Volume 2010, Article ID 123524,17pages doi:10.1155/2010/123524
Research Article
A System of Random Nonlinear Variational
Inclusions Involving Random Fuzzy Mappings and
H
·
,
·
-Monotone Set-Valued Mappings
Xin-kun Wu and Yun-zhi Zou
College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Yun-zhi Zou,zouyz@scu.edu.cn Received 8 June 2010; Accepted 24 July 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq2010 X.-k. Wu and Y.-z. Zou. 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.
We introduce and study a new system of random nonlinear generalized variational inclusions involving random fuzzy mappings and set-valued mappings withH·,·-monotonicity in two Hilbert spaces and develop a new algorithm which produces four random iterative sequences. We also discuss the existence of the random solutions to this new kind of system of variational inclusions and the convergence of the random iterative sequences generated by the algorithm.
1. Introduction
The classic variational inequality problem VIF,Kis to determine a vector x∗ ∈ K ⊂ Rn,
such that
Fx∗T, x−x∗≥0, ∀x∈K, 1.1
whereFis a given continuous function fromKtoRnandKis a given closed convex subset
of then-dimensional Euclidean spaceRn. This is equivalent to find anx∗∈K, such that
0∈Fx∗ N⊥x∗, 1.2
whereN⊥is normal cone operator.
equilibrium, and linear or nonlinear programming etcetera, variational inequality and its generalizations have been extensively studied during the past 40 years. For details, we refer readers to1–7and the references therein.
It is not a surprise that many practical situations occur by chance and so variational inequalities with random variables/mappings have also been widely studied in the past decade. For instance, some random variational inequalities and random quasivariational
inequalities problems have been introduced and studied by Chang8, Chang and Huang
9,10, Chang and Zhu11, Huang12,13, Husain et al.14, Tan et al.15, Tan16, and Yuan7.
It is well known that one of the most important and interesting problems in the theory
of variational inequalities is to develop efficient and implementable algorithms for solving
variational inequalities and its generalizations. The monotonic properties of associated operators play essential roles in proving the existence of solutions and the convergence of
sequences generated by iterative algorithms. In 2001, Huang and Fang 17were the first
to introduce the generalizedm-accretive mapping and give the definition of the resolvent
operator for generalizedm-accretive mappings in Banach spaces. They also showed some
properties of the resolvent operator for generalized m-accretive mappings. Recently, Fang
and Huang, Verma, and Cho and Lan investigated many generalized operators such asH
-monotone, H-accretive, H, η-monotone, H, η-accretive, and A, η-accretive mappings.
For details, we refer to 6, 17–22 and the references therein. In 2008, Zou and Huang
23 introduced the H·,·-accretive operator in Banach spaces which provides a unified
framework for the existingH-monotone,H, η-monotone, andA, η-monotone operators
in Hilbert spaces andH-accretive,H, η-accretive, andA, η-accretive operators in Banach
spaces.
In 1965, Zadeh24introduced the concept of fuzzy sets, which became a cornerstone
of modern fuzzy mathematics. To explore connections among VIs, fuzzy mapping
and random mappings, in 1997, Huang 25 introduced the concept of random fuzzy
mappings and studied the random nonlinear quasicomplementarity problem for random
fuzzy mappings. Later, Huang 26 studied the random generalized nonlinear variational
inclusions for random fuzzy mappings. In 2005, Ahmad and Baz´an27studied a class of
random generalized nonlinear mixed variational inclusions for random fuzzy mappings and constructed an iterative algorithm for solving such random problems. For related work in
this hot area, we refer to Ahmad and Farajzadeh28, Ansari and Yao29, Chang and Huang
9,10, Cho and Huang30, Cho and Lan31, Huang25,26,32, Huang et al.33, and the references therein.
Motivated and inspired by recent research work mentioned above in this field, in this paper, we try to inject some new energy into this interesting field by studying on a new kind of random nonlinear variational inclusions in two Hilbert spaces. We will prove the existence of random solutions to the system of inclusions and propose an algorithm which produces a convergent iterative sequence. For a suitable choice of some mappings, we can obtain several known results10,11,21,23,31,34as special cases of the main results of this paper.
2. Preliminaries
Throughout this paper, letΩ,Abe a measurable space, whereΩis a set andAis aσ-algebra
overΩ. LetX1 be a separable real Hilbert space endowed with a norm · X1 and an inner
product ·,·X1. LetX2be a separable real Hilbert space endowed with a norm · X2and an
We denote byD·,·the Hausdorff metric between two nonempty closed bounded
subsets, where the Hausdorffmetric betweenAandBis defined by
DA, B max
sup
a∈A
inf
b∈Bda, b,supb∈Bainf∈Ada, b
. 2.1
We denote byBX1, 2X1, andCBX1the class of Borelσ-fields inX1, and the family
of all nonempty subsets ofX1, the family of all nonempty closed bounded subsets ofX1.
In this paper, to make it self-contained, we start with the following basic definitions and similar definitions can also be found in26,32,34.
Definition 2.1. A mappingx1:Ω → X1is said to be measurable if for anyB∈ BX1,
{t∈Ω:xt∈B} ∈ A. 2.2
Definition 2.2. A mappingT1 :Ω×X1 → X1is called a random mapping if for anyx∈ X1,
z1t T1t, xis measurable.
Definition 2.3. A random mappingT1:Ω×X1 → X1is said to be continuous if for anyt∈Ω,
T1t,·:X1 → X1is continuous.
Definition 2.4. A set-valued mappingV1 : Ω → 2X1 is said to be measurable if for anyB ∈
BX1,
V1−1B {v∈Ω:V1v∩B /∅} ∈ A. 2.3
Definition 2.5. A mapping u : Ω → X1 is called a measurable selection of a set-valued
measurable mappingU:Ω → 2X1ifuis measurable and for anyt∈Ω,ut∈Ut.
Definition 2.6. A set-valued mappingW1 :Ω×X1 → 2X1is called random set-valued if for
anyx1∈X1,W1·, x1:Ω → 2X1is a measurable set valued mapping.
Definition 2.7. A random set-valued mappingW1 :Ω×X1 → CBX1is said to beξEt-D
-continuous if there exists a measurable functionξE:Ω → 0,∞, such that
DW1t, x1t, W1t, x2t≤ξEtx1t−x2tX1, 2.4
for allt∈Ωandx1t, x2t∈X1.
Definition 2.8. A set-valued mappingA:X1 → 2X1is said to be monotone if for allx1, y1∈X1
andu1∈Ax1,v1∈Ay1,
u1−v1, x1−y1
X1≥0. 2.5
Definition 2.9. Letf1, g1:X1 → X1andH1 :X1×X1 → X1be three single-valued mappings
andA:X1 → 2X1be a set-valued mapping.Ais said to beH1·,·-monotone with respect to
Definition 2.10. The inverses of A : X1 → 2X1 and B : X2 → 2X2 are defined as follows,
respectively,
A−1y x∈X1 :y∈Ax
, ∀y∈X1,
B−1y x∈X2 :y∈Bx
, ∀y∈X2.
2.6
Definition 2.11. p:Ω×X1 → X1is said to be
1monotone if
pt, x1t−pt, x2t, x1t−x2t
X1≥0, ∀t∈Ω, ∀x1t, x2t∈X1, 2.7
2strictly monotone ifpis monotone and
pt, x1t−pt, x2t, x1t−x2t
X10 ⇐⇒ x1t x2t, ∀t∈Ω, ∀x1t, x2t∈X1,
2.8
3δpt-strongly monotone if there exists some measurable function δp : Ω →
0,∞, such that
pt, x1t−pt, x2t, x1t−x2t
X1≥δptx1t−x2t 2
x1, ∀t∈Ω, ∀x1t, x2t∈X1,
2.9
4σpt-Lipschitz continuous if there exists some measurable function σp : Ω →
0,∞, such that
pt, x1t−pt, x2tX1≤σptx1t−x2tX1, ∀t∈Ω, ∀x1t, x2t∈X1. 2.10
Definition 2.12. A single-valued mappingM:X1×X1×X2 → X1is said to be
1ζAt-strongly monotone with respect to the random single-valued mappingsM :
Ω×X1 → X1 in the first argument if there exists some measurable functionζA :
Ω → 0,∞, such that
MsMt, u1t,·,·−MsMt, u2t,·,·, u1t−u2tX1 ≥ζAtu1t−u2t 2
X1, 2.11
for allt∈Ωandu1t, u2t∈X1,
2ξMt-Lipschitz continuous with respect to the random single-valued mappingsM:
Ω ×X1 → X1 in its first argument if there exists some measurable functionξM :
Ω → 0,∞, such that
MsMt, u1t,·,·−MsMt, u2t,·,·X1≤ξMtu1t−u2tX1, 2.12
3βMt-Lipschitz continuous with respect to its second argument if there exists some
measurable functionβM:Ω → 0,∞, such that
M·, x1t,·−M·, x2t,·X1 ≤βMtx1t−x2tX1, 2.13
for allt∈Ωandx1t, x2t∈X1,
4ηMt-Lipschitz continuous with respect to its third argument if there exists some
measurable functionηM:Ω → 0,∞such that
M·,·, y1t−M
·,·, y2t X1 ≤ηMty1t−y2tX2, 2.14
for allt∈Ωandy1t, y2t∈X2;
Definition 2.13. Assume thatp:Ω×X1 → X1is a random single-valued mapping,f1:X1 →
X1,g1 :X1 → X1, andH1f1, g1:X1 → X1are three single-valued mappings,H1f1, g1is
said to be
1μAt-strongly monotone with respect to the mapping p if there exists some
measurable functionμA :Ω → 0,∞such that
H1
f1
pt, x1t , g1
pt, x1t
−H1
f1
pt, y1t
, g1
pt, y1t
, x1t−y1t
X1
≥μAtx1t−y1t2X1,
2.15
for allt∈Ωandx1t, y1t∈X1,
2aAt-Lipschitz continuous with respect to the mapping p if there exists some
measurable functionaA:Ω → 0,∞such that
H1
f1
pt, x1t , g1
pt, x1t
−H1
f1
pt, y1t
, g1
pt, y1t X1
≤aAtx1t−y1tX1,
2.16
for allt∈Ωandx1t, y1t∈X1.
3αA-strongly monotone with respect to f1 in the first argument if there exists a
positive constantαA, such that
H1
f1x1, u1 −H1
f1
y1 , u1 , x1−y1
X1≥αAx1−y1 2
X1, 2.17
4βA-relaxed monotone with respect to g1 in the second argument if there exists a
positive constantβA, such that
H1
u1, g1x1 −H1
u1, g1
y1
, x1−y1
X1 ≥ −βAx1−y1 2
X1, 2.18
for allx1, y1, u1∈X1.
LetFX1be a collection of all fuzzy sets overX1. A mappingFfromΩintoFX1is
called a fuzzy mapping. IfFis a fuzzy mapping onX1, then for any givent∈Ω,Ftdenote
it byFtin the sequelis a fuzzy set onX1andFtyis the membership function ofyinFt.
LetA∈ FX1,α∈0,1, then the set
Aα{x∈X1:Ax≥α} 2.19
is called anα-cut set of fuzzy setA.
Definition 2.14. A random fuzzy mappingF :Ω → FX1is said to be measurable if for any
givenα∈0,1,F·α:Ω → 2X1is a measurable set-valued mapping.
Definition 2.15. A fuzzy mappingE:Ω×X1 → FX1is called a random fuzzy mapping if
for any givenx1 ∈X1,E·, x1:Ω → FX1is a measurable fuzzy mapping.
Remark 2.16. The above is mainly about some definitions inX1. There are similar definitions
and notations for operators inX2.
LetE: Ω×X1 → FX1andF : Ω×X2 → FX2be two random fuzzy mappings
satisfying the following condition∗∗:
∗∗there exist two mappingsα:X1 → 0,1andβ:X2 → 0,1, such that
Et,x1αx1∈CBX1, ∀t, x1∈Ω×X1,
Ft,x2βx2∈CBX2, ∀t, x2∈Ω×X2.
2.20
By using the random fuzzy mappings E and F, we can define the two set-valued
mappingsE∗andF∗as follows, respectively,
E∗:Ω×X1−→CBX1, t, x1−→Et,x1αx1, ∀t, x1∈Ω×X1,
F∗:Ω×X2−→CBX2, t, x2−→Et,x2αx2, ∀t, x2∈Ω×X2.
2.21
It follows that
E∗t, x1 Et,x1αx1{z1∈X1:Et,x1z1≥αx1},
F∗t, x2 Ft,x2βx2
z2∈X2:Ft,x2z2≥βx2
.
2.22
It is easy to see thatE∗ andF∗ are two random set-valued mappings. We callE∗andF∗the
Problem 1. Letf1, g1:X1 → X1be two single-valued mappings andsM, p:Ω×X1 → X1be
two random single-valued mappings. Letf2, g2 :X2 → X2be two single-valued mappings
andsN, q:Ω×X2 → X2be two random single-valued mappings. LetH1 :X1×X1 → X1,
H2 : X2×X2 → X2,M :X1×X1×X2 → X1andN : X2×X1×X2 → X2 be four
single-valued mappings. Suppose thatA:X1 → 2X1is anH1·,·-monotone mapping with respect
tof1andg1andB :X2 → 2X2 is anH2·,·-monotone mapping with respect tof2 andg2.
E:Ω×X1 → FX1andF:Ω×X2 → FX2are two random fuzzy mappings,α,β,E∗, and
F∗are the same as the above. Assume thatpt, ut∩domA/∅andqt, vt∩domB/∅
for allt∈Ω. We consider the following problem.
Find four measurable mappingsu, x:Ω → X1andv, y:Ω → X2, such that
Et,utxt≥αut,
Ft,vtyt ≥βvt,
0∈MsMt, ut, xt, yt Apt, ut ,
0∈NsNt, vt, xt, yt Bqt, vt ,
2.23
for allt∈Ω.
Problem1 is called a system of generalized random nonlinear variational inclusions
involving random fuzzy mappings and set-valued mappings with H·,·-monotonicity in
two Hilbert spaces. A set of the four measurable mappingsx, y, u,andvis called one solution
of Problem1.
3. Random Iterative Algorithm
In order to prove the main results, we need the following lemmas.
Lemma 3.1see23. Let H1,f1,g1, and A be defined as in Problem1. Let H1f1, g1 beαA
-strongly monotone with respect to f1, βA-relaxed monotone with respect to g1, where αA > βA.
Suppose that A : X1 → 2X1 is anH1·,·-monotone set-valued mapping with respect to f1 and
g1, then the resolvent operatorRA,λH1·,· Hf, g λA−1is a single-valued mapping.
Lemma 3.2see23. LetH1,f1,g1,Abe defined as in Problem1. LetH1f1, g1beαA-strongly
monotone with respect tof1,βA-relaxed monotone with respect tog1, whereαA > βA. Suppose that A : X1 → 2X1 is an H1·,·-monotone set-valued mapping with respect to f1 and g1. Then, the
resolvent operatorRH1·,·
A,λ is1/αA−βA-Lipschitz continuous.
Remark 3.3. Some interesting examples concerned with the H1·,·-monotone mapping and
the resolvent operatorRH1·,·
A,λ can be found in23.
Lemma 3.4see Chang8. LetV : Ω×X1 → CBX1be aD-continuous random set-valued
mapping. Then for any given measurable mappingu:Ω → X1, the set-valued mappingV·, u·:
Lemma 3.5see Chang8. LetV, W : Ω → CBX1be two measurable set-valued mappings,
and let ε > 0 be a constant and u : Ω → X1 a measurable selection of V. Then there exists a
measurable selectionv:Ω → X1ofW, such that for allt∈Ω,
ut−vt ≤1εDVt, Wt. 3.1
Lemma 3.6. The four measurable mappingsx, u : Ω → X1 and y, v : Ω → X2 are solution of
Problem1if and only if, for allt∈Ω,
xt∈E∗t, ut,
xt∈F∗t, vt,
pt, ut RH1·,·
A,λ
H1
f1
pt, ut , g1
pt, ut −λMsMt, ut, xt, yt ,
qt, ut RH2·,·
B,ρ
H2
f2
qt, vt , g2
qt, vt −ρNsNt, vt, xt, yt ,
3.2
whereRH1·,·
A,λ H1f1, g1 λA
−1andRH2·,·
B,ρ H2f2, g2 ρB
−1are two resolvent operators.
Proof. From the definitions ofRH1·,·
A,λ andR H2·,·
B,ρ , one has
H1
f1
pt, ut , g1
pt, ut −λMsMt, ut, xt, yt
∈H1
f1
pt, ut , g1
pt, ut λApt, ut , ∀t∈Ω,
H2
f2
qt, vt , g2
qt, vt −ρNsNt, vt, xt, yt
∈H2
f2
qt, vt , g2
qt, vt ρBqt, vt , ∀t∈Ω.
3.3
Hence,
0∈MsMt, ut, xt, yt Apt, ut , ∀t∈Ω,
0∈NsNt, vt, xt, yt Bqt, vt , ∀t∈Ω. 3.4
Thus,x, y, u, vis a set of solution of Problem1. This completes the proof.
Now we use Lemma3.6to construct the following algorithm.
Letu0:Ω → X1andv0:Ω → X2be two measurable mappings, then by Himmelberg
35, there existx0 : Ω → X1, a measurable selection ofE∗·, u0· : Ω → CBX1and
y0 : Ω → X2, a measurable selection of F∗·, v0· : Ω → CBX2. We now propose the
Algorithm 3.7. For any given measurable mappingsu0:Ω → X1andv0 :Ω → X2, iterative
sequences that attempt to solve Problem1are defined as follows:
un1t unt−pt, unt
RH1·,·
A,λ
H1
f1
pt, unt , g1
pt, unt −λMsMt, unt, xnt, ynt ,
vn1t vnt−qt, vnt
RH2·,·
B,ρ
H2
f2
qt, vnt , g2
qt, vnt −ρNsNt, vnt, xnt, ynt . 3.5
Choosexn1t∈E∗t, un1tandyn1t∈F∗t, vn1t, such that
xn1t−xntX1 ≤1εn1DE ∗t, u
n1t, E∗t, unt,
yn1t−ynt
X2 ≤1εn1DF ∗t, v
n1t, F∗t, vnt,
3.6
for anyt∈Ωandn0,1,2,3, . . . .
Remark 3.8. The existence ofxnandynis guaranteed by Lemmas3.4and3.5.
4. Existence and Convergence
Theorem 4.1. LetX1andX2be two separable real Hilbert spaces. Suppose thatsM, p:Ω×X1 → X1
and sN, q : Ω×X2 → X2 are four random mappings. Suppose thatf1, g1 : X1 → X1, H1 :
X1×X1 → X1,f2, g2 :X2 → X2,H2 :X2×X2 → X2are six single-valued mappings. Assume
that
1A:X1 → 2X1is anH1·,·-monotone with respect to operatorsf1andg1,
2B:X2 → 2X2is anH2·,·-monotone with respect to operatorsf2andg2,
3pt, ut∩domA/∅andqt, vt∩domB/∅for allt∈Ω,
4M :X1×X1×X2 → X1 isζAt-monotone with respect to the mappingsMin the first
argument,ξMt-Lipschitz continuous with respect to mappingsMin the first argument, βMt-Lipschitz continuous with respect to the second argument and ηMt-Lipschitz continuous with respect to the third argument,
5N :X2×X1×X2 → X2 isζBt-monotone with respect to the mappingsNin the first
argument,ξNt-Lipschitz continuous with respect to mappingsN in the first argument, βNt-Lipschitz continuous with respect to the second argument and ηNt-Lipschitz continuous with respect to the third argument,
6LetE : Ω×X1 → FX1andF : Ω×X2 → FX2be two random fuzzy mappings
satisfying the condition∗∗,α,β,E∗andF∗are four mappings induced byEandF.E∗
andF∗areξEt-D-Lipschitz andξFt-D-Lipschitz continuous, respectively;
with respect to the mappingpandaAt-Lipschitz continuous with respect to the mapping
p,
8H1f1, g1 is αA-strongly monotone with respect to f1, and βA-relaxed monotone with
respect tog1, whereαA > βA,
9q is δqt-strongly monotone with respect to its second argument and σqt-Lipschitz
continuous with respect to its second argument, H2f2, g2is μBt-strongly monotone
with respect to the mappingq, andaBt-Lipschitz continuous with respect to the mapping
q,
10H2f2, g2 is αB-strongly monotone with respect to f2 and βB-relaxed monotone with
respect tog2, whereαB> βB,
If
At λ
αA−βA
βMtξEt 2
1−2δpt σpt2
1
αA−βA
2
1−2μAt aAt2
1
αA−βA
2
1−2λζAt λ2ξMt2,
Bt λ
αA−βA
ηMtξFt,
Ct ρ
αB−βB
βNtξEt,
Dt ρ
αB−βB
ηNtξFt 2
1−2δqt σqt2
1
αB−βB
2
1−2μBt aBt2
1
αB−βB
2
1−2ρζBt ρ2ξNt2,
0< At Ct<1, ∀t∈Ω,
0< Bt Dt<1, ∀t∈Ω,
4.1
then there exist four measurable mappingsx, u:Ω → X1andy, v:Ω → X2, such
thatx, y, u, vis a set of solution of Problem1. Moreover,
lim
n→ ∞xnt xt, nlim→ ∞ynt yt, nlim→ ∞unt ut, nlim→ ∞vnt vt,
4.2
Proof. To simplify calculations, for anyn∈N, we let
snt H1
f1
pt, unt , g1
pt, unt −λMsMt, unt, xnt, ynt ,
tnt H2
f2
qt, vnt , g2
qt, vnt −ρNsNt, vnt, xnt, ynt ,
4.3
Snt −pt, unt RH1·,·
A,λ snt, Tnt −qt, vnt R H2·,·
B,ρ tnt. 4.4
We use · 1to replace · X1and · 2to replace · X2.
Thus,
un1t unt Snt, 4.5
vn1t vnt Tnt. 4.6
By the definition ofSnt,Tnt,sn, andtn, we have the following
Snt1Sn−1t Snt−Sn−1t1
unt−un−1t Snt−Sn−1t1
≤unt−un−1t−pt,unt−pt, un−1t1
RH1·,·
A,λ snt−R H1·,·
A,λ sn−1t
1,
4.7
Tnt2Tn−1t Tnt−Tn−1t2
vnt−vn−1t Tnt−Tn−1t2
≤vnt−vn−1t−qt, vnt−qt, vn−1t2
RH2·,·
B,ρ tnt−R H2·,·
B,ρ tn−1t
2.
4.8
We first prove that for the two sequencesunt andvnt, there exist two sequences
{Ant}and{Bnt}in0,1, such that
un1t−unt1≤Antunt−un−1t1Bntvnt−vn−1t2. 4.9
In fact, for the first term in4.7, from theδpt-strongly monotonicity andσpt-Lipschitz
continuity of the functionp, we have the following:
unt−un−1t−pt, unt−pt, un−1t21
unt−un−1t21−2
pt, unt−pt, un−1t, unt−un−1t1
pt, unt−pt, un−1t21
≤1−2δpt σpt2unt−un−1t21.
For the second term in4.7, it follows from Lemma3.2that
RH1·,·
A,λ snt−R H1·,·
A,λ sn−1t1
≤ 1
αA−βAsn
t−sn−1t1
≤ 1
αA−βA
H1
f1
pt, unt , g1
pt, unt −λMsMt, unt, xnt, ynt
−H1
f1
pt, un−1t , g1
pt, un−1t −λMsMt, un−1t, xn−1t, yn−1t 1
≤ 1
αA−βA
×unt−un−1t−H1
f1
pt, unt , g1
pt, unt
−H1
f1
pt, un−1t , g1
pt, un−1t 1
−unt−un−1t−λMsMt, unt, xnt, ynt−MsMt, un−1t, xnt, ynt 1
λMsMt, un−1t, xnt, ynt−MsMt, un−1t, xn−1t, ynt 1
λMsMt, un−1t, xn−1t, ynt −M
sMt, un−1t, xn−1t, yn−1t 1
.
4.11
There are four terms in 4.11. Since H1f1, g1 is μAt-strongly monotone and aAt
-Lipschitz continuous with respect to the mappingp, then for the first term in4.11, we can
obtain the following
unt−un−1t−H1
f1
pt, unt , g1
pt, unt H1
f1
pt, un−1t , g1
pt, un−1t 21
unt−un−1t21−2
H1
f1
pt, unt , g1
pt, unt
−H1
f1
pt, un−1t , g1
pt, un−1t , unt−un−1t1
H1
f1
pt, unt , g1
pt, unt −H1
f1
pt, un−1t , g1
pt, un−1t 21
≤1−2μAt aAt2unt−un−1t21.
4.12
For the second term, SinceMisζAt-monotone with respect to the mappingsMin the first
so
unt−un−1t−λMsMt, unt, xnt, ynt −MsMt, un−1t, xnt, ynt 21
unt−un−1t21−2λ
MsMt, unt, xnt, ynt
−MsMt, un−1t, xnt, ynt , unt−un−1t1
λ2MsMt, unt, xnt, ynt −MsMt, un−1t, xnt, ynt 21
≤ unt−un−1t12−2λζAtunt−un−1t21λ2ξMt2unt−un−1t21
≤1−2λζAt λ2ξMt2unt−un−1t21.
4.13
For the third term,MisβMt-Lipschitz continuous with respect to the second argument and
E∗isξEt-D-Lipschitz continuous, therefore, we must have the following:
MsMt, un−1t, xnt, ynt−MsMt, un−1t, xn−1t, ynt1
≤βMtxnt−xn−1t1
≤βMt1εnDE∗t, unt, E∗t, un−1t
≤βMt1εnξEtunt−un−1t1.
4.14
Similarly, becauseηMt-Lipschitz continuous with respect to the third argument andF∗ is
ξFt-D-Lipschitz continuous, so we can derive
MsMt, un−1t, xn−1t, ynt −M
sMt, un−1t, xn−1t, yn−1t 1 ≤ηMtynt−yn−1t2
≤ηMt1εnDF∗t, vnt, F∗t, vn−1t
≤ηMt1εnξFtvnt−vn−1t2.
4.15
If we let
Ant λ
αA−βA
βMt1εnξEt 2
1−2δpt σpt2
1
αA−βA
2
1−2μAt aAt2
1
αA−βA
2
1−2λζAt λ2ξMt2
4.16
and
Bnt λ
αA−βA
then, from4.5to4.17, we can have
un1t−unt1≤Antunt−un−1t1Bntvnt−vn−1t2. 4.18
Under the assumptions of this theorem, in a similar way, we can also show that, for the other two sequences{vnt}and{unt}, there exist two sequences{Cnt}and{Dnt}
in0,1, such that
vn1t−vnt2≤Cntunt−un−1t1Dntvnt−vn−1t2. 4.19
We now can claim easily thatunt,xntare two Cauchy sequences inX1 andvnt,
ynt are two Cauchy sequences in X2. In fact, from 4.18 and 4.19, we can obtain the
following inequalities:
un1t−unt1vn1t−vnt2
≤Ant Cntunt−un−1t1 Bnt Dntvnt−vn−1t2
≤maxAnt Cnt, Bnt Dntunt−un−1t1vnt−vn−1t2.
4.20
If we let
θnt maxAnt Cnt, Bnt Dnt,
θt maxAt Ct, Bt Dt, 4.21
then we have the following:
lim
n→ ∞Ant At, nlim→ ∞Bnt Bt, nlim→ ∞Cnt Ct,
lim
n→ ∞Dnt Dt, nlim→ ∞θnt θt.
4.22
It follows from the assumptions of Theorem4.1that 0< θt<1, for allt∈Ω, and so
{unt}and{vnt}are both Cauchy sequences. For sequences{xn}and{yn}, since
xn1t−xnt1 ≤1εn1DE∗t, un1t, E∗t, unt≤2ξEtun1t−unt1,
yn1t−ynt2≤1εn1DF∗t, vn1t, F∗t, vnt≤2ξFtvn1t−vnt2,
4.23
thus,{xnt}and{ynt}are also Cauchy sequences in Hilbert spacesX1andX2, respectively.
We now show that there exist four measurable mappingsx, u : Ω → X1 andy, v :
Ω → X2such thatx, y, u, vis a set of solution of Problem1and
lim
where {xnt}, {ynt}, {unt}, and {vnt} are four iterative sequences generated by
Algorithm3.7.
BecauseX1,X2are two Hilbert spaces and{xnt},{ynt},{unt}, and{vnt}are four
Cauchy sequences, thus, there exist four elements{xt},{yt},{ut}, and{vt}such that
lim
n→ ∞xnt xt, nlim→ ∞ynt yt, nlim→ ∞unt ut, nlim→ ∞vnt vt. 4.25
Furthermore,
dxt, E∗t, ut inf{xt−a:a∈E∗t, ut} ≤ xt−xnt1dxnt, E∗t, ut ≤ xt−xnt1DE∗t, unt, E∗t, ut ≤ xt−xnt1ξEtunt−ut1.
4.26
Since limn→ ∞xnt xt, limn→ ∞unt ut, and E∗t, ut ∈ CBX1, we have the
following:
xt∈E∗t, ut. 4.27
Similar argument leads to the fact that
yt∈F∗t, vt. 4.28
By the continuity of p, q, H1, f1, g1, H2, f2, g2, E∗, F∗, RA,λH1·,·, and RB,ρH2·,·, we have the
following:
pt, ut RH1·,·
A,λ
H1
f1
pt, ut , g1
pt, ut −λMsMt, ut, xt, yt ,
qt, ut RH2·,·
B,ρ
H2
f2
qt, vt , g2
qt, vt −ρNsNt, vt, xt, yt .
4.29
So, by Lemma3.6,x, y, u, vis a set of solution to Problem1.
This completes the proof.
Remark 4.2. By some suitable choices of mappings in Theorem4.1, the main results in this paper extend many existing ones, for instance, the main results in10,11,21,23,26,31,34.
Acknowledgment
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