Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 609353, 9pages
doi:10.1155/2009/609353
Research Article
The Solvability of a New System of Nonlinear
Variational-Like Inclusions
Zeqing Liu,1Min Liu,1Jeong Sheok Ume,2and Shin Min Kang3
1Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian Liaoning 116029, China
2Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea
3Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University,
Jinju 660-701, South Korea
Correspondence should be addressed to Jeong Sheok Ume, jsume@changwon.ac.kr
Received 23 November 2008; Accepted 1 April 2009
Recommended by Marlene Frigon
We introduce and study a new system of nonlinear variational-like inclusions involving s-G, η-
maximal monotone operators, strongly monotone operators, η-strongly monotone operators,
relaxed monotone operators, cocoercive operators, λ, ξ-relaxed cocoercive operators, ζ, ϕ, -
g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the
resolvent operator technique associated with s-G, η-maximal monotone operators and Banach
contraction principle, we demonstrate the existence and uniqueness of solution for the system of
nonlinear variational-like inclusions. The results presented in the paper improve and extend some
known results in the literature.
Copyright q2009 Zeqing Liu 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
It is well known that the resolvent operator technique is an important method for solving
various variational inequalities and inclusions 1–20. In particular, the generalized resolvent
operator technique has been applied more and more and has also been improved intensively.
For instance, Fang and Huang 5introduced the class of H-monotone operators and defined
the associated resolvent operators, which extended the resolvent operators associated with η-
subdifferential operators of Ding and Luo 3and maximal η-monotone operators of Huang
and Fang 6, respectively. Later, Liu et al. 17researched a class of general nonlinear
implicit variational inequalities including the H-monotone operators. Fang and Huang 4
created a class of H, η-monotone operators, which offered a unifying framework for the
classes of maximal monotone operators, maximal η-monotone operators and H-monotone
operators. Recently, Lan 8introduced a class of A, η-accretive operators which further
2 Fixed Point Theory and Applications
enriched and improved the class of generalized resolvent operators. Lan 10studied a
system of general mixed quasivariational inclusions involving A, η-accretive mappings in
q-uniformly smooth Banach spaces. Lan et al. 14constructed some iterative algorithms
for solving a class of nonlinear A, η-monotone operator inclusion systems involving
nonmonotone set-valued mappings in Hilbert spaces. Lan 9investigated the existence of
solutions for a class of A, η-accretive variational inclusion problems with nonaccretive set-
valued mappings. Lan 11analyzed and established an existence theorem for nonlinear
parametric multivalued variational inclusion systems involving A, η-accretive mappings
in Banach spaces. By using the random resolvent operator technique associated with A, η-
accretive mappings, Lan 13established an existence result for nonlinear random multi-
valued variational inclusion systems involving A, η-accretive mappings in Banach spaces.
Lan and Verma 15studied a class of nonlinear Fuzzy variational inclusion systems with
A, η-accretive mappings in Banach spaces. On the other hand, some interesting and classical
techniques such as the Banach contraction principle and Nalder’s fixed point theorems have
been considered by many researchers in studying variational inclusions.
Inspired and motivated by the above achievements, we introduce a new system
of nonlinear variational-like inclusions involving s-G, η-maximal monotone operators in
Hilbert spaces and a class of ζ, ϕ, -g-relaxed cocoercive operators. By virtue of the Banach’s
fixed point theorem and the resolvent operator technique, we prove the existence and
uniqueness of solution for the system of nonlinear variational-like inclusions. The results
presented in the paper generalize some known results in the field.
2. Preliminaries
In what follows, unless otherwise specified, we assume that Hiis a real Hilbert space
endowed with norm ·
iand inner product ·,·i,and2
Hidenotes the family of all nonempty
subsets of Hifor i∈{1,2}.Now let’s recall some concepts.
Definition 2.1. Let A:H1→H2,f,g :H1→H1,η:H1×H1→H1be mappings.
1Ais said to be Lipschitz continuous, if there exists a constant α>0 such that
Ax −Ay
2≤α
x−y
1,∀x, y ∈H1;2.1
2Ais said to be r-expanding, if there exists a constant r>0 such that
Ax −Ay
2≥r
x−y
1,∀x, y ∈H1;2.2
3fis said to be δ-strongly monotone, if there exists a constant δ>0 such that
fx −fy,x −y1≥δ
x−y
2
1,∀x, y ∈H1;2.3
4fis said to be δ-η-strongly monotone, if there exists a constant δ>0 such that
fx −fy,ηx, y1≥δ
x−y
2
1,∀x, y ∈H1;2.4
Fixed Point Theory and Applications 3
5fis said to be ζ, ϕ, -g-relaxed cocoercive, if there exist nonnegtive constants ζ, ϕ
and such that
fx −fy,gx −gy1≥−ζ
fx −fy
2
1−ϕ
gx −gy
2
1
x−y
2
1,∀x, y ∈H1;2.5
6gis said to be ζ-relaxed Lipschitz, if there exists a constant ζ>0 such that
gx −gy,x −y1≤−ζ
x−y
2
1,∀x, y ∈H1.2.6
Definition 2.2. Let N:H2×H1×H2→H1,A,C :H1→H2,B :H2→H1be mappings. N
is called
1λ, ξ-relaxed cocoercive with respect to Ain the first argument, if there exist
nonnegative constants λ, ξ such that
NAu, x, y−NAv,x,y
,u−v1
≥−λAu −Av2
2ξu−v2
1,∀u, v, x ∈H1,y ∈H2;
2.7
2θ-cocoercive with respect to Bin the second argument, if there exists a constant θ>0
such that
Nx, Bu, y−Nx, Bv, y,u−v1≥θBu −Bv2
1,∀u, v, x, y ∈H2;2.8
3τ-relaxed Lipschitz with respect to Cin the third argument, if there exists a constant
τ>0 such that
Nx, y, Cu−Nx, y, Cv,u−v1≤−τu−v2
1,∀u, v, y ∈H1,x∈H2;2.9
4τ-relaxed monotone with respect to Cin the third argument, if there exists a constant
τ>0 such that
Nx, y, Cu−Nx, y, Cv,u−v1≥−τu−v2
1,∀u, v, y ∈H1,x ∈H2;2.10
5Lipschitz continuous in the first argument, if there exists a constant μ>0 such that
Nu, x, y−Nv, x, y
1≤μu−v1,∀u, v, y ∈H2,x ∈H1.2.11
Similarly, we can define the Lipschitz continuity of Nin the second and third
arguments, respectively.
4 Fixed Point Theory and Applications
Definition 2.3. For i∈{1,2},j ∈{1,2}\{i},letMi:Hj×Hi→2Hi,η
i:Hi×Hi→Hibe
mappings. For each given x2,x
1∈H1×H2and i∈{1,2},M
ixi,·:Hi→2Hiis said to be
si-ηi-relaxed monotone, if there exists a constant si>0 such that
x∗−y∗,η
ix, yi≥−si
x−y
2
i,∀x, x∗,y, y∗∈graphMixi,·.2.12
Definition 2.4. For i∈{1,2},j ∈{1,2}\{i},letMi:Hj×Hi→2Hi,G
i:Hi→Hibe mappings.
For any given x2,x
1∈H1×H2and i∈{1,2},M
ixi,·:Hi→2Hiis said to be si-Gi,η
i-
maximal monotone, if B1Mixi,·is si-ηi-relaxed monotone; B2GiρiMixi,·HiHi
for ρi>0.
Lemma 2.5 see 8.Let Hbe a real Hilbert space, η:H×H→Hbe a mapping, G:H→Hbe a
d-η-strongly monotone mapping and M:H→2Hbe a s-G, η-maximal monotone mapping. Then
the generalized resolvent operator RG,η
M,ρ GρM−1:H→His singled-valued for d>ρs>0.
Lemma 2.6 see 8.Let Hbe a real Hilbert space, η:H×H→Hbe a σ-Lipschitz continuous
mapping, G:H→Hbe a d-η-strongly monotone mapping, and M:H→2Hbe a s-G, η-
maximal monotone mapping. Then the generalized resolvent operator RG,η
M,ρ :H→His σ/d−ρs-
Lipschitz continuous for d>ρs>0.
For i∈{1,2}and j∈{1,2}\{i}, assume that Ai,C
i:Hi→Hj,B
i:Hj→Hi,η
i:
Hi×Hi→Hi,N
i:Hj×Hi×Hj→Hi,f
i,g
i:Hi→Hiare single-valued mappings, Mi:
Hj×Hi→2Hisatisfies that for each given xi∈Hj,M
ixi,·is si-Gi,η
i-maximal monotone,
where Gi:Hi→Hiis di-ηi-strongly monotone and Rangefi−gidomMixi,·/
∅.We
consider the following problem of finding x, y∈H1×H2such that
x∈N1A1x, B1y, C1xM1y, f1−g1x,
y∈N2A2y, B2x, C2yM2x, f2−g2y,
2.13
where fi−gixfix−gixfor x∈Hiand i∈{1,2}. The problem 2.13is called the
system of nonlinear variational-like inclusions problem.
Special cases of the problem 2.13are as follows.
If A1B1B2C2f1−g1f2−g2I,N1x, y, zN1x, yx,N2u, v, w
N2v, ww,M1x, yM1y,M2u, vM2vfor each x, z, v ∈H2,y,u,w ∈H1, then
the problem 2.13collapses to finding x, y∈H1×H2such that
0∈N1x, yM1x,
0∈N2x, yM2y,
2.14
which was studied by Fang and Huang 4with the assumption that Miis Gi,η
i-monotone
fori∈{1,2}.
Fixed Point Theory and Applications 5
If HiH, AiA, BiB, CiC, MiM, fif, gig, and Niu, v, wNu, v,for
all u, v, w ∈Hfor i∈{1,2}, then the problem 2.13reduces to finding x∈Hsuch that
0∈NAx, BxMx, f−gx,2.15
which was studied in Shim et al. 19.
It is easy to see that the problem 2.13includes a number of variational and
variational-like inclusions as special cases for appropriate and suitable choice of the
mappings Ni,A
i,B
i,C
i,M
i,f
i,g
ifor i∈{1,2}.
3. Existence and Uniqueness Theorems
In this section, we will prove the existence and uniqueness of solution of the problem 2.13.
Lemma 3.1. Let ρ1and ρ2be two positive constants. Then x, y∈H1×H2is a solution of the
problem 2.13if and only if x, y∈H1×H2satisfies that
f1xg1xRG1,η1
M1y,·,ρ1xG1f1−g1x−ρ1N1A1x, B1y, C1x,
f2yg2yRG2,η2
M2x,·,ρ2yG2f2−g2y−ρ2N2A2y, B2x, C2y,
3.1
where RG1,η1
M1y,·,ρ1uG1ρ1M1y, ·−1u,R
G2,η2
M2x,·,ρ2vG2ρ2M2x, ·−1v, for all
u, v∈H1×H2.
Theorem 3.2. For i∈{1,2},j ∈{1,2}\{i},let ηi:Hi×Hi→Hibe Lipschitz continuous
with constant σi,Ai,C
i:Hi→Hj,B
i:Hj→Hi,f
i,g
i:Hi→Hibe Lipschitz continuous
with constants αi,γ
i,β
i,ϑ
fi,ϑ
girespectively, Ni:Hj×Hi×Hj→Hibe Lipschitz continuous in
the first, second and third arguments with constants μi,ν
i,ω
irespectively, let Nibe λi,ξ
i-relaxed
cocoercive with respect to Aiin the first argument, and τi-relaxed Lipschitz with respect to Ciin the
third argument, fibe ζi,ϕ
i,
i-gi-relaxed cocoercive, fi−gibe δfi,gi-strongly monotone, Gi:Hi→
Hibe ti-Lipschitz continuous and di-ηi-strongly monotone, and Gifi−gibe ζi-relaxed Lipschitz,
Mi:Hj×Hi→2Hisatisfy that for each fixed xi∈Hj,M
ixi,·:Hi→2Hiis si-Gi,η
i-maximal
monotone, Rangefi−gi∩dom Mixi,·/
∅and
RGi,ηi
Miyi,·,ρi
x−RGi,ηi
Mizi,·,ρix
i
≤r
yi−zi
j,∀x∈Hi,y
i,z
i∈Hj,i∈{1,2},j∈{1,2}\{i}.
3.2
If there exist positive constants ρ1,ρ
2, and ksuch that
di>ρ
isi,i∈{1,2},3.3
kmaxm1σ1
d1−ρ1s1c1ρ1l1σ2
d2−ρ2s2
χ2,m
2σ2
d2−ρ2s2c2ρ2l2σ1
d1−ρ1s1
χ1r<1,
3.4