Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 02, 1996, pp. 34–41. Submitted: June 17, 1996. Published: September 9, 1996.

ON THE CONVERGENCE OF PERTURBED NEWTON-LIKE

METHODS IN BANACH SPACE AND APPLICATIONS

Ioannis K. Argyros

Abstract. In this study we use disturbed Newton-like methods to approximate a locally unique solution of a nonlinear equation containing a nondifferentiable term. We choose the linear operators to be inverted at every step of the computation of the Newton iterates to be associated not only with the differentiable part of the equation as it has happened so far but also with the nondifferentiable part. This way the upper error bounds on the distances are smaller than before.

1991 A.M.S. (MOS) Subject Classification Codes. 65J15, 65B05, 65L50, 47H17, 49D15.

Key Words and Phrases. Disturbed Newton-like methods, Banach space, Mysovskii theorem, divided difference of order one, Fréchet-derivative.

I. INTRODUCTION

In this study we are concerned with the problem of approximating a locally unique solution x^{∗} of the nonlinear equation

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

where the operators F, Q are defined on an open, convex subset D of a Banach space E_{1} with values in a Banach space E_{2}, and are continuous on D.

We introduce disturbed Newton-like methods of the form

(2)  x_{n+1} = x_{n} − A_{n}^{−1}(F (x_{n}) + Q(x_{n})) − z_{n} (n ≥ 0),

where

A_{n} = A(x_{n−1}, x_{n}) = L_{1}(x_{n−1}, x_{n}) + L_{2}(x_{n−1}, x_{n}) + L_{3}(x_{n−1}, x_{n}) (n ≥ 0)

for x_{−1}, x_{0} ∈ D and A, L_{1}, L_{2}, L_{3} : D × D → L(E_{1}, E_{2}).

Typeset by AMS-TEX

Cameron University, Lawton, Oklahoma

E-mail Address: ioannisa@cameron.edu

\begin{matrix}c \\ ◯ \end{matrix}1996 Cameron University

The points z_{n} ∈ E_{1} (n ≥ 0) are determined in such a way that the iteration {x_{n}} (n ≥ 0) converges to a solution x^{∗} of equation (1). The importance of studying disturbed 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. Note that if for example an equation on the real line is solved, F (x_{n}) + Q(x_{n}) ≥ 0, and A_{n} overestimates the derivative, then x_{n} − A_{n}^{−1}(F (x_{n}) + Q(x_{n})) (n ≥ 0) is always larger than the corresponding Newton iterate. In such cases, a positive z_{n} (n ≥ 0) correction term is appropriate. We assume that L_{1}(x, y) is used to approximate the difference F (z) − F (y) for x, y, z ∈ D; L_{2}(x, y) is used to approximate the difference Q(z) − Q(y) for all x, y, z ∈ D and L_{3}(x_{n−1}, x_{n}) is a linear operator depending on F, Q, x_{n} (n ≥ −1) and z_{n} (n ≥ 0). Special choices of operators A_{n} (n ≥ 0) and points z_{n} (n ≥ 0) lead to iterations that have already been examined by many authors. Indeed: (a) for Q(x) = 0 (x ∈ D), L_{1} = F′(·), L_{2} = L_{3} = 0 (linear operator), and z_{n} = 0 (n ≥ 0) iteration (2) reduces to Newton’s method [3], [5], [10]; (b) for Q(x) = 0 (x ∈ D), L_{1}(x, y) = [x, y; F] (divided difference of order one of F on D at the points x, y ∈ D), L_{2} = L_{3} = 0, and z_{n} = 0 (n ≥ 0) iteration (2) reduces to the secant method [3], [5], [6]; (c) for Q(x) = 0 (x ∈ D), L_{1}(x, y) = [g_{1}(x), g_{2}(y); F], E_{1} = E_{2} = E, g_{1}, g_{2} : D → E, L_{2} = L_{3} = 0 and z_{n} (n ≥ 0) we get the Steffensen–Aitken method [3], [8], [9]; (d) for L_{2} = L_{3} = 0, L_{1} = F′(·), and z_{n} = 0 (n ≥ 0) we obtain the Zincenko iteration [3], [9], [10]; (e) for E_{1} = E_{2} = E, L_{1}(x_{n−1}, x_{n}) = F′(x_{n}) (n ≥ 0), L_{2}(x_{n−1}, x_{n}) = [x_{n−1}, x_{n}; Q] (n ≥ 0), L_{3} = 0 and z_{n} = 0 (n ≥ 0) we obtain the iteration introduced in [4]; (f) for L_{2} = L_{3} = 0, L′_{1}(x, y) = A(x), (A approximating F′(·)) and z_{n} = 0 (n ≥ 0) we get iterations considered in [1], [2], [3], [9].

We show under more general and weaker conditions that our iteration (2) is faster than the ones mentioned above.

II. CONVERGENCE ANALYSIS

Let x_{−1}, x_{0} ∈ D, z_{n} ∈ E_{1} (n ≥ 0), and a_{1}, a_{2}, a_{3} ≥ 0, r ∈ [0, R], R ≥ 0, U(x_{0}, R) = {x ∈ E_{1} | ∥x − x_{0}∥ ≤ R} with U(x_{0}, R) ⊆ D, C_{1}, C_{2}, C_{3} : [0, +∞) → [0, +∞) be continuous nondecreasing functions that vanish at the origin, and A, L_{1}, L_{2}, L_{3} : D × D → L(E_{1}, E_{2}) such that

(3) ∥A(y, z)^{−1}[F (z) − F (y) − L_{1}(x, y)(z − y)]∥ ≤ [C_{1}(∥x − x_{0}∥, ∥y − x_{0}∥, ∥z − x_{0}∥, ∥x − y∥, ∥z − y∥) + a_{1}]∥z − y∥,

(4) ∥A(y, z)^{−1}[Q(z) − Q(y) − L_{2}(x, y)(z − y)]∥ ≤ [C_{2}(∥x − x_{0}∥, ∥y − x_{0}∥, ∥z − x_{0}∥, ∥x − y∥, ∥z − y∥) + a_{2}]∥z − y∥,

and

\begin{matrix}∥A(x_{n+1}, x_{n})^{−1}[A_{n+1}(z_{n+1}) − A_{n}(z_{n}) − L_{3}(x_{n−1}, x_{n})(x_{n+1} − x_{n})]∥ \\ ≤ [C_{3}(∥x_{n−1} − x_{0}∥, ∥x_{n} − x_{0}∥, ∥x_{n+1} − x_{0}∥, \\ ∥x_{n−1} − x_{n}∥, ∥x_{n+1} − x_{n}∥) + a_{3}]∥x_{n+1} − x_{n}∥ (n ≥ 0) \end{matrix} (5)

for all x, y, z ∈ U(x_{0}, r) where x_{n} (n ≥ −1) are given by (2).

Let a_{4}, a_{5} be constants such that

(6)  ∥x_{0} − x_{−1}∥ ≤ a_{4} and ∥A_{0}^{−1}(F (x_{0}) + Q(x_{0})) − z_{0}∥ ≤ a_{5}.

It is convenient to introduce the iterations

h̄_{n+1} = C(∥x_{n−1} − x_{0}∥, ∥x_{n} − x_{0}∥, ∥x_{n+1} − x_{0}∥,
∥x_{n−1} − x_{n}∥, ∥x_{n+1} − x_{n}∥)∥x_{n+1} − x_{n}∥ (n ≥ 0)

where

C(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) = (C_{1} + C_{2} + C_{3})(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) + a_{1} + a_{2} + a_{3}

for all s_{1}, s_{2}, s_{3}, s_{4}, s_{5} ∈ [0, +∞),

(7)  t_{n+1} = t_{n+1} + h_{n+1}, t_{−1} = 0 (n ≥ 0),

where

h_{n+1} = C(|t_{n−1} − t_{0}|, t_{n} − t_{0}, t_{n+1} − t_{0}, t_{n} − t_{n−1}, t_{n+1} − t_{n})(t_{n+1} − t_{n}) (n ≥ 0),

and the functions T, C_{4} : [0, +∞) → [0, +∞) by

(8) T (r) = t_{1} + C(r, r, r, r, r)(r − t_{1}) + C(t_{0}, 0, t_{1} − t_{0}, t_{0}, t_{1} − t_{0})(t_{1} − t_{0})

and

(9)  C_{4}(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) = (C_{1} + C_{2})(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) + a_{1} + a_{2}.

We can now state and prove the following Newton–Mysovskii [6] semilocal convergence theorem for the disturbed Newton like method generated by (2).

Theorem. Assume:

(a) operators F, Q are defined on some open convex subset D of a Banach space E_{1} with values in E_{2} and continuous on D;

(b) there exist r, R with r ∈ [0, R] such that A(x, y) ∈ L(E_{1}, E_{2}) for all x, y ∈ U(x_{0}, r) is boundedly invertible, U(x_{0}, R) ⊆ D and conditions (3)–(5) are satisfied;

(c) sequence {z_{n}} (n ≥ 0) ∈ E_{1} is null;

(d) there exist numbers t_{0} and t_{1} such that

(10)  a_{5} ≤ t_{1} − t_{0} ≤ ∥x_{0} − x_{−1}∥ ≤ a_{4} ≤ t_{0} − t_{−1};

(e) there exists a minimum nonnegative number r^{∗} satisfying

(11)  r^{∗} ≤ R

and

(12)  T (r^{∗}) ≤ r^{∗}

where function T is given by (8);

(f) moreover r^{∗}, R satisfy

(13)  C_{4}(r^{∗}, r^{∗}, R, 2r^{∗}, r^{∗} + R) < 1

where function C_{4} is given by (9).

Then

(i) Scalar iteration {t_{n}} (n ≥ −1) generated by (7) is monotonically increasing, bounded above by r^{∗} and lim_{n→∞} t_{n} = r^{∗};

(ii) disturbed Newton-like method {x_{n}} (n ≥ −1) generated by (2) is well defined, remains in U(x_{0}, r^{∗}) for all n ≥ −1, and converges to a unique solution x^{∗} of equation F (x) + Q(x) = 0 in U(x_{0}, R).

Moreover the following error estimates are true for all n ≥ 0

(14)  ∥x_{n+1} − x_{n}∥ ≤ h̄_{n+1} ≤ h_{n+1} = t_{n+1} − t_{n}

and

(15)  ∥x_{n} − x^{∗}∥ ≤ r^{∗} − t_{n}.

Proof. (i) It follows from (7), (10) and (12) that t_{−1} ≤ t_{0} ≤ t_{1} ≤ r^{∗}. Let us assume that t_{k} ≤ t_{k+1} ≤ r^{∗} for k = −1, 0, 1, 2, …, n. The C function is nonnegative and nondecreasing, hence h_{k+1} ≥ 0. That is t_{k+1} ≤ t_{k+2} (by (7)). Moreover from (7),

(12) and the induction hypothesis we obtain in turn

t_{k+2} = t_{k+1} + h_{k+1} ≤ t_{k+1} + C(r^{∗}, r^{∗}, r^{∗}, r^{∗}, r^{∗})(t_{k+1} − t_{k})
≤ ⋯ ≤ t_{1} + C(r^{∗}, r^{∗}, r^{∗}, r^{∗}, r^{∗})(t_{k+1} − t_{1})
+ C(t_{0}, 0, t_{1} − t_{0}, t_{0}, t_{1} − t_{0})(t_{1} − t_{0})
= T (r^{∗}) ≤ r^{∗}.

Hence, iteration {t_{n}} (n ≥ −1) is also bounded above by r^{∗}. Since r^{∗} is the minimum nonnegative number on [0, r^{∗}] satisfying (12) it follows that lim_{n→∞} t_{n} = r^{∗}.

(ii) It follows from (10), (12) that x_{−1}, x_{1} ∈ U(x_{0}, r^{∗}), and that (14) is satisfied for n = −1, 0. Assume that x_{k} ∈ U(x_{0}, r^{∗}) and that (14) is satisfied for k = −1, 0, …, n. Using (14), the triangle inequality and the induction hypothesis we obtain

∥x_{k+1} − x_{0}∥ ≤ ∑_{i=0}^{k} ∥x_{k+1} − x_{i}∥ ≤ ∑_{i=0}^{k}(t_{i+1} − t_{i}) = t_{k+1} − t_{0} ≤ r^{∗}.

That is x_{k+1} ∈ U(x_{0}, r^{∗}). We will show that (14) is true for k = n + 1. Using (2)–(5), (7), the induction hypothesis we can obtain the approximation

(16) \begin{aligned}F (x_{k+1}) & + Q(x_{k+1}) + A_{k+1}(z_{k+1}) = F (x_{k+1}) + Q(x_{k+1}) \\ & + A_{k+1}(z_{k+1}) − F (x_{k}) − Q(x_{k}) − A_{k}(z_{k}) − A_{k}(x_{k+1} − x_{k}) \\ & = [F (x_{k+1}) − F (x_{k}) − L_{1}(x_{k−1}, x_{k})(x_{k+1} − x_{k})] \\ & + [Q(x_{k+1}) − Q(x_{k}) − L_{2}(x_{k−1}, x_{k})(x_{k+1} − x_{k})] \\ & + [A_{k+1}(z_{k+1}) − A_{k}(z_{k}) − L_{3}(x_{k−1}, x_{k})(x_{k+1} − x_{k})], \end{aligned}

and

(17) \begin{matrix}∥x_{k+2} − x_{k+1}∥ = ∥A_{k+1}^{−1}[F (x_{k+1}) + Q(x_{k+1}) − A_{k+1}(z_{k+1})]∥ \\ ≤ [C_{1}(∥x_{k−1} − x_{0}∥, ∥x_{k} − x_{0}∥, ∥x_{k+1} − x_{0}∥, \\ ∥x_{k−1} − x_{k}∥, ∥x_{k+1} − x_{k}∥) + a_{1}]∥x_{k+1} − x_{k}∥ \\ + [C_{2}(∥x_{k−1} − x_{0}∥, ∥x_{k} − x_{0}∥, ∥x_{k+1} − x_{0}∥, ∥x_{k−1} − x_{k}∥, \\ ∥x_{k+1} − x_{k}∥) + a_{2}]∥x_{k+1} − x_{k}∥ \\ + [C_{3}(∥x_{k−1} − x_{0}∥, ∥x_{k} − x_{0}∥, ∥x_{k+1} − x_{0}∥, ∥x_{k−1} − x_{k}∥, \\ ∥x_{k+1} − x_{k}∥) + a_{3}]∥x_{k+1} − x_{k}∥ \\ = C(∥x_{k−1} − x_{0}∥, ∥x_{k} − x_{0}∥, ∥x_{k+1} − x_{0}∥, ∥x_{k−1} − x_{k}∥, \\ ∥x_{k+1} − x_{k}∥)∥x_{k+1} − x_{k}∥ = h̄_{k+1} \\ ≤ C(|t_{k−1} − t_{0}|, t_{k} − t_{0}, t_{k+1} − t_{0}, t_{k} − t_{k−1}, t_{k+1} − t_{k})(t_{k+1} − t_{k}) \\ = h_{k+1} = t_{k+2} − t_{k+1}, \end{matrix}

which shows (14) for all n ≥ 0.

It now follows from part (i) and (14) that the inexact Newton-like iteration {x_{n}} (n ≥ −1) is Cauchy in a Banach space E_{1} and as such it converges to some x^{∗} ∈ U(x_{0}, r^{∗}) (since U(x_{0}, r^{∗}) is a closed set). By letting n → ∞ in (16) and using hypothesis (c) we deduce that F (x^{∗}) + Q(x^{∗}) = 0. That is x^{∗} is a solution of equation (1). Estimate (15) follows immediately from (14) by using standard majorization techniques [3], [5].

To show uniqueness, we reason as follows: under the hypotheses of the theorem iteration (2) for z_{n} = 0 (n ≥ 0) becomes

(18)  v_{n+1} = v_{n} − A_{n}^{−1}(F (v_{n}) + Q(v_{n})), v_{−1} = x_{−1}, v_{0} = x_{0} (n ≥ 0)

and converges to some x_{1}^{∗} ∈ U(x_{0}, r^{∗}). Let us assume that there exists y^{∗} ∈ U(x_{0}, R) such that F (y^{∗}) + Q(y^{∗}) = 0. Using (18), (3), (4) and the approximation

v_{n+1} − y^{∗} = v_{n} − A_{n}^{−1}(F (v_{n}) + Q(v_{n})) − y^{∗}
= A_{n}^{−1}[F (v_{n}) − F (y^{∗}) − L_{1}(v_{n−1}, v_{n})(v_{n} − y^{∗})]
+ A_{n}^{−1}[Q(v_{n}) − Q(y^{∗}) − L_{2}(v_{n−1}, v_{n})(v_{n} − y^{∗})]

we obtain

\begin{aligned}∥v_{n+1} − y^{∗}∥ & ≤ [C_{1}(∥v_{n−1} − v_{0}∥, ∥v_{n} − v_{0}∥, ∥y^{∗} − v_{0}∥, ∥v_{n−1} − v_{n}∥, \\ & ∥v_{n} − y^{∗}∥) + a_{1}]∥v_{n} − y^{∗}∥ \\ & + [C_{2}(∥v_{n−1} − v_{0}∥, ∥v_{n} − v_{0}∥, ∥y^{∗} − v_{0}∥, ∥v_{n−1} − v_{n}∥, \\ & ∥v_{n} − y^{∗}∥) + a_{2}]∥v_{n} − y^{∗}∥ \\ & ≤ C_{4}(∥v_{n−1} − v_{0}∥, ∥v_{n} − v_{0}∥, ∥y^{∗} − v_{0}∥, ∥v_{n−1} − v_{n}∥, \\ & ∥v_{n} − y^{∗}∥)∥v_{n} − y^{∗}∥. \end{aligned} (19)

By (13) there exist r^{∗}, q ∈ [0, 1] such that

(20)  C(r^{∗}, r^{∗}, R, 2r^{∗}, r^{∗} + R) ≤ q < 1.

It follows from (19) and (20) that

∥v_{n+1} − y^{∗}∥ ≤ q^{n+1}∥v_{0} − y^{∗}∥ ≤ q^{n+1}R → 0 as n → ∞.

That is, y^{∗} = lim_{n→∞} v_{n+1} = x_{1}^{∗}. Hence we deduce x^{∗} = x_{1}^{∗} = y^{∗}.

The proof of the Theorem is now complete.

Remarks. (1) Hypotheses (3)–(5) can be replaced by weaker (3)′–(5)′ if A(y, z)^{−1} is replaced by A_{0}^{−1} for all y, z ∈ U(x_{0}, r). In this case we need to add a hypothesis of the form

(21)  ∥A_{0}^{−1}(A(x, y) − A_{0})∥ ≤ C_{5}(∥x − x_{−1}∥, ∥y − x_{0}∥) + a_{6},

with C_{5} : [0, +∞)^{2} → [0, +∞) be a continuous nondecreasing function vanishing at the origin and a_{6} ≥ 0. Parameter r^{∗} must satisfy the additional hypothesis

C_{5}(r^{∗}, r^{∗}) + a_{6} < 1.

It then follows from the Banach Lemma on invertible operators [5] that

∥A_{n+1}^{−1}A_{0}∥ ≤ b̄_{n+1} ≤ b_{n+1} (n ≥ 0)

where

b̄_{n+1} = {1 − [a_{6} + C_{5}(∥x_{n} − x_{−1}∥, ∥x_{n+1} − x_{0}∥)]}^{−1}

and

b_{n+1} = {1 − [a_{6} + C_{5}(t_{n}, t_{n+1} − t_{0})]}^{−1}.

The following modifications must also take place: iterations h̄_{n+1}, h_{n+1} must be multiplied by b̄_{n+1}, b_{n+1} (n ≥ 0) respectively, and functions C, C_{4} must be multiplied by

b(r) = {1 − [a_{6} + C_{5}(r, r)]}^{−1}.

It can then be easily seen that with the above modifications the conclusions of the Theorem under the above-mentioned weaker assumptions will also hold.

(2) Several choices for the “C” functions appearing in conditions (3)–(5) can be suggested. To compare with conditions suggested by Yamamoto and Chen in [9] under the special choice (f) stated at the introduction of this study, we choose (22)

C_{1}(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) = ∫_{0}^{1}[w_{1}(s_{1} + ts_{4}) − w_{2}(s_{1}) + w_{3}(s_{2} + ts_{5}) − w_{4}(s_{2}) + a_{1}]s_{5}dt,

(23) C_{2}(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) = [w_{5}(s_{1} + s_{4}) − w_{6}(s_{1}) + w_{7}(s_{2} + s_{5}) − w_{8}(s_{2}) + a_{2}]s_{5}

and

(24) C_{3}(s_{1}, s_{2}, s_{3}, s_{4}, s_{5}) = [w_{9}(s_{1} + s_{4}) − w_{10}(s_{1}) + w_{11}(s_{2} + s_{5}) − w_{12}(s_{2}) + a_{3}]s_{5}

where w_{m} : [0, +∞) → [0, +∞), m = 1, 2, …, 12 are continuous functions vanishing at the origin and satisfying

0 ≤ w_{i}(d_{1} + d_{2}) − w_{i+1}(d_{1}) ≤ w_{i}(d_{3} + d_{4}) − w_{i+1}(d_{3}), i = 1, 3, 5, 7, 9, 11

for all r_{1} ≤ r_{3} and r_{2} ≤ r_{4}. Yamamoto–Chen conditions (4), (5) and (6) in [9] are given respectively by (following their notation)

∥A(x_{0})^{−1}(A(x) − A(x_{0}))∥ ≤ w_{0}(∥x − x_{0}∥) + b

∥A(x_{0})^{−1}(F′(x + t(y − x)) − A(x))∥ ≤ w(∥x − x_{0}∥ + (t∥y − x∥) − w_{0}(∥x − x_{0}∥) + c

and

∥A(x_{0})^{−1}(Q(x) − Q(y))∥ ≤ e(r)∥x − y∥.

These conditions correspond to our conditions (21), (3) and (4) respectively. It can easily be seen from (22)–(24) that for special choices of the w_{i} functions our conditions (21) and (4) reduce to Yamamoto–Chen (4) and (6), whereas our condition (4) is weaker than Yamamoto–Chen (5). Moreover since the left-hand side of Yamamoto–Chen condition (5) (times (y − x) inside the norm) is larger than the left-hand side of our condition (3), our upper bounds on the distances ∥x_{n+1} − x_{n}∥ and ∥x_{n} − x^{∗}∥ (n ≥ 0) are smaller than Yamamoto–Chen upper bounds.

(3) Favorable comparisons can also be made with the rest of the cases (a)–(e) mentioned at the introductions of this study. The details are left to the motivated reader. Numerical examples for the usefulness of iteration (2) can also be found in [1]–[4].

(4) Define the residuals r_{n} = −A_{n}(z_{n}) and set Δ_{n} = x_{n+1}−x_{n} (n ≥ 0). Since the sequence {z_{n}} (n ≥ 0) is null and the linear operators {A_{n}} (n ≥ 0) are bounded it follows that the sequence {r_{n}} (n ≥ 0) is null also. Consequently the conclusions of the Theorem hold for the system

x_{n+1} − x_{n} = Δ_{n},

and

A_{n}Δ_{n} = −(F (x_{n}) + Q(x_{n})) + r_{n} (n ≥ 0) [1], [3].

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