Table of Contents
International Journal of Analysis
Volume 2018, Article ID 9576137, 12 pages
https://doi.org/10.1155/2018/9576137
Research Article

On the Equivalence of Stochastic Fixed Point Iterations for Generalized -Contractive-Like Operators

1Department of Mathematics, Covenant University, Canaanland, KM 10, Idiroko Road, PMB 1023, Ota, Ogun State, Nigeria
2Department of Mathematics, Adeniran Ogunsanya College of Education, Otto-Ijanikin, Lagos, Nigeria

Correspondence should be addressed to Victoria Olisama; moc.oohay@amasilociv

Received 4 February 2018; Accepted 2 April 2018; Published 20 May 2018

Academic Editor: Shamsul Qamar

Copyright © 2018 Hudson Akewe et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present the equivalence of some stochastic fixed point iterative algorithms by proving the equivalence between the convergence of random implicit Jungck-Kirk-multistep, random implicit Jungck-Kirk-Noor, random implicit Jungck-Kirk-Ishikawa, and random implicit Jungck-Kirk-Mann iterative algorithms for generalized -contractive-like random operators defined on separable Banach spaces.

1. Introduction

Probabilistic functional analysis is an aspect of mathematics that deals with probabilistic models to solve uncertainties and ambiguities that exist in real-world problems. Random nonlinear analysis is a vital area of probabilistic functional analysis that deals with various classes of random operator equations and related problems and solutions. The development of various random methods has transformed the financial markets. Random fixed point theorems for random contraction mappings on separable complete metric spaces were first proved by Spacek [1] and Hans (see [2, 3]). Random fixed point theory has become the full-fledged research area and various ideas associated with the theory are applied to obtain the solutions to a class of stochastic integral equations (see [4, 5]). Random fixed theorems are well known stochastic generalizations of classical fixed point theorems and are usually needed in the theory of random equations, random matrices, random differential equations, and different classes of random operators emanating in physical systems [6, 7]. The origin and various generalizations of random fixed point theorem exist in the literature; for complete survey, see [3] and several related references therein.

The concept of employing various iterative schemes in approximating fixed points of contractive-like operators is very useful in fixed point theory and applications and other relevant fields like numerical analysis, operation research, and so forth (see [815]) This is due to the close relationship that exists between the problem of solving nonlinear equations and that of approximating fixed points of corresponding contractive-like operator.

However, while many researchers have proved useful results on the equivalence of the various iterations, that is, the convergence of any of the iterative methods to the unique fixed point of the contractive operator for single map is equivalent to the convergence of the other iterations (see [1624]), it is observed that little result is known of the equivalence of implicit schemes for a pair of maps [25]. This work will address these areas.

2. Preliminary

Definition 1. Let be a complete probability measure space and a nonempty subset of a separable Banach space . For two random mappings with and being a nonempty closed convex subset of a separable Banach space , there exists a real number and a monotone increasing function with , and for all , one hasand for and , one has For , ,For and since is a monotone increasing sequence with , then

Remark 2. Observe that the contractive conditions imply one another but not conversely: (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6).

Definition 3. Let be a measurable space and be a nonempty closed convex sunset of a separable Banach space . A function is said to be measurable if for each Borel set of . A function is called a random operator if is measurable for every . A measurable function is called a random fixed point for the operator if .

Definition 4. Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . A measurable function is called a random coincidence for two random mappings if for all . The maps are said to be random weakly compatible if they commute at their random coincidence; that is, if for every , then or

We define the random explicit Jungck-Kirk-multistep hybrid iterative algorithm as follows.

Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . Let be two random mappings with . Let be an arbitrary measurable mapping for , for .

(i) The random explicit Jungck-Kirk-multistep hybrid iterative algorithm is a sequence defined iteratively bywhere , , for each and are measurable sequences in for each while are fixed integers (for each ).

(ii) The random explicit Jungck-Kirk-Noor hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.

(iii) The random explicit Jungck-Kirk-Ishikawa hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.

(iv) The random explicit Jungck-Kirk-Mann hybrid iterative algorithm is a sequence defined iteratively bywhere , and is a measurable sequence in and is a fixed integer.

Next, we define the random implicit Jungck-Kirk-multistep hybrid iterative algorithms as follows.

Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . Let be two random mappings with . Let be an arbitrary measurable mapping for .

(i) The random implicit Jungck-Kirk-multistep hybrid iterative algorithm is a sequence defined iteratively bywhere are fixed integers (for each ) with , , and (for each ) are measurable sequences in ; (11) is called random implicit Jungck-Kirk-multistep hybrid iterative algorithm.

(ii) The random implicit Jungck-Kirk-Noor hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.

(iii) The random implicit Jungck-Kirk-Ishikawa hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.

(iv) The random implicit Jungck-Kirk-Mann hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and is a fixed integer.

Next, we introduce the random Jungck-Kirk-SP multiple iterative algorithm.

Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . Let be two random mappings with . Let be an arbitrary measurable mapping for .

(i) The random Jungck-Kirk-multistep-SP hybrid iterative algorithm is a sequence defined iteratively bywhere are fixed integers (for each ) with , and (for each ) are measurable sequences in .

(ii) The random Jungck-Kirk Noor-SP hybrid iterative algorithm is a sequence defined iteratively bywhere are fixed integers with , and (for each ) are measurable sequences in

Lemma 5. If is a real number such that and is a sequence of positive numbers such that , then, for any sequence of positive numbers satisfying one has

Lemma 6. Let be a normed linear space and be a non-self-random commuting operator on an arbitrary set with values in satisfying (6) such that : Let be a sublinear, monotone increasing function such that and for all . Then, for every and , we have

Proof. Lemma 6 is proved by mathematical induction in [17].

3. Main Results

Theorem 7. Let be a separable Banach space and be two random commuting mappings satisfying (19) such that Assume that and are random weakly compatible. Let be the random common point of (i.e., If defined by as sequences satisfying (14) and (11), respectively, then the following are equivalent:(i)Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to (ii)Random implicit Jungck-Kirk-multistep iteration (11) converges strongly to

Proof. We first prove that (i) ⇒ (ii).
Assume , and then using (14) and (11) and generalized contractive condition (19), we get From (20), Substituting (21) into (20), we get Using (11) and (14) and contractive condition (19), we get Following the method of proof in (21), we can writeSubstituting (24) into (23) and simplifying, we obtain Also, using (11) and (14) and contractive condition (19), we get Substituting (25) and (26) into (22), we get Continuing this process to and simplifying, we get the following: Recall that
Let ; then, Substituting (29) into (28), we get where ,Applying Lemma 5 in (30), it follows that
By assumption , then as , that is,
Next, we show that (ii) (i). Assume ; then, using (11) and (14) and contractive condition (19), we have From (32), Substituting (33) into (32) and simplifying, we obtain Using (11) and (14) and contractive condition (19), we have