . . . . . . . . . . . . . . . . . 37–46

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. 01, 1995, pp. 1–12. Submitted: March 8, 1995. Published: June 20, 1995.

PERTURBED NEWTON-LIKE METHODS AND

NONDIFFERENTIABLE OPERATOR EQUATIONS ON

BANACH SPACES WITH A CONVERGENCE STRUCTURE

Ioannis K. Argyros

Abstract. In this study, we use perturbed Newton-like methods to find solutions of nonlinear, nondifferentiable operator equations on Banach spaces with a convergence structure. This technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semi-local results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, our results are used to solve integral equations that cannot be solved with existing methods.

(1991) A.M.S. (MOS) Subject Classification Codes. 65J15, 65B05, 47D15, 47H17, 49D15.

Key Words and Phrases. Banach space, perturbed Newton-like methods, nondifferentiable operator.

I. INTRODUCTION

In this study, we are concerned with approximating a solution x^{∗} of the nonlinear operator equation

F (x) + Q(x) = 0,  (1)

where F is a Fréchet-differentiable operator defined on a convex subset D of a Banach space X with values in X, and Q is a nondifferentiable nonlinear operator with the same domain and values in X.

We introduce the perturbed Newton-like method

x_{n+1} = x_{n} + A(x_{n})^{∗}[−(F (x_{n}) + Q(x_{n}))] − z_{n}, x_{0} = 0 (n ≥ 0)  (2)

to approximate a solution x^{∗} of equation (1). Here A(x_{n})^{∗} (n ≥ 0) denotes a linear operator which is an approximation for F′(x_{n})^{−1} (n ≥ 0).

The importance of studying perturbed Newton-like methods comes from the fact that many commonly used variants of Newton’s method can be considered procedures of this type. Indeed, approximation (2) characterizes any iterative process in which the corrections are taken as approximate solutions of the Newton equations. In particular for A(x) = F′(x), Q(x) = 0 and z_{n} = 0 (n ≥ 0) we obtain the ordinary Newton-Kantorovich method [4].

The notion of a Banach space with a convergence structure was used in the elegant paper in [5] (see also [1], [2], [3], [8]) to solve equation (1), when A(x) = F′(x), Q(x) = 0 for all x ∈ D and z_{n} = 0 for all n ≥ 0. However, there are many interesting real life applications already in the literature, where

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Department of Mathematics, Cameron University, Lawton, OK 73505, U.S.A.

E-mail Address: ioannisa@cua.cameron.edu

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University

equation (1) contains a nondifferentiable term. See for example the applications at the end of this study. Moreover, since the Newton-like iterates can rarely be computed exactly, suitable choices of the points z_{n} will “correct” the sequence x_{n} and force it to converge to a solution x^{∗} of equation (1). The case when Q(x) = 0 for all x ∈ D and F′(x_{n})^{∗} = F′(x_{n})^{−1} (n ≥ 0) has already been considered but on a Banach space without generalized structure [3], [4], [7], [8], [9].

As in [6], we provide semi-local results of Kantorovich-type and global results based on monotonicity considerations from the same general theorem. Moreover, we show that our results can be reduced to the one obtained in [6], when Q(x) = 0 for all x ∈ D and z_{n} = 0 (n ≥ 0), and furthermore to the ones obtained in [1], [2], [3], [5], [6], [7], [8] by further relaxing the requirements on X.

Finally, our results apply to solve a nonlinear integral equation involving a nondifferentiable term that cannot be solved with existing methods.

II. PRELIMINARIES

We will need the definitions:

Definition 1. The triple (X, V, E) is a Banach space with a convergence structure if (C_{1}) (X, ∥ · ∥) is a real Banach space; (C_{2}) (V, C, ∥ ∥_{v}) is a real Banach space which is partially ordered by the closed convex cone C; the norm

∥ ∥_{v} is assumed to be monotone on C; (C_{3}) E is a closed convex cone in X × V satisfying {0} × C ⊆ E ⊆ X × C; (C_{4}) the operator | | : D → C is well defined:

|x| = inf{q ∈ C|(x, q) ∈ E}

for

x ∈ D = {x ∈ X|∃q ∈ C : (x, q) ∈ E};

and (C_{5}) for all x ∈ D ∥x∥ ≤ ∥|x|∥_{v}.

The set

U(a) = {x ∈ X|(x, a) ∈ E}

defines a sort of generalized neighborhood of zero.

Definition 2. An operator L ∈ C^{1}(V_{1} → V) defined on an open subset V_{1} of an ordered Banach space V is order convex on [a, b] ⊆ V_{1} if

c, d ∈ [a, b], c ≤ d ⇒ L′(d) − L′(c) ∈ L_{+}(V),

where for m ≥ 0

L_{+}(V^{m}) = {L ∈ L(V^{m})|0 ≤ x_{i} ⇒ 0 ≤ L(x_{1}, x_{2}, …, x_{m})}

and L(V^{m}) denotes the space of m-linear, symmetric, bounded operators on V.

Definition 3. The set of bounds for an operator H ∈ L(X^{m}) is defined to be

B(H) = {L ∈ L_{+}(V^{m})|(x_{i}, q_{i}) ∈ E ⇒ (H(x_{1}, …, x_{m}), L(q_{1}, …, q_{m})) ∈ E}

Definition 4. Let H ∈ L(X) and y ∈ X be given, then

H^{∗}(y) = z ⇔ z = T^{∞}(0) = lim_{n→∞} T^{n}(0),

T (x) = (I − H)(x) + y ⇔ z = ∑_{i=0}^{∞}(I − H)^{i}y,

if this limit exists.

We will also need the Lemmas [2], [3], [5], [6]:

Lemma 1. Let L ∈ L_{+}(V) and a, q ∈ C be given such that:

L(q) + a ≤ q and L^{n}(q) → 0 as n → ∞.  (3)

Then the operator

(I − L)^{∗} : [0, a] → [0, a]

is well defined and continuous.

The following is a generalization of Banach’s lemma [3], [5], [6].

Lemma 2. Let H ∈ L(X), L ∈ B(H), y ∈ D and q ∈ C be such that

L(q) + |y| ≤ q and L^{n}(q) → 0 as n → ∞.

Then the point x = (I − H)^{∗}(y) is well defined, x ∈ S and

|x| ≤ (I − L)^{∗}|y| ≤ q.

Moreover, the sequence

b_{n+1} = L(b_{n}) + |y|, b_{0} = 0

is well defined and

b_{n+1} ≤ q, lim_{n→∞} b_{n} = b = (I − L)^{∗}|y| ≤ q.

III. CONVERGENCE ANALYSIS

Let K, L, M, M_{1} be operators such that L ∈ C^{1}(V_{1} → V), K, M_{1} ∈ L_{+}(V), M ∈ C(V_{1} → V), and x_{n} be points in D. It is convenient for us to define the sequences c_{n}, d_{n}, a_{n}, b_{n} (n ≥ 0) by

d_{n+1} = (L + K + M + M_{1})(d_{n}) + L′(|x_{n}|)c_{n}, d_{0} = 0,  (4)
c_{n} = |x_{n+1} − x_{n}|,  (5)
a_{n} = (L + K + M + M_{1})^{n}(a) for some a ∈ C,  (6)
b_{n} = (L + K + M + M_{1})^{n}(0),  (7)

and the point b by

b = (L + K + M + M_{1})^{∞}(0).  (8)

We can now state and prove the main result of this section.

\underline{Theorem 1}. Let X be a Banach space with convergence structure (X, V, E) with V = (V, C, ∥ ∥_{v}), an operator F ∈ C^{1}(D → X) with D ⊆ X, an operator Q ∈ C(D → X), an operator A(x) ∈ L(X), operators L, L_{1} ∈ C^{1}(V_{1} → V) with V_{1} ⊆ V, operators K, M_{1}(V_{1} → V) with K, M_{1} ∈ L_{+}(V), M ∈ C(V_{1} → V), a point a ∈ C, and a null sequence {z_{n}} ∈ D such that the following conditions are satisfied:

(C_{6}) the inclusions U(a) ⊆ D and [0, a] ⊆ V_{1} are true; (C_{7}) L is order-convex on [0, a] and satisfies

K + L′(|x| + |y|) − L′(|x|) ∈ B(A(x) − F′(x + y))

for all x, y ∈ U(a) with |x| + |y| ≤ a; (C_{8}) L_{1} is order-convex on [0, a] and satisfies

L′_{1}(|x|) − L′_{1}(0) ∈ B(A(0) − A(x))

for all x, y ∈ U(a) with |x| + |y| ≤ a; (C_{9}) the estimate L′_{1} ≤ L′ is true; (C_{10}) M satisfies the conditions

0 ≤ (Q(x) − Q(x + y), M(|x| + |y|) − M(|x|) ∈ E

for all x, y ∈ U(a) with |x| + |y| ≤ a,

M(w_{1}) − M(w_{2}) ≤ M(w_{3}) − M(w_{4}) and M(w) ≥ 0

for all w, w_{1}, w_{2}, w_{3}, w_{4} ∈ [0, a] with w_{1} ≤ w_{3} and w_{2} ≤ w_{4}; (C_{11}) M_{1}, x_{n}, z_{n} satisfy the inequality

0 ≤ (A(x_{n})(z_{n}) − A(x_{n−1})(z_{n−1}), M_{1}(d_{n} − d_{n−1})) ∈ E for all n ≥ 1;

(C_{12}) L′_{1}(0) ∈ B(I − A(0)) and (−(F (0) + Q(0) + A(0)(z_{0})), L(0) + M(0) + M_{1}(0)) ∈ E; (C_{13}) (L + K + M + M_{1})(a) ≤ a with 0 ≤ L + K + M + M_{1};

and (C_{14}) (M + M_{1} + L′(a) + K)^{n}a → 0 as n → ∞.

Then

(i) The sequence (x_{n}, d_{n}) ∈ (X × V)^{N} is well defined, remains in E^{N}, is monotone and satisfies for all

n ≥ 0

d_{n} ≤ b

where d_{n}, b are given by (4) and (8) respectively and b is the smallest fixed point of L+K +M +M_{1} in [0, a].

(ii) Moreover, the iteration {x_{n}} n ≥ 0 generated by (2) converges to a solution x^{∗} ∈ U(b) of the equation

F (x) + Q(x) = 0, which is unique in U(a). (iii) Furthermore, the following estimates are true for all n ≥ 0:

b_{n} ≤ d_{n} ≤ b,  (9)
b_{n} ≤ a_{n},  (10)
|x_{n+1} − x_{n}| ≤ d_{n+1} − d_{n},  (11)
|x_{n} − x^{∗}| ≤ b − d_{n}  (12)

and

|x_{n} − x^{∗}| ≤ a_{n} − b_{n} for M_{1} = 0 and z_{n} = 0 (n ≥ 0)  (13)

where d_{n}, a_{n}, b_{n} and b are given by (4), (6), (7) and (8) respectively.

Proof. We first note that b replacing a also satisfies the conditions of the theorem. Using conditions (C_{9}) and (C_{10}) we obtain

|I − A(0)|(b) + | − (F (0) + Q(0) + A(0)(z_{0})| ≤ L′_{1}(0)(b) + L(0) + M(0) + M_{1}(0)
≤ L′_{1}(0)(b − 0) + L(0) + M(0) + M_{1}(0)
≤ (L′_{1}(0) + M + M_{1})(b − 0) + L(0) + M(0) + M_{1}(0)
≤ L(b) − L(0) + M(b) − M(0) + M_{1}(b) − M_{1}(0) + L(0) + M(0) + M_{1}(0)
≤ (L + K + M + M_{1})(b) = b.

Hence, by Lemma 2, x_{1} is well defined and (x_{1}, b) ∈ E. We also get

x_{1} = (I − A(0))(x_{1}) + [−(F (0) + Q(0) + A(0)(z_{0}))]

⟹

|x_{1}| ≤ L′(0)|x_{1}| + M(0) + L(0) + M_{1}(0) = d_{1}

and by the order convexity of L

d_{1} = L′(0)|x_{1}| + (M + M_{1} + L)(0) ≤ L′(0)b + (M + M_{1} + L)(0)
≤ (L′(0) + M + M_{1})(b − 0) + (M + M_{1} + L)(0)
≤ L(b) − L(0) + M(b) − M(0) + M_{1}(b) − M_{1}(0) + M(0) + M_{1}(0) + L(0)
≤ (M + M_{1} + L + K)(b) = b.

That is, we get |x_{1} − x_{0}| ≤ d_{1} − d_{0} or

0 ≤ (x_{0}, d_{0}) ≤ (x_{1}, d_{1}).

We now assume that

0 ≤ (x_{k−1}, d_{k−1}) ≤ (x_{k}, d_{k}) for k = 1, 2, …, n and d_{k} ≤ b.

We need to find a bound for I − A(x_{n}). We will show that L′_{1}(|x_{n}|) ∈ B(I − A(x_{n})).

This fact follows from (C_{7}), (C_{12}) and the estimate

|I − A(x_{n})| ≤ |I − A(0)| + |A(0) − A(x_{n})|
≤ L′_{1}(0) + L′_{1}(0 + |x_{n}|) − L′_{1}(0) = L′_{1}(|x_{n}|).

Using (2) we obtain the approximation

−[F (x_{n}) + Q(x_{n}) + A(x_{n})(z_{n})] = −F (x_{n}) − Q(x_{n}) − A(x_{n})(z_{n})

+ A(x_{n−1})(x_{n} − x_{n−1}) + F (x_{n−1}) + Q(x_{n−1}) + A(x_{n−1})(z_{n−1}).

Hence, by (C_{7}), (C_{10}), (C_{11}) and the induction hypotheses, we obtain in turn

| − F (x_{n}) + F (x_{n−1}) + A(x_{n−1})(x_{n} − x_{n−1})| + |Q(x_{n−1}) − Q(x_{n})| + |A(x_{n})(z_{n}) − A(x_{n−1})(z_{n−1})|

≤ ∫_{0}^{1} [L′(|x_{n−1}| + tc_{n−1}) − L′(|x_{n−1}|) + K] c_{n−1} dt + M(|x_{n−1}| + |x_{n} − x_{n−1}|) − M(|x_{n−1}|)

+ M_{1}(d_{n} − d_{n−1})

≤ L(|x_{n−1}| + c_{n−1}) − L(|x_{n−1}|) − L′(|x_{n−1}|)c_{n−1} + Kc_{n−1} + M(|x_{n−1}| + |x_{n} − x_{n−1}|) − M(|x_{n−1}|) + M_{1}(d_{n} − d_{n−1})

≤ L(d_{n−1} + d_{n} − d_{n−1}) − L(d_{n−1}) − L′(|x_{n−1}|)c_{n−1} + K(d_{n} − d_{n−1})

+ M(d_{n−1} + d_{n} − d_{n−1}) − M(d_{n−1}) + M_{1}(d_{n} − d_{n−1}) = (M + M_{1} + L + K)(d_{n}) − d_{n}. (14)

We can now obtain that

L′_{1}(|x_{n}|)(b − d_{n}) + | − (F (x_{n}) + Q(x_{n}) + A(x_{n})(z_{n}))| + d_{n}
≤ L′(d_{n})(b − d_{n}) + L(d_{n}) + M(d_{n}) + M_{1}(d_{n}) + K(d_{n}) − d_{n} + d_{n}
≤ (L′(d_{n}) + M + M_{1} + K)(b − d_{n}) + (L + K + M + M_{1})(d_{n})

≤ K(b − d_{n}) + L(b) − L(d_{n}) + M(b) − M(d_{n}) + M_{1}(b) − M_{1}(d_{n}) + (L + K + M + M_{1})(d_{n})

= (M + M_{1} + L + K)(b) = b.

That is, x_{n+1} is also well defined by Lemma 2 and c_{n} ≤ b − d_{n}. Hence d_{n+1} is well defined too and we obtain:

d_{n+1} ≤ (M + M_{1} + L + K)(d_{n}) + L′(d_{n})(b − d_{n}) ≤ (M + M_{1} + L + K)(d_{n})
+ (L′(d_{n}) + M + M_{1})(b − d_{n})

≤ (M + M_{1} + L + K)(d_{n}) + L(b) − L(d_{n}) + M(b) − M(d_{n}) + M_{1}(b) − M_{1}(d_{n}) + K(b − d_{n})

= (M + M_{1} + L + K)(b) = b.

The monotonicity (x_{n}, d_{n}) ≤ (x_{n+1}, d_{n+1}) can be derived from

c_{n} + d_{n} ≤ L′(|x_{n}|)c_{n} + | − (F (x_{n}) + Q(x_{n}) + A(x_{n})(z_{n}))| + d_{n}

≤ L′(|x_{n}|)c_{n} + (M + M_{1} + L + K)d_{n} = d_{n+1}.

The induction has now been completed. We need to show that

b_{n} ≤ d_{n} for all n ≥ 1.

For n = 1 and from the definitions of b_{n}, d_{n}

b_{1} = (L′(0) + K + M + M_{1})^{1}(0) ≤ d_{1}.

Assume that

b_{k} ≤ d_{k} for k = 1, 2, …, n.

Then, we obtain in turn

b_{n+1} = (L′(0) + K + M + M_{1})^{n+1}(0) = (L′(0) + K + M + M_{1})(L′(0) + K + M + M_{1})^{n}(0)
≤ (L′(0) + K + M + M_{1})(d_{n})
≤ L′(0)(d_{n}) + (M + K + M_{1})(d_{n})
≤ L(d_{n}) − L(0) + (M + K + M_{1})(d_{n})
≤ (L + K + M + M_{1})d_{n} ≤ d_{n} ≤ d_{n+1}.

Since d_{n} ≤ b, we have b_{n} ≤ d_{n} ≤ b. We also need to show that

0 ≤ a_{n} − b_{n} ≤ (L′(a) + K + M + M_{1})^{n}(a)  (n ≥ 1).  (15)

For n = 1, we get

a_{1} − b_{1} = (L + K + M + M_{1})(a) − (L + K + M + M_{1})(0) = (L(a) − L(0)) + (M(a) − M(0))

+ K(a − 0) + (M_{1}(a) − M_{1}(0)) = ∫_{0}^{1}[L′(ta) + K + M + M_{1}](a) dt

≤ ∫_{0}^{1}(L′(a) + K + M + M_{1})(a) dt = (L′(a) + K + M + M_{1})^{1}(a),

which shows (15) for n = 1. Let us now assume that (15) is true for k = 1, 2, …, n, then we can obtain in turn

a_{n+1} − b_{n+1} = (L + K + M + M_{1})^{n+1}(a) − (L + K + M + M_{1})^{n}(0)
= (L + K + M + M_{1})[(L + K + M + M_{1})^{n}(a)] − (L + K + M + M_{1})
· [(L + K + M + M_{1})^{n}(0)]
= L[(L + K + M + M_{1})^{n}(a)] − L[(L + K + M + M_{1})^{n}(0)]
+ (M + K + M_{1})[(L + K + M + M_{1})^{n}(a)] − (M + K + M_{1})[(L + K + M + M_{1})^{n}(0)]
= ∫_{0}^{1} L′[(L + K + M + M_{1})^{n}(0) + t((L + K + M + M_{1})^{n}(a) − (L + K + M + M_{1})^{n}(0))]
· [(L + K + M + M_{1})^{n}(a) − (L + K + M + M_{1})^{n}(0)] dt
+ (M + K + M_{1})[(L + K + M + M_{1})^{n}(a) − (L + K + M + M_{1})^{n}(0)]
≤ ∫_{0}^{1} L′((L + K + M + M_{1})^{n}(a))(L′(a) + K + M + M_{1})^{n}(a) dt
+ (M + K + M_{1})((L′(a) + K + M + M_{1})^{n}(a))
≤ L′(a)(L′(a) + K + M + M_{1})^{n}(a) + (M + K + M_{1})(L′(a) + K + M + M_{1})(a)
= (L′(a) + K + M + M_{1})(a).

By condition (C_{14}) and the above, we deduce that the sequence {b_{n}} n ≥ 0 is Cauchy in a Banach space C, and as such, it converges to some b = (L + K + M + M_{1})^{∞}(0). From (L + K + M + M_{1})(b) = (L + K + M + M_{1})(lim_{n→∞}(L + K + M + M_{1})^{n}(0)) = lim_{n→∞}(L + K + M + M_{1})^{n+1}(0) = b, we obtain

(L + K + M + M_{1})(b) = b ≤ a,

which makes b smaller than any solution of the inequality

(L + K + M + M_{1})(p) ≤ p. (see also [2], [3], [5]).

It also follows that the sequence {x_{n}} n ≥ 0 is Cauchy in X, and as such, it converges to some x^{∗} ∈ U(b). By letting n → ∞ in (14) and using the hypotheses that lim_{n→∞} z_{n} = 0, we deduce that x^{∗} is a solution of the equation F (x) + Q(x) = 0.

To show uniqueness, let us assume that there exists another solution y^{∗} of the equation F (x)+Q(x) = 0 in U(a). Then, exactly as in [1], [2], [3], [5], [6], by considering the modified Newton-process

x_{n+1} = x_{n} − (F (x_{n}) + Q(x_{n})),

we can show that this sequence converges, under the hypotheses of the theorem. Moreover, as above, we can easily show (see also [1], [2], [3], [5]) that

|y^{∗} − x_{n}| ≤ a_{n} − b_{n} for M_{1} = 0 and z_{n} = 0 (n ≥ 0),

from which follows that x_{n} → y^{∗} as n → ∞. Finally, estimates (12) and (13) are obtained by using standard majorization techniques [3], [4], [9].

That completes the proof of the theorem.

We will now introduce results on a posteriori estimates. It is convenient to define the operator

R_{n}(q) = (I − L′(|x_{n}|))^{∗}S_{n}(q) + c_{n}

where

S_{n}(q) = (L + K + M + M_{1})(|x_{n}| + q) − (L + K + M + M_{1})(|x_{n}|) − L′(|x_{n}|)(q)

and the interval

I_{n} = [0, a − |x_{n}|].

It can easily be seen that the operators S_{n} are monotone on I_{n}. Moreover, the operators R_{n} : [0, a−d_{n}] → [0, a − d_{n}] are well defined and monotone. This fact follows from lemma 1 and the scheme

d_{n} +c_{n} ≤ d_{n+1} ⇒ R(a)−d_{n+1} ≤ a−d_{n} −c_{n} ⇒ S_{n}(a−d_{n})+L′(|x_{n}|)(a−d_{n} −c_{n}) ≤ a−d_{n} −c_{n} (n ≥ 0).

Then, exactly as in [6], we can show:

Proposition. The following implications are true:

(i) if q ∈ I_{n} satisfy R_{n}(q) ≤ q, then

c_{n} ≤ R_{n}(q) = p ≤ q

and

R_{n+1}(p − c_{n}) ≤ p − c_{n} for all n ≥ 0;

(ii) under the hypotheses of Theorem 1, let q_{n} ∈ I_{n} be a solution of R_{n}(q) ≤ q, then

|x^{∗} − x_{m}| ≤ a_{m} (m ≥ n)

where

a_{n} = q_{n} and a_{m+1} = R_{m}(a_{m}) − c_{m};

and (iii) under the hypotheses of Theorem 1, any solution q ∈ I_{n} of R_{n}(q) ≤ q is such that

|x^{∗} − x_{n}| ≤ R_{n}^{∞}(0) ≤ q.

Remark 1.

(a) The results obtained in Theorem 1 and the proposition reduce immediately to the corresponding ones in [6] (Theorem 5 and Lemmas 10–12) when A(x) = F′(x_{n})^{−1}, Q(x) = 0 for all x ∈ D and z_{n} = 0 for all n ≥ 0 (M, M_{1}, K will be 0 then and L = L_{1}).

(b) As in [2], [3], [5], we can also show that if conditions (C_{6})–(C_{12}) are satisfied and there exists t ∈ (0, 1) such that (L + K + M + M_{1})(a) ≤ ta, then there exists a_{1} ∈ [0, ta] satisfying conditions (C_{6})–(C_{14}). The solution x^{∗} ∈ U(a_{1}) is unique in U(a).

(c) From the approximation

A(x_{n})(z_{n}) − A(x_{n−1})(z_{n−1}) = (A(x_{n})(z_{n}) − z_{n}) + (−A(x_{n−1})(z_{n−1}) + z_{n−1}) + I(z_{n} − z_{n−1})

we observe that (C_{11}) will be true if

M_{1} = 2L′_{0}(b) + I with L_{0} = L_{1} or L

and z_{n} are chosen so that

|z_{n}| + |z_{n−1}| + |z_{n} − z_{n−1}| ≤ d_{n} − d_{n−1} (n ≥ 1).

(d) Another choice for M_{1}, z_{n} can be M_{1} = |ε|I and z_{n} = z_{n−1} + ε_{n}(x_{n} − x_{n−1}) (n ≥ 1) and |ε_{n}| ≤ |ε| (n ≥ 0) provided that F′(x) = I for all x ∈ D. It can then easily be seen that (C_{11}) is satisfied. The sequence z_{n} must still be chosen to be null. At the end of this paper, in Part V, we have given examples to cover this case. Several other choices are also possible.

(e) From the proof of the theorem, it can easily be seen that condition (C_{9}) can be replaced by the weaker condition (which is what we actually need)

L′_{1}(d_{n})(b − d_{n}) ≤ L(b) − L(d_{n}) (n ≥ 0).

This condition however is not easy to verify in general.

(f) It can easily be seen that the conclusion of Theorem 1 remains valid if in the hypotheses L_{1} replaces L everywhere except in condition (C_{7}).

(g) Condition (C_{7}) can be replaced by the set of conditions

K + L′_{2}(|x| + |y|) − L′_{2}(|x|) ∈ B(F′(x) − F′(x + y))

and

L′_{3}(|x|) − L′_{3}(|0|) ∈ B(A(x) − F′(x))

for all x, y ∈ [0, a] with |x| + |y| ≤ a as long as we can find L such that

L′(|x| + |y|) = L′_{3}(|x|) + L′_{2}(|x| + |y|) + K and L′(|x|) = L′_{2}(|x|) + L′_{3}(|0|),

for all x, y ∈ [0, a] with |x| + |y| ≤ a. The operators L, L_{2}, L_{3} ∈ C^{1}(V_{1} → V) defined here must also be order convex.

(h) It can easily be seen from the proof of theorem 1 that the results obtained in theorem 1 remain valid if (C_{11}) is replaced by the weaker condition:

0 ≤ (A(x_{n})(z_{n}) − A(x_{n−1})(z_{n−1}), M_{2}|x_{n} − x_{n−1}|) ∈ E (n ≥ 1)

for some M_{2} ∈ L_{+}(V). This condition seems to be more helpful since it relates z_{n} with x_{n} and not an unknown sequence depending on M_{1}. Under this condition the term M_{1}(b − b_{n}) appearing in the Proposition can be replaced by M_{2}|x_{n} − x_{n−1}| (n ≥ 1). Other choices involving M_{2} are possible.

(i) Define the residuals r_{n} = −A(x_{n})(z_{n}) (n ≥ 0) and set δ_{n} = x_{n+1} − x_{n} (n ≥ 0). Then from the

approximation

r_{n} = [(I − A(x_{n})) − I] (z_{n})

we obtain

|r_{n}| ≤ (L′_{1}(|x_{n}|) + I) |z_{n}|

which shows that r_{n} → 0 as n → ∞ if z_{n} → 0 as n → ∞. Consequently, the conclusions of theorem 1 remain true for the system

x_{n+1} − x_{n} = δ_{n}, A(x_{n})δ_{n} = −(F (x_{n}) + Q(x_{n})) + r_{n} (n ≥ 0).

(j) (C_{11}) is equivalent to the condition that (A(x_{n})(z_{n}), M_{1}(d_{n})) ∈ E^{N} (n ≥ 1) is an increasing sequence. Similar observations can be made for (C_{10}).

IV. THE MONOTONE CASE

Let A ∈ L(X → X) be a given operator. Define the operators P, T (D → X) by

P (x) = AT (x + u)
T (x) = G(x) + R(x), P (x) = F (x) + Q(x)

and

F (x) = AG(x + u), Q(x) = AR(x + u),

where G, R are as F, Q respectively. We deduce immediately that under the hypotheses of Theorem 1, the zero x^{∗} of P is a zero of AT also, if u = 0.

We will now provide a monotonicity result to find a zero x^{∗} of AT. The space X is assumed to be partially ordered and satisfies the conditions for V given in (C_{1})–(C_{5}). Moreover, we set X = V, D = C^{2} so that | | turns out to be I.

We can now state and prove the main result of this section.

\underline{Theorem 2}. Let V be a partially ordered Banach space satisfying conditions (C_{1})–(C_{5}), Y be a Banach space, G ∈ C^{1}(D → Y), R ∈ C(D → Y) with D ⊆ V, A, A(x) ∈ L(X → V), K, M_{1}(D → V) with K, M_{1} ∈ L_{+}(V), M ∈ C(V_{1} → V) and u, v ∈ V such that:

(C_{15}) [u, v] ⊆ D; (C_{16}) 2I − AG′(u) − A(v − u) + M + K + M_{1} ∈ L_{+}(V); (C_{17}) for all w, w_{1}, w_{2} ∈ [u, v] : w_{1} ≤ w_{2} ⇒ AG′(u + w) ≤ I,

A(w_{2}) ≤ A(w_{1})
A(w_{2}) ≤ AG′(w_{1} + u) + K
A(w_{1}) − A(w_{2}) + AG′(u + w_{1}) − AG′(u + w_{2}) ≥ 0

and

−(Q(0) + A(0)(z_{0})) ≤ (M + K + M_{1})(0);

(C_{18}) AT (u)+A(u)(z_{0}) ≤ 0, AT (v)+A(v)(z_{0}) ≥ 0 and v −u ≤ ∫_{0}^{v−u} A(z)dz +AG(v)−(M +K +M_{1})(v −u); (C_{19}) condition (C_{10}) is satisfied; (C_{20}) condition (C_{11}) is satisfied; and (C_{21}) (2I − AG′(v) − A(v − u) + M + K + M_{1})^{n}(v − u) → 0 as n → ∞.

Then the Newton sequence

y_{0} = u, y_{n+1} = y_{n} + (A(y_{n}))^{∗}[−AT (y_{n})] − z_{n} (n ≥ 0)

is well defined for all n ≥ 0, monotone and converges to a unique zero x^{∗} of AT in [u, v].

Proof. The proof follows immediately by setting P (x) = AT (x + u), L_{1}(e) = e − ∫_{0}^{e} A(z)dz, L_{2}(e) = e − F (e), L = L_{1} + L_{2} and a = v − u.

Remark 2.

(a) By setting R(x) = 0 for all x ∈ D and z_{n} = 0 (n ≥ 0), we immediately deduce that our results reduce to the ones obtained in [6, thm 13].

(b) If M, R are linear operators, then condition (C_{19}) will certainly be satisfied if we set M(e) = −Q(e).

(c) The operators L, L_{1}, L_{2} can be chosen in general as L_{1}(e) = z_{1} + R_{1}(e) − ∫_{0}^{e} A(z)dz, L_{2} = z_{2} + R_{2}(e) − F (e) and L = L_{1} + L_{2} with z_{1}, z_{2} ∈ [u, v] and R_{1}, R_{2} ∈ L(D → Y). Conditions similar to the ones in Theorem 2 can then be easily produced for the above choices of L, L_{1}, L_{2}. For example, conditions (C_{16}) and (C_{21}) will become

R_{1} + R_{2} + K − AG′(u) − A(v − u) + M_{1} + M ∈ L_{+}(V)

and

(R_{1} + R_{2} − AG′(u) − A(v − u) + M_{1} + K + M)^{n}(v − u) → 0 as n → ∞

respectively.

V. APPLICATIONS

We will complete this study with two examples that show how to choose L, M, M_{1}, z_{n} in practical applications. For simplicity, we assume that A(x) = F′(x) for all x ∈ D.

Example 1. We discuss the case of a real Banach space with norm ∥ ∥. Assume for simplicity that F′(0) = I and there exists a monotone operator

f : [0, a] → IR

such that

∥F′′(x)∥ ≤ f(∥x∥) for all x ∈ U(a)

and a continuous nondecreasing function g on [0, r], r ≤ a such that

∥Q(x) − Q(y)∥ ≤ g(r)∥x − y∥  (16)

for all x, y ∈ U(\frac{r}{2}).

We showed in [3], (see also [9]) that (16) implies that

∥Q(x + ℓ) − Q(x)∥ ≤ h(r + ∥ℓ∥) − h(r), x ∈ U(a), ∥ℓ∥ ≤ a − r,  (17)

where

h(r) = ∫_{0}^{r} g(t) dt.

Conversely, it is not hard to see that we may assume, without loss of generality, that the function h and all functions h(r + t) − h(r) are monotone in r. Hence, we may assume that h(r) is convex and hence differentiable from the right. Then, as in [9], we show that (17) implies (16) and g(r) = h′(r + 0). Hence, we can set

L_{1}(q) = L(q) = ∥F (0) + Q(0)∥ + ∫_{0}^{q} ds ∫_{0}^{s} f(t) dt  (18)

and

M(q) = ∫_{0}^{q} g(t) dt.  (19)

In Remark 1 (c) and (d) we have already provided some choices for M_{1}, z_{n}. Here, however, for simplicity let us choose z_{n} = 0 (n ≥ 0) and M_{1} = 0.

Then condition (C_{13}) will be true if

\frac{1}{2}f(a)a^{2} − (1 − g(a))a + ∥F (0) + Q(0)∥ ≤ 0.  (20)

If we set Q = 0 and g = 0, (20) is true if ∥F (0)∥f(a) ≤ \frac{1}{2}, which is a well known condition due to Kantorovich [4, Ch. 18]. If Q ≠ 0, condition (20) is the same condition with the one found in [3], [9] for the Zincenko iteration.

In the example that follows, we show that our results can apply to solve nonlinear integral equations involving a nondifferentiable term, whereas the results obtained in [6] (or [5], [7], [8]) cannot apply.

Example 2. Let X = V = C[0, 1], and consider the integral equation

x(t) = ∫_{0}^{1} k(t, s, x(s)) ds on X,  (21)

where the kernel k(t, s, x(s)) with (t, s) ∈ [0, 1] × [0, 1] is a nondifferentiable operator on X. Consider (21) in the form (1) where F, Q : X → X are given by

F (x)(t) = Ix(t)

and

Q(x)(t) = − ∫_{0}^{1} k(t, s, x(s)) ds.

The operator | | is defined by considering the sup-norm. We assume that V is equipped with natural partial ordering, and there exists α, a ∈ [0, +∞), and a real function α(t, s) such that

∥k(t, s, x) − k(t, s, y)∥ ≤ α(t, s)∥x − y∥

for all t, s ∈ [0, 1], x, y ∈ U(\frac{a}{2}), and

α ≥ sup_{t∈[0,1]} ∫_{0}^{1} α(t, s) ds.

Define the real functions h, f, g on [0, a] by h(r) = αr, f(r) = 0 and g(r) = α for all r ∈ [0, a]. By choosing L, M, M_{1} as in (18) and (19) and Remark 1 (d) respectively, and by setting K = 0, we can easily see that the conditions (C_{1})–(C_{12}) of Theorem 1 are satisfied. In particular, condition (C_{13}) becomes

(1 − α − |ϵ|)a − ∥Q(0)∥ ≥ 0  (22)

which is true in the following cases: if 0 ≤ α < 1−|ϵ|, choose a ≥ β = \frac{∥Q(0)∥}{1−α−|ϵ|}; if α = 1−|ϵ| and Q(0) = 0, choose a ≥ 0; if α > 1 − |ϵ| and Q(0) = 0, choose a = 0. If in (22) strict inequality is valid, then there exists a solution a^{∗} of equation (22) satisfying condition (C_{14}). Note that if we choose α ∈ (0, 1 − |ϵ|), a ∈ (β, +∞) and ϵ ∈ (−1, 1), condition (22) is valid as a strict inequality. Finally, we remark that the results obtained in [6] (or [5], [7], [8]) cannot apply here to solve equation (21), since Q is nondifferentiable on X and the z_{n}’s are not necessarily zero.

CONCLUSION

In this study, we used perturbed Newton-like methods to find solutions of nonlinear, nondifferentiable operator equations on Banach spaces with a generalized structure. This technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. This way the metric properties of the examined problem have been shown to be analyzed more precisely. Convergence results and error estimates have been improved compared with the real norm theory. Moreover, this approach also allowed us to derive, on the one hand, semi-local results of Kantorovich-type and, on the other hand, global results base on monotonicity considerations, from the same general theorem. All our convergence theorems were derived from metric estimates. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature [1], [2], [5], [6], [7]. Finally, we used our results to solve applied problems that cannot be solved with existing methods.

REFERENCES

[1] Argyros, I.K. The secant method in generalized Banach spaces, Appl. Math. Comp. 39, (1990), 111–121.

[2] . Newton-like methods in generalized Banach spaces, Functiones et Approximatio Commentarii Mathematici, XXII, (1993), 13–20.

[3] Argyros, I.K. and Szidarovszky, F. The Theory and Applications of iteration methods, C.R.C. Press, Inc. Boca Raton Florida, U.S.A. 1993.

[4] Kantorovich, L.V. and Akilov, G.P. Functional Analysis in normed spaces, Academic Press, New York, 1978.

[5] Meyer, P.W. Newton’s method in generalized Banach spaces, Numer. Funct. Anal. Optimiz. 9 (3 and 4), (1987), 249–259.

[6] . A unifying theorem for Newton’s method, Numer. Funct. Anal. Optimiz. 13 (5 and 6), (1992), 463–473.

[7] Pandian, M.C. A convergence test and component-wise error estimates for Newton-type methods, SIAM J. Numer. Anal. 22, (1985), 779–799.

[8] Vandergraft, J.A. Newton’s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432.

[9] Zabrejko, P.P. and Nguen, D.F. The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optimiz. 9 (5 and 6), (1987), 671–684.

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu ISSN 1083-0464 Vol. 01, 1995, pp. 13–16. Submitted: March 16, 1995. Published: June 30, 1995.

DIFFERENTIATED SHIFT-INVARIANT

MULTIVARIATE INTEGRAL OPERATORS

George A. Anastassiou

Abstract. In a recent paper [1] the author, together with H. Gonska, introduced some wavelet-type, very general multivariate integral operators over ℝ^{d}, d ≥ 1, and studied their various properties such as shift-invariance, global smoothness preservation and optimality, convergence to the unit, differentiation, and preservation of continuous multivariate probability distribution functions. These operators are introduced through a convolution-like iteration of another very general multivariate operator with a general multivariate scaling type function. In this paper the author gives sufficient conditions, so that the partial derivatives of the above described multivariate operators possess most of the above listed nice properties of their originals. In particular, a sufficient condition is given so that a corresponding “global smoothness preservation” sharp multivariate inequality is attained. The corresponding convergence of differentiated operators to the differentiated unit operator is given with rates of convergence.

(1991) A.M.S. (MOS) Subject Classification Codes. 60, 62, 28.

Key Words and Phrases. Wavelet integral operators, Probability distribution functions.

Main Results.

Let X := C_{U}(ℝ^{d}), d ≥ 1 be the space of uniformly continuous functions from ℝ^{d} into ℝ.

For f ∈ X, define the first order modulus of continuity of f by

ω_{1}(f, δ) := sup_{\begin{matrix}x,y∈ℝ^{d} \\ ∥x−y∥≤δ \end{matrix}} |f(x) − f(y)|, δ > 0,

where ∥ · ∥ is an arbitrary norm in ℝ^{d}. It is known [1] that ω_{1}(f, δ) < ∞, ∀δ > 0. Here C(ℝ^{d}) denotes the space of continuous functions from ℝ^{d} into ℝ. Let {ℓ_{k}}_{k∈ℤ} be a sequence of linear operators from C(ℝ^{d}) into itself such that

ℓ_{k}(f, x) = ℓ_{0}(f(2^{−k}·), x), all x ∈ ℝ^{d}, f ∈ C(ℝ^{d}).  (1)

Assume that

ℓ_{0} : C^{(ρ)}(ℝ^{d}) → C^{(ρ)}(ℝ^{d}), ρ ∈ ℕ be fixed,  (2)

where C^{(ρ)}(ℝ^{d}) is the space of ρ-times continuously differentiable functions.

In this article we consider only f ∈ C^{(ρ)}(ℝ^{d}) such that \frac{∂^{ρ}f}{∂x_{i_{1}}^{j_{1}} ···∂x_{im}^{jm}} ∈ X, where fixed i_{1}, …, i_{m} ∈ \{1, …, d\}

be such that i_{1} < ⋯ < i_{m}, and fixed j_{1}, …, j_{m} ∈ ℕ be such that ∑_{r=1}^{m} j_{r} = ρ.

From now on we will denote by

f^{∂} := \frac{∂^{ρ}f}{∂x_{i_{1}}^{j_{1}} ⋯ ∂x_{i_{m}}^{j_{m}}} , f ∈ C^{(ρ)}(ℝ^{d}),  (3)

Department of Mathematics, Memphis State University, Memphis, TN

Email: anastasg@ kermes.msci.memst.edu

Copyright \begin{matrix}c \\ ◯ \end{matrix}1995 by Cameron University

where the mixed partial is defined as above.

Let φ be a bounded real valued function of compact support ⊆ ×_{i=1}^{d}[−a_{i}, a_{i}], a_{i} > 0. We assume that φ ≥ 0, φ is Lebesgue measurable and

\underbrace{∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞}}_{d−fold} φ(x_{1} − u_{1}, x_{2} − u_{2}, …, x_{d} − u_{d}) du_{1} … du_{d} = 1,

all (x_{1}, …, x_{d}) ∈ ℝ^{d}  (4)

((4) same as

∫_{ℝ^{d}} φ(x − u) du = 1, all x ∈ ℝ^{d};

where x = (x_{1}, …, x_{d}), u = (u_{1}, …, u_{d})). One can easily prove that

∫_{ℝ^{d}} φ(u)du = 1.  (5)

Example. For i = 1, …, d take the characteristic function

φ_{i}(x_{i}) := χ_{[− \frac{1}{2} , \frac{1}{2} )}(x_{i}) = \{ \begin{matrix}1, & x_{i} ∈ [−\frac{1}{2}, \frac{1}{2}) \\ 0, & else. \end{matrix}

Define

φ^{∗}(x) := ∏_{i=1}^{d} φ_{i}(x_{i}), all x = (x_{1}, …, x_{d}) ∈ ℝ^{d}.

Note that φ^{∗} is bounded, supp φ^{∗} ⊆ [−\frac{1}{2}, \frac{1}{2}]^{d}, φ^{∗} ≥ 0 and φ^{∗} is Lebesgue measurable. Also

\underbrace{∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞}}_{d−fold} ∏_{i=1}^{d} φ_{i}(x_{i} − u_{i}) du_{1} … du_{d} = 1,

i.e.,

∫_{ℝ^{d}} φ^{∗}(x − u)du = 1, all x ∈ ℝ^{d}.

Let {L_{k}}_{k∈ℤ} be the sequence of linear operators acting on C^{(ρ)}(ℝ^{d}) and defined as follows

L_{0}(f, x) := ∫_{ℝ^{d}} ℓ_{0}(f, u)φ(x − u) du  (6)

and

L_{k}(f, x) := L_{0}(f(2^{−k}·); 2^{k}x),  (7)

all x ∈ ℝ^{d}, k ∈ ℤ. I.e.,

L_{k}(f, x) = ∫_{ℝ^{d}} ℓ_{k}(f, u)φ(2^{k}x − u) du.  (8)

For our convenience we shall use the notation

∫ a⃗ ; −a⃗ • = ∫_{×_{i=1}^{d}[−a_{i},a_{i}]} • .  (9)

Furthermore, we obtain that

L_{k}^{∂}(f, x) = 2^{kρ} · ∫ a⃗ ; −a⃗ ℓ_{k}^{θ}(f, 2^{k}x − u)φ(u) du

= 2^{kρ} · ∫_{ℝ^{d}} ℓ_{k}^{∂}(f, 2^{k}x − u)φ(u) du
= 2^{kρ} · ∫_{ℝ^{d}} ℓ_{k}^{∂}(f, u)φ(2^{k}x − u) du, all x ∈ ℝ^{d}, k ∈ ℤ.  (10)

Assumption. Take f ∈ C^{(ρ)}(ℝ^{d}), ρ ∈ ℕ fixed, such that f^{∂} ∈ X. For fixed α > 0 we assume that

sup_{\begin{matrix}u,y∈ℝ^{d} \\ ∥u−y∥∞≤α \end{matrix}} |ℓ_{0}^{∂}(f, u) − f^{∂}(y)| ≤ ω_{1,∞} (f^{∂}, \frac{mα + n}{2^{r}}) ,  (11)

is true for f as above, m ∈ ℕ, n ∈ ℤ_{+}, r ∈ ℤ, where ω_{1,∞} is the modulus of continuity ω_{1} defined with respect to ∥ · ∥_{∞}.

Here we present our results without proofs due to their length. We intend to show proofs elsewhere. Definition 1. Let f_{α}(·) := f(· + α), α ∈ ℝ^{d} and ϕ be an operator. If ϕ(f_{α}) = (ϕf)_{α}, then ϕ is called a shift invariant operator. Proposition 1. Assume that

ℓ_{0}^{∂} (f(2^{−k} · +α); 2^{k}u) = ℓ_{0}^{∂} (f(2^{−k}·); 2^{k}(u + α)) ,  (12)

all k ∈ ℤ, α ∈ ℝ^{d} fixed, all u ∈ ℝ^{d}; any f ∈ C^{(ρ)}(ℝ^{d}); ρ ∈ ℕ be given. Then L_{k}^{∂}, any k ∈ ℤ is a shift invariant operator.

Next we establish the property of “global smoothness preservation” of operators L_{k}^{∂}, k ∈ ℤ. Theorem 1. For any f ∈ C^{(ρ)}(ℝ^{d}), such that f^{∂} ∈ X, ρ ∈ ℕ fixed, and any u ∈ ℝ^{d} assume that

|ℓ_{0}^{∂}(f, x − u) − ℓ_{0}^{∂}(f, y − u)| ≤ ω_{1}(f^{∂}, ∥x − y∥), all x, y ∈ ℝ^{d},  (13)

where ∥ · ∥ is an arbitrary norm in ℝ^{d}. Then

ω_{1}((L_{k}f)^{∂}, δ) ≤ ω_{1}(f^{∂}, δ),  (14)

any δ > 0.

Optimality of (14) follows. Theorem 2. Let i_{1}, …, i_{m} ∈ {1, …, d} be such that i_{1} < ⋯ < i_{m}, and j_{1}, …, j_{m} ∈ ℕ be such that ∑_{r=1}^{m} j_{r} = ρ. Consider g_{j_{1}}(x) := \frac{x^{j_{1}+1}}{(j_{1}+1)!}, and g_{j_{r}}(x) := \frac{x^{jr}}{j_{r}!} , r = 2, …, m; all x ∈ ℝ. Denote by f^{∂} := \frac{∂^{ρ}f}{∂x_{i_{1}}^{j_{1}} ...∂x_{im}^{jm}}, for any f ∈ C^{(ρ)}(ℝ^{d}). Denote also by pr_{i_{r}} : ℝ^{d} ∋ (x_{1}, …, x_{d}) → x_{i_{r}}, r = 1, …, m the projection onto the i_{r} coordinate. For all x, y, u ∈ ℝ^{d} assume that

(ℓ_{0} ((∑_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}}) (2^{−k}·)))^{∂} (2^{k}x − u)

− (ℓ_{0} ((∏_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}}) (2^{−k}·)))^{∂} (2^{k}y − u) = 2^{−kρ} · (x_{i_{1}} − y_{i_{1}}).  (15)

Then

ω_{1} ((L_{k} (∏_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}}))^{∂} ; δ) = ω_{1} ((∏_{r=1}^{m} g_{j_{r}} ∘ pr_{i_{r}})^{∂} ; δ) ,  (16)

for any δ > 0; k ∈ ℤ. That is establishing that (14) is sharp!

The convergence of \frac{∂^{ρ}L_{k}}{∂x_{i_{1}}^{j_{1}} ...∂x_{im}^{jm}} (∑_{r=1}^{m} j_{r} = ρ) to \frac{∂^{ρ}}{∂x_{i_{1}}^{j_{1}} ...∂x_{im}^{jm}} operator, as k → +∞, with rates given in the following. Theorem 3. For f ∈ C^{(ρ)}(ℝ^{d}), d ≥ 1, such that f^{∂} ∈ C_{U}(ℝ^{d}), under the assumption (11), it holds

|L_{k}^{∂}(f, x) − f^{∂}(x)| ≤ ω_{1,∞} (f^{∂}, \frac{ma + n}{2^{k+r}}) ,  (17)

where m ∈ ℕ, n ∈ ℤ_{+}, k, r ∈ ℤ, and a := max(a_{i}), i = 1, …, d.

Again, due to their length we intend to present applications of the above results elsewhere.

References

1. G.A. Anastassiou and H.H. Gonska, On some shift-invariant integral operators, multivariate case, Proceedings of International Conference in Approximation Theory and Probability, Santa Barbara, California, May 20-22, 1993 (1994), Plenum, New York, 41–64.

2. H. Bauer, Maβ-und Integrationstheorie (1990), de Gruyter, Berlin.

3. C. Cottin and H.H. Gonska, Simultaneous approximation and global smoothness preservation, Proceedings of the Second International Conference in Functional Analysis and Approximation Theory Acquafredda di Maratea-Potenza-Italy, 1992. Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Numero 33, (1993), 259–279..

4. A.N. Shiryayev, Probability, Springer-Verlag, New York, Berlin, 1984.

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 01, 1995, pp. 17–26 Submitted: June 11, 1995. Published: September 5, 1995.

STOCHASTIC CALCULUS AND INITIAL VALUE

ANALYSIS ON THE ONE DIMENSIONAL DIFFUSIONS

Dražen Pantić

Abstract. Stochastic calculus and initial value analysis are used for complete description of distribution type of diffusion processes with Lipschitz coefficients. Sufficient conditions so that the solutions of stochastic differential equations posses absolutely continuous one dimensional distribution are given. For stochastic differential equations with uniformly elliptic coefficients probability density is investigated in detail. Also, the distribution of inverse process is given.

(1991) A.M.S. Subject Classification Codes. 60H07, 60H10.

Key Words and Phrases. Malliavin calculus, solutions of stochastic differential equations, point distributions.

INTRODUCTION

Let {X_{t,x}, t ≥ 0}, x ∈ ℝ, be solution of SDE

X_{t,x} − x = ∫_{0}^{t} a(X_{s,x})dB_{s} + ∫_{0}^{t} b(X_{s,x})ds  (1)

where coefficients a and b have bounded and continuous first order derivatives (a, b ∈ ℂ_{b}^{1}). Under those conditions for p > 1 there exists L_{p} derivative DX_{t,x} of X_{t,x} w.r.t. initial condition x, so that, as in [2, §8],

DX_{t,x} − 1 = ∫_{0}^{t} a′(X_{s,x})DX_{s,x}dB_{s} + ∫_{0}^{t} b′(X_{s,x})DX_{s,x} ds.  (2)

Using Ito formula, DX_{t,x} could be computed explicitly as

DX_{t,x} = exp\{∫_{0}^{t} a′(X_{s,x})dB_{s} + ∫_{0}^{t}(b′(X_{s,x}) − \frac{1}{2}a′(X_{s,x}))^{2}ds\}.  (3)

The boundeness of derivatives implies that, for p > 1, DX_{t,x} as well as its inverse (DX_{t,x})^{−1} are in L_{p}, [6], while (DX_{t,x})^{−1} satisfies following SDE, [10, Lemma (3.7)]:

(DX_{t,x})^{−1} − 1 = − ∫_{0}^{t} a′(X_{s,x})(DX_{s,x})^{−1}dB_{s} + ∫_{0}^{t}(a′(X_{s,x})^{2} − b′(X_{s,x}))(DX_{s,x})^{−1} ds.

Department of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Yugoslavia

E-mail Address: epantidr@ubbg.etf.bg.ac.yu

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University

Also, for a Skorohod integrable process u = {u_{s}, 0 ≤ s ≤ T } a directional derivative D_{u}X_{t,x} of functional X_{t,x} could be expressed in terms of derivatives {DX_{t,x}, t ≥ 0} in the following way, [12],

D_{u}X_{t,x} = DX_{t,x} ∫_{0}^{t}(DX_{s,x})^{−1}a(X_{s,x}) u_{s} ds.  (4)

First we shall treat the general case. Defining process \underline{u} = \{\underline{u}_{s}, 0 ≤ s ≤ t\} as

\underline{u}_{s} = a(X_{s,x})DX_{s,x}

we have

D\underline{u}X_{t,x} = DX_{t,x} ∫_{0}^{t} a(X_{s,x})^{2} ds.  (5)

Process \underline{u} is Ito integrable, so for a continuously differentiable function with compact support f it follows, by integration by parts formula, [5], and relation (5), that

𝔼\{f′(X_{t,x})DX_{t,x} ∫_{0}^{t} a(X_{s,x})^{2}ds\} = 𝔼\{D\underline{u}(f(X_{t,x}))\} = 𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})DX_{s,x}dB_{s}\}.  (6)

Knowing that DX_{t,x} is a.s. positive, we have to find conditions so that ∫_{0}^{t} a(X_{s,x})^{2}ds is a.s. positive in order to apply next Lemma.

Lemma 1. Let X be a random variable such that there exists integrable and almost sure positive random variable Y and a constant C > 0 so that for each f ∈ ℂ_{b}^{1} we have

| 𝔼{f′(X) · Y } |≤ C sup{| f(x) |: x ∈ ℝ}.

Then probability law generated by the random variable X is absolutely continuous with respect to Lebesgue measure.

Proof. Let us define positive bounded linear functional on the space of the continuously differentiable functions with compact support with

L(f) = 𝔼{f(X)Y } = 𝔼{f(X)𝔼{Y |X}} = ∫ f(x)𝔼{Y |X = x}G(dx)

where G is probability measure generated by X. Then, we have

| ∫ f′(x)𝔼{Y |X = x}G(dx) |≤ C sup{| f(x) |: x ∈ ℝ}

so using [4, §1] it follows that there exists density g so that

∫ f(x)𝔼{Y |X = x}dG(x) = ∫ f(x)g(x) dx.

Clearly, the previous relation could be extended to the space of bounded measurable functions, so for a Borel set B we have

𝔼{𝕀{X ∈ B}Y } = ∫ 𝕀{x ∈ B}g(x) dx.

If B is of Lebesgue measure zero then

𝔼{𝕀{X ∈ B}Y } = 0.

Knowing that Y > 0 almost sure one easily gets that ℙ{X ∈ B} = 0, i.e. probability law generated by the random variable X is absolutely continuous with respect to Lebesgue measure. □

GENERAL CASE

In the next theorem we shall completely describe the first moment τ from which ∫_{0}^{t} a(X_{s,x})^{2}ds is a.s. positive, as well as the behavior of diffusion process (1) up to that moment.

Theorem 1. Let {X_{t,x}, t ≥ 0}, x ∈ ℝ, be solution of SDE (1) with coefficients in ℂ_{1}^{b}, and {Y_{t,x}, t ≥ 0} deterministic process defined by

Y_{t,x} − x = ∫_{0}^{t} b(Y_{s,x}) ds.

Define

τ_{x} = inf{t ≥ 0 :| a(X_{t,x}) |> 0} and D_{x} = inf{t ≥ 0 :| a(Y_{t,x}) |> 0}.

Then, with probability one, τ_{x} = D_{x} and X_{t,x} = Y_{t,x} for t ≤ D_{x} .

Proof. Assume first that 0 < D_{x} < ∞, and let 0 < t < D_{x}. Then

X_{t,x} − Y_{t,x} = ∫_{0}^{t}(a(X_{s,x}) − a(Y_{s,x}))dB_{s} + ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds, and

𝔼{(X_{t,x} − Y_{t,x})^{2}}^{1/2} ≤
≤ 𝔼{(∫_{0}^{t}(a(X_{s,x}) − a(Y_{s,x}))dB_{s})^{2}}^{1/2} + 𝔼{( ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds)^{2}}^{1/2} ≤
≤ 𝔼{∫_{0}^{t}(a(X_{s,x}) − a(Y_{s,x}))^{2}ds}^{1/2} + 𝔼{t ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))^{2}ds}^{1/2} ≤
≤ K(∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2}}ds)^{1/2} + t^{1/2}K( ∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2}}ds)^{1/2} ≤
≤ K(1 + D_{x}^{1/2})( ∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2}}ds)^{1/2},

where K is upper bound for first derivatives of a and b. Then, using Gronwall inequality, we can conclude that with probability one X ≡ Y on [0, D_{x}], and hence D_{x} ≤ τ_{x} a.s.

Assume now that 0 < τ_{x} < ∞, and let 0 < t < τ_{x}. As τ_{x} is optional time the set

A = {ω : t < τ_{x}(ω)}

is F_{t} measurable, where {F_{t}, t ≥ 0} is standard filtration generated by {B_{t}, t ≥ 0}. Then we have

𝔼{∫_{0}^{t} a(X_{s,x})^{2}ds 𝕀_{A}} = 0.

It is obvious that

(∫_{0}^{t} a(X_{s,x}) dB_{s})^{2} 𝕀{t < τ_{x}} ≤ (∫_{0}^{t∧τ_{x}} a(X_{s,x}) dB_{s})^{2}.

Having in mind that t ∧ τ_{x} is optional time we have

𝔼{(∫_{0}^{t} a(X_{s,x}) dB_{s})^{2} 𝕀{t < τ_{x}}} ≤ 𝔼{(∫_{0}^{t∧τ_{x}} a(X_{s,x}) dB_{s})^{2}} = 𝔼{∫_{0}^{t∧τ_{x}} a(X_{s,x})^{2} ds} = 0,

and consequently ∫_{0}^{t} a(X_{s,x})dB_{s} = 0 a.s. on set A. Hence for each t > 0 on the set A we have

X_{t,x} − Y_{t,x} = ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds, and
𝔼{(X_{t,x} − Y_{t,x})^{2} 𝕀{t < τ_{x}}} =
= 𝔼{(∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds)^{2} 𝕀{t < τ_{x}}} ≤
≤ t 𝔼{∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))^{2}ds 𝕀{t < τ_{x}}} ≤
≤ t 𝔼{∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))^{2} 𝕀{s < τ_{x}} ds} ≤
≤ tK^{2} ∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2} 𝕀{s < τ_{x}}} ds

Let us mark Δ_{t} = 𝔼{(X_{t,x} − Y_{t,x})^{2}𝕀{t < τ_{x}}}. Then previous relation could be written as

Δ_{t} ≤ tK^{2} ∫_{0}^{t} Δ_{s} ds,

from where it follows that Δ_{t} = 0 for each t > 0. Hence we can conclude that for each t > 0

ℙ{ω : t < τ_{x}(ω), X_{t,x} ≠ Y_{t,x}} = 0.

Using the continuity of paths we have that

ℙ{ω : (∃t > 0)(t < τ_{x}(ω), X_{t,x} ≠ Y_{t,x})} = 0,

so with probability one X ≡ Y on [0, τ_{x}], i.e. τ_{x} ≤ D_{x} a.s. and the proof is complete. □

The following theorem completely describes the type of one dimensional distributions of the solutions of SDE :

Theorem 2. Under conditions from Theorem 1. the distribution of random variable X_{t,x} is either degenerate or absolutely continuous, whether t ≤ D_{x} or t > D_{x}.

Proof. The first part when t ≤ D_{x} is described in Theorem 1 so let t > D_{x}. Then ∫_{0}^{t} a(X_{s,x})^{2}ds is positive a.s. and from relation (6) and Lemma 1 we can reach our conclusion. □

UNIFORMLY ELLIPTIC CASE

Now, we shall treat an important special case when that coefficient a is uniformly elliptic in sense that

inf{| a(x) |: x ∈ ℝ} > 0.  (EL)

Theorem 3. Let the diffusion process {X_{t,x}, t ≥ 0} defined by SDE (1) has continuously differentiable coefficients a and b with bounded derivatives and let

inf{|a(x)| : x ∈ ℝ} > 0.

Then we have :

1^{∘} for each t > 0 and x ∈ ℝ the probability law of random variable X_{t,x} is absolutely continuous w.r.t. Lebesgue measure with transition density p_{t}(x, ·); 2^{∘} for any bounded, measurable function f and t > 0 the mapping x → 𝔼{f(X_{t,x})} is continuously

differentiable and

\frac{∂}{∂x}𝔼\{f(X_{t,x})\} = t^{−1} 𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}\};  (7)

3^{∘} for each y ∈ ℝ and t > 0 transition density p_{t}(x, y) is differentiable w.r.t. initial condition x and

\frac{∂}{∂x}p_{t}(x, y) = t^{−1} 𝔼\{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x} = y\} p_{t}(x, y).  (8)

Proof. 1^{∘} Taking ũ_{s} = a(X_{s,x})^{−1}DX_{s,x}, one has, from (4)

DũX_{t,x} = DX_{t,x} t.  (9)

Process {ũ} is Ito integrable, so for a continuously differentiable function with compact support f it follows by integration by parts formula and relation (9), that

𝔼{f′(X_{t,x})DX_{t,x} t} = 𝔼{Dũ(f(X_{t,x}))} = 𝔼{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}}.  (10)

Now, DX_{t,x} is a.s. positive random variable and its inverse (DX_{t,x})^{−1} is in L_{p} for each p > 1. Also the assumptions we have made on coefficient a assure, via a Burgholder theorem, that ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s} is in L_{2}, hence by Lemma 1 assertion 1^{∘} holds. 2^{∘} First, take f ∈ ℂ_{b}^{1}. Then by [2] and relation (10) one has

\frac{∂}{∂x}𝔼\{f(X_{t,x})\} = 𝔼\{f′(X_{t,x})DX_{t,x})\} = t^{−1}𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}\}.

Now, for x, y ∈ ℝ we have

𝔼\{f(X_{t,y})\} − 𝔼\{f(X_{t,x})\} = ∫_{x}^{y} 𝔼\{f(X_{t,z}) ∫_{0}^{t} a(X_{s,z})^{−1}DX_{s,z} dB_{s}\} \frac{dz}{t} .  (11)

Both sides of relation (11) are continuous linear functionals on the set M_{b} of measurable and bounded functions, so we can conclude that (11) is valid for each f ∈ M_{b}. Also, for f ∈ M_{b} and t > 0 mapping x → 𝔼{f(X_{t,x})} is absolutely continuous.

If we define semigroup {P_{t}, t ≥ 0} of operators on the space M_{b} of bounded measurable functions as P_{t}f(x) = 𝔼{f(X_{t,x})}, then Markov property of diffusion {X_{t,x}} yields, for δ > 0

P_{t+δ}f(x) = 𝔼{P_{δ}f(X_{t,x})}.

Substituting t with t + δ and f with P_{δ}f, relation (11) reads as

P_{t+δ}f(y) − P_{t+δ}f(x) = ∫_{x}^{y} 𝔼\{P_{δ}f(X_{t,z}) ∫_{0}^{t} a(X_{s,z})^{−1}DX_{s,z} dB_{s}\} \frac{dz}{t} .

Then, from continuity of the function P_{δ}f(x) it follows that for each x ∈ ℝ there exists

lim_{y→x} \frac{P_{t+δ}f(y) − P_{t+δ}f(x)}{y − x}, and

\frac{∂}{∂x}P_{t+δ}f(x) = t^{−1} 𝔼\{P_{δ}(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}\}

hence \frac{∂}{∂x}P_{t+δ}f(x) is continuous function in x.

Now relation (11) yields that, for each x

\frac{∂}{∂x}P_{t}f(x) = t^{−1} 𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x} dB_{s}\}.

3^{∘} Let us mark

W_{t,x} = t^{−1} ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}.  (12)

Then relation (10) reads as

𝔼{f′(X_{t,x})DX_{t,x}} = 𝔼{f(X_{t,x})W_{t,x}},

or, having in mind 1^{∘},

∫_{−∞}^{∞} f′(u) 𝔼{DX_{t,x}|X_{t,x} = u}p_{t}(x, u) du = ∫_{−∞}^{∞} f(u) 𝔼{W_{t,x}|X_{t,x} = u}p_{t}(x, u) du.

Taking, for ϵ > 0 and y ∈ ℝ,

f_{ϵ}(x) = ∫_{−∞}^{x} e^{−(t−y)^{2}/2ϵ} \frac{dt}{\sqrt{2πϵ}},

we get, using partial integration

∫_{−∞}^{∞} e^{−(u−y)^{2}/2ϵ} 𝔼\{DX_{t,x}|X_{t,x} = u\}p_{t}(x, u) \frac{du}{\sqrt{2πϵ}} =

= ∫_{−∞}^{∞}(∫_{−∞}^{u} e^{−(t−y)^{2}/2ϵ} \frac{dt}{\sqrt{2πϵ}})𝔼\{W_{t,x}|X_{t,x} = u\}p_{t}(x, u) du =

= − ∫_{−∞}^{∞} e^{−(t−y)^{2}/2ϵ}(∫_{−∞}^{t} 𝔼\{W_{t,x}|X_{t,x} = u\}p_{t}(x, u) du)\frac{dt}{\sqrt{2πϵ}}. (13)

¿From relation (11) follows, via [3, Theorem 1.28.], that 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) is continuous and bounded function so letting ϵ ↓ 0 one can conclude that

𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) = − ∫_{−∞}^{y} 𝔼{W_{t,x}|X_{t,x} = u}p_{t}(x, u)du = −𝔼{W_{t,x}𝕀{X_{t,x} ≤ y}}.  (14)

It is obvious that there exist

lim_{y→±∞} 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y)

and as 𝔼{DX_{t,x}|X_{t,x} = ·}p_{t}(x, ·) is nonnegative, integrable function it follows that

lim_{y→±∞} 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) = 0.

Relation (7) implies that

𝔼\{W_{t,x}𝕀\{X_{t,x} ≤ y\}\} = \frac{∂}{∂x}𝔼\{𝕀\{X_{t,x} ≤ y\}\} = \frac{∂}{∂x}ℙ\{X_{t,x} ≤ y\},

so relation (14) read as

𝔼\{DX_{t,x}|X_{t,x} = y\}p_{t}(x, y) = −\frac{∂}{∂x}ℙ\{X_{t,x} ≤ y\}.  (15)

Now, as ℙ{X_{t,x} ≤ y} is continuously differentiable in x, we have for x_{1} < x_{2}

ℙ{X_{t,x_{2}} ≤ y} − ℙ{X_{t,x_{1}} ≤ y} = − ∫_{x_{1}}^{x_{2}} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz
= ∫_{x_{1}}^{x_{2}} ∫_{−∞}^{y} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dudz,

and by Fubini theorem for almost all u function 𝔼{W_{t,·}|X_{t,·} = u}p_{t}(·, u) is integrable and

∫_{x_{1}}^{x_{2}} ∫_{−∞}^{y} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dudz = ∫_{−∞}^{y} ∫_{x_{1}}^{x_{2}} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dzdu.

Finally, as ℙ{X_{t,x} ≤ y} = ∫_{−∞}^{y} p_{t}(x, u) du we have that, for each y ∈ ℝ,

∫_{−∞}^{y} p_{t}(x_{2}, u) du − ∫_{−∞}^{y} p_{t}(x_{1}, u) du = ∫_{−∞}^{y} ∫_{x_{1}}^{x_{2}} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dzdu

so for almost all y we have

p_{t}(x_{2}, y) − p_{t}(x_{1}, y) = ∫_{x_{1}}^{x_{2}} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u)dz

hence p_{t}(x, y) is absolutely continuous function in x and

\frac{∂}{∂x}p_{t}(x, y) = 𝔼\{W_{t,x}|X_{t,x} = y\} p_{t}(x, y). □

As an appendix to the previous theorem we shall study in detail partial derivative \frac{∂}{∂x}p_{t}(x, y). Precisely, we shall show that the process t^{−1}𝔼{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x}} is a backwards martingale.

As {X_{t,x}, t ≥ 0} is a Markov process, for a f ∈ M_{b} and s < t we have

𝔼_{x}{f(X_{t})} = 𝔼_{x}{𝔼{f(X_{t})|F_{s}}} = 𝔼_{x}{𝔼_{X_{s}}{f(X_{t−s})}},

where {F_{t}, t ≥ 0} is standard filtration generated by {B_{t}, t ≥ 0}, and 𝔼_{x} is expectation operator generated by initial value X_{0} = x. Now, 1^{∘} of Theorem 3 yields

\frac{∂}{∂x}𝔼_{x}\{f(X_{t})\} = 𝔼_{x}\{f(X_{t})W_{t}\} = 𝔼_{x}\{f(X_{t})𝔼\{W_{t}|X_{t}\} \} as well as
\frac{∂}{∂x}𝔼_{x}\{f(X_{t})\} = \frac{∂}{∂x}𝔼_{x}\{𝔼_{X_{s}}\{f(X_{t−s})\}\} = 𝔼_{x}\{𝔼_{X_{s}}\{f(X_{t−s})\} W_{s} \} =
= 𝔼_{x}{ 𝔼_{X_{s}}{f(X_{t−s})} 𝔼{W_{s}|X_{s}} } = 𝔼_{x}{ f(X_{t}) 𝔼{W_{s}|X_{s}} },

hence, for s < t

𝔼{W_{t,x}|X_{t,x}} = 𝔼{ 𝔼{W_{s,x}|X_{s,x}} |X_{t,x}}.

If we mark

W̃_{t,x} = 𝔼{W_{t,x}|X_{t,x}}

then previous relation could be rewritten as

W̃_{t,x} = 𝔼{W̃_{s,x}|X_{s,x}}, s < t.

Let us define backwards flow of σ-algebras as

F^{←}_{t,x} = σ{X_{u,x}, u ≥ t}, t > 0, x ∈ ℝ,

and let for ϕ(x_{1}, …, x_{n}) be n-dimensional bounded Borel function. For s < t ≤ t_{1} < ⋯ < t_{n} we have

𝔼_{x}{W̃_{s,x}ϕ(X_{t_{1},x}, …, X_{t_{n},x})} = 𝔼_{x}{ 𝔼{W̃_{s,x}ϕ(X_{t_{1},x}, …, X_{t_{n},x})|F_{t}} } =
= 𝔼_{x}{ W̃_{s,x} 𝔼{ϕ(X_{t_{1},x}, …, X_{t_{n},x})|F_{t}} } =
= 𝔼_{x}{ W̃_{s,x} 𝔼_{X_{t,x}}{ϕ(X_{t_{1}−t,x}, …, X_{t_{n}−t,x})} } =
= 𝔼_{x}{ 𝔼{W̃_{s,x}|X_{t,x}} 𝔼_{X_{t,x}}{ϕ(X_{t_{1}−t,x}, …, X_{t_{n}−t,x})} } =
= 𝔼_{x}{ 𝔼{W̃_{s,x}|X_{t,x}} ϕ(X_{t_{1},x}, …, X_{t_{n},x}) },

hence

𝔼{W̃_{s,x}|F^{←}_{t,x}} = 𝔼{W̃_{s,x}|X_{t,x}} = W̃_{t,x}, i.e.

process

W̃_{t,x} = t^{−1}𝔼{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x}}

is a backward martingale, relative to flow of σ-algebras

F^{←}_{t,x} = σ{X_{u,x}, u ≥ t}, t > 0, x ∈ ℝ.

Having in mind the definition of W̃_{t,x} and 3^{∘} of Theorem 3, we deduce

∫_{−∞}^{∞}|\frac{∂}{∂x}p_{t}(x, y)|dy = 𝔼\{|W̃_{t,x}|\} < ∞.

Moreover, for s < t

\begin{aligned}∫_{−∞}^{∞}|\frac{∂}{∂x}p_{t}(x, y)|dy & = 𝔼\{|W̃_{t,x}|\} = 𝔼\{ |𝔼\{W̃_{s,x}|F^{←}_{t,x}\}| \} \\ & ≤ 𝔼\{ 𝔼\{|W̃_{s,x}||F^{←}_{t,x}\} \}= 𝔼\{ |W̃_{s,x}| \}= \end{aligned}

= ∫_{−∞}^{∞}|\frac{∂}{∂x}p_{s}(x, y)|dy

We have proved the following Corollary: Corollary 1. For each t > 0 and x ∈ ℝ process

W̃_{t,x} = t^{−1}𝔼{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x}}

is a backward martingale, relative to flow of σ-algebras

F^{←}_{t,x} = σ{X_{u,x}, u ≥ t}, t > 0, x ∈ ℝ.

Consequently, function

∫_{−∞}^{∞}|\frac{∂}{∂x}p_{t}(x, y)|dy

is finite and decreasing with t. □

POINT CUT AS SPATIAL HOMEOMORPHISM

It is well known [10, Theorem 2.20] that, for each t > 0, the mapping x → X_{t,x} is a.s. homeomorphism of ℝ onto ℝ, hence we can define inverse random transformation

X_{t}^{−1}(·) : ℝ → ℝ

such that

X_{t}^{−1}(X_{t,x}) = x, x ∈ ℝ.

In the next theorem we shall investigate the distribution of random variable X_{t}^{−1}(y), i.e. the random function X_{t}^{−1}(·) applied to the point y.

Theorem 4. Let everything be as in Theorem 3. Then:

1^{∘} for each t > 0 and y ∈ ℝ function 𝔼{DX_{t,·}|X_{t,·} = y}p_{t}(·, y) is integrable and

∫_{−∞}^{+∞} 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) dx = 1;  (16)

2^{∘} for each t > 0 and x, y ∈ ℝ

ℙ{X_{t}^{−1}(y) ≤ x} = ∫_{−∞}^{x} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz;  (17)

3^{∘} probability density

ρ_{t}(y, x) = 𝔼{DX_{t,x}|X_{t,x} = y} p_{t}(x, y)  (18)

of the random variable X_{t}^{−1}(y) satisfy Chapman-Kolmogorov equations:

ρ_{t+s}(y, x) = ∫_{−∞}^{+∞} ρ_{s}(y, z) ρ_{t}(z, x) dz  (19)

for t, s > 0 and x, y ∈ ℝ.

Proof. 1^{∘} As for each t > 0, X_{t,·} is a.s. homeomorphism of ℝ onto ℝ and DX_{t,x}, partial derivative in L_{2}(Ω) sense of random variable X_{t,x} w.r.t. x, is a.s. positive, we have

lim_{x→±∞} X_{t,x} = ±∞,

and from relation (15) we have, for x_{1} < x_{2}

ℙ{X_{t,x_{2}} ≤ y} − ℙ{X_{t,x_{1}} ≤ y} = − ∫_{x_{1}}^{x_{2}} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz.  (20)

Now, letting x_{1} → −∞ and x_{2} → +∞ one can conclude that

∫_{−∞}^{+∞} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz = 1,

what was to be proved in 1^{∘}. 2^{∘} ¿From the fact that a.s. X_{t,x} is homeomorphism and that X_{t}^{−1}(y) is its inverse we have the following equivalence

X_{t}^{−1}(y) ≤ x ⇔ X_{t,x} ≥ y,

so, using relation (20), with x_{1} = −∞ and x_{2} = x

ℙ{X_{t}^{−1}(y) ≤ x} = ℙ{X_{t,x} ≥ y} = 1 − ℙ{X_{t,x} < y} = ∫_{−∞}^{x} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y)dz.

3^{∘} Denote with ℙ_{x}{·} the probability measure generated by the Markov process {X_{t,x}, t ≥ 0} starting from x. Then for t, s > 0 the Markov property yields

ℙ{X_{t+s,x} ≤ y} = ℙ_{x}{X_{t+s} ≤ y} = ∫_{−∞}^{+∞} ℙ_{z}{X_{s} ≤ y}p_{t}(x, z) dz = 𝔼{ℙ_{X_{t,x}}{X_{s,z} ≤ y}}.

Using relation (15) we have

ρ_{t+s}(y, x) = −\frac{∂}{∂x}ℙ\{X_{t+s,x} ≤ y\} = −\frac{∂}{∂x}𝔼\{ℙ_{X_{t,x}}\{X_{s,z} ≤ y\}\} =

= −𝔼\{\frac{∂}{∂x}ℙ_{X_{t,x}}\{X_{s,z} ≤ y\} DX_{t,x}\} =

= −𝔼{−𝔼_{X_{t,x}}{DX_{s}|X_{s} = y}p_{s}(X_{t,x}, y) 𝔼{D_{t,x}|X_{t,x}}} =

= ∫_{−∞}^{+∞} 𝔼_{z}{DX_{s}|X_{s} = y}p_{s}(z, y) 𝔼{D_{t,x}|X_{t,x} = z}p_{t}(x, z) dz =

= ∫_{−∞}^{+∞} 𝔼{DX_{s,z}|X_{s,z} = y}p_{s}(z, y) 𝔼{D_{t,x}|X_{t,x} = z}p_{t}(x, z) dz =

= ∫_{−∞}^{+∞} ρ_{s}(y, z) ρ_{t}(z, x) dz,

and the theorem is completely proved. □

References

1. N. Bouleau and F. Hirsch, Formes de Dirichlet generales et densite des variables aleatoires reelles sur l’espace de Wiener., J. Func. Anal. 69 (1986), 229–259.

2. I.I. Gihman and A.V. Skorohod, Stochastic differential equations, Naukova Dumka, Kiev, 1968. (Russian)

3. S. Kusuoka and D. Stroock, Applications of the Malliavin Calculus. I , Proc of the Taniguchi Inter. Symp. on Stoch. Analysis. Katata, North-Holland, 1982, pp. 271–308.

4. P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976, pp. 195–263.

5. D. Nualart, Noncausal stochastic integrals and calculus, Proc. of Workshop on Stoch. Anal. and Rel. Topics. Silivri 1986, 1988, pp. 80–129.

6. D. Nualart and M. Zakai, Generalized stochastic integrals and Malliavin Calculus, Probability Theory and Rel. Fields 73 (1986), 255–280.

7. D. Pantić, Stochastic Calculus on Distorted Brownian Motion, J.Math.Phys. 29 (1988), 207-210.

8. D. Pantić, Stochastic Calculus and Schroedinger Semigroups Admitting Ground State, J. Math. Phys. 31 (1990), 1221-1225.

9. D. Pantić, Toward the Existence of Density for the Distribution of a Brownian Functional, submitted for publication.

10. D. W. Stroock, Topics in stochastic differential equations, Tata Institute of Fundamental Research, 1982.

11. S. Watanabe, Lectures on stochastic differential equations and Malliavin Calculus, Tata Institute of Fundamental Research, 1984.

12. M. Zakai, The Malliavin Calculus, Acta Appl. Math 3 (1985), 175–207.

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. 01, 1995, pp. 27–29. Submitted: July 19, 1995. Published: September 18, 1995.

A LOGICAL ALTERNATIVE TO THE

EXISTING POSITIONAL NUMBER SYSTEM

Robert R. Forslund

Abstract. This article introduces an alternative positional number system. The advantages of this alternative system over the existing one are discussed, and an illustration of the use of the system to re-interpret apparent errors in ancient archaeological documents is presented.

(1991) A.M.S. (MOS) Subject Classification Codes. 11, 01.

Key Words and Phrases. positional number system, archaeology, zero, placeholder.

Introduction

To simplify this article, the history of positional number systems will not be discussed, and interested readers can find this information in the references listed. In addition, the discussion is restricted to the set of positive integers; however, the extension to decimal and negative values, if necessary, is easily accomplished.

The existing positive integer number system in a selected base (b), is represented by the following mathematical notation :

N = ∑(a_{j} ∗ b^{j}) where 0 ≤ a_{j} ≤ (b − 1), b > 1, and the sum is over j

with 0 ≤ j ≤ (n − 1), and a_{j} are the n digits of the positive integer N in base b.

For example:

10203 = 1 ∗ 10^{4} + 0 ∗ 10^{3} + 2 ∗ 10^{2} + 0 ∗ 10^{1} + 3

Traditionally we have selected base : b = 10, as in the example, and commonly use other bases such as binary (b = 2), octal (b = 8) and hexadecimal (b = 16). Ancient civilizations introduced other bases such as 60 used by the Babylonians [1], and 18 and 20 used by the Mayans [1].

The alternative system

There is a problem with the existing number system. Base 1 is illogical since the digits are equal to zero, and therefore we must select base, b > 1 when defining the system. In addition, the digit integers starting at one are concrete concepts, while the concept of zero (nothingness) is abstract. Therefore, I sought a system that did not require the use of zero as a placeholder. As you will see shortly, there is such a system, and the concept of a zero placeholder adds unnecessary complexity.

As a result, I reasoned as follows. First of all, and most importantly, why not use digits from 1 to b rather than 0 to (b − 1)? In base 10, this simply means introducing a 10th symbol for the digit “ten”. Any symbol will do, however the symbol “A” borrowed from the hexadecimal system provides some familiarity. This is the only change to the notation used in the introduction for N : viz : 1 ≤ a_{j} ≤ b and b ≥ 1. This alternative system has no need for a placeholder (zero) as I illustrate below. I will use bolded symbols for digits in the alternative system to distinguish between the two systems.

Ministry of Natural Resources Centre for Northern Forest Ecosystem Research c/o Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1

E-mail Address: mjohnsto@sky.Lakehead.ca

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University

In the alternative system (base A), the number 10 is now represented by the digit A. The number 20 now becomes : 1A = 1 ∗ A + A, and 90 is 8A. The number 100 now becomes : 9A = 9 ∗ A + A, and 1000 = 99A = 9 ∗ A^{2} + 9 ∗ A + A = 900 + 90 + 10, and so on. Notice that, with the exception of integers that have digits equal to zero in the existing system, all other numbers remain the same as in the alternative system as long as the digits lie between 1 and (b − 1) inclusive. Therefore the smaller the base, the more the differences will be between the two systems. Base 2 for example will remain the same as the existing system only when all digits are 1. An example in base 2 is : 21 = 10101 in the existing system, and 1221 in the alternative system. In comparison, the number 31 in both systems remains at 11111 in base 2.

Advantages of the alternative system

Base 1 is now permissible and completely logical and is simply a tally system. For example, 7 = 1111111 (base 1) in the alternative system. Secondly, for number theorists, the re-arrangement of digits in the existing system causes digits to be lost. For example the reversal of 12300 in the existing system is 321 and two zero digits are “lost”. In the alternative system this does not occur. For example, 12300 = 1229A, which is A9221 upon reversal. As this simple example illustrates, this alternative system lends itself to further research in number theory. In addition, in the existing system, powers of the base are identified by the number of zeros in the number - for example 10000 = 10^{4}. Powers are similar in the alternative system and are identified as in this example by 999A = A^{4}. More generally, in base b, powers = … (b − 1)(b − 1) … (b − 1)b. Notice that fewer digits are often needed to represent the numbers - this too is an advantage, and one of the main reasons positional notation was first introduced. Also, in base 10 for example, instead of having 9 numbers with one digit, 90 numbers with 2 digits, and so on, the alternative system now has the first 10 numbers having a single digit, the next 100 numbers having 2 digits, and so on. This too is simpler since the number of integers with n digits in base b, is b^{n} rather than b^{n} − b^{(n−1)} in the existing system.

Application of the alternative system to archaeology

It appears that this alternative system would have been a slightly more logical first step in the development of a number system in ancient times. One possible use of this system might be in the re-interpretation of ancient calculations found in archaeological documents. History tends to identify the introduction of the zero placeholder into the number system, as an advancement in mathematics. However, the alternative system, with its inherent simplicity, illustrates that the introduction of zero may have unnecessarily complicated the use of integers, and numbers in general.

Perhaps by re-interpreting ancient mathematical documents using this alternative system, mathematical archaeologists will reveal that some of the ancient calculations, previously thought to be in error, really were correct. I will illustrate using the commonly used digit symbols in base ten, realizing that ancient documents used other symbols for digits. The ancients may have used 10 symbols (if base 10 was used); however, the two symbols “10” may really have meant 1A - which is twenty (not ten). In other words, it may be that the zero symbol represented the digit ten - not zero. To detect such discrepancies, ancient documents must contain calculations such as addition. For example : 10 + 11 = 31 - this appears to be incorrect. However, if the symbol “0” actually represents the digit ten (A) and not zero, then the equation is correct and equivalent to 1A + 11 = 31 = 20 + 11 in the existing system. Another example : 101 + 9 = 100. Once again, this appears incorrect, however, if “0” symbol is re-interpreted as “A”, then this is: 1A1 + 9 = 1AA, which is 201 + 9 = 210 in the existing system. In base 10, these apparent errors would only exist when the symbol for ten (and interpreted as zero) occurred in the digits of numbers. Larger bases such as 60 used by the Mesopotamians [1], would be even more void of such “errors”, and therefore it would be harder to determine if the alternative number system was used. I leave it to archaeologists specializing in the interpretation of these ancient documents to examine this usage of the alternative positional number system.

In summary, the alternative system appears to be a slightly more logical system relative to the existing one. As illustrated, it has immediate application to the field of archaeology. Readers should have little problem finding other fields for the application of this system especially in the theory of numbers. If this article motivates others to explore the possibilities of this alternative system, then it has succeeded.

References

1. Boyer, C.B. and Merzbach, U.T. : A History of Mathematics (Second Edition), John Wiley & Sons Inc., New York, 1991.

2. Hellemans, A. and Bunch, B. : The Timetables of Science - A Chronology of the Most Important People and Events in the History of Science, Simon & Schuster Inc., New York, 1991.

3. Swetz, F.J. : From Five Fingers to Infinity, Open Court, Chicago and La Salle Illinois, 1995.

Acknowledgements

I wish to thank Dr. Mark Johnston and Mr. Luke Dalla Bona at the Centre for Northern Forest Ecosystem Research for providing me with access to the use of electronic communication via the Internet and for their encouragement and comments regarding this paper.

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. 01, 1995, pp. 30-36. Submitted: May 12, 1995. Published: October 16, 1995.

RESULTS ON CONTROLLING THE RESIDUALS OF

PERTURBED NEWTON-LIKE METHODS ON BANACH

SPACES WITH A CONVERGENCE STRUCTURE

Ioannis K. Argyros

Abstract. In this study, we provide sufficient conditions for controlling the residuals of perturbed Newton-like methods, in such a way that the convergence of such methods is ensured. In order to achieve this, we apply a theorem due to Kantorovich on a setting of a regular space with a convergence structure.

(1991) A.M.S. (MOS) Subject Classification Codes. 65J15, 65B05, 47D15, 47H17, 49D15.

Key Words and Phrases. Newton-like methods, residuals.

I. INTRODUCTION

In this study, we are concerned with approximating a solution x^{∗} of the nonlinear operator equation

F (x) + Q(x) = 0,  (1)

where F is a Frechet-differentiable operator defined on a convex subset D of a Banach space X with values in X, and Q is a non-differentiable nonlinear operator with the same domain and values in X.

We generate a sequence {x_{n}} (n ≥ 0) using the perturbed Newton-like method scheme given by

x_{n+1} = x_{n} + δ_{n} (n ≥ 0)  (2)

where the correction δ_{n} satisfies

A(x_{n})δ_{n} = −(F (x_{n}) + Q(x_{n})) + r_{n} (n ≥ 0)  (3)

for a suitable r_{n} ∈ X (n ≥ 0) called residual. The importance of studying perturbed Newton-like methods comes from the fact that variants of Newton’s method can be considered as procedures of this type. In fact, the approximations (2) and (3) characterize any iterative process in which the corrections are taken as approximate solutions of the Newton equations. This happens when approximation (2) is solved by any iterative method or when therein the derivative is replaced by a suitable approximation. In [4] we provided sufficient conditions for the convergence of iteration (2) to a solution x^{∗} of equation (1), by assuming that X is a Banach space with a convergence structure (to be precised later).

Here we derive sufficient conditions for controlling the residuals r_{n} in such a way that the convergence of the sequence {x_{n}} (n ≥ 0) to a solution of equation (1) is ensured.

We also refer the reader to [5], [6], [8] and the references there for relevant work, which however is valid on a Banach space X without a convergence structure. The advantages of working on a Banach space with a convergence structure have been explained in some detail in [4], [6], [7].

Department of Mathematics, Cameron University, Lawton, OK 73505, U.S.A.

E-mail Address: ioannisa@cua.cameron.edu

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University

II. PRELIMINARIES

We will need the definitions:

\underline{Definition 1}. The triple (X, V, E) is a Banach space with a convergence structure if (C_{1}) X, ∥ · ∥) is a real Banach space; (C_{2}) (V, C, ∥ ∥_{v}) is a real Banach space which is partially ordered by the closed convex cone C; the norm

∥ ∥_{v} is assumed to be monotone on C; (C_{3}) E is a closed convex cone in X × V satisfying {0} × C ⊆ E ⊆ X × C; (C_{4}) the operator | | : D → C is well defined:

|x| = inf{q ∈ C|(x, q) ∈ E}

for

x ∈ D = {x ∈ X|∃q ∈ C : (x, q) ∈ E};

and (C_{5}) for all x ∈ D ∥x∥ ≤ ∥|x|∥_{v}.

The set

U(a) = {x ∈ X|(x, a) ∈ E}

defines a sort of generalized neighborhood of zero.

\underline{Definition 2}. An operator L ∈ C^{1}(V_{1} → V) defined on an open subset V_{1} of an ordered Banach space V is order convex on [a, b] ⊆ V_{1} if

c, d ∈ [a, b], c ≤ d ⇒ L′(d) − L′(c) ∈ L_{+}(V),

where for m ≥ 0

L_{+}(V^{m}) = {L ∈ L(V^{m})|0 ≤ x_{i} ⇒ 0 ≤ L(x_{1}, x_{2}, …, x_{m})}

and L(V^{m}) denotes the space of m-linear, symmetric, bounded operators on V.

\underline{Definition 3}. The set of bounds for an operator H ∈ L(X^{m}) is defined to be

B(H) = {L ∈ L_{+}(V^{m})|(x_{i}, q_{i}) ∈ E ⇒ (H(x_{1}, …, x_{m}), L(q_{1}, …, q_{m})) ∈ E}.

\underline{Definition 4}. A partially ordered topological space V is called regular if every order bounded increasing sequence has a limit.

We will need the following proposition due to Kantorovich [3], [5].

Proposition. Let V be a regular partially ordered topological space (POTL-space) and let x, y be two points of V such that x ≤ y. If H : [x, y] → V is a continuous isotone operator having the property that x ≤ Hx and y ≥ Hy, then there exists a point z ∈ [x, y] such that z = Hz.

III. CONVERGENCE ANALYSIS

We will need the following basic result:

\underline{Lemma}. Let V be a regular partially ordered topological space, an operator L ∈ C^{1}(V_{1} → V) with [0, a] ⊆ V_{1} ⊆ V for some a ∈ V, an operator M ∈ C(V_{1} → V), operators R < I, B > I, T, K ∈ L_{+}(V), a point c ∈ V with c > 0 and a point p ∈ [0, a]. Assume:

(a) The equation

g(q) = BT [L(p + q) − L(p) − L′(p)q + M(p + q) − M(p) + K(p + q) − K(p)]
− (I − R)q + c = 0  (4)

has solutions in the interval [0, a] and denote by q^{∗} the least of them.

(b) Let G ∈ L_{+}(V) be given and R_{+} ∈ L_{+}(V), c_{+}, p_{+} ∈ V be such that the following conditions are satisfied:

R_{+} < min {(G − 2I)BT L′(p) + BT GL′(p_{+}) + G(R − I) + I, I} = α  (5)
0 < c_{+} ≤ BT G(L(p_{+}) + M(p_{+}) + K(p_{+})) + GBT (L(p) + M(p) + K(p))
−2BT (L(p) + M(p) + K(p)) + (R_{+} + G − I − BT GL′(p_{+}))c = β  (6)

and

0 ≤ p_{+} ≤ p + c,  (7)

where α and β are functions of the operators and points involved.

(c) The following estimate is true

M(p) ≤ M(p + q)  (8)

for all p, q ∈ [0, a].

Then the equation

g_{+}(q) = BGT [L(p_{+} + q) − L(p_{+}) − L′(p_{+})q + M(p_{+} + q) − M(p_{+}) + K(p_{+} + q) − K(p_{+})]
− (I − R_{+})q + c_{+} = 0  (9)

has nonnegative solutions and the least of them, denoted by q_{+}^{∗} lies in the interval [c_{+}, q^{∗} − c].

Proof. Using the hypothses g(q^{∗}) = 0 and R < I we deduce from (4) that c ≤ q^{∗}. We will show that

g_{+}(q^{∗} − c) ≤ 0.  (10)

From equation (9) and using (6), we obtain in turn

g_{+}(q^{∗} − c) ≤ BGT [L(p + c + q^{∗} − c) − L(p_{+}) − L′(p_{+})(q^{∗} − c)
+ M(p + c + q^{∗} − c) − M(p_{+}) + K(p + c + q^{∗} − c) − K(p_{+})] − (I − R_{+})(q^{∗} − c) + c_{+}
= g(q^{∗}) − BT [L(p + q^{∗}) − L(p) − L′(p)q^{∗} + M(p + q^{∗}) − M(p) + K(p + q^{∗}) − K(p)]

+ BT G[L(p + q^{∗}) − L(p_{+}) − L′(p_{+})(q^{∗} − c) + M(p + q^{∗}) − M(p_{+}) + K(p + q^{∗}) − K(p_{+})]

+ (I − R)q^{∗} − c − (I − R_{+})(q^{∗} − c) + c_{+}
= [BT L′(p) − BT GL′(p_{+}) + (I − R) − (I − R_{+})]q^{∗}
+ (G − I){(I − R)q^{∗} − c − BT [L(p) + M(p) + K(p) + L′(p)q^{∗}]}
+ BT [L(p) + M(p) + K(p) − GL(p_{+}) − GM(p_{+})]
+ BT GL′(p_{+})c + (I − R_{+})c + c_{+} − c
= [(2I − G)BT L′(p) − BT GL′(p_{+}) + R_{+} + G(I − R) − I]q^{∗}
+ 2BT (L(p) + M(p) + K(p)) − BT G(L(p_{+}) + M(p_{+}) + K(p_{+}))
− GBT (L(p) + M(p) + K(p)) + (BT GL′(p_{+}) − G + I − R_{+})c + c_{+} ≤ 0

since (5) and (6) are satisfied. Moreover from (9) for q = c_{+}, we obtain

g_{+}(c_{+}) ≥ 0.  (11)

By inequalities (10), (11), the fact that g is continuous and isotone on [c_{+}, q^{∗} − c], and since V is a regular partially ordered topological space, from the proposition, we deduce that there exists a point q_{+}^{∗} with

g_{+}(q_{+}^{∗}) = 0  (12)

and

c_{+} ≤ q_{+}^{∗} ≤ q^{∗} − c.  (13)

We can assume that q_{+}^{∗} denotes the least of the solutions of equation (9).

That completes the proof of the lemma.

The following result is a consequence of the above lemma.

\underline{Theorem 1}. Let \{c_{n}\} ∈ V, \{T_{n}\}, \{R_{n}\}, \{G_{n}\} ∈ L_{+}(V) (n ≥ 0) be sequences and V as in the above lemma. Assume:

(a) There exists a sequence {p_{n}} ∈ [0, a] ⊆ V_{1} ⊆ V for some a ∈ V with p_{0} = 0 and

p_{n+1} ≤ ∑_{j=0,1,...,n} c_{j} for n ≥ 0.  (14)

(b) R_{0} < I and the function

g_{0}(q) = BT_{0}[L(p_{0} + q) − L(p_{0}) − L′(p_{0})q + M(p_{0} + q) − M(p_{0}) + K(p_{0} + q) − K(p_{0})]
− (I − R_{0})q + c_{0} = 0  (15)

has roots on [0, a], where B, L, M, K are as in the above lemma. Denote by q_{0}^{∗} the least of them.

(c) The following conditions are satisfied for all n ≥ 0

R_{n+1} < α_{n+1},  (16)
0 < c_{n+1} ≤ β_{n+1}  (17)

and

0 ≤ p_{n+1} ≤ p_{n} + c_{n}.  (18)

(d) The linear operators T_{n} are boundedly invertible for all n ≥ 0, and set G_{n} = T_{n+1}T_{n}^{−1} (n ≥ 0).

(e) Condition (8) is satisfied.

Then, the equation

g_{n}(q) = BG_{n}T_{n}[L(p_{n} + q) − L(p_{n}) − L′(p_{n})q + M(p_{n} + q) − M(p_{n}) + K(p_{n} + q) − K(p_{n})]
− (I − R_{n})q + c_{n} = 0  (19)

has solutions in [0, a] for every n ≥ 0 and denoting by q_{n}^{∗} the least of them, we have

∑_{j=n,...,∞} c_{j} ≤ q_{n}^{∗} (n ≥ 0).  (20)

Proof. Let us assume that for some nonnegative integer n, I −R_{n} > 0, g_{n}(q) has roots on [0, a] and denote by q_{n}^{∗} the least of them. We use induction on n. We also observe that this is true by hypothesis (b) for n = 0. Using (14), (16), (17), (18), the lemma, and setting c = c_{n}, c_{+} = c_{n+1}, R = R_{n}, R_{+} = R_{n+1}, G = G_{n} we deduce that q_{n+1}^{∗} exists, and

c_{n+1} ≤ q_{n+1}^{∗} ≤ q_{n}^{∗} − c_{n}.  (21)

The induction is now complete and (20) follows immediately from (21).

That completes the proof of the theorem. From now on we assume that X is a Banach space with a convergence structure in the sense of [4].

The following result is an immediate consequence of Theorem 1.

\underline{Theorem 2}. Assume:

(a) the hypotheses of Theorem 1 are satisfied;

(b) there exists a sequence {x_{n}} (n ≥ 0) in a Banach space X with a convergence structure such that |x_{n+1} − x_{n}| ≤ c_{n}. Then

(i) the sequence {x_{n}} (n ≥ 0) converges to some point x^{∗}; (ii) moreover the following error bounds are true

|x^{∗} − x_{n}| ≤ q_{n}^{∗}  (22)

and

|x^{∗} − x_{n+1}| ≤ q_{n}^{∗} − c_{n} for all n ≥ 0.  (23)

We can introduce the main result:

\underline{Theorem 3}. Let X be a Banach space with convergence structure (X, V, E) with V = (V, C, ∥ ∥_{v}), an operator F ∈ C^{1}(D → X) with D ⊆ X, an operator Q ∈ C(D → X), an operator A(x) ∈ C(X → D), an operator L ∈ C^{1}(V_{1} → V) with V_{1} ⊆ V, an operator M ∈ C(V_{1} → V), an operator K ∈ L_{+}(V), and a point a ∈ C such that the following conditions are satisfied:

(a) the inclusions U(a) ⊆ D and [0, a] ⊆ V_{1} are true;

(b) L is order-convex on [0, a] and satisfies

K + L′(|x| + |y|) − L′(|x|) ∈ B(A(x) − F′(x + y))  (24)

for all x, y ∈ U(a) with |x| + |y| ≤ a;

(c) M satisfies the condition

0 ≤ (Q(x) − Q(x + y), M(|x| + |y|) − M(|x|) ∈ E, M(0) = 0  (25)

for all x, y ∈ U(a) with |x| + |y| ≤ a;

(d) for the sequences {c_{n}}, {T_{n}}, {R_{n}}, {G_{n}}, {p_{n}} (n ≥ 0) the hypotheses (14), (b), (16), (18) and (d) of Theorem 1 are satisfied;

(e) the following conditions are also satisfied

|δ_{n}| ≤ c_{n} ≤ T_{n}| − (F (x_{n}) + Q(x_{n}))| ≤ γ_{n} ≤ β_{n} (n ≥ 1),  (26)
| − r_{n}| ≤ T_{n}^{−1}R_{n}c_{n}  (27)

where

γ_{n} = T_{n}[L(p_{n} +c_{n})−L(p_{n})−L′(p_{n})c_{n} +M(p_{n} +c_{n})−M(p_{n})+K(p_{n} +c_{n})−K(p_{n})]+R_{n}c_{n} (n ≥ 1).

(28)

Then

(i) the sequence {x_{n}} (n ≥ 0) generated by

x_{n+1} = x_{n} + δ_{n} with x_{0} = 0

remains in \overline{U}(x_{0}, t_{0}^{∗}) and converges to a solution x^{∗} of equation F (x) = 0; (ii) moreover, the error estimates (22) and (23) are true where q_{n}^{∗} is the least root in [0, a] of the function

g_{n}(q) defined in (19), with p_{n} = ∥x_{n} − x_{0}∥ (n ≥ 0).

Proof. Let us assume that x_{n}, x_{n+1} ∈ U(x_{0}, q_{0}^{∗}), where the existence of q_{0}^{∗} is guaranteed from hypotheses (d). We note that |δ_{0}| ≤ c_{0}. Using the approximation

−(F (x_{n+1}) + Q(x_{n+1})) = (F (x_{n}) − F (x_{n+1}) + A(x_{n})(x_{n+1} − x_{n})) + (Q(x_{n}) − Q(x_{n+1})) − r_{n}, (29)

(24), (25), (27) and setting p_{n} = |x_{n} − x_{0}|, we obtain in turn

| − (F (x_{n+1}) + Q(x_{n+1}))| ≤ |F (x_{n}) − F (x_{n+1}) + A(x_{n})(x_{n+1} − x_{n})| + |Q(x_{n}) − Q(x_{n+1})| + | − r_{n}|

≤ ∫_{0}^{1} [L′(p_{n} + t|x_{n+1} − x_{n}|) − L′(p_{n}) + K]|x_{n+1} − x_{n}| dt + M(p_{n} + |x_{n+1} − x_{n}|) − M(p_{n}) + | − r_{n}|

≤ L(p_{n} + c_{n}) − L(p_{n}) − L′(p_{n})c_{n} + Kc_{n} + M(p_{n} + c_{n}) − M(p_{n}) + T_{n}^{−1}R_{n}c_{n}.

Hence, by (26) we get

c_{n+1} ≤ T_{n+1}| − (F (x_{n+1}) + Q(x_{n+1}))| ≤ γ_{n} ≤ β_{n} (n ≥ 1),

which shows (17).

It can easily be seen that by using induction on n, the hypotheses of Theorem 2 are satisfied. Hence, by (24) and (27) the iteration {x_{n}} (n ≥ 0) remains in U(x_{0}, q_{0}^{∗}) and converges to x^{∗} so that (22) and (23) are satisfied. Moreover, from the estimate

| − (F (x_{n}) + Q(x_{n}))| ≤ |A(x_{n}) − F′(x_{n})|c_{n} + |F′(x_{n})|c_{n} + | − r_{n}|,

(24), (27), the continuity of F, F′, A, T_{n}, R_{n} and c_{n} → 0 as n → ∞, we deduce that

F (x^{∗}) + Q(x^{∗}) = 0.

That completes the proof of the theorem.

IV. APPLICATIONS

We will complete this study with an example that shows how to choose L, M, in practical applications. For simplicity, we assume that A(x) = F′(x) for all x ∈ D.

Example. We discuss the case of a real Banach space with norm ∥ ∥. Assume that F′(0) = I and there exists a monotone operator

f : [0, a] → IR

such that

∥F′′(x)∥ ≤ f(∥x∥) for all x ∈ U(a)  (30)

and a continuous nondecreasing function ℓ on [0, r], r ≤ a such that

∥Q(x) − Q(y)∥ ≤ ℓ(r)∥x − y∥  (31)

for all x, y ∈ U(\frac{r}{2}).

We showed in [3], (see also [9]) that (18) implies that

∥Q(x + e) − Q(x)∥ ≤ h(r + ∥e∥) − h(r), x ∈ U(a), ∥e∥ ≤ a − r,  (32)

where

h(r) = ∫_{0}^{r} ℓ(t) dt.  (33)

Conversely, it is not hard to see that we may assume, without loss of generality, that the function h and all functions h(r + t) − h(r) are monotone in r. Hence, we may assume that h(r) is convex and hence differentiable from the right. Then, as in [9], we show that (32) implies (31) and ℓ(r) = h′(r + 0). Hence, we can set

L_{1}(q) = L(q) = ∥F (0) + Q(0)∥ + ∫_{0}^{q} ds ∫_{0}^{s} f(t) dt  (34)

and

M(q) = ∫_{0}^{q} ℓ(t) dt.  (35)

Define the functions f_{1}, f_{2}, f_{3} on [0, a] by

f_{1}(q) = L(q) − q, f_{2}(q) = f_{1}(q) + h(q) and f_{3}(q) = ∫_{0}^{q} f(t) dt.

Choose B = T_{0} = 1, R_{0} = K = 0 and p_{0} = 0. Then by (15) we get

g_{0}(q) = f_{2}(q).

It can easily be seen that with the above choices of L and M conditions (24) and (25) are satisfied. Suppose that the function g_{0} has a unique zero q_{0}^{∗} in [0, a] and g_{0}(a) ≤ 0. It is then known [3], [9] that

equation (1) admits a solution x^{∗} in U(q_{0}^{∗}), this solution is unique in U(a), and the iteration {x_{n}} (n ≥ 0) given by (2) is well defined, remains in U(q_{0}^{∗}) for all n ≥ 0 and converges to x^{∗}. By applying the Banach lemma on invertible operators we can show that A(x_{n}) is invertible and ∥A(x_{n})^{−1}∥ ≤ −f′_{1}(∥x_{n}∥)^{−1} = T_{n} (n ≥ 0).

Moreover, suppose that (30) is satisfied. Furthermore, assume that instead of conditions (16) and (17), the weaker condition (10) is satisfied. Using (30) and the approximation

r_{n} = [(F′(x_{n}) − F′(x_{0})) + F′(x_{0})]δ_{n} + F (x_{n}) + Q(x_{n})

we obtain

∥r_{n}∥ ≤ (f_{3}(∥x_{n}∥) + 1)c_{n} + T_{n}^{−1}γ_{n} (n ≥ 1).

The above estimate provides us with a possible (but not the only) choice for the R_{n}’s. Indeed, (27) will be true if the R_{n}’s (n ≥ 0) can be chosen so that

(f_{3}(∥x_{n}∥) + 1)c_{n} + T_{n}^{−1}γ_{n} ≤ T_{n}^{−1}R_{n}c_{n} (n ≥ 1).

Using the above choices and Theorem 3, it finally also follows that estimates (22) and (23) are satisfied for all n ≥ 0.

REFERENCES

[1] Argyros, I.K. The secant method in generalized Banach spaces, Appl. Math. Comp. 39, (1990), 111–121.

[2] . Newton-like methods in generalized Banach spaces, Functiones et Approximatio Commentarii Mathematici, XXII, (1993), 13–20.

[3] Argyros, I.K. and Szidarovszky, F. The Theory and Applications of iteration methods, C.R.C. Press, Inc. Boca Raton Florida, U.S.A. 1993.

[4] Argyros, I.K. Perturbed Newton-like methods and nondifferentiable operator equations on Banach spaces with a convergence structure, Southwest Journal of Pure and Applied Mathematics (SWJPAM) Vol. 1, No. 1 (1995), 1–12.

[5] Kantorovich, L.V. and Akilov, G.P. Functional Analysis in normed spaces, Academic Press, New York, 1978.

[6] Meyer, P.W. Newton’s method in generalized Banach spaces, Numer. Funct. Anal. Optimiz. 9 (3 and 4), (1987), 249–259.

[7] . A unifying theorem for Newton’s method, Numer. Funct. Anal. Optimiz. 13 (5 and 6), (1992), 463–473.

[8] Moret, I. A Kantorovich-type theorem for inexact Newton methods, Numer. Funct. Anal. and Optimiz. 10 (3 and 4), (1989), 351–365.

[9] Zabrejko, P.P. and Nguen, D.F. The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optimiz. 9 (5 and 6), (1987), 671–684.

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 01, 1995, pp. 37-45. Submitted: June 8, 1995. Published: October 26, 1995.

A PROBABILISTIC APPROACH TO THE ESTIMATION OF THE DISTRIBUTION

OF THE TARGET LOCATION POINT

M.A. Tabatabai

Abstract. A probabilistic approach is used to estimate the probability distribution of the target location point. The results can be used in missile-testing and target acquisition fire support model simulation. The target may be a mobile multiple launcher rocket, an antiaircraft gun or a ship in mid-ocean. In each case, there is a time lag between the detection of the target and the arrival of the projectile.

(1991) A.M.S. (MOS) Subject Classification Codes. 60, 62.

Key Words and Phrases. Probability density function, Gamma function.

I. Introduction

The target location problem frequently appears in missile-testing and artillery field programs. In this article, interest focuses on the probability distribution of the target location point for a multiple launcher rocket and a Howitzer. Our main concern is to find the probability density function of a mobile MLR location which is deliberately moving so as to confound prediction of its position. Let us now describe the situation by calling the two parties Red and Blue. The mission of the Red multiple launcher rocket is to strike by firing and move as quickly as possible in a specific direction with an average speed R. The reaction of the Blue commander is to predict the target location and launch a round of artillery at an anticipated target location.

Let the initial target location point (the point at which the Red initially fires) be represented by a bivariate normal random vector (X, Y), where X denotes the horizontal coordinate, Y denotes the vertical coordinate and X and Y are assumed to be independent random variables with a mean vector

(μ_{X}, μ_{Y}) = (0, 0)

and a standard deviation vector

(σ_{X}, σ_{Y}) = (100, 100)

where the mean and standard deviation for both X and Y are in meters.

Let T denote the Blue artillery reaction time. This random variable is the total response time from the Red multiple launch rocket firing to the first rounds of the Blue artillery landing on target. The distribution of the random variable T will be derived in Section 2. The Red displacement time using daylight values ranges from 1.5 minutes to 2 minutes.

Department of Mathematical Sciences, Cameron University, Lawton, OK 73505

E-mail Address: mohammad@tiger.cameron.edu

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University

II. Distribution of Reaction Time

A sample of 103 response times from the Red firing to the first rounds of the Blue artillery landing on target is included in the analysis. The data was provided by the Fire Support Center at Cameron University. The goal of this section is to find the distribution of Blue reaction time.

Before making a specificaiton of the distribution of response time, it is a good practice to look at the data in a graphical form in order to see if they could have come from one of a known family of distributions.

One of the simplest ways of looking at the data is to make a stem-and-leaf plot. Figure 1 displays the stem-and-leaf plot for Blue reaction time (in minutes). In this display we immediately see the asymmetry of the main part of the data. The distribution is positively skewed and the figure indicates the presence of some extreme outliers. In Figure 2, we have the Boxplot. It provides a visual impression of several important aspects of the underlying distribution of the Blue reaction time. From a Boxplot we can pick out the following features of a data set:

Location Skewness Spread Tail length Outlying data points

Stem-and-leaf of Blue Re N=103 Leaf Unit=0.10

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | \begin{matrix}1246777789999 \\ 00001112222233333444555566666666666777777888999 \\ 001111222334444446668899 \\ 00014 \\ 2 \\ 24578 \\ 79 \\ 49 \\ 44 \\ 1 \\ 7 \end{matrix}

Figure 1. Stem-and-leaf Plot of Blue Reaction Time

The mean reaction time for this data set is 4.554 minutes with a standard deviation of 2.396 minutes. The data ranges from a low of 2.160 minutes to a high of 16.770 minutes. The first, second and third quartiles are 3.300, 3.740, and 4.630 respectively and the trimmed mean is 4.238 minutes.

−−−−−
−−−−I + I −−−−−− ∗ ∗ ∗ 0 0 0 0 0 0
−−−−−
−−−−−−−−+−−−−−−−−−+−−−−−−−−−+−−−−−−−−−+−−−−−−−−−+−−−−−−−−−
3 . 0  6 . 0  9 . 0  1 2 . 0  1 5 . 0

Figure 2. Boxplot of Blue Reaction Time

Let T be the Blue reaction time and M be the Red movement time. More specifically, M is the difference between Blue reaction time and Red displacement time. If we use 2 minutes for the Red displacement time, then we can write the random variable M as M = T − 2. Figure 3 displays the stem-and-leaf graph for Red movement time.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 | \begin{matrix}1246777789999 \\ 00001112222233333444555566666666666777777888999 \\ 001111222334444446668899 \\ 00014 \\ 2 \\ 24578 \\ 79 \\ 49 \\ 44 \\ 1 \\ 7 \end{matrix}

Figure 3. Stem-and-leaf Plot for Red Movement Time

To have a more reliable estimate for the parameters of underlying distrubtion of random variable M, we examine the Boxplot for the Blue reaction time. Figure 4 shows the Boxplot for M.

−−−−−  0
−−−I + I −−−−−− ∗ ∗ ∗ 0 0 0 0 0 0 0
−−−−−
−−+−−−−−−−−−+−−−−−−−−−+−−−−−−−−−+−−−−−−−−−+−−−−−−−−−−+−−−
0 . 0  3 . 0  6 . 0  9 . 0  1 2 . 0  1 5 . 0

Figure 4. Boxplot of Red Movement Time

The small hollow circles in the Boxplot indicate the presence of eight extreme outliers. There is a widespread awareness of the dangers posed by the occurrence of extreme outliers, which may be a result of keypunch errors, recording or transmission errors, misplaced decimal points, exceptional phenomena such as a lack of adequate training in firing rockets, or members of a different population slipping into the sample. Outliers occur frequently in real data. When the data contain outliers, one approach is to remove the outliers and then calculate your sample statistics. A second approach is to use robust and resistant estimators to estimate the population parameters. Robust estimators limit the influence of extreme observations and are insensitive to outliers. For more details about robust statistical techniques, see Hoaglin, Mosteller, and Tukey (1983); Roussaauw and Leroy (1987); Tabatabai and Argyros (1993). In this article we first remove the extreme outliers and then calculate some statistics.

Table 1 shows a summary of some relevant estimates of the parameters of Red movement time.

N  MEAN  MEADIAN TRMEAN STDEV  SEMEAN
95  1.960  1.690  1.838  1.124  0.115
MIN  MAX  Q1  Q3
0.160  5.850  1.270  2.410

Table 1. Summary of Red Movement Time

Let us now turn our attention to the theory underlying estimation of the density function of M. Let the range be partitioned into k class intervals I_{j}, j = 1, 2, …, k and I_{j} = (a_{j−1}, a_{j}), j = 1, 2, …, k where a_{0} < a_{1} < ⋯ < a_{k}. Here we assume that a_{j} − a_{j−1} = 1 minute for j = 1, 2, …, k. Define

f̂_{M}(m) = \{ \begin{matrix}f_{j}/[n(a_{j} − a_{j−1})] & for a_{j−1} < m ≤ a_{j}, j = 1, 2, …, k \\ 0, & elsewhere \end{matrix}

where f_{j} is the frequency of the class interval I_{j} and n = ∑_{j=1}^{k} f_{j} is the total number of observations. The

function f̂_{M} defined on (−∞, ∞) is the relative frequency histogram for M. We note that

f̂_{M}(m) ≥ 0 for all m ∈ (−∞, ∞),  (1)

∫_{−∞}^{∞} f̂_{M}(m) dm = ∫_{a_{0}}^{a_{k}} f̂_{M}(m) dm = ∑_{j=1}^{k} ∫_{a_{j−1}}^{a_{j}} \frac{f_{j}}{n(a_{j} − a_{j−1})} dm = ∑_{j=1}^{k} \frac{f_{j}}{n} = 1.  (2)

Hence f̂_{M} defines a density function and can be used as an estimate of the underlying density function f_{M}. In particular, an estimate of the probability of an event A is given by

P̂(A) = ∫_{A} f̂_{M}(m)dm.

Moreover, P̂(A) is an unbiased and consistent estimate of P (A).

For our data on Red movement time, the estimate of probability distribution of M is given by

\underline{m}  \underline{f̂_{M}(m)}  \underline{m}  \underline{f̂_{M}(m)}
.5  13/103  8.5  2/103
1.5  47/103  9.5  2/103
2.5  24/103  10.5  1/103
3.5  5/103  11.5  0/103
4.5  1/103  12.5  0/103
5.5  5/103  13.5  0/103
6.5  0/103  14.5  1/103
7.5  2/103

where m_{j} = (a_{j−1} + a_{j})/2 is the class mark of class interval I_{j}. The mean of f̂_{M}(m) is given by

μ̂_{M} = ∫_{a_{0}}^{a_{k}} mf̂_{M}(m) dm = ∑_{j=1}^{k} ∫_{a_{j−1}}^{a_{j}} \frac{mf_{j}}{n(a_{j} − a_{j−1})} dm = ∑_{j=1}^{k} \frac{f_{j}}{n} (\frac{a_{j} + a_{j−1}}{2} ) = ∑_{j=1}^{k} \frac{f_{j}m_{j}}{n} .

So far we have answered th following questions: What does the underlying probability density function f_{M} look like? How do we estimate f_{M}? The next question that one needs to answer is: Does f_{M} look like a known or specified distribution? The probability density function of M is Gamma distribution with parameters α and β. We have

f_{M}(m) = \{ \begin{matrix}\frac{m^{α−1} exp(−m/β)}{Γ(α)β^{α}}, & m > 0 \\ 0, & m ≤ 0 \end{matrix}

where Γ(α) = ∫_{0}^{∞}x^{α−1}e^{−x} dx.

The mean μ_{M} = αβ and standard deviation σ_{M} = β\sqrt{α}. To estimate the parameters α and β, we use the method of moments technique. Hence, the moment estimators α̂ and β̂ are

α̂ = \frac{(sample mean)^{2}}{sample variance} and β̂ = \frac{sample mean}{α̂}.

If we use Table 1, then the moment estimates are

α̂ = \frac{(1.960)^{2}}{(1.124)^{2}} = 3.0419 and β̂ = \frac{1.960}{3.0419} = 0.6443.

Now, we use the chi-square goodness-of-fit test to test the following hypothesis?

H_{0} : f_{M}(m) = \frac{m^{3.0419−1} exp(−m/0.6443)}{Γ(3.0419)(0.6443)^{3.0419}}

vs

H_{1} : f_{M}(m) ≠ \frac{m^{3.0419−1} exp(−m/0.6443)}{Γ(3.0419)(0.6443)^{3.0419}}.

The chi-square test statistic is 4.9153. The associated degrees of freedom is 3 and the P − value is 0.1781. Therefore, we have to conclude that the Gamma distribution with parameters α = 3.0419 and β = 0.6443 does provide an excellent fit for the data. Similar conclusions were reached when we used the Kolmogorov-Smirnov test.

The maximum likelihood estimates of α and β are solutions of the following equations

ln α − (\frac{d}{dα}Γ(α)) /Γ(α) = ln \overline{m} − \frac{1}{n} ∑_{j=1}^{n} ln m_{j} and β = \frac{\overline{m}}{α}

where \overline{m} is the sample mean.

These type of equations are not easily solvable and it is necessary to resort to numerical methods. Let F_{T} (t) denote the distribution function of Blue reaction time, then we have

F_{T} (t) = P (T ≤ t) = P (M + 2 ≤ t)

= P (M ≤ t − 2) = ∫_{−∞}^{t−2} f_{M}(m) dm

= ∫_{−∞}^{t−2} \frac{m^{α−1} exp(−m/β)}{Γ(α)β^{α}} dm.

Then, the probability density function f_{T} (t) of Blue reaction time is given by

f_{T} (t) = \frac{d}{dt}F_{T} (t) = \{ \begin{matrix}\frac{(t − 2)^{α−1} exp(−(t − 2)/β)}{Γ(α)β^{α}}, & t > 2 \\ 0, & elsewhere \end{matrix}

where the values of α and β can be estimated by 3.0419 and 0.6443 respectively. Then mean for Blue reaction time is estimated by μ̂_{T} = 3.960 and the standard deviation is estimated by σ̂_{T} = 1.124.

III. Distribution of Target Location Point

Let the random vector (D_{1}, D_{2}) be the target location point and be defined by

D_{1} = X + RM cos θ
D_{2} = Y + RM sin θ

where R is the average speed of Red, M is the Red movement time, θ is the Red angle of movement and X, Y are independent normal random variables with μ_{X} = μ_{Y} = 0 meters and σ_{X} = σ_{Y} = 100 meters.

Our goal is to find the joint probability density function of (D_{1}, D_{2}). We begin this section by first finding the probability distribution of RM.

Let W = RM and F_{W} (w) denote the distribution function of the random variable W. Then

F_{W} (w) = P (W ≤ w) = P (RM ≤ w) = P (M ≤ \frac{1}{R}w)

= ∫_{−∞}^{\frac{1}{R} w} f_{M}(m) dm = ∫_{−∞}^{\frac{1}{R} w} \frac{m^{α−1} exp(−m/β)}{Γ(α)β^{α}} dm.

Then the probability density function f_{W} (w) is given by

f_{W} (w) = \frac{d}{dW} F_{W} (w) = \{ \begin{matrix}\frac{W^{α−1} exp(−w/(Rβ))}{Γ(α)(Rβ)^{α}}, & W > 0 \\ 0, & elsewhere. \end{matrix}

Thus, we can write

D_{1} = X + W cos θ
D_{2} = Y + W sin θ

where the random variable θ is uniformly distributed on the interval [0, 2π) with the probability density function g(θ) defined by

g(θ) = \{ \begin{matrix}\frac{1}{2π} & 0 ≤ θ < 2π \\ 0, & elsewhere \end{matrix}

and the random variables X, Y, θ, and W are independent reandom variables with joint probability density function given by

f(x, y, θ, w) = f_{X}(x) f_{Y} (y) g(θ) f_{W} (w)

= \{ \begin{matrix}\frac{w^{α−1} exp [ − w/(Rβ) − \frac{1}{2}(x^{2}/100^{2} + y^{2}/100^{2})]}{40000π^{2}Γ(α)(Rβ)^{α}}, & \begin{matrix}−∞ < & y < ∞ \\ −∞ < & x < ∞ \\ 0 ≤ & θ < 2π \\ 0 < & w < ∞ \end{matrix} \\ 0, & elsewhere \end{matrix}

Let

\begin{aligned}D_{1} & = g_{1}(X, Y, θ, W) = X + W cos θ, \\ D_{2} & = g_{2}(X, Y, θ, W) = Y + W sin θ, \\ D_{3} & = g_{3}(X, Y, θ, W) = θ, \\ D_{4} & = g_{4}(X, Y, θ, W) = W, \end{aligned}

A = {(x, y, θ, w) : −∞ < x < ∞, −∞ < y < ∞, 0 ≤ θ < 2π, 0 < w < ∞},
B = {(D_{1}, D_{2}, D_{3}, D_{4}) : −∞ < D_{1} < ∞, −∞ < D_{2} < ∞, 0 ≤ D_{3} < 2π, 0 < D_{4} < ∞}.

Define the transformation g : A ⟶_{onto}^{1−1} B as

g = (g_{1}(x, y, θ, w), g_{2}(x, y, θ, w), g_{3}(x, y, θ, w), g_{4}(x, y, θ, w))
= (D_{1}, D_{2}, D_{3}, D_{4})

then the inverse transformation is g^{−1} : B ⟶_{onto}^{1−1} A which can be written as

g^{−1} = (g_{1}^{−1}(D_{1}, D_{2}, D_{3}, D_{4}), g_{2}^{−1}(D_{1}, D_{2}, D_{3}, D_{4}), g_{3}^{−1}(D_{1}, D_{2}, D_{3}, D_{4}), g_{4}^{−1}(D_{1}, D_{2}, D_{3}, D_{4}))
= (x, y, θ, w)

where

X = D_{1} − D_{4} cos D_{3}
Y = D_{2} − D_{4} sin D_{3}
θ = D_{3}
W = D_{4}.

Let

J (\frac{x, y, θ, w}{D_{1}, D_{2}, D_{3}, D_{4}} ) = | \begin{matrix}\frac{∂x}{∂D_{1}} & · & · & · & \frac{∂x}{∂D_{4}} \\ \frac{∂y}{∂D_{1}} & · & · & · & \frac{∂y}{∂D_{4}} \\ \frac{∂θ}{∂D_{1}} & · & · & · & \frac{∂θ}{∂D_{4}} \\ \frac{∂w}{∂D_{1}} & · & · & · & \frac{∂w}{∂D_{4}} \end{matrix} |

be the Jacobian of inverse transformation g^{−1}. Then

J (\frac{x, y, θ, w}{D_{1}, D_{2}, D_{3}, D_{4}} ) = | \begin{matrix}1 & 0 & D_{4} sin D_{3} & − cos D_{3} \\ 0 & 1 & −D_{4} cos D_{3} & − sin D_{3} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} | = 1.

The joint probability density function of random variables D_{1}, D_{2}, D_{3}, D_{4} is denoted by h(D_{1}, D_{2}, D_{3}, D_{4}) and is given by

h(D_{1}, D_{2}, D_{3}, D_{4}) = |J| · f (g_{1}^{−1}, g_{2}^{−1}, g_{3}^{−1}, g_{4}^{−1})

= \{ \begin{matrix}\frac{D_{4}^{α−1} exp \{−D_{4}/(Rβ) − [D_{1}^{2} + D_{2}^{2} + D_{4}^{2} − 2D_{4}(D_{1} cos D_{3} + D_{2} sin D_{3})] /20000\}}{40000π^{2}Γ(α)(Rβ)^{α}}, & (D_{1}, D_{2}, D_{3}, D_{4}) ∈ B \\ 0, & elsewhere. \end{matrix}

Finally, the distribution of the target location point is denoted by h(D_{1}, D_{2}) and is given by h(D_{1}, D_{2})

= \{ \frac{exp \{−(D_{1}^{2} + D_{2}^{2}) /20000\}}{40000π^{2}(Rβ)^{α}Γ(α)} ∫_{0}^{2π}∫_{0}^{∞}exp \{−[D_{4}^{2} − 2D_{4}(D_{1} cos D_{3} + D_{2} sin D_{3}) + \frac{20000}{Rβ}D_{4}] /20000\} dD_{4}dD_{3}, \begin{matrix}−∞ < & D_{1} < ∞ \\ −∞ < & D_{1} < ∞ \end{matrix} 0, elsewhere.

IV. Some Special Cases

Case 1. Suppose the initial point (X, Y) is predetermined and is equal to X = a, and Y = b. Then

D_{1} = a + W cos θ
D_{2} = b + W sin θ

Define the transformation S : A_{1} ⟶_{onto}^{1−1} B_{1} as

S = (S_{1}(w, θ), S_{2}(w, θ)) = (D_{1}, D_{2})

where

A_{1} = {(w, θ) : 0 < w < ∞, 0 ≤ θ < 2π}

and

B_{1} = {(D_{1}, D_{2}) : −∞ < D_{1} < ∞, −∞ < D_{2} < ∞}.

Then the inverse transformation S^{−1} : B_{1} → A_{1} is a one-to-one and onto function defined by

S^{−1} = (S_{1}^{−1}(D_{1}, D_{2}), S_{2}^{−1}(D_{1}, D_{2})) = (W, θ)

where

W = [(D_{1} − a)^{2} + (D_{2} − b)^{2}]^{\frac{1}{2}}

and

θ = arctan (\frac{D_{2} − b}{D_{1} − a}) .

The Jacobian of inverse transformation is

J (\frac{W, θ}{D_{1}, D_{2}} ) = | \begin{matrix}\frac{∂W}{∂D_{1}} & \frac{∂W}{∂D_{2}} \\ \frac{∂θ}{∂D_{1}} & \frac{∂θ}{∂D_{2}} \end{matrix} | = \frac{1}{[(D_{1} − a)^{2} + (D_{2} − b)^{2}]^{\frac{1}{2}}} .

The joint probability density function of the target location point conditioned upon knowledge of (X, Y) = (a, b) is denoted by h_{1}(D_{1}, D_{2}) and is given by

h_{1}(D_{1}, D_{2}) = |J (\frac{W, θ}{D_{1}, D_{2}} )| · f_{W} (S_{1}^{−1}) · g(S_{2}^{−1})

= \{ \begin{matrix}\frac{[(D_{1} − a)^{2} + (D_{2} − b)^{2}]^{\frac{α}{2} −1}}{2π(Rβ)^{α}Γ(α)} · exp \{− [(D_{1} − a)^{2} + (D_{2} − b)^{2}]^{\frac{1}{2}} /(Rβ)\} , & if (D_{1}, D_{2}) ∈ B_{1} \\ 0, & elsewhere. \end{matrix}

Case 2. If θ = 0, then the joint probability density function of D_{1} and D_{2} is denoted by h_{0}(D_{1}, D_{2}) and is equal to

h_{0}(D_{1}, D_{2})

= \{ \begin{matrix}\frac{exp \{−[D_{1}^{2} + D_{2}^{2}]\}}{20000π(Rβ)^{α}Γ(α)} ∫_{0}^{∞} D_{4}^{α−1} exp \{− [D_{4}^{2} − 2D_{1}D_{4} + \frac{20000}{Rβ}D_{4}] /20000\} dD_{4}, & \begin{matrix}−∞ < & D_{1} < ∞ \\ −∞ < & D_{2} < ∞ \end{matrix} \\ 0, & elsewhere. \end{matrix}

In general, if θ = C, then the joint probability density function of D_{1} and D_{2} is denoted by h_{C}(D_{1}, D_{2}) and is given by

h_{C}(D_{1}, D_{2})

= \{ \frac{exp \{−[D_{1}^{2} + D_{2}^{2}]\}}{20000π(Rβ)^{α}Γ(α)} ∫_{0}^{∞} D_{4}^{α−1} exp \{− [D_{4}^{2} − 2D_{4}(D_{1} cos C + D_{2} sin C) + \frac{20000}{Rβ}D_{4}] /20000\} dD_{4}, \begin{matrix}−∞ < & D_{1} < ∞ \\ −∞ < & D_{2} < ∞ \end{matrix} 0, elsewhere.

References

[1] Hoaglin, D.C., Mosteller, F., Tukey, J., Understanding Robust and Exploratory Data Analysis, John Wiley & Sons, Inc. (1983).

[2] Rousseeuw, P.J., and Leroy, A.M., Robust Regression and Outlier Detection, John Wiley & Sons, Inc. (1987).

[3] Tabatabai, M.A., and Argyros, I.K., Robust Estimation and Testing for General Nonlinear Regression Models, Applied Math and Comp 58/1 (1993), 85–101.

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SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS VOLUME 1, JULY–DECEMBER 1995

CONTENTS

I. Argyros  Perturbed Newton-Like Methods and Nondifferentiable Operator Equations
on Banach Spaces with a Convergence Structure . . . . . . . . . . 1–12
G. Anastassiou Differentiated Shift-Invariant Multivariate Integral Operators . . . . . . 13–16
D. Pantić  Stochastic Calculus and Initial Value Analysis on the One Dimensional
Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17–26
R. Forslund  A Logical Alternative to the Existing Positional Number System …. 27–29
I. Argyros  Results on Controlling the Residuals of Perturbed Newton-Like Methods
on Banach Spaces with a Convergence Structure . . . . . . . . . . 30–36
M. Tabatabai A Probabilistic Approach to the Estimation of the Distribution of the
Target Location Point . . . . . . . . . . . . . . . . . . . . . . 37–46