Volume 2009, Article ID 865093,17pages doi:10.1155/2009/865093
Research Article
General Nonlinear Random Equations with
Random Multivalued Operator in Banach Spaces
Heng-You Lan,
1Yeol Je Cho,
2and Wei Xie
11Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China
2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea
Correspondence should be addressed to Yeol Je Cho,[email protected]
Received 16 December 2008; Accepted 27 February 2009
Recommended by Jewgeni Dshalalow
We introduce and study a new class of general nonlinear random multivalued operator equations involving generalizedm-accretive mappings in Banach spaces. By using the Chang’s lemma and the resolvent operator technique for generalizedm-accretive mapping due to Huang and Fang2001, we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations inq-uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in iterature.
Copyrightq2009 Heng-You Lan 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.
1. Introduction and Preliminaries
The variational principle has been one of the major branches of mathematical sciences for more than two centuries. It is a tool of great power that can be applied to a wide variety of problems in pure and applied sciences. It can be used to interpret the basic principles of mathematical and physical sciences in the form of simplicity and elegance. During this period, the variational principles have played an important and significant part as a unifying influence in pure and applied sciences and as a guide in the mathematical interpretation of many physical phenomena. The variational principles have played a fundamental role in the development of the general theory of relativity, gauge field theory in modern particle physics and soliton theory. In recent years, these principles have been enriched by the discovery of the variational inequality theory, which is mainly due to Hartman and Stampacchia1.
various fields of mathematics, physics, economics, regional, and engineering sciences. The ideas and techniques are being applied in a variety of diverse areas of sciences and prove to be productive and innovative. In fact, many researchers have shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems.
Variational inclusion is an important generalization of variational inequality, which has been studied extensively by many authorssee, e.g.,2–14and the references therein.
In 2001, Huang and Fang15introduced the concept of a generalizedm-accretive mapping, which is a generalization of anm-accretive mapping, and gave the definition of the resolvent operator for the generalizedm-accretive mapping in Banach spaces. Recently, Huang et al. 6, 7, Huang 8, Jin and Liu 9 and Lan et al. 11 introduced and studied some new classes of nonlinear variational inclusions involving generalized m-accretive mappings in Banach spaces. By using the resolvent operator technique in6, they constructed some new
iterative algorithms for solving the nonlinear variational inclusions involving generalized
m-accretive mappings. Further, they also proved the existence of solutions for nonlinear variational inclusions involving generalized m-accretive mappings and convergence of sequences generated by the algorithms.
On the other hand, It is well known that the study of the random equations involving the random operators in view of their need in dealing with probabilistic models in applied sciences is very important. Motivated and inspired by the recent research works in these fascinating areas, the random variational inequality problems, random variational inequality problems, random variational inclusion problems and random quasi-complementarity problems have been introduced and studied by Ahmad and Baz´an16,
Chang17, Chang and Huang18, Cho et al.19, Ganguly and Wadhwa20, Huang21,
Huang and Cho22, Huang et al.23, and Noor and Elsanousi24.
Inspired and motivated by recent works in these fields see 3, 11, 12, 16, 25–
28, in this paper, we introduce and study a new class of general nonlinear random
multivalued operator equations involving generalized m-accretive mappings in Banach spaces. By using the Chang’s lemma and the resolvent operator technique for generalized
m-accretive mapping due to Huang and Fang15, we also prove the existence theorems
of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations inq-uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in literature.
Throughout this paper, we suppose thatΩ, A, μis a completeσ-finite measure space andEis a separable real Banach space endowed with dual spaceE∗, the norm · and the dual pair·,·betweenEandE∗. We denote byBEthe class of Borelσ-fields inE. Let 2E andCBEdenote the family of all the nonempty subsets ofE, the family of all the nonempty bounded closed sets ofE, respectively. The generalized duality mapping Jq : E → 2E
∗
is defined by
Jqx
f∗∈E∗:x, f∗xq, f∗xq−1 1.1
for allx ∈ E, where q > 1 is a constant. In particular, J2 is the usual normalized duality
identity mapping ofH. In what follows we will denote the single-valued generalized duality mapping byjq.
Suppose thatA : Ω×E×E → 2E is a random multivalued operator such that for each fixedt ∈ Ωand s ∈ E,At,·, s : E → 2E is a generalizedm-accretive mapping and RangepdomAt,·, s/∅. LetS, p: Ω×E → E,η: Ω×E×E → EandN: Ω×E×E×E → Ebe single-valued operators, and letM, T, G: Ω×E → 2Ebe three multivalued operators.
Now, we consider the following problem.
Findx, v, w: Ω → Esuch thatvt∈Tt, xt,wt∈Gt, xt,and
0∈Nt, St, xt, ut, vt At, pt, xt, wt 1.2
for all t ∈ Ω and u ∈ Mt, xt. The problem 1.2 is called the general nonlinear random equationwith multivalued operator involving generalized m-accretive mapping in Banach spaces.
Some special cases of the problem1.2are as follows.
1 IfG is a single-valued operator,p ≡ I, the identity mapping andNt, x, y, z
ft, z gt, x, yfor all t ∈ Ω andx, y, z ∈ E, then problem 1.2is equivalent to finding
x, v: Ω → Esuch thatvt∈Tt, xtand
0∈ft, vt gt, St, xt, ut At, xt, Gt, xt 1.3
for allt ∈Ωandu∈ Mt, xt. The determinate form of the problem1.3was considered and studied by Agarwal et al.2whenG≡I.
2IfAt, x, s At, xfor allt ∈ Ω,x, s ∈ Eand, for allt ∈ Ω,At,· : E → 2E
is a generalized m-accretive mapping, then the problem 1.2 reduces to the following generalized nonlinear random multivalued operator equation involving generalized m -accretive mapping in Banach spaces.
Findx, v: Ω → Esuch thatvt∈Tt, xtand
0∈Nt, St, xt, ut, vt At, pt, xt 1.4
for allt∈Ωandu∈Mt, xt.
3IfE E∗ His a Hilbert space andAt,· ∂φt,·for allt∈ Ω, where∂φt,·
denotes the subdifferential of a lower semicontinuous andη-subdifferetiable functionφ : Ω×H → R∪ { ∞}, then the problem1.4becomes the following problem.
Findx, v: Ω → Hsuch thatvt∈Tt, xtand
Nt, St, xt, ut, vt, ηt, z, pt, xt ≥φt, pt, xt−φt, z 1.5
for allt ∈ Ω,u ∈ Mt, xt,and z ∈ H, which is called thegeneralized nonlinear random variational inclusionsfor random multivalued operators in Hilbert spaces. The determinate form of the problem1.5was studied by Agarwal et al.3whenNSx, u, v px− Bu, vfor allx, u, v∈H, whereB: H×H → His a single-valued operator.
4Ifηt, ut, vt ut−vtfor allt ∈Ω,ut, vt ∈H, then the problem1.5
Findx, v, w: Ω → Hsuch thatvt∈Tt, xt,u∈Mt, xt,and
Nt, St, xt, ut, vt, z−pt, xt ≥φt, pt, xt−φt, z 1.6
for allt∈Ωandz∈H, whose determinate form is a generalization of the problem considered in4,5,29.
5If, in the problem1.6,φis theindictor functionof a nonempty closed convex setK
inHdefined in the form
φy
⎧ ⎨ ⎩
0 if y∈K,
∞ otherwise, 1.7
then1.6becomes the following problem.
Findx, u, v: Ω → Hsuch thatvt∈Tt, xt,u∈Mt, xt, and
Nt, St, xt, ut, vt, z−pt, xt ≥0 1.8
for all t ∈ Ω and z ∈ K. The problem 1.8 has been studied by Cho et al. 19 when
Nt, x, ut, vt ut−vtfor allt∈Ω,xt, ut, vt∈H.
Remark 1.1. For appropriate and suitable choices ofS,p,N,η,M,G,T,Aand for the space
E, a number of known classes of random variational inequality, random quasi-variational inequality, random complementarity, and random quasi-complementarity problems were studied previously by many authorssee, e.g.,17–20,22–24and the references therein.
In this paper, we will use the following definitions and lemmas.
Definition 1.2. An operatorx: Ω → Eis said to bemeasurableif, for anyB∈ BE,{t∈Ω:
xt∈B} ∈ A.
Definition 1.3. An operatorF: Ω×E → Eis called arandom operatorif for anyx∈E,Ft, x ytis measurable. A random operatorFis said to becontinuousresp.,linear, boundedif, for anyt∈Ω, the operatorFt,·: E → Eis continuousresp., linear, bounded.
Similarly, we can define a random operatora: Ω×E×E → E. We will writeFtx
Ft, xtandatx, y at, xt, ytfor allt∈Ωandxt, yt∈E.
It is well known that a measurable operator is necessarily a random operator.
Definition 1.4. A multivalued operatorG : Ω → 2E is said to bemeasurableif, for anyB ∈ BE,G−1B {t∈Ω: Gt∩B /∅} ∈ A.
Definition 1.5. An operator u : Ω → E is called a measurable selection of a multivalued measurable operatorΓ: Ω → 2Eifuis measurable and for anyt∈Ω,ut∈Γt.
is said to beH-continuousif, for anyt∈Ω,Ft,·is continuous inH·,·, whereH·,·is the Hausdorffmetric onCBEdefined as follows: for any givenA, B∈CBE,
HA, B max
sup
x∈A inf y∈Bd
x, y,sup
y∈B inf x∈Ad
x, y
. 1.9
Definition 1.7. A random operatorg:Ω×E → Eis said to be
aα-strongly accretiveif there existsj2xt−yt∈J2xt−ytsuch that
gtx−gt
y, j2xt−yt ≥αtxt−yt2 1.10
for allxt, yt∈Eandt∈Ω, whereαt>0 is a real-valued random variable; bβ-Lipschitz continuousif there exists a real-valued random variableβt>0 such that
gtx−gt
y≤βtxt−yt 1.11
for allxt, yt∈Eandt∈Ω.
Definition 1.8. LetS: Ω×E → Ebe a random operator. An operatorN:Ω×E×E×E → E
is said to be
a-strongly accretive with respect toS in the first argument if there existsj2xt− yt∈J2xt−ytsuch that
NtStx,·,·−Nt
St
y,·,·, j2
xt−yt ≥txt−yt2 1.12
for allxt, yt∈E,andt∈Ω, wheret>0 is a real-valued random variable; b -Lipschitz continuous in the first argument if there exists a real-valued random
variableεt>0 such that Ntx,·,·−Nt
y,·,·≤ txt−yt 1.13
for allxt, yt∈Eandt∈Ω.
Similarly, we can define the Lipschitz continuity in the second argument and third argument ofN·,·,·.
Definition 1.9. Letη : Ω×E×E → E∗be a random operator andM : Ω×E → 2E be a
random multivalued operator. ThenMis said to be aη-accretiveif
ut−vt, ηt
x, y≥0 1.14
for all xt, yt ∈ E,ut ∈ Mtx,vt ∈ Mty,and t ∈ Ω, where Mtz
bstrictlyη-accretiveif
ut−vt, ηt
x, y ≥0 1.15
for allxt, yt∈E,ut∈Mtx,vt∈Mty,andt∈Ωand the equality holds if and only ifut vtfor allt∈Ω;
cstronglyη-accretiveif there exists a real-valued random variablert>0 such that
ut−vt, ηt
x, y ≥rtxt−yt2 1.16
for allxt, yt∈E,ut∈Mtx,vt∈Mty,andt∈Ω;
dgeneralizedm-accretiveifMisη-accretive andI λtMt,·E Efor allt ∈ Ω
andequivalently, for someλt>0.
Remark 1.10. IfE E∗ H is a Hilbert space, thena–dofDefinition 1.9 reduce to the definition ofη-monotonicity, strictη-monotonicity, strongη-monotonicity, and maximalη -monotonicity, respectively; if E is uniformly smooth and ηx, y j2x−y ∈ J2x−y,
thena–dofDefinition 1.9reduces to the definitions of accretive, strictly accretive, strongly accretive, andm-accretive operators in uniformly smooth Banach spaces, respectively.
Definition 1.11. The operatorη: Ω×E×E → E∗is said to be
amonotoneif
xt−yt, ηt
x, y≥0 1.17
for allxt, yt∈Eandt∈Ω;
bstrictly monotoneif
xt−yt, ηt
x, y ≥0 1.18
for allxt, yt∈E,andt∈Ωand the equality holds if and only ifxt ytfor allt∈Ω;
cδ-strongly monotoneif there exists a measurable functionδ: Ω → 0,∞such that
xt−yt, ηt
x, y ≥δtxt−yt2 1.19
for allxt, yt∈Eandt∈Ω;
dτ-Lipschitz continuousif there exists a real-valued random variableτt>0 such that ηt
x, y≤τtxt−yt 1.20
Definition 1.12. A multivalued measurable operator T : Ω ×E → CBE is said to be
γ-H-Lipschitz continuousif there exists a measurable functionγ : Ω → 0, ∞such that, for anyt∈Ω,
HTtx, Tt
y≤γtxt−yt 1.21
for allxt, yt∈E.
Themodules of smoothnessofEis the functionρE: 0,∞ → 0,∞defined by
ρEt sup
1
2x y x−y−1 : x ≤1, y≤t
. 1.22
A Banach space Eis calleduniformly smoothif lim
t→0ρEt/t 0 andEis calledq-uniformly smoothif there exists a constantc >0 such thatρEt≤ctq, whereq >1 is a real number.
It is well known that Hilbert spaces,Lporlpspaces, 1< p <∞and the Sobolev spaces
Wm,p, 1< p <∞, are allq-uniformly smooth.
In the study of characteristic inequalities inq-uniformly smooth Banach spaces, Xu 30proved the following result.
Lemma 1.13. Letq > 1be a given real number and letEbe a real uniformly smooth Banach space.
ThenEisq-uniformly smooth if and only if there exists a constantcq >0such that, for allx, y∈E
andjqx∈Jqx, the following inequality holds:
x yq ≤ xq qy, jqx cqyq. 1.23
Definition 1.14. LetA: Ω×E → 2Ebe a generalizedm-accretive mapping. Then theresolvent
operatorJAρtforAis defined as follows:
JAρtz I ρtA−1z 1.24
for allt∈Ωandz∈E, whereρ: Ω → 0,∞is a measurable function andη: Ω × E× E → E∗is a strictly monotone mapping.
From Huang et al.6,15, we can obtain the following lemma.
Lemma 1.15. Letη : Ω×E×E → E∗beδ-strongly monotone andτ-Lipschitz continuous. Let A : Ω×E → 2E be a generalizedm-accretive mapping. Then the resolvent operatorJAρt forAis Lipschitz continuous with constantτt/δt, that is,
JAρtx−JAρty≤ τt
δtx−y 1.25
2. Random Iterative Algorithms
In this section, we suggest and analyze a new class of iterative methods and construct some new random iterative algorithms with errors for solving the problems 1.2–1.4,
respectively.
Lemma 2.131. LetM: Ω×E → CBEbe anH-continuous random multivalued operator. Then, for any measurable operatorx: Ω → E, the multivalued operatorM·, x·: Ω → CBE is measurable.
Lemma 2.231. LetM, V : Ω×E → CBEbe two measurable multivalued operators, let >0
be a constant, and letx : Ω → Ebe a measurable selection ofM. Then there exists a measurable selectiony: Ω → EofVsuch that, for anyt∈Ω,
xt−yt≤1 HMt, Vt. 2.1
Lemma 2.3. Measurable operatorsx, u, v, w : Ω → Eare a solution of the problem 1.2if and only if
ptx JAρtt·,w
ptx−ρtNtStx, u, v
, 2.2
whereJAρt
t·,w I ρtAt·, w
−1andρt>0is a real-valued random variable.
Proof. The proof directly follows from the definition ofJAρt
t·,wand so it is omitted.
Based onLemma 2.3, we can develop a new iterative algorithm for solving the general nonlinear random equation1.2as follows.
Algorithm 2.4. Let A : Ω×E×E → 2E be a random multivalued operator such that for
each fixedt ∈ Ωand s ∈ E,At,·, s : E → 2E is a generalizedm-accretive mapping, and RangepdomAt,·, s/∅. LetS, p: Ω×E → E,η: Ω×E×E → EandN: Ω×E×E×E → E
be single-valued operators, and letM, T, G: Ω×E → 2Ebe three multivalued operators, and letλ: Ω → 0,1be a measurable step size function. Then, byLemma 2.1and Himmelberg 32, it is known that, for givenx0· ∈ E, the multivalued operatorsM·, x0·, T·, x0·,
andG·, x0·are measurable and there exist measurable selectionsu0·∈M·, x0·, v0·∈ T·, x0·,andw0·∈G·, x0·. Set
x1t x0t−λtptx0−JAρtt·,w0
ptx0−ρtNtStx0, u0, v0
λte0t, 2.3
Ttx0∈CBE,andw0t∈Gtx0∈CBE, byLemma 2.2, there exist measurable selections
u1t∈Mtx1, v1t∈Ttx1,andw1t∈Gtx1such that, for allt∈Ω,
u0t−u1t ≤
1 1
1
HMtx0, Mtx1,
v0t−v1t ≤
1 1
1
HTtx0, Ttx1,
w0t−w1t ≤
1 1
1
HGtx0, Gtx1.
2.4
By induction, one can define sequences {xnt}, {unt}, {vnt},and {wnt} inductively satisfying
xn 1t xnt−λt
ptxn−JAρtt·,wn
ptxn−ρtNtStxn, un, vn
λtent,
unt∈Mtxn, unt−un 1t ≤
1 1
n 1
HMtxn, Mtxn 1,
vnt∈Ttxn, vnt−vn 1t ≤
1 1
n 1
HTtxn, Ttxn 1,
wnt∈Gtxn, wnt−wn 1t ≤
1 1
n 1
HGtxn, Gtxn 1,
2.5
whereentis an error to take into account a possible inexact computation of the resolvent operator point, which satisfies the following conditions:
lim
n→ ∞ent0,
∞
n1
ent−en−1t<∞ 2.6
for allt∈Ω.
FromAlgorithm 2.4, we can get the following algorithms.
Algorithm 2.5. Suppose that E,A,η,S,M, T and λ are the same as inAlgorithm 2.4. Let
G:Ω×E → Ebe a random single-valued operator,p≡IandNt, x, y, z ft, z gt, x, y
for allt∈Ωandx, y, z∈E. Then, for given measurablex0 : Ω → E, one has
xn 1t 1−λtxnt λtJAρtt·,Gtxn
xnt−ρt
ftvn gtStxn, un
λtent,
unt∈Mtxn, unt−un 1t ≤
1 1
n 1
HMtxn, Mtxn 1,
vnt∈Ttxn, vnt−vn 1t ≤
1 1
n 1
HTtxn, Ttxn 1,
2.7
Algorithm 2.6. LetA:Ω×E → 2Ebe a random multivalued operator such that for each fixed
t∈Ω,At,·:E → 2Eis a generalizedm-accretive mapping, and RangepdomAt,·/∅.
IfS,p,η,N,M, T,andλare the same as inAlgorithm 2.4, then, for given measurablex0 :
Ω → E, we have
xn 1t xnt−λt
ptxn−JAρtt·
ptxn−ρtNtStxn, un, vn
λtent,
unt∈Mtxn, unt−un 1t ≤
1 1
n 1
HMtxn, Mtxn 1,
vnt∈Ttxn, vnt−vn 1t ≤
1 1
n 1
HTtxn, Ttxn 1,
2.8
whereentis the same as inAlgorithm 2.4.
Remark 2.7. Algorithms2.4–2.6include several known algorithms of2,4–9,12,17–23,25,26,
29as special cases.
3. Existence and Convergence Theorems
In this section, we will prove the convergence of the iterative sequences generated by the algorithms in Banach spaces.
Theorem 3.1. Suppose thatEis aq-uniformly smooth and separable real Banach space,p: Ω×E →
Eisα-strongly accretive andβ-Lipschitz continuous, andA:Ω×E×E → 2Eis a random multivalued
operator such that for each fixedt ∈Ωands ∈ E,At,·, s :E → 2Eis a generalizedm-accretive
mapping andRangepdomAt,·, s/∅. Letη : Ω×E×E → Ebeδ-strongly monotone and
τ-Lipschitz continuous, and letS : Ω×E → Ebe aσ-Lipschitz continuous random operator, and
letN : Ω×E×E×E → Ebe-strongly accretive with respect toSand -Lipschitz continuous
in the first argument, and μ-Lipschitz continuous in the second argument,ν-Lipschitz continuous in the third argument, respectively. Let multivalued operatorsM, T, G : Ω×E → CBEbe
γ-H-Lipschitz continuous,ξ-H-Lipschitz continuous,ζ-H-Lipschitz continuous, respectively. If there
exist real-valued random variablesρt>0andπt>0such that, for anyt∈Ω,x, y, z∈E,
JAρt
t·,xz−J
ρt At·,yz
≤πtx−y, 3.1
kt πtζt 1 τtδt−11−qαt cqβtq
1/q
<1,
ρtμtγt νtξt 1−qρtt cqρtq tqσtq 1/q
< δt1−kt
τt ,
3.2
wherecqis the same as inLemma 1.13, then, for anyt∈Ω, there existx∗t∈E,u∗t∈Mtx∗,
v∗t∈Ttx∗,andw∗t∈Gtx∗such thatx∗t, u∗t, v∗t, w∗tis a solution of the problem 1.2and
as n → ∞, where {xnt}, {unt}, {vnt} and {wnt} are iterative sequences generated by Algorithm 2.4.
Proof. It follows from2.5,Lemma 1.15and3.1that
xn 1t−xnt
xnt−λt
ptxn−JAρtt·,wn
ptxn−ρtNtStxn, un, vn
λtent
−xn−1t λt
ptxn−1−JAρtt·,wn−1
ptxn−1−ρtNtStxn−1, un−1, vn−1
−λten−1t
≤xnt−xn−1t−λt
ptxn−ptxn−1
λtJAρt t·,wn
ptxn−ρtNtStxn, un, vn
−JAρt t·,wn−1
ptxn−1−ρtNtStxn−1, un−1, vn−1 λtent−en−1t
≤xnt−xn−1t−λt
ptxn−ptxn−1
λtJAρt t·,wn
ptxn−ρtNtStxn, un, vn
−JAρt t·,wn
ptxn−1−ρtNtStxn−1, un−1, vn−1
λtJAρt t·,wn
ptxn−1−ρtNtStxn−1, un−1, vn−1
−JAρt t·,wn−1
ptxn−1−ρtNtStxn−1, un−1, vn−1 λtent−en−1t
≤xnt−xn−1t−λt
ptxn−ptxn−1
λt·τt
δtptxn−ρtNtStxn, un, vn
−ptxn−1−ρtNtStxn−1, un−1, vn−1
λtπtwn−wn−1 λtent−en−1t
≤xnt−xn−1t−λt
ptxn−ptxn−1
λtτt
δt xnt−xn−1t−
ptxn−ptxn−1
xnt−xn−1t−ρtNtStxn, un, vn−NtStxn−1, un, vn
ρtNtStxn−1, un, vn−NtStxn−1, un−1, vn
ρtNtStxn−1, un−1, vn−NtStxn−1, un−1, vn−1
≤1−λtxnt−xn−1t
λt
1 τt
δt
xnt−xn−1t−
ptxn−ptxn−1
λtτt
δt xnt−xn−1t−ρtNtStxn, un, vn−NtStxn−1, un, vn
ρtNtStxn−1, un, vn−NtStxn−1, un−1, vn
ρtNtStxn−1, un−1, vn−NtStxn−1, un−1, vn−1
λtπtwn−wn−1 λtent−en−1t.
3.4
Sincepis strongly accretive and Lipschitz continuous, xnt−xn−1t−ptxn−ptxn−1q
≤ xnt−xn−1tq−qptxn−ptxn−1, jqxnt−xn−1t
cqptxn−ptxn−1q
≤1−qαt cqβtq
xnt−xn−1tq,
3.5
that is,
xnt−xn−1t−
ptxn−ptxn−1
≤1−qαt cqβtq 1/q
xnt−xn−1t,
3.6
wherecq is the same as inLemma 1.13. Also from the strongly accretivity ofNwith respect toSand the Lipschitz continuity ofNin the first argument, we have
xnt−xn−1t−ρtNtStxn, un, vn−NtStxn−1, un, vn
≤1−qρtt cqρtq tqσtq 1/q
xnt−xn−1t.
3.7
By Lipschitz continuity ofNin the second and third argument, andH-Lipschitz continuity
ofT, M, G, we obtain
NtStxn−1, un, vn−NtStxn−1, un−1, vn
≤μtun−un−1 ≤μt
1 1
n
HMtxn−1−Mtxn
≤
1 1
n
μtγtxnt−xn−1t,
NtStxn−1, un−1, vn−NtStxn−1, un−1, vn−1
≤νtvn−vn−1 ≤νt
1 1
n
HTtxn−1−Ttxn
≤
1 1
n
νtξtxnt−xn−1t,
3.9
wn−wn−1 ≤
1 1
n
HGtxn−1−Gtxn
≤
1 1
n
ζtxnt−xn−1t.
3.10
Using3.6–3.10in3.4, we have, for allt∈Ω,
xn 1t−xnt ≤θt, nxnt−xn−1t λtent−en−1t, 3.11
where
θt, n 1−λt λtκt, n,
κt, n 1 τtδt−11−qαt cqβtq
1/q
τtδt−11−qρtt cqρtq tqσtq 1/q
1 1
n
ρtμtγt νtξt
1 1
n
πtςt.
3.12
Let
κt πtζt 1 τtδt−11−qαt cqβtq
1/q
τtδt−1ρtμtγt νtξt 1−qρtt cqρtqεtqσtq 1/q
,
θt 1−λt λtκt.
3.13
Thenκt, n → κt,θt, n → θtas n → ∞. From the condition 3.2, we know that
such thatθt, n≤ θtfor alln ≥n0andt∈Ω. Therefore, for alln > n0, by3.11, we now
know that, for allt∈Ω,
xn 1t−xnt
≤θtxnt−xn−1t λtent−en−1t
≤θtθtxn−1t−xn−2t λten−1t−en−2t
λtent−en−1t
θt2xn−1−xn−2 λt
θten−1t−en−2t ent−en−1t
≤ · · ·
≤θtn−n0xn0 1−xn0 λt n−n0
i1
θti−1en−i−1−en−i,
3.14
which implies that, for anym≥n > n0,
xmt−xnt ≤ m−1
jn
xj 1t−xjt
≤m−1 jn
θtj−n0xn0 1t−xn0t
λt
m−1
jn j−n0
i1
θti−1en−i−1t−en−it.
3.15
Since 0< λt≤1 andθt<1 for allt∈Ω, it follows from2.6and3.15that lim
n→ ∞xmt−
xnt0 and so{xnt}is a Cauchy sequence. Settingxnt → x∗tasn → ∞for allt∈Ω. From3.8–3.10, we know that{unt},{vnt},{wnt}are also Cauchy sequences. Hence there existu∗t, v∗t, w∗t∈Esuch thatunt → u∗t,vnt → v∗t,wnt → w∗tas
n → ∞.
Now, we show thatu∗t∈Mtx∗. In fact, we have
du∗t, Mtx∗ infu∗t−y:y∈Mtx∗
≤ u∗t−unt dunt, Mtx∗ ≤ u∗t−unt HMtxn, Mtx∗ ≤ u∗t−unt γtxnt−x∗t −→0.
3.16
This implies that u∗t ∈ Mtx∗. Similarly, we havev∗t ∈ Ttx∗ and w∗t ∈ Gtx∗. Therefore, from2.5,2.6and the continuity ofJAρ
t·,wt,p, N,andS, we have
ptx∗ JAρtt·,w∗
ptx∗−ρtNtStx∗, u∗, v∗
ByLemma 2.3, now we know thatx∗t, u∗t, v∗t, w∗tis a solution of the problem1.2.
This completes the proof.
Remark 3.2. IfEis a 2-uniformly smooth Banach space and there exists a measurable function
ρ: Ω → 0,∞such that
kt πtζt 1 τtδt−11−2αt C2βt2<1,
ht μtγt νtξt<C2 tσt,
ρt< δt1−kt
τtht ,
ρt− tτt−htδt1−kt τtC2 t2σt2−ht2
<
!
tτt−htδt1−kt2−C2 t2σt2−ht2
τt2−δt21−kt2
τtC2 t2σt2−ht2
,
tτt> htδt1−kt !C2 t2σt2−ht2τt2−δt21−kt2,
3.18
then3.2holds. We note that Hilbert spaces andLporlpspaces, 2≤p <∞, are 2-uniformly smooth.
FromTheorem 3.1, we can get the following theorems.
Theorem 3.3. LetE,η,S,M, T,andλbe the same as inTheorem 3.1. Assume thatA:Ω×E×E →
2Eis a random multivalued operator such that, for each fixedt∈Ωands∈E,At,·, s:E → 2Eis a
generalizedm-accretive mapping. Letf: Ω×E → Ebeν-Lipschitz continuous, letS: Ω×E → E
be aσ-Lipschitz continuous random operator, letG: Ω×E → Ebeζ-Lipschitz continuous, and let
g: Ω×E×E → Ebe-strongly accretive with respect toSand -Lipschitz continuous in the first
argument andμ-Lipschitz continuous in the second argument, respectively. If there exist real-valued random variablesρt>0andπt>0such that3.1holds and
ρtμtγt νtξt 1−qρtt cqρtq tqσtq 1/q
< δt1−πtζt
τt 3.19
for all t ∈ Ω, where cq is the same as in Lemma 1.13, then, for any t ∈ Ω, the iterative
sequences{xnt},{unt},and{vnt}defined byAlgorithm 2.5converge strongly to the solution x∗t, u∗t, v∗tof the problem1.3.
is a generalized m-accretive mapping andRangepdomAt,·/∅. If there exists a real-valued random variableρt>0such that, for anyt∈Ω,
kt 1 τtδt−11−qαt cqβtq
1/q
<1,
ρtμtγt νtξt 1−qρtt cqρtq tqσtq 1/q
< δtτt−11−kt,
3.20
wherecq is the same as inLemma 1.13, then, for anyt∈Ω, the iterative sequences{xnt},{unt},
and {vnt}defined by Algorithm 2.6converge strongly to the solutionx∗t, u∗t, v∗tof the
problem1.4.
Remark 3.5. For an appropriate choice of the mappings S, p, A, M, T, G, N, η and the
spaceE, Theorems3.1–3.4include many known results of generalized variational inclusions as special casessee2,4–9,12,17–23,25,26,29and the references therein.
Acknowledgment
This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department 2006A106 and the Sichuan Youth Science and Technology Foundation 08ZQ026-008. This work was supported by the Korea Research Foundation Grant funded by the Korean GovernmentKRF-2008-313-C00050.
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