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

Authors: Hudson Akewe; Kanayo Stella Eke; Victoria Olisama
Journal: International Journal of Analysis
 (2018)

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## 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 generalizedφ-contractive-like random operators defined on separable Banach spaces.


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## Body

## 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 [8–15]) 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 mapT is equivalent to the convergence of the other iterations (see [16–24]), 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(Ω,Σ,μ) be a complete probability measure space and Y a nonempty subset of a separable Banach space E. For two random mappings S,T:Ω×Y→E with Tω,Y⊆S(ω,Y) and C being a nonempty closed convex subset of a separable Banach space E, there exists a real number δ∈[0,1) and a monotone increasing function φ:R+→R+ with φ(0)=0, and for all x,y∈C, one has(1)Tω,x-Tω,y≤δSω,x-Sω,y+φSω,x-Tω,y,and for x,y∈C and h∈(0,1), one has(2)Tω,x-Tω,y≤hmax⁡Sω,x-Sω,y,Sω,x-Tω,x+Sω,y-Tω,y2,Sω,x-Tω,y+Sω,y-Tω,x2,(3)Tω,x-Tω,y≤hmax⁡Sω,x-Sω,y,Sω,x-Tω,x+Sω,y-Tω,y2,Sω,x-Tω,y,Sω,y-Tω,x. For x,y∈C, L>0,0<δ<1,(4)Tω,x-Tω,y≤δSω,x-Sω,y+LSω,x-Tω,x.For x,y∈C,0≤δ<1,M≥0 and since φ:R+→R+ is a monotone increasing sequence with φ(0)=0, then(5)Tω,x-Tω,y≤δSω,x-Sω,y+φSω,x-Tω,x1+MSω,x-Tω,x,(6)Tω,x-Tω,y≤δSω,x-Sω,y+φSω,x-Tω,x.Remark 2.
Observe that the contractive conditions imply one another but not conversely: (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6).Definition 3.
Let(Ω,Σ) be a measurable space and C be a nonempty closed convex sunset of a separable Banach space E. A function T:Ω→C is said to be measurable if T-1B∩C∈Σ for each Borel set B of E. A function T:Ω×C→C is called a random operator if T(·,x):Ω→C is measurable for every x∈C. A measurable function f:Ω→C is called a random fixed point for the operator T:Ω×C→C if T(ω,f(ω))=f(ω).Definition 4.
Let(Ω,Σ) be a measurable space and C be a nonempty closed convex subset of a separable Banach space E. A measurable function f:Ω→C is called a random coincidence for two random mappings S,T:Ω×C→C if T(ω,f(ω))=S(ω,f(ω)) for all ω∈Ω. The maps S,T are said to be random weakly compatible if they commute at their random coincidence; that is, if S(ω,f(ω))=T(ω,f(ω)) for every ω∈Ω, then S(T(ω,f(ω)))=T(S(ω,f(ω))) or S(ω,T(ω,f(ω)))=T(ω,S(ω,f(ω))).We define the random explicit Jungck-Kirk-multistep hybrid iterative algorithm as follows.Let(Ω,Σ) be a measurable space and C be a nonempty closed convex subset of a separable Banach space E. Let S,T:Ω×C→E be two random mappings with T(ω,C)⊆S(ω,C). Let x0,y0,z0,u0:Ω→C be an arbitrary measurable mapping for ω∈Ω, for n=1,2,….(i) The random explicit Jungck-Kirk-multistep hybrid iterative algorithm is a sequence{S(ω,xn(ω))}n=0∞ defined iteratively by(7)Sω,xn+1ω=αn,0Sω,xnω+∑i=1k1αn,iTiω,yn1ω,∑i=1k1αn,i=1Sω,ynjω=βn,0jSω,xnω+∑i=1kj+1βn,ijTiω,ynj+1ω,∑i=1kj+1βn,ij=1,j=1,2,…,q-2,Sω,ynq-1ω=βn,0q-1Sω,xnω+∑i=1kqβn,iq-1Tiω,xnω,∑i=1kqβn,iq-1=1,q≥2,where k1≥k2≥k3≥⋯≥kq, αn,i≥0,αn,0≠0,βn,ij≥0,βn,0j≠0, for each j and αn,i,βn,ij are measurable sequences in 0,1 for each j while k1,kj are fixed integers (for each j).(ii) The random explicit Jungck-Kirk-Noor hybrid iterative algorithm is a sequence{S(ω,yn(ω))}n=0∞ defined iteratively by(8)Sω,yn+1ω=αn,0Sω,ynω+∑i=1k1αn,iTiω,yn1ω,∑i=1k1αn,i=1Sω,yn1ω=βn,01Sω,ynω+∑i=1k2βn,i1Tiω,yn2ω,∑i=1k2βn,i1=1,Sω,yn2ω=βn,02Sω,ynω+∑i=1k3βn,i2Tiω,ynω,∑i=1k3βn,i2=1,where k1≥k2≥k3,αn,i≥0,αn,0≠0,βn,i1≥0,βn,01≠0,βn,i2≥0,βn,02≠0, and αn,i,βn,i1,βn,i2 are measurable sequences in [0,1] and k1,k2,k3 are fixed integers.(iii) The random explicit Jungck-Kirk-Ishikawa hybrid iterative algorithm is a sequence{S(ω,zn(ω))}n=0∞ defined iteratively by(9)Sω,zn+1ω=αn,0Sω,znω+∑i=1k1αn,iTiω,zn1ω,∑i=1k1αn,i=1Sω,zn1ω=βn,01Sω,znω+∑i=1k2βn,i1Tiω,znω,∑i=1k2βn,i1=1,where k1≥k2αn,i≥0,αn,0≠0,βn,i1≥0,βn,01≠0, and αn,i,βn,i1 are measurable sequences in [0,1] and k1,k2 are fixed integers.(iv) The random explicit Jungck-Kirk-Mann hybrid iterative algorithm is a sequence{S(ω,un(ω))}n=0∞ defined iteratively by(10)Sω,un+1ω=αn,0Sω,unω+∑i=1k1αn,iTiω,un1ω,∑i=1k1αn,i=1,where αn,i≥0,αn,0≠0, and αn,i is a measurable sequence in [0,1] and k1 is a fixed integer.Next, we define the random implicit Jungck-Kirk-multistep hybrid iterative algorithms as follows.Let(Ω,Σ) be a measurable space and C be a nonempty closed convex subset of a separable Banach space E. Let S,T:Ω×C→E be two random mappings with T(ω,C)⊆S(ω,C). Let x0,y0,z0,u0:Ω→C be an arbitrary measurable mapping for ω∈Ω,n=0,1,2,….(i) The random implicit Jungck-Kirk-multistep hybrid iterative algorithm is a sequence{S(ω,xn(ω))}n=0∞ defined iteratively by(11)Sω,xn+1ω=αn,0Sω,xn1ω+∑i=1k1αn,iTiω,xn+1ω,∑i=1k1αn,i=1Sω,xnjω=βn,0jSω,xnj+1ω+∑i=1kj+1βn,ijTiω,xnjω,∑i=1kj+1βn,ij=1,j=1,2,…,q-2,Sω,xnq-1ω=βn,0q-1Sω,xnω+∑i=1kqβn,iq-1Tiω,xnω,∑i=1kqβn,iq-1=1,q≥2,where k1,kj are fixed integers (for each j) with k1≥k2≥k3≥⋯≥kq,αn,i≥0, αn,0≠0,βn,i≥0,βn,0≠0, and αn,i,βn,i (for each j) are measurable sequences in [0,1]; (11) is called random implicit Jungck-Kirk-multistep hybrid iterative algorithm.(ii) The random implicit Jungck-Kirk-Noor hybrid iterative algorithm is a sequence{S(ω,yn(ω))}n=0∞ defined iteratively by(12)Sω,yn+1ω=αn,0Sω,yn1ω+∑i=1k1αn,iTiω,yn+1ω,∑i=1k1αn,i=1Sω,yn1ω=βn,01Sω,yn2ω+∑i=1k2βn,i1Tiω,yn1ω,∑i=1k2βn,i1=1,Sω,yn2ω=βn,02Sω,ynω+∑i=1k3βn,i2Tiω,yn2ω,∑i=1k3βn,i2=1,where k1≥k2≥k3,αn,i≥0,αn,0≠0,βn,i1≥0,βn,01≠0,βn,i2≥0,βn,02≠0, and αn,i,βn,i1,βn,i2 are measurable sequences in [0,1] and k1,k2,k3 are fixed integers.(iii) The random implicit Jungck-Kirk-Ishikawa hybrid iterative algorithm is a sequence{S(ω,zn(ω))}n=0∞ defined iteratively by(13)Sω,zn+1ω=αn,0Sω,zn1ω+∑i=1k1αn,iTiω,zn+1ω,∑i=1k1αn,i=1Sω,zn1ω=βn,01Sω,znω+∑i=1k2βn,i1Tiω,zn1ω,∑i=1k2βn,i1=1,where k1≥k2,αn,i≥0,αn,0≠0,βn,i1≥0,βn,01≠0, and αn,i,βn,i1 are measurable sequences in [0,1] and k1,k2 are fixed integers.(iv) The random implicit Jungck-Kirk-Mann hybrid iterative algorithm is a sequence{S(ω,un(ω))}n=0∞ defined iteratively by(14)Sω,un+1ω=αn,0Sω,unω+∑i=1k1αn,iTiω,un+1ω,∑i=1k1αn,i=1,where αn,i≥0,αn,0≠0, and αn,i are measurable sequences in [0,1] and k1 is a fixed integer.Next, we introduce the random Jungck-Kirk-SP multiple iterative algorithm.Let(Ω,Σ) be a measurable space and C be a nonempty closed convex subset of a separable Banach space E. Let S,T:Ω×C→E be two random mappings with T(ω,C)⊆S(ω,C). Let x0,y0:Ω→C be an arbitrary measurable mapping for ω∈Ω,n=0,1,2,….(i) The random Jungck-Kirk-multistep-SP hybrid iterative algorithm is a sequence{S(ω,xn(ω))}n=0∞ defined iteratively by(15)Sω,xn+1ω=αn,0Sω,yn1ω+∑i=1k1αn,iTiω,yn1ω,∑i=1k1αn,i=1Sω,ynjω=βn,0jSω,ynj+1ω+∑i=1kj+1βn,ijTiω,ynj+1ω,∑i=1kj+1βn,ij=1,j=1,2,…,q-2Sω,ynq-1ω=βn,0q-1Sω,xnω+∑i=1kqβn,iq-1Tiω,xnω,∑i=1kqβn,iq-1=1,q≥2,where k1,kj are fixed integers (for each j) with k1≥k2≥k3≥⋯≥kq,αn,i≥0,αn,0≠0,βn,i≥0,βn,0≠0, and αn,i,βn,i (for each j) are measurable sequences in [0,1].(ii) The random Jungck-Kirk Noor-SP hybrid iterative algorithm is a sequence{S(ω,yn(ω))}n=0∞ defined iteratively by(16)Sω,yn+1ω=αn,0Sω,yn1ω+∑i=1k1αn,iTiω,yn1ω,∑i=1k1αn,i=1Sω,yn1ω=βn,01Sω,yn2ω+∑i=1k2βn,i1Tiω,yn2ω,∑i=1k2βn,i1=1Sω,yn2ω=βn,02Sω,ynω+∑i=1k3βn,i2Tiω,ynω,∑i=1k3βn,i2=1,where k1,k2,k3 are fixed integers with k1≥k2≥k3,αn,i≥0,αn,0≠0,βn,i≥0,βn,0≠0, and αn,i,βn,i (for each j) are measurable sequences in [0,1].Lemma 5.
Ifδ is a real number such that 0≤δ<1 and {εn}n=0∞ is a sequence of positive numbers such that limn→∞=0, then, for any sequence of positive numbers {ϱ}n=0∞ satisfying (17)ϱn+1≤δϱn+εn,n=0,1,2,…,one has limn→∞ϱn=0.Lemma 6.
LetX· be a normed linear space and S,T:Y→X be a non-self-random commuting operator on an arbitrary set Y with values in X satisfying (6) such that Tω,Y⊆Sω,Y: (18)
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Lemma6 is proved by mathematical induction in [17].


## 3. Main Results

Theorem 7.
LetE be a separable Banach space and S,T:Ω×E→E be two random commuting mappings satisfying (19) such that T(ω,E)⊆S(ω,E). Assume that S and T are random weakly compatible. Let p(ω) be the random common point of S,T,Si,Ti (i.e., S(ω,p(ω))=T(ω,p(ω))=p(ω)). If u0,x0∈E defined by {S(ω,un(ω))}n=0∞,{S(ω,xn(ω))}n=0∞ as sequences satisfying (14) and (11), respectively, then the following are equivalent:(i)
Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to pω.(ii)
Random implicit Jungck-Kirk-multistep iteration (11) converges strongly to p(ω).Proof.
We first prove that (i) ⇒ (ii).
Assumelimn→∞S(ω,un(ω))=p(ω), and then using (14) and (11) and generalized contractive condition (19), we get (20)Sω,un+1ω-Sω,xn+1ω≤αn,0Sω,unω-Sω,xn1ω+∑i=1k1αn,iTiω,un+1ω-Tiω,xn+1ω≤αn,0Sω,unω-Sω,xn1ω+∑i=1k1αn,iδiSω,un+1ω-Sω,xn+1ω+∑i=1k1αn,i∑j=0iijδi-jφjSω,un+1ω-Tiω,un+1ω.From (20), (21)
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. Substituting (21) into (20), we get (22)Sω,un+1ω-Sω,xn+1ω≤αn,01-∑i=1k1αn,iδiSω,unω-Sω,xn1ω+∑i=1k1αn,i∑j=0iijδi-jφj1+δiSω,un+1ω-pω1-∑i=1k1αn,iδi. Using (11) and (14) and contractive condition (19), we get (23)Sω,unω-Sω,xn1ω≤βn,01Sω,unω-Sw,xn2ω+∑i=1q2βn,i1Sω,unω-Tiω,unω+Tiω,unω-Tiω,xn1ω≤βn,01Sω,unω-Sω,xn2ω+∑i=1q2βn,i1Sω,unω-Tiω,unω+∑i=1k2βn,i1Tiω,unω-Tiω,xn1ω≤βn,01Sω,unω-Sw,xn2ω+∑i=1q2βn,i1Sω,unω-Tiω,unω+∑i=1k2βn,i1δiSω,unω-Sω,xn1ω+∑i=1k2βn,i1∑j=0iijδi-jφjSω,unω-Tiω,unω. Following the method of proof in (21), we can write(24)Sω,unω-Tiω,unω≤1+δiSω,unω-pω.Substituting (24) into (23) and simplifying, we obtain (25)Sω,unω-Sω,xn1ω≤βn,011-∑i=1k2βn,i1δiSω,unω-Sω,xn2ω+∑i=1k2βn,i11+δi1-∑i=1q2βn,i1δiSω,unω-pω+∑i=1k2βn,i1∑j=0iijδi-jφj1+δiSω,unω-pω1-∑i=1k2βn,i1δi. Also, using (11) and (14) and contractive condition (19), we get (26)Sω,unω-Sω,xn2ω≤βn,021-∑i=1k3βn,i2δiSω,unω-Sω,xn3ω+∑i=1k3βn,i21+δi1-∑i=1q3βn,i2δiSω,unω-pω+∑i=1k3βn,i2∑j=0iijδi-jφj1+δiSω,unω-pω1-∑i=1k3βn,i2δi. Substituting (25) and (26) into (22), we get(27)Sω,un+1ω-Sω,xn+1ω≤αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δiSω,unω-Sω,xn3ω+αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δi∑i=1k3βn,i21+δi1-∑i=1k3βn,i2δiSω,unω-pω+αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δi∑i=1k3βn,i2∑j=0iijδi-jφj1+δiSω,unω-pω1-∑i=1k3βn,i2δi+αn,01-∑i=1k1αn,iδi∑i=1k2βn,i11+δi1-∑i=1k2βn,i1δiSω,unω-pω+αn,01-∑i=1k1αn,iδi∑i=1q2βn,i1∑j=0iijδi-jφj1+δiSω,unω-pω1-∑i=1k2βn,i1δi+∑i=1k1αn,i∑j=0iijδi-jφj1+δiSω,un+1ω-pω1-∑i=1k1αn,iδi. Continuing this process to (q-1) and simplifying, we get the following: (28)Sω,un+1ω-Sω,xn+1ω≤αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δiβn,0q-11-∑i=1kqβn,iq-1δiSω,unω-Sω,xnω+βn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δi∑i=1kqβn,iq-11+δi1-∑i=1kqβn,iq-1δi+βn,011-∑i=1k2βn,i1δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δi∑i=1kq-1βn,iq-21+δi1-∑i=1kq-1βn,iq-2δi+∑i=1k2βn,i11+δi1-∑i=1k2βn,i1δiαn,01-∑i=1k1αn,iδiSω,unω-pω+αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δi∑i=1kqβn,iq-1∑j=0iijδi-jφj1+δiSω,unω-pω1-∑i=1kqβn,iq-1δi+∑i=1k1αn,i∑j=0iijδi-jφj1+δiSω,un+1ω-pω1-∑i=1k1αn,iδi.Recall that αn,0/1-∑i=1q1αn,iδi≤∑i=1q1αn,iδi+αn,0.
Letδi≤δ<1; then, (29)∑i=1q1αn,iδi+αn,0≤1-αn,0δ+αn,0.Substituting (29) into (28), we get (30)Sω,un+1ω-Sω,xn+1ω≤1-λnSω,unω-Sω,xnω+en,where λn=1-αn,01-δ,(31)en=1-βn,01δ+βn,011-βn,02δ+βn,02⋯1-βn,0k-2δ+βn,0q-2∑i=1kqβn,iq-11-∑i=1kqβn,iq-1δi+1-βn,01δ+βn,01⋯∑i=1kq-1βn,iq-21-∑i=1kq-1βn,ik-2δi+∑i=1k2βn,i11-∑i=1k2βn,i1δ11-αn,0δ+αn,01+δiSω,unω-pω+1-αn,0δ+αn,01-βn,01δ+βn,01⋯1-βn,0q-2δ+βn,0q-2∑i=1kqβn,ik-1∑j=0iijδi-j1-∑i=1kqβn,iq-1δiφj1+δiSω,unωpω.Applying Lemma 5 in (30), it follows that limn→∞Sω,unω-Sω,xnω=0.
By assumptionlimn→∞Sω,unω=pω, then Sω,xnω-pω≤Sω,unω-Sω,xnω+Sω,unω-pω→0 as n→∞, that is, limn→∞Sω,xnω=pω.
Next, we show that (ii)→ (i). Assume limn→∞Sω,xnω=p(ω); then, using (11) and (14) and contractive condition (19), we have (32)Sω,xn+1ω-Sω,un+1ω≤αn,0Sω,xn1ω-Sω,unω+∑i=1k1αn,iTiω,xn+1ω-Tiω,un+1ω≤αn,0Sω,xn1ω-Sω,unω+∑i=1k1αn,iδiSω,xn+1ω-Sω,un+1ω+∑i=1k1αn,i∑j=0iijδi-jφjSω,xn+1ω-Tiω,xn+1ω. From (32), (33)Sω,xn+1ω-Tiω,xn+1ω=Sω,xn+1ω-Sω,pω+Tω,pω-Tiω,xn+1ω≤Sω,xn+1ω-pω+Tω,pω-Tiω,xn+1ω≤Sω,xn+1ω-pω+δipω-Sω,xn+1ω+∑i=1k1αn,i∑j=0iijδi-jφjSω,pω-Tω,pω=1+δiSω,xn+1ω-pω. Substituting (33) into (32) and simplifying, we obtain (34)Sω,xn+1ω-Sω,un+1ω≤αn,01-∑i=1k1αn,iδiSω,xn1ω-Sω,unω+∑i=1k1αn,i∑j=0iijδi-jφj1+δiSω,xn+1ω-pω1-∑i=1k1αn,iδi. Using (11) and (14) and contractive condition (19), we have (35)
S
ω
,
x
n
1
ω
-
S
ω
,
u
n
ω
≤
β
n
,
0
1
S
ω
,
x
n
2
ω
-
S
ω
,
u
n
ω
+
∑
i
=
1
k
2
β
n
,
i
1
T
i
ω
,
x
n
1
ω
-
S
ω
,
x
n
1
ω
+
∑
i
=
1
k
2
β
n
,
i
1
S
ω
,
x
n
1
ω
-
S
ω
,
u
n
ω
≤
β
n
,
0
1
1
-
∑
i
=
1
k
2
β
n
,
i
1
S
ω
,
x
n
2
ω
-
S
ω
,
u
n
ω
+
∑
i
=
1
k
2
β
n
,
i
1
1
-
∑
i
=
1
k
2
β
n
,
i
1
T
i
ω
,
x
n
1
ω
-
S
ω
,
x
n
1
ω
. From (35), (36)Sω,xn1ω-Tiω,xn1ω=Sω,xn1ω-Sω,pω+Tω,pω-Tiω,xn1ω≤Sω,xn1ω-pω+Tω,pω-Tiω,xn1ω≤Sω,xn1ω-pω+δiSω,xn1ω-pω+∑j=0iijδi-jφjSω,pω-Tω,pω=1+δiSω,xn1ω-pω,(37)Sω,xn1ω-pω≤βn,01Sω,xn2ω-pω+∑i=1k2βn,i1Tiω,xn1ω-pω≤βn,011-∑i=1k2βn,i1δiSω,xn2ω-pω,(38)Sω,xn2ω-pω≤βn,021-∑i=1k3βn,i2δiβn,031-∑i=1k4βn,i3δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δiβn,0q-11-∑i=1kqβn,iq-1δiSω,xnω-pω,(39)Sω,xnq-1ω-pω≤βn,0q-11-∑i=1kqβn,iq-1δiSω,xnω-pω.Substituting (37) and (38) into (36) yields(40)
S
ω
,
x
n
1
ω
-
T
i
ω
,
x
n
1
ω
≤
1
+
δ
i
β
n
,
0
1
1
-
∑
i
=
1
k
2
β
n
,
i
1
δ
i
β
n
,
0
2
1
-
∑
i
=
1
k
3
β
n
,
i
2
δ
i
⋯
β
n
,
0
q
-
2
1
-
∑
i
=
1
k
q
-
1
β
n
,
i
q
-
2
δ
i
β
n
,
0
q
-
1
1
-
∑
i
=
1
k
q
β
n
,
i
q
-
1
δ
i
S
ω
,
x
n
ω
-
p
ω
.Substituting (40) into (35), we obtain (41)Sω,xn1ω-Sω,unω≤βn,011-∑i=1k2βn,i1Sω,xn2ω-Sω,unω+1+δiβn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δiβn,0q-11-∑i=1kqβn,iq-1δi∑i=1k2βn,i11-∑i=1k2βn,i1Sω,xnω-pω.Sω,xn2ω-Sω,unω,…,S(ω,xnq-2(ω))-Sω,unω, and Sω,xnq-1ω-Sω,unω are similarly obtained and substituted into (41) to get (42)
S
ω
,
x
n
1
ω
-
S
ω
,
u
n
ω
≤
β
n
,
0
1
1
-
∑
i
=
1
k
2
β
n
,
i
1
δ
i
β
n
,
0
2
1
-
∑
i
=
1
k
3
β
n
,
i
2
δ
i
⋯
β
n
,
0
q
-
2
1
-
∑
i
=
1
k
q
-
1
β
n
,
i
q
-
2
δ
i
∑
i
=
1
k
q
β
n
,
i
k
-
1
1
-
∑
i
=
1
k
q
β
n
,
i
q
-
1
δ
i
S
ω
,
x
n
ω
-
S
ω
,
u
n
ω
+
∑
i
=
1
k
2
β
n
,
i
1
1
-
∑
i
=
1
k
2
β
n
,
i
1
+
∑
i
=
1
k
3
β
n
,
i
2
1
-
∑
i
=
1
k
3
β
n
,
i
2
δ
i
+
⋯
+
∑
i
=
1
k
q
β
n
,
i
k
-
1
1
-
∑
i
=
1
k
q
β
n
,
i
q
-
1
δ
i
β
n
,
0
1
1
-
∑
i
=
1
k
2
β
n
,
i
1
δ
i
β
n
,
0
2
1
-
∑
i
=
1
k
3
β
n
,
i
2
δ
i
…
β
n
,
0
q
-
2
1
-
∑
i
=
1
k
q
-
1
β
n
,
i
q
-
2
δ
i
β
n
,
0
q
-
1
1
-
∑
i
=
1
k
q
β
n
,
i
q
-
1
δ
i
1
+
δ
i
S
ω
,
x
n
ω
-
p
ω
. Substituting (42) into (34), we have (43)Sω,xn+1ω-Sω,un+1ω≤αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δi·∑i=1kqβn,iq-11-∑i=1kqβn,iq-1δiSω,xnω-Sω,unω+∑i=1k2βn,i11-∑i=1k2βn,i1+∑i=1k3βn,i21-∑i=1k3βn,i2δi+⋯+∑i=1kqβn,iq-11-∑i=1kqβn,iq-1δi·αn,01-∑i=1k1αn,iδiβn,011-∑i=1k2βn,i1δiβn,021-∑i=1k3βn,i2δi⋯βn,0q-21-∑i=1kq-1βn,iq-2δiβn,0q-11-∑i=1kqβn,iq-1δi1+δiSω,xnω-pω+∑i=1k1αn,iδi∑j=0iijδi-jφj1+δiSω,xn+1ω-pω1-∑i=1k1αn,iδi. Recall that αn,0/1-∑i=1k1‍αn,iδi≤∑i=1k1‍αn,iδi+αn,0.Letδi≤δ<1; then, (44)∑i=1k1αn,iδi+αn,0≤1-αn,0δ+αn,0.Using (44) in (43), we have (45)Sω,xn+1ω-Sω,un+1ω≤1-λnSω,xnω-Sω,unω+en,where λn=(1-αn,0)(1-δ), (46)en=1+δi1-αn,0δ+αn,01-βn,01δ+βn,011-βn,02δ+βn,02⋯1-βn,0q-2δ+βn,0q-2∑i=1q2βn,i11-∑i=1k2βn,i1+∑i=1k3βn,i21-∑i=1k3βn,i2δi+⋯+∑i=1kqβn,iq-11-∑i=1kqβn,iq-1δiSω,xnω-pω+∑i=1k1αn,iδi∑j=0iijδi-jφj1+δiSω,xn+1ω-pω1-∑i=1k1αn,iδi. Using Lemma 5 in (45), it follows that limn→∞Sω,xnω-Sω,unω=0.Since, by assumption,limn→∞Sω,xnω=pω, then S(ω,un(ω))-p(ω)≤Sω,xnω-Sω,unω+Sω,xnω-pω→0 as n→∞; that is, limn→∞Sω,unω=pω.Since(i)→(ii) and (ii)→(i), it is shown that the convergence of random implicit Jungck-Kirk-Mann iteration (14) is equivalent to the convergence of random implicit Jungck-Kirk-multistep iteration (11) when applied to a pair of random weakly compatible generalized φ-contractive-like conditions (19). This ends the proof.Since the random implicit Jungck-Kirk-multistep iteration (11) generalizes other random implicit Jungck-Kirk-type iterations (12), (13), and (14), then Theorem 7 leads to the following corollaries.Corollary 8.
LetE be a separable Banach space and S,T:Ω×E→E be two random commuting mappings satisfying (19) such that T(ω,E)⊆S(ω,E). Assume that S and T are random weakly compatible. Let p(ω) be the random common point of S,T,Si,Ti (i.e., S(ω,p(ω))=Tω,pω=pω). If u0,z0,y0∈E defined by {Sω,unω}n=0∞,{Sω,znω}n=0∞ and {S(ω,yn(ω))}n=0∞ as sequences satisfying (14), (13), and (12), respectively, then the following are equivalent: (a)
(i)
Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to p(ω).(ii)
Random implicit Jungck-Kirk-Ishikawa iteration (13) converges strongly to p(ω).(b)
(i)
Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to p.(ii)
Random implicit Jungck-Kirk-Noor iteration (12) converges strongly to p.Proof.
The proof of Corollary8 is similar to that of Theorem 7.Corollary 9.
LetE be a separable Banach space and S,T:Ω×E→E be two random commuting mappings satisfying (19) such that T(ω,E)⊆S(ω,E). Assume that S and T are random weakly compatible. Let p(ω) be the random common point of S,T,Si,Ti (i.e., S(ω,p(ω))=T(ω,p(ω))=p(ω)). If u0,z0,y0,x0∈E defined by {S(ω,un(ω))}n=0∞,{S(ω,zn(ω))}n=0∞,{S(ω,yn(ω))}n=0∞ and {S(ω,xn(ω))}n=0∞ as sequences satisfying (14), (13), (12), and (11), respectively, then the following are equivalent:(i)
Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to p(ω).(ii)
Random implicit Jungck-Kirk-Ishikawa iteration (13) converges strongly to p(ω).(iii)
Random implicit Jungck-Kirk-Noor iteration (12) converges strongly to p(ω).(iv)
Random implicit Jungck-Kirk-multistep iteration (11) converges strongly to p(ω).Example 10.
Consider the equationf(ω,x(ω))=0, where f is a random real function defined on interval [0,π/2] by f(ω,x(ω))=x2(ω)-π/22cos(x(ω)).
Note thatf can be decomposed as f=π/2(S-T), where the maps S and T are the self-mappings in [0,π/2] defined by S((ω,x(ω))≔2/πx2(ω) and T((ω,x(ω))≔π/2cos(x(ω)). Clearly, S(ω,xω and T((ω,x(ω)) satisfy the generalized φ-contractive condition (19). They coincide at a(ω)≈1.0792 and we have p(ω)=S(ω,a(ω))=T(ω,a(ω))≈0.7415. Thus, a(ω) is a solution to f(ω,x(ω))=0. However, if S and T are random weakly compatible, then p(ω)=S(ω,p(ω))=T(ω,p(ω))≈0.7415.Using MATLAB, we show in Table1 that (11), (12), (13), and (14) are equivalently used to approximate the stochastic common fixed point p(ω).Table 1
Approximation of stochastic common fixed point.
n x n y n z n u n S ( ω , x n ( ω ) ) S ( ω , y n ( ω ) ) S ( ω , z n ( ω ) ) S ( ω , u n ( ω ) ) 0 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 1 1.0472 1.0468 1.0462 1.0459 0.6982 0.6926 0.6912 0.6908 2 1.0739 1.0711 1.0692 1.0673 0.7343 0.7337 0.7332 0.7328 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 5 1.0790 1.0783 1.0740 1.0702 0.7411 0.7407 0.7404 0.7401 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 7 1.0791 1.0791 1.0769 1.0766 0.7414 0.7413 0.7412 0.7411 8 1.0792 1.0792 1.0791 1.0791 0.7415 0.7415 0.7414 0.7414 9 1.0792 1.0792 1.0792 1.0791 0.7415 0.7415 0.7415 0.7415


## 4. Conclusion

In this work, we have investigated the equivalence of convergence of various stochastic implicit Jungck-Kirk-type iterations for generalized random contractive-like operators in separable Banach spaces. We also gave an example to show the equivalence results. We hope that the findings in this paper will help researchers enhance and promote the further study on iterative schemes for two or more maps in well known spaces to carry out a general framework for their applications in real life.



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*Source: 9576137-2018-05-20.xml*