A NEW CLASS OF GENERAL RANDOM IMPLICIT
QUASI-VARIATIONAL INEQUALITIES
HENG-YOU LAN
Received 9 November 2005; Accepted 21 January 2006
We introduce a class of projection-contraction methods for solving a class of general random implicit quasi-variational inequalities with random multivalued mappings in Hilbert spaces, construct some random iterative algorithms, and give some existence the-orems of random solutions for this class of general random implicit quasi-variational inequalities. We also discuss the convergence and stability of a new perturbed Ishikawa iterative algorithm for solving a class of generalized random nonlinear implicit quasi-variational inequalities involving random single-valued mappings. The results presented in this paper improve and extend the earlier and recent results.
Copyright © 2006 Heng-You Lan. 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.
1. Introduction
Throughout this paper, we suppose thatRdenotes the set of real numbers and (Ω,Ꮽ,μ) is a completeσ-finite measure space. LetEbe a separable real Hilbert space endowed with the norm · and inner product·,·, letDbe a nonempty subset ofE, and letCB(E) denote the family of all nonempty bounded closed subsets ofE. We denote byᏮ(E) the class of Borelσ-fields inE.
In this paper, we will consider the following new general random implicit quasi-variational inequality with random multivalued mappings by using a new class of pro-jection method: findx:Ω→Dandu,v,ξ:Ω→Esuch thatg(t,x(t))∈J(t,v(t)),u(t)∈
T(t,x(t)),v(t)∈F(t,x(t)),ξ(t)∈S(t,x(t)), and
at,u(t),g(t,y)−gt,x(t)+Nt,ft,x(t),ξ(t),g(t,y)−gt,x(t)≥0 (1.1)
for allt∈Ωandg(t,y)∈J(t,v(t)), whereg:Ω×D→E, f :Ω×E→E, andN:Ω×E×
E→Eare measurable single-valued mappings,T,S:Ω×D→CB(E) andF:Ω×D→
CB(D) are random multivalued mappings, J:Ω×D→P(E) is a random multivalued
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 81261, Pages1–17
mapping such that for eacht∈Ωandx∈D,P(E) denotes the power set ofE,J(t,x) is closed convex, anda:Ω×E×E→Ris a random function.
Some special cases of the problem (1.1) are presented as follows.
Ifg≡I, the identity mapping, then the problem (1.1) is equivalent to the problem of findingx:Ω→Dandu,v,ξ:Ω→Esuch thatx(t)∈J(t,v(t)),v(t)∈F(t,x(t)),u(t)∈
T(t,x(t)),ξ(t)∈S(t,x(t)), and
at,u(t),y−x(t)+Nt,ft,x(t),ξ(t),y−x(t)≥0 (1.2)
for allt∈Ωandy∈J(t,v(t)). The problem (1.2) is called a generalized random strongly nonlinear multivalued implicit quasi-variational inequality problem and appears to be a new one.
IfJ(t,x(t))=m(t,x(t)) +K, wherem:Ω×D→EandK is a nonempty closed con-vex subset ofE, then the problem (1.1) becomes to the following generalized nonlin-ear random implicit quasi-variational inequality for random multivalued mappings: find x:Ω→Dandu,v,ξ:Ω→Esuch thatg(t,x(t))−m(t,v(t))∈K,u(t)∈T(t,x(t)),v(t)∈
F(t,x(t)),ξ(t)∈S(t,x(t)), and
at,u(t),g(t,y)−gt,x(t)+Nt,ft,x(t),ξ(t),g(t,y)−gt,x(t)≥0 (1.3)
for allt∈Ωandg(t,y)∈m(t,v(t)) +K.
IfN(t,z,w)=w+z for allt∈Ω,z,w∈E, then the problem (1.1) reduces to find-ingx:Ω→Dandu,v,ξ:Ω→Esuch thatg(t,x(t))∈J(t,v(t)),u(t)∈T(t,x(t)),v(t)∈
F(t,x(t)),ξ(t)∈S(t,x(t)), and
at,u(t),g(t,y)−gt,x(t)+ft,x(t)+ξ(t),g(t,y)−gt,x(t)≥0 (1.4)
for allt∈Ωandg(t,y)∈J(t,v(t)).
The problem (1.4) is called a generalized random implicit quasi-variational inequality for random multivalued mappings, which is studied by Cho et al. [8] whenT≡I and f ≡0, and includes various known random variational inequalities. For details, we refer the reader to [8,11,12,19] and the references therein.
IfT,S:Ω×D→EandF:Ω×D→Dare random single-valued mappings, andF=I, then the problem (1.4) becomes to the following random generalized nonlinear implicit quasi-variational inequality problem involving random single-valued mappings: findx: Ω→Dsuch thatg(t,x(t))∈J(t,x(t)) and
at,Tt,x(t),g(t,y)−gt,x(t)+ft,x(t)+St,x(t),g(t,y)−gt,x(t)≥0 (1.5)
for allt∈Ωandg(t,y)∈J(t,x(t)).
The study of such types of problems is inspired and motivated by an increasing interest in the nonlinear random equations involving the random operators in view of their need in dealing with probabilistic models, which arise in biological, physical, and system sci-ences and other applied scisci-ences, and can be solved with the use of variational inequalities (see [21]). Some related works, we refer to [2,4] and the references therein. Further, the recent research works of these fascinating areas have been accelerating the random vari-ational and random quasi-varivari-ational inequality problems to be introduced and studied by Chang [4], Chang and Zhu [7], Cho et al. [8], Ganguly and Wadhwa [11], Huang [14], Huang and Cho [15], Huang et al. [16], Noor and Elsanousi [19], and Yuan et al. [24].
On the other hand, in [22], Verma studied an extension of the projection-contraction method, which generalizes the existing projection-contraction methods, and applied the extended projection-contraction method to the solvability of a general monotone varia-tional inequalities. Very recently, Lan et al. [17,18] introduced and studied some new iter-ative algorithms for solving a class of nonlinear variational inequalities with multivalued mappings in Hilbert spaces, and gave some convergence analysis of iterative sequences generated by the algorithms.
In this paper, we introduce a class of projection-contraction methods for solving a new class of general random implicit quasi-variational inequalities with random multivalued mappings in Hilbert spaces, construct some random iterative algorithms, and give some existence theorems of random solutions for this class of general random implicit quasi-variational inequalities. We also discuss the convergence and stability of a new perturbed Ishikawa iterative algorithm for solving a class of generalized random nonlinear implicit quasi-variational inequalities involving random single-valued mappings. Our results im-prove and generalize many known corresponding results in the literature.
2. Preliminaries
In the sequel, we first give the following concepts and lemmas which are essential for this paper.
Definition 2.1. A mappingx:Ω→Eis said to be measurable if for anyB∈Ꮾ(E),{t∈
Ω:x(t)∈B} ∈Ꮽ.
Definition 2.2. A mappingF:Ω×E→Eis said to be
(i) a random operator if for anyx∈E,F(t,x)=y(t) is measurable;
(ii) Lipschitz continuous (resp., monotone, linear, bounded) if for any t∈Ω, the mappingF(t,·) :E→Eis Lipschitz continuous (resp., monotone, linear, bound-ed).
Definition 2.3. A multivalued mappingΓ:Ω→P(E) is said to be measurable if for any B∈Ꮾ(E),Γ−1(B)= {t∈Ω:Γ(t)∩B=∅} ∈Ꮽ.
Definition 2.4. A mappingu:Ω→Eis called a measurable selection of a multivalued measurable mappingΓ:Ω→P(E) ifuis measurable and for anyt∈Ω,u(t)∈Γ(t).
T:Ω×D→CB(E) be a multivalued measurable mapping. Then
(i) f is said to bec(t)-strongly monotone onΩ×Dwith respect to the second argu-ment ofNandg, if for allt∈Ωandx,y∈D, there exists a measurable function c:Ω→(0, +∞) such that
g(t,x)−g(t,y),Nt,f(t,x),·−Nt,f(t,y),·≥c(t)g(t,x)−g(t,y)2; (2.1)
(ii) f is said to beν(t)-Lipschitz continuous onΩ×Dwith respect to the second argument ofN andg, if for anyt∈Ωandx,y∈D, there exists a measurable functionν:Ω→(0, +∞) such that
N
t,f(t,x),·−Nt,f(t,y),·≤ν(t)g(t,x)−g(t,y) (2.2)
and if N(t,f(t,x),y)= f(t,x) for allt∈Ωandx,y∈D, then f is said to be Lipschitz continuous onΩ×Dwith respect togwith measurable functionν(t); (iii)Nis said to beσ(t)-Lipschitz continuous onEwith respect to the third argument
if there exists a measurable functionσ:Ω→(0, +∞) such that
N(t,·,x)−N(t,·,y)≤σ(t)x−y, ∀x,y∈E; (2.3)
(iv)T is said to beH-Lipschitz continuous onΩ×Dwith respect tog with measur-able functionγ(t) if there exists a measurable functionγ:Ω→(0, +∞) such that for anyt∈Ωandx,y∈D,
HT(t,x),T(t,y)≤γ(t)g(t,x)−g(t,y), (2.4)
whereH(·,·) is the Hausdorffmetric onCB(E) defined as follows: for any given A,B∈CB(E),
H(A,B)=max
sup x∈A
inf y∈B
d(x,y), sup y∈B
inf x∈A
d(x,y)
. (2.5)
Definition 2.6. A random multivalued mappingS:Ω×E→P(E) is said to be (i) measurable if, for anyx∈E,S(·,x) is measurable;
(ii)H-continuous if, for anyt∈Ω,S(t,·) is continuous in the Hausdorffmetric.
Lemma2.7 [3]. LetM:Ω×E→CB(E)be anH-continuous random multivalued map-ping. Then for any measurable mappingx:Ω→E, the multivalued mappingM(·,x) :Ω→
CB(E)is measurable.
measurable selectiony:Ω→EofV such that x(t)−y(t)≤(1 +)H
M(t),V(t), ∀t∈Ω. (2.6)
Definition 2.9. An operatora:Ω×E×E→Eis called a randomβ(t)-bounded bilinear function if the following conditions are satisfied:
(1) for anyt∈Ω,a(t,·,·) is bilinear and there exists a measurable functionβ:Ω→ (0, +∞) such that
a(t,x,y) ≤β(t)x · y, ∀t∈Ω,x,y∈E; (2.7)
(2) for anyx,y∈E,a(·,x,y) is a measurable function.
Lemma2.10 [15]. Ifais a random bilinear function, then there exists a unique random bounded linear operatorA:Ω×E→Esuch that
A(t,x),y=a(t,x,y), A(t,·)=a(t,·,·), (2.8)
for allt∈Ωandx,y∈E, where
A(t,·)=supA(t,x):x ≤1 , a(t,·,·)=sup a(t,x,y) :x ≤1,y ≤1
. (2.9)
Lemma2.11 [4]. LetK be a closed convex subset ofE. Then for an elementz∈K,x∈K satisfies the inequalityx−z,y−x ≥0for ally∈Kif and only if
x=PK(z), (2.10)
wherePKis the projection ofEonK.
It is well known that the mappingPKdefined by (2.10) is nonexpansive, that is,
PK(x)−PK(y)≤ x−y, ∀x,y∈E. (2.11)
Definition 2.12. LetJ:Ω×D→P(E) be a random multivalued mapping such that for eacht∈Ωandx∈E,J(t,x) is a nonempty closed convex subset ofE. The projection PJ(t,x)is said to be aτ(t)-Lipschitz continuous random operator onΩ×Dwith respect to gif
(1) for any givenx,z∈E,PJ(t,x)(z) is measurable;
(2) there exists a measurable functionτ:Ω→(0, +∞) such that for allx,y,z∈Eand t∈Ω,
PJ(t,x)(z)−PJ(t,y)(z)≤τ(t)g(t,x)−g(t,y). (2.12)
Lemma2.13 [5]. LetK be a closed convex subset ofEand letm:Ω×E→Ebe a random operator. IfJ(t,x)=m(t,x) +Kfor allt∈Ωandx∈E, then
(i)for anyz∈E,PJ(t,x)(z)=m(t,x) +PK(z−m(t,x))for allt∈Ωandx∈E; (ii)PJ(t,x)is a2κ(t)-Lipschitz continuous operator whenmis aκ(t)-Lipschitz
continu-ous random operator.
3. Existence and convergence theorems
In this section, we suggest and analyze a new projection-contraction iterative method for solving the random multivalued variational inequality (1.1). Firstly, from the proof of Theorem 3.2 in Cho et al. [9], we have the following lemma.
Lemma 3.1. Let D be a nonempty subset of a separable real Hilbert spaceE and letJ: Ω×D→P(E)be a random multivalued mapping such that for eacht∈Ωandx∈E,J(t,x) is a closed convex subset inEandJ(Ω×D)⊂g(Ω×D). Then the measurable mappings x:Ω→Dandu,v,ξ:Ω→Eare the solutions of (1.1) if and only if for anyt∈Ω,u(t)∈
T(t,x(t)),v(t)∈F(t,x(t)),ξ(t)∈S(t,x(t)), and
gt,x(t)=PJ(t,v(t))
gt,x(t)−ρ(t)Nt,ft,x(t),ξ(t)+At,u(t), (3.1)
whereρ:Ω→(0, +∞)is a measurable function andA(t,x),y =a(t,x,y)for allt∈Ωand x,y∈E.
Based onLemma 3.1, we are now in a position to propose the following generalized and unified new projection-contraction iterative algorithm for solving the problem (1.1).
Algorithm 3.2. Let Dbe a nonempty subset of a separable real Hilbert spaceEand let λ:Ω→(0, 1) be a measurable step size function. Letg:Ω×D→E, f :Ω×E→E, and N:Ω×E×E→Ebe measurable single-valued mappings. LetT,S:Ω×D→CB(E) and F:Ω×D→CB(D) be multivalued random mappings. LetJ:Ω×D→P(E) be a ran-dom multivalued mapping such that for eacht∈Ωandx∈D,J(t,x) is closed convex andJ(Ω×D)⊂g(Ω×D). Then byLemma 2.7and Himmelberg [12], we know that for given x0(·)∈D, the multivalued mappings T(·,x0(·)),F(·,x0(·)), and S(·,x0(·)) are measurable, and there exist measurable selectionsu0(·)∈T(·,x0(·)),v0(·)∈F(·,x0(·)), andξ0(·)∈S(·,x0(·)). Setg(t,x1(t))=g(t,x0(t))−λ(t){g(t,x0(t))−PJ(t,v0(t))[g(t,x0(t))−
ρ(t)(N(t,f(t,x0(t)),ξ0(t)) +A(t,u0(t)))]}, whereρandAare the same as inLemma 3.1. Then it is easy to know thatx1:Ω→Eis measurable. Sinceu0(t)∈T(t,x0(t))∈CB(E), v0(t)∈F(t,x0(t))∈CB(D), andξ0(t)∈S(t,x0(t))∈CB(E), by Lemma 2.8, there exist measurable selectionsu1(t)∈T(t,x1(t)),v1(t)∈F(t,x1(t)), andξ1(t)∈S(t,x1(t)) such that for allt∈Ω,
u0(t)−u1(t)≤1 +1 1
HTt,x0(t),Tt,x1(t),
v0(t)−v1(t)≤1 +1 1
HFt,x0(t),Ft,x1(t),
ξ0(t)−ξ1(t)≤1 +1 1
HSt,x0(t),St,x1(t).
By induction, we can define sequences{xn(t)},{un(t)},{vn(t)}, and{ξn(t)}inductively satisfying
gt,xn+1(t)=gt,xn(t)−λ(t)gt,xn(t)−PJ(t,vn(t))
gt,xn(t)
−ρ(t)Nt,ft,xn(t),ξn(t)+At,un(t),
un(t)∈T
t,xn(t)
,
un(t)−un+1(t)≤1 + 1 n+ 1
HTt,xn(t),Tt,xn+1(t),
vn(t)∈Ft,xn(t),
vn(t)−vn+1(t)≤1 + 1 n+ 1
HFt,xn(t),Ft,xn+1(t),
ξn(t)∈St,xn(t),
ξn(t)−ξn+1(t)≤1 + 1 n+ 1
HSt,xn(t)
,St,xn+1(t).
(3.3)
Similarly, we have the following algorithms.
Algorithm 3.3. Suppose thatD,λ,g, f,N,T,S,F,Aare the same as inAlgorithm 3.2
and J(t,z(t))=m(t,z(t)) +K for allt∈Ωand measurable operatorz:Ω→D, where m:Ω×D→Eis a single-valued mapping andKis a closed convex subset ofE. For an ar-bitrarily chosen measurable mappingx0:Ω→D, the sequences{xn(t)},{un(t)},{vn(t)}, and{ξn(t)}are generated by a random iterative procedure
gt,xn+1(t)=gt,xn(t)−λ(t)gt,xn(t)−mt,vn(t)−PK
gt,xn(t)
−ρ(t)Nt,ft,xn(t)
,ξn(t)
+At,un(t)
+mt,vn(t)
,
un(t)∈Tt,xn(t),
un(t)−un+1(t)≤1 + 1 n+ 1
HTt,xn(t),Tt,xn+1(t),
vn(t)∈Ft,xn(t),
vn(t)−vn+1(t)≤1 + 1 n+ 1
HFt,xn(t),Ft,xn+1(t),
ξn(t)∈S
t,xn(t)
,
ξn(t)−ξn+1(t)≤1 + 1 n+ 1
HSt,xn(t),St,xn+1(t).
Algorithm 3.4. LetD,λ,g, f,N,J,Abe the same as inAlgorithm 3.2and letT,S:Ω×
D→Ebe two measurable single-valued mappings. For an arbitrarily chosen measurable mappingx0:Ω→D, we can define the sequence{xn(t)}of measurable mappings by
gt,xn+1(t)
=gt,xn(t)−λ(t)gt,xn(t)−PJ(t,xn(t))
gt,xn(t)−ρ(t)Nt,ft,xn(t),St,xn(t)
+At,Tt,xn(t).
(3.5)
It is easy to see that if we suppose that the mappingg:Ω×D→Eis expansive, then the inverse mappingg−1ofgexists and eachxn(t) is computable for allt∈Ω.
Now, we prove the existence of solutions of the problem (1.1) and the convergence of
Algorithm 3.2.
Theorem 3.5. Let D be a nonempty subset of a separable real Hilbert spaceE and let g:Ω×D→Ebe a measurable mapping such thatg(Ω×D)is a closed set inE. Suppose thatN:Ω×E×E→Eis a randomκ(t)-Lipschitz continuous single mapping with respect to the third argument andJ:Ω×D→P(E)is a random multivalued mapping such that J(Ω×D)⊂g(Ω×D), for each t∈Ωand x∈E,J(t,x)is nonempty closed convex and PJ(t,x) is anη(t)-Lipschitz continuous random operator on Ω×D. Let f :Ω×E→E be δ(t)-strongly monotone and σ(t)-Lipschitz continuous on Ω×D with respect to the sec-ond argument ofN andg, and leta:Ω×E×E→Rbe a randomβ(t)-bounded bilinear function. Let random multivalued mappings T,S:Ω×D→CB(E), F:Ω×D→CB(D) beH-Lipschitz continuous with respect tog with measurable functionsσ(t),τ(t), andζ(t), respectively. If for anyt∈Ω,
ν(t)=κ(t)τ(t) +β(t)σ(t)< σ(t),
0< ρ(t)<1−η(t)ζ(t) ν(t) ,
δ(t)>ν(t)1−η(t)ζ(t)+
η(t)ζ(t)2−η(t)ζ(t)σ(t)2−ν(t)2,
ρ(t)−δ(t)−ν(t)
1−η(t)ζ(t) σ(t)2−ν(t)2
<
δ(t)−ν(t)1−η(t)ζ(t)2−η(t)ζ(t)2−η(t)ζ(t)σ(t)2−ν(t)2
σ(t)2−ν(t)2 ,
(3.6)
then for any t∈Ω, there existx∗(t)∈D,u∗(t)∈T(t,x∗(t)),v∗(t)∈F(t,x∗(t)), and ξ∗(t)∈S(t,x∗(t)) such that(x∗(t),u∗(t),v∗(t),ξ∗(t))is a solution of the problem (1.1) and
gt,xn(t)−→gt,x∗(t), un(t)−→u∗(t), vn(t)−→v∗(t), ξn(t)−→ξ∗(t) asn−→ ∞, (3.7)
Proof. It follows from (3.3),Lemma 2.11, andDefinition 2.12that g
t,xn+1(t)−gt,xn(t)≤1−λ(t)gt,xn(t)−gt,xn−1(t)
+λ(t)PJ(t,vn(t))
gt,xn(t)−ρ(t)Nt,ft,xn(t),ξn(t)+At,un(t)
−PJ(t,vn(t))
gt,xn−1(t)−ρ(t)Nt,ft,xn−1(t),ξn−1(t)+At,un−1(t)
+λ(t)PJ(t,vn(t))
gt,xn−1(t)−ρ(t)Nt,ft,xn−1(t),ξn−1(t)+At,un−1(t)
−PJ(t,vn−1(t))
gt,xn−1(t)−ρ(t)Nt,ft,xn−1(t),ξn−1(t)+At,un−1(t)
≤1−λ(t)gt,xn(t)−gt,xn−1(t)+λ(t)η(t)vn(t)−vn−1(t)
+λ(t)gt,xn(t)−gt,xn−1(t)−ρ(t)Nt,ft,xn(t),ξn(t)−Nt,ft,xn−1(t),ξn(t)
+ρ(t)λ(t)Nt,ft,xn−1(t),ξn(t)
−Nt,ft,xn−1(t),ξn−1(t)
+ρ(t)λ(t)At,un(t)−At,un−1(t).
(3.8)
Since f :Ω×E→Eisδ(t)-strongly monotone andσ(t)-Lipschitz continuous with re-spect to the second argument ofNandg,T,S, andFareH-Lipschitz continuous with respect togwith measurable functionsσ(t),τ(t), andζ(t), respectively,Nisκ(t)-Lipschitz continuous with respect to the third argument, andais a randomβ(t)-bounded bilinear function, we get
g
t,xn(t)−gt,xn−1(t)−ρ(t)Nt,ft,xn(t),ξn(t)−Nt,ft,xn−1(t),ξn(t)
≤1−2ρ(t)δ(t) +ρ(t)2σ(t)2gt,xn(t)−gt,x
n−1(t),
(3.9) vn(t)−vn−1(t)≤(1 +ε)H
Ft,xn(t),Ft,xn−1(t)
≤(1 +ε)ζ(t)gt,xn(t)−gt,xn−1(t), (3.10)
N
t,ft,xn−1(t),ξn(t)
−Nt,ft,xn−1(t),ξn−1(t)
≤κ(t)ξn(t)−ξn−1(t)≤κ(t)(1 +ε)HSt,xn(t),St,xn−1(t)
≤(1 +ε)κ(t)τ(t)gt,xn(t)−gt,xn−1(t),
(3.11)
A
t,un(t)−At,un−1(t)≤β(t)un(t)−un−1(t)
≤(1 +ε)β(t)HTt,xn(t)
,Tt,xn−1(t)
≤(1 +ε)β(t)σ(t)gt,xn(t)
−gt,xn−1(t).
(3.12)
Using (3.9)–(3.12) in (3.8), for allt∈Ω, we have g
t,xn+1(t)−gt,xn(t)
≤1−λ(t) +λ(t)η(t)ζ(t)(1 +ε) +1−2ρ(t)δ(t) +ρ(t)2σ(t)2
+ρ(t)(1 +ε)κ(t)τ(t) +β(t)σ(t)
×gt,xn(t)
−gt,xn−1(t)=ϑ(t,ε)gt,xn(t)
−gt,xn−1(t),
where
ϑ(t,ε)=1−λ(t)1−k(t,ε),
k(t,ε)=η(t)ζ(t)(1 +ε) +1−2ρ(t)δ(t) +ρ(t)2σ(t)2
+ρ(t)(1 +ε)κ(t)τ(t) +β(t)σ(t).
(3.14)
Let k(t)=η(t)ζ(t) +1−2ρ(t)δ(t) +ρ(t)2σ(t)2+ρ(t)(κ(t)τ(t) +β(t)σ(t)), ϑ(t)=1− λ(t)(1−k(t)). Thenk(t,ε)→k(t),ϑ(t,ε)→ϑ(t) asε→0. From condition (3.6), we know that 0< ϑ(t)<1 for allt∈Ωand so 0< ϑ(t,ε)<1 forεsufficiently small and allt∈Ω. It follows from (3.13) that{g(t,xn(t))}is a Cauchy sequence ing(Ω×D). Sinceg(Ω×D) is closed inE, there exists a measurable mappingx∗:Ω→Dsuch that fort∈Ω,
gt,xn(t)−→gt,x∗(t) asn−→ ∞. (3.15)
By (3.10)–(3.12), we know that{un(t)},{vn(t)}, and{ξn(t)}are also Cauchy sequences inE. Let
un(t)−→u∗(t), vn(t)−→v∗(t), ξn(t)−→ξ∗(t) asn−→ ∞. (3.16)
Now we show thatu∗(t)∈T(t,x∗(t)). In fact, we have
du∗(t),Tt,x∗(t)=infu∗(t)−y:y∈Tt,x∗(t)
≤u∗(t)−un(t)+dun(t),Tt,x∗(t)
≤u∗(t)−un(t)+HTt,xn(t),Tt,x∗(t)
≤u∗(t)−un(t)+σ(t)gt,xn(t)−gt,x∗(t)−→0.
(3.17)
This implies thatu∗(t)∈T(t,x∗(t)). Similarly, we havev∗(t)∈F(t,x∗(t)) andξ∗(t)∈
S(t,x∗(t)).
Next, we prove that
gt,x∗(t)=PJ(t,v∗(t))gt,x∗(t)−ρ(t)Nt,ft,x∗(t),ξ∗(t)+At,u∗(t). (3.18)
Indeed, from (3.3), we know that it is enough to prove that
lim
n→∞PJ(t,vn(t))
gt,xn(t)−ρ(t)Nt,ft,xn(t),ξn(t)+At,un(t)
=PJ(t,v∗(t))gt,x∗(t)−ρ(t)Nt,ft,x∗(t),ξ∗(t)+At,u∗(t).
It follows fromLemma 2.11andDefinition 2.12that PJ(t,v
n(t))
gt,xn(t)−ρ(t)Nt,ft,xn(t),ξn(t)+At,un(t)
−PJ(t,v∗(t))gt,x∗(t)−ρ(t)Nt,ft,x∗(t),ξ∗(t)+At,u∗(t)
≤PJ(t,vn(t))
gt,xn(t)
−ρ(t)Nt,f(t,xn(t)
,ξn(t)
+At,un(t)
−PJ(t,vn(t))
gt,x∗(t)−ρ(t)Nt,ft,x∗(t),ξ∗(t)+At,u∗(t)
+PJ(t,vn(t))
gt,x∗(t)−ρ(t)Nt,ft,x∗(t),ξ∗(t)+At,u∗(t)
−PJ(t,v∗(t))gt,x∗(t)−ρ(t)Nt,ft,x∗(t),ξ∗(t)+At,u∗(t)
≤gt,xn(t)−gt,x∗(t)−ρ(t)Nt,ft,xn(t),ξn(t)−Nt,ft,x∗(t),ξn(t)
+ρ(t)Nt,ft,x∗(t),ξn(t)
−Nt,ft,x∗(t),ξ∗(t)
+ρ(t)At,un(t)−At,u∗(t)+η(t)vn(t)−v∗(t)
≤1−2ρ(t)δ(t) +ρ(t)2σ(t)2gt,xn(t)−gt,x∗(t)
+ρ(t)κ(t)ξn(t)−ξ∗(t)+ρ(t)β(t)un(t)−u∗(t)
+η(t)vn(t)−v∗(t).
(3.20)
This implies that (3.19) holds and so (3.18) is true. This completes the proof.
FromTheorem 3.5, we can obtain the following results.
Theorem3.6. LetD,g,N, f,T,F,S, andabe the same as inTheorem 3.5. Suppose that K is a nonempty closed convex subset ofEand thatm:Ω×D→Eis anη(t)/2-Lipschitz continuous random operator. If there exists a measurable mappingρ:Ω→(0, +∞)such that condition (3.6) holds, then for anyt∈Ωthere existx∗(t)∈D,u∗(t)∈T(t,x∗(t)),v∗(t)∈
F(t,x∗(t)), andξ∗(t)∈S(t,x∗(t))such that(x∗(t),u∗(t),v∗(t),ξ∗(t))is a solution of the problem (1.3) and
gt,xn(t)−→gt,x∗(t), un(t)−→u∗(t), vn(t)−→v∗(t), ξn(t)−→ξ∗(t) (3.21)
asn→ ∞, where{xn(t)},{un(t)},{vn(t)}, and{ξn(t)}are iterative sequences generated by
Algorithm 3.3.
Proof. The conclusion follows fromLemma 2.13andTheorem 3.5. Remark 3.7. IfJ(t,x)≡Kfor allt∈Ωandx∈D, then fromTheorem 3.5, we can obtain the corresponding result.
Theorem3.8. LetD,g,N, f, andabe the same as inTheorem 3.5. Suppose thatJ:Ω×
D→P(E)is a random multivalued mapping such thatJ(Ω×D)⊂g(Ω×D), for eacht∈
onΩ×Dwith respect tog with measurable functionsσ(t)andτ(t), respectively. If there exists a measurable mappingρ:Ω→(0, +∞)such that
ν(t)=κ(t)τ(t) +β(t)σ(t)< σ(t),
ρ(t)<1−η(t) ν(t) ,
δ(t)>ν(t)1−η(t)+η(t)2−η(t)σ(t)2−ν(t)2,
ρ(t)−δ(t)−ν(t)
1−η(t) σ(t)2−ν(t)2
<
δ(t)−ν(t)1−η(t)2−η(t)2−η(t)σ(t)2−ν(t)2 σ(t)2−ν(t)2 ,
(3.22)
then there exists a unique measurable mappingx∗:Ω→D, which is the solution of the problem (1.5) andlimn→∞g(t,xn(t))=g(t,x∗(t))for allt∈Ω, where{xn(t)}is the iterative
sequence generated byAlgorithm 3.4.
Proof. FromTheorem 3.5, we know that there exists a measurable mappingx∗:Ω→D such that it is the solution of the problem (1.5) and limn→∞g(t,xn(t))=g(t,x∗(t)) for all
t∈Ω. Now we prove thatx∗:Ω→Dis a unique solution of the problem (1.5). In fact, if x:Ω→Dis also a solution of the problem (1.5), then
gt,x(t)=PJ(t,x(t))gt,x(t)−ρ(t)Nt,ft,x(t),St,x(t)+At,Tt,x(t). (3.23) By the proof of (3.13) inTheorem 3.5, we have
g
t,x(t)−gt,x∗(t)
≤gt,x(t)−gt,x∗(t)−ρ(t)Nt,ft,x(t),St,x(t)−Nt,ft,x∗(t),St,x(t) +ρ(t)Nt,ft,x∗(t),St,x(t)−Nt,ft,x∗(t),St,x∗(t)
+ρ(t)At,Tt,x(t)−At,Tt,x∗(t)+η(t)gt,x(t)−gt,x∗(t)
≤(t)gt,x(t)−gt,x∗(t),
(3.24)
where(t)=η(t) +1−2ρ(t)δ(t) +ρ(t)2σ(t)2+ρ(t)(κ(t)τ(t) +β(t)σ(t)). It follows from (3.22) that 0< (t)<1 and sox∗(t)=x(t) for allt∈Ω. This completes the proof.
Remark 3.9. From Theorems3.5–3.8, we get several known results of [4,5,10,13,15,17,
20] as special cases.
4. Random perturbed algorithm and stability
convergence and stability of the iterative sequence generated by the perturbed iterative algorithm.
For our purpose, we first review the following concept and lemma.
Definition 4.1. LetSbe a self-map ofE,x0∈E, and letxn+1=h(S,xn) define an iteration procedure which yields a sequence of points{xn}∞
n=0 in E. Suppose that{x∈E:Sx= x} =∅and that {xn}∞
n=0 converges to a fixed pointx∗ ofS. Let{un} ⊂Eand letn=
un+1−h(S,un). If limn=0 implies thatun→x∗, then the iteration procedure defined byxn+1=h(S,xn) is said to beS-stable or stable with respect toS.
Lemma4.2 [23]. Let{γn}be a nonnegative real sequence and let{λn}be a real sequence in [0, 1]such that∞n=0λn= ∞. If there exists a positive integern1such that
γn+1≤
1−λn
γn+λnσn, ∀n≥n1, (4.1)
whereσn≥0for alln≥0andσn→0asn→ ∞, thenlimn→∞γn=0.
Algorithm 4.3. LetD be a nonempty subset of a separable real Hilbert spaceE, letg: Ω×D→E, f :Ω×E→E,N:Ω×E×E→E, andT,S:Ω×D→Ebe measurable single-valued mappings, and letJ:Ω×D→P(E) be a random multivalued mapping such that for eacht∈Ωandx∈D,J(t,x) is closed convex andJ(Ω×D)⊂g(Ω×D). For a given measurable mappingx0:Ω→D, the perturbed iterative sequence{xn(t)}is defined by
gt,xn+1(t)=1−αn(t)gt,xn(t)
+αn(t)PJ(t,yn(t))
gt,yn(t)−ρ(t)Nt,ft,yn(t),St,yn(t)
+At,Tt,yn(t),
gt,yn(t)=1−βn(t)gt,xn(t)
+βn(t)PJ(t,xn(t))
gt,xn(t)−ρ(t)Nt,ft,xn(t),St,xn(t)
+At,Tt,xn(t),
(4.2)
forn=0, 1, 2,. . ., whereρandAare the same as inLemma 3.1, and{αn(t)}and{βn(t)}
satisfy the following conditions:
0≤αn(t), βn(t)≤1 (n≥0),
∞
n=0
αn(t)= ∞, ∀t∈Ω. (4.3)
Let{un(t)}be any sequence inEand define{n(t)}by
n(t)=gt,un+1(t)
−1−αn(t)gt,un(t)+αn(t)PJ(t,vn(t))
gt,vn(t)
−ρ(t)Nt,ft,vn(t),St,vn(t)+At,Tt,vn(t), gt,vn(t)=1−βn(t)gt,un(t)
+βnPJ(t,un(t))
gt,un(t)−ρ(t)Nt,ft,un(t),St,un(t)
+At,Tt,un(t),
(4.4)
Theorem4.4. LetDbe a nonempty subset of a separable real Hilbert spaceEand letg: Ω×D→Ebe a measurable mapping such thatg(Ω×D)is a closed set inE. Suppose that N:Ω×E×E→Eis a randomκ(t)-Lipschitz continuous single mapping with respect to the third argument, and f :Ω×E→Eisδ(t)-strongly monotone andσ(t)-Lipschitz continuous onΩ×Dwith respect to the second argument ofNandg. LetJ:Ω×D→P(E)be a random multivalued mapping such that J(Ω×D)⊂g(Ω×D)for each t∈Ωandx∈E,J(t,x) is nonempty closed convex, andPJ(t,x)is anη(t)-Lipschitz continuous random operator on Ω×Dwith respect tog, and letT,S:Ω×D→Ebe Lipschitz continuous with respect to g with measurable functionsσ(t)andτ(t), respectively. Ifa:Ω×E×E→Ris a random β(t)-bounded bilinear function such that (3.22) holds, then
(i)if limn→∞ni=0ln[1−αi(1−q)]= −∞, where q=1−τ+ (1 +λρα)−1(δ+ρν), then the sequence{xn(t)}generated by (4.2) converges strongly to the unique so-lutionx(t)of the problem (1.5) for allt∈Ω;
(ii)moreover, if for any t∈Ω,0< a≤αn(t), then limun(t)=x(t) if and only if limn(t)=0, wheren(t)is defined by (4.4).
Proof. FromTheorem 3.8, we know that there exists a unique solutionx(t) of the prob-lem (1.5) and so
gt,x(t)=PJ(t,x(t))
gt,x(t)−ρ(t)Nt,ft,x(t),St,x(t)+At,Tt,x(t). (4.5)
From (4.2), (4.5),Lemma 2.11, andDefinition 2.12, it follows that
g
t,xn+1(t)−gt,x(t)
≤1−αn(t)gt,xn(t)−gt,x(t)
+αn(t)PJ(t,yn(t))
gt,yn(t)−ρ(t)Nt,ft,yn(t),St,yn(t)
+At,Tt,yn(t)
−PJ(t,yn(t))
gt,x(t)−ρ(t)Nt,ft,x(t),St,x(t)+At,Tt,x(t)
+PJ(t,yn(t))
gt,x(t)−ρ(t)Nt,ft,x(t),St,x(t)+At,Tt,x(t)
−PJ(t,x(t))
gt,x(t)−ρ(t)Nt,ft,x(t),St,x(t)+At,Tt,x(t)
≤1−αn(t)g
t,xn(t)
−gt,x(t)
+αn(t)gt,yn(t)−gt,x(t)−ρ(t)Nt,ft,yn(t),St,yn(t)
−Nt,ft,x(t),St,yn(t)
+ρ(t)Nt,ft,x(t),St,yn(t)−Nt,ft,x(t),St,x(t)
+ρ(t)At,Tt,yn(t)−At,Tt,x(t)
+η(t)gt,yn(t)−gt,x(t).
By (4.6) and the assumption ofN, f,a,S,T, we obtain g
t,xn+1(t)−gt,x(t) ≤1−αn(t)gt,xn(t)−gt,x(t)
+αn(t)q(t)gt,yn(t)−gt,x(t), (4.7)
where
q(t)=1−2ρ(t)δ(t) +ρ(t)2σ(t)2+ρ(t)κ(t)τ(t) +β(t)σ(t)+η(t). (4.8)
Similarly, we have g
t,yn(t)−gt,x(t)≤1−βn(t)gt,xn(t)−gt,x(t)
+βn(t)q(t)gt,xn(t)−gt,x(t). (4.9)
Combining (4.7) and (4.9), we obtain g
t,xn+1(t)−gt,x(t)≤1−αn(t)1−q(t)1 +βn(t)gt,xn(t)−gt,x(t). (4.10)
Condition (3.22) implies that 0< q(t)<1, and so from (4.10), we have g
t,xn+1(t)−gt,x(t)
≤1−αn(t)1−q(t)gt,xn(t)−gt,x(t)
≤ ··· ≤
n
i=0
1−αi(t)
1−q(t)gt,x0(t)−gt,x(t).
(4.11)
Since
lim n→∞
n
i=0
ln1−αi(t)1−q(t)= −∞, i.e., lim n→∞
n
i=0
1−αi(t)1−q(t)=0, (4.12)
we know thatg(t,xn(t))→g(t,x(t)) asn→ ∞. Now we prove conclusion (ii). By (4.4), we obtain g
t,un+1(t)−gt,x(t)
≤1−αn(t)gt,un(t) +αn(t)PJ(t,vn(t))
gt,vn(t)−ρ(t)Nt,ft,vn(t),St,vn(t)+At,Tt,vn(t)
−gt,x(t)+n(t).
(4.13)
As the proof of inequality (4.10), we have 1−αn(t)
gt,un(t)
+αn(t)PJ(t,vn(t))
gt,vn(t)−ρ(t)Nt,ft,vn(t),St,vn(t)
+At,Tt,vn(t)−gt,x(t)
≤1−αn(t)
1−q(t)gt,un(t)
−gt,x(t).
Since 0< a≤αn(t) for allt∈Ω, by (4.13) and (4.14), we have g
t,un+1(t)−gt,x(t)≤1−αn(t)1−q(t)gt,un(t)−gt,x(t)
+αn(t)1−q(t)· n(t) a1−q(t).
(4.15)
Suppose that for anyt∈Ω, limn(t)=0. Then from∞n=0αn(t)= ∞andLemma 4.2, we have limg(t,un(t))=g(t,x(t)) and so limun(t)=x(t).
Conversely, if limun(t)=x(t) for allt∈Ω, then we get
n(t)≤gt,un+1(t)−gt,x(t) +1−αn(t)gt,un(t)
+αn(t)PJ(t,vn(t))
gt,vn(t)−ρ(t)Nt,ft,vn(t),St,vn(t)
+At,Tt,vn(t)−g(t,x(t)
≤gt,un+1(t)−gt,x(t)
+1−αn(t)1−q(t)gt,un(t)−gt,x(t)−→0,
(4.16)
asn→ ∞. This completes the proof.
Remark 4.5. IfJ(t,x)=m(t,x) +K for allt∈Ωandx∈D, wherem:Ω×D→Eis an η(t)/2-Lipschitz continuous random mapping andKis a closed convex subset ofE, then we can obtain the same results as inTheorem 4.4.
Acknowledgments
The author acknowledges the referees for their valuable suggestions and the support of the Educational Science Foundation of Sichuan, Sichuan, China (2004C018).
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Heng-You Lan: Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China