Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 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.

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