Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 609353, 9 pages
doi:10.1155/2009/609353
Research Article
The Solvability of a New System of Nonlinear
Variational-Like Inclusions
Zeqing Liu,1 Min Liu,1 Jeong Sheok Ume,2 and Shin Min Kang3
1
Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian Liaoning 116029, China
Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea
3
Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University,
Jinju 660-701, South Korea
2
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, ηmaximal 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 q 2009 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 5 introduced the class of H-monotone operators and defined
the associated resolvent operators, which extended the resolvent operators associated with ηsubdifferential operators of Ding and Luo 3 and maximal η-monotone operators of Huang
and Fang 6, respectively. Later, Liu et al. 17 researched 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 8 introduced 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 10 studied a
system of general mixed quasivariational inclusions involving A, η-accretive mappings in
q-uniformly smooth Banach spaces. Lan et al. 14 constructed some iterative algorithms
for solving a class of nonlinear A, η-monotone operator inclusion systems involving
nonmonotone set-valued mappings in Hilbert spaces. Lan 9 investigated the existence of
solutions for a class of A, η-accretive variational inclusion problems with nonaccretive setvalued mappings. Lan 11 analyzed 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, ηaccretive mappings, Lan 13 established an existence result for nonlinear random multivalued variational inclusion systems involving A, η-accretive mappings in Banach spaces.
Lan and Verma 15 studied 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 Hi is a real Hilbert space
endowed with norm · i and inner product ·, ·i , and 2Hi denotes the family of all nonempty
subsets of Hi for i ∈ {1, 2}. Now let’s recall some concepts.
Definition 2.1. Let A : H1 → H2 , f, g : H1 → H1 , η : H1 × H1 → H1 be mappings.
1 A is said to be Lipschitz continuous, if there exists a constant α > 0 such that
Ax − Ay ≤ αx − y ,
2
1
∀x, y ∈ H1 ;
2.1
2 A is said to be r-expanding, if there exists a constant r > 0 such that
Ax − Ay ≥ r x − y ,
2
1
∀x, y ∈ H1 ;
2.2
3 f is said to be δ-strongly monotone, if there exists a constant δ > 0 such that
2
fx − fy, x − y 1 ≥ δx − y1 ,
∀x, y ∈ H1 ;
2.3
4 f is said to be δ-η-strongly monotone, if there exists a constant δ > 0 such that
2
fx − fy, ηx, y 1 ≥ δx − y1 ,
∀x, y ∈ H1 ;
2.4
Fixed Point Theory and Applications
3
5 f is said to be ζ, ϕ, -g-relaxed cocoercive, if there exist nonnegtive constants ζ, ϕ
and such that
fx − fy, gx − gy
1
2
2
2
≥ −ζfx − fy1 − ϕgx − gy1 x − y1 ,
∀x, y ∈ H1 ;
2.5
6 g is said to be ζ-relaxed Lipschitz, if there exists a constant ζ > 0 such that
2
gx − gy, x − y 1 ≤ −ζx − y1 ,
∀x, y ∈ H1 .
2.6
Definition 2.2. Let N : H2 × H1 × H2 → H1 , A, C : H1 → H2 , B : H2 → H1 be mappings. N
is called
1 λ, ξ-relaxed cocoercive with respect to A in the first argument, if there exist
nonnegative constants λ, ξ such that
N Au, x, y − N Av, x, y , u − v 1
≥ −λAu − Av22 ξu − v21 ,
∀u, v, x ∈ H1 , y ∈ H2 ;
2.7
2 θ-cocoercive with respect to B in the second argument, if there exists a constant θ > 0
such that
N x, Bu, y − N x, Bv, y , u − v 1 ≥ θBu − Bv21 ,
∀u, v, x, y ∈ H2 ;
2.8
3 τ-relaxed Lipschitz with respect to C in the third argument, if there exists a constant
τ > 0 such that
N x, y, Cu − N x, y, Cv , u − v 1 ≤ −τu − v21 ,
∀u, v, y ∈ H1 , x ∈ H2 ;
2.9
4 τ-relaxed monotone with respect to C in the third argument, if there exists a constant
τ > 0 such that
N x, y, Cu − N x, y, Cv , u − v 1 ≥ −τu − v21 ,
∀u, v, y ∈ H1 , x ∈ H2 ;
2.10
5 Lipschitz continuous in the first argument, if there exists a constant μ > 0 such that
N u, x, y − N v, x, y ≤ μu − v ,
1
1
∀u, v, y ∈ H2 , x ∈ H1 .
2.11
Similarly, we can define the Lipschitz continuity of N in the second and third
arguments, respectively.
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Fixed Point Theory and Applications
Definition 2.3. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, let Mi : Hj × Hi → 2Hi , ηi : Hi × Hi → Hi be
mappings. For each given x2 , x1 ∈ H1 × H2 and i ∈ {1, 2}, Mi xi , · : Hi → 2Hi is said to be
si -ηi -relaxed monotone, if there exists a constant si > 0 such that
2
x∗ − y∗ , ηi x, y i ≥ −si x − yi ,
∀x, x∗ , y, y∗ ∈ graphMi xi , ·.
2.12
Definition 2.4. For i ∈ {1, 2}, j ∈ {1, 2}\{i}, let Mi : Hj ×Hi → 2Hi , Gi : Hi → Hi be mappings.
For any given x2 , x1 ∈ H1 × H2 and i ∈ {1, 2}, Mi xi , · : Hi → 2Hi is said to be si -Gi , ηi maximal monotone, if B1 Mi xi , · is si -ηi -relaxed monotone; B2 Gi ρi Mi xi , ·Hi Hi
for ρi > 0.
Lemma 2.5 see 8. Let H be a real Hilbert space, η : H×H → H be a mapping, G : H → H be a
d-η-strongly monotone mapping and M : H → 2H be a s-G, η-maximal monotone mapping. Then
G,η
the generalized resolvent operator RM,ρ G ρM−1 : H → H is singled-valued for d > ρs > 0.
Lemma 2.6 see 8. Let H be a real Hilbert space, η : H × H → H be a σ-Lipschitz continuous
mapping, G : H → H be a d-η-strongly monotone mapping, and M : H → 2H be a s-G, ηG,η
maximal monotone mapping. Then the generalized resolvent operator RM,ρ : H → H is σ/d − ρsLipschitz continuous for d > ρs > 0.
For i ∈ {1, 2} and j ∈ {1, 2} \ {i}, assume that Ai , Ci : Hi → Hj , Bi : Hj → Hi , ηi :
Hi × Hi → Hi , Ni : Hj × Hi × Hj → Hi , fi , gi : Hi → Hi are single-valued mappings, Mi :
Hj ×Hi → 2Hi satisfies that for each given xi ∈ Hj , Mi xi , · is si -Gi , ηi -maximal monotone,
∅. We
where Gi : Hi → Hi is di -ηi -strongly monotone and Rangefi − gi domMi xi , · /
consider the following problem of finding x, y ∈ H1 × H2 such that
x ∈ N1 A1 x, B1 y, C1 x M1 y, f1 − g1 x ,
y ∈ N2 A2 y, B2 x, C2 y M2 x, f2 − g2 y ,
2.13
where fi − gi x fi x − gi x for x ∈ Hi and i ∈ {1, 2}. The problem 2.13 is called the
system of nonlinear variational-like inclusions problem.
Special cases of the problem 2.13 are as follows.
If A1 B1 B2 C2 f1 − g1 f2 − g2 I, N1 x, y, z N1 x, y x, N2 u, v, w N2 v, w w, M1 x, y M1 y, M2 u, v M2 v for each x, z, v ∈ H2 , y, u, w ∈ H1 , then
the problem 2.13 collapses to finding x, y ∈ H1 × H2 such that
0 ∈ N1 x, y M1 x,
0 ∈ N2 x, y M2 y ,
2.14
which was studied by Fang and Huang 4 with the assumption that Mi is Gi , ηi -monotone
fori ∈ {1, 2}.
Fixed Point Theory and Applications
5
If Hi H, Ai A, Bi B, Ci C, Mi M, fi f, gi g, and Ni u, v, w Nu, v, for
all u, v, w ∈ H for i ∈ {1, 2}, then the problem 2.13 reduces to finding x ∈ H such that
0 ∈ NAx, Bx M x, f − g x ,
2.15
which was studied in Shim et al. 19.
It is easy to see that the problem 2.13 includes a number of variational and
variational-like inclusions as special cases for appropriate and suitable choice of the
mappings Ni , Ai , Bi , Ci , Mi , fi , gi for 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 ρ1 and ρ2 be two positive constants. Then x, y ∈ H1 × H2 is a solution of the
problem 2.13 if and only if x, y ∈ H1 × H2 satisfies that
G ,η1
f1 x g1 x RM11 y,·,ρ
x G1 f1 − g1 x − ρ1 N1 A1 x, B1 y, C1 x ,
1
3.1
G ,η2
y G2 f2 − g2 y − ρ2 N2 A2 y, B2 x, C2 y ,
f2 y g2 y RM22 x,·,ρ
2
1
2
u G1 ρ1 M1 y, ·−1 u, RM22 x,·,ρ
v G2 ρ2 M2 x, ·−1 v, for all
where RM11 y,·,ρ
1
2
u, v ∈ H1 × H2 .
G ,η
G ,η
Theorem 3.2. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, let ηi : Hi × Hi → Hi be Lipschitz continuous
with constant σi , Ai , Ci : Hi → Hj , Bi : Hj → Hi , fi , gi : Hi → Hi be Lipschitz continuous
with constants αi , γi , βi , ϑfi , ϑgi respectively, Ni : Hj × Hi × Hj → Hi be Lipschitz continuous in
the first, second and third arguments with constants μi , νi , ωi respectively, let Ni be λi , ξi -relaxed
cocoercive with respect to Ai in the first argument, and τi -relaxed Lipschitz with respect to Ci in the
third argument, fi be ζi , ϕi , i -gi -relaxed cocoercive, fi − gi be δfi ,gi -strongly monotone, Gi : Hi →
Hi be ti -Lipschitz continuous and di -ηi -strongly monotone, and Gi fi − gi be ζi -relaxed Lipschitz,
Mi : Hj × Hi → 2Hi satisfy that for each fixed xi ∈ Hj , Mi xi , · : Hi → 2Hi is si -Gi , ηi -maximal
∅ and
monotone, Rangefi − gi ∩ dom Mi xi , · /
Gi ,ηi
Gi ,ηi
≤ r yi − zi ,
R
x−R
x
Mi yi ,·,ρi
j
Mi zi ,·,ρi
i
∀x ∈ Hi , yi , zi ∈ Hj , i ∈ {1, 2}, j ∈ {1, 2} \ {i}.
3.2
If there exist positive constants ρ1 , ρ2 , and k such that
di > ρi si ,
k max m1
i ∈ {1, 2},
3.3
σ2
σ1
σ1
σ2
c1 ρ1 l1 c2 ρ2 l2 χ2 , m2 χ1 r < 1,
d1 − ρ1 s1
d2 − ρ2 s2
d2 − ρ2 s2
d1 − ρ1 s1
3.4
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Fixed Point Theory and Applications
where
mi 1 − 2δfi ,gi ϑf2i 2 ζi ϑfi ϕi ϑgi − i ϑg2i ,
2
1 − 2ζi t2i ϑfi ϑgi ,
li μ2i α2i 2λi αi − ξi 1 ωi2 γi2 − 2τi 1,
ci χi ρi νi βi ,
3.5
i ∈ {1, 2},
then the problem 2.13 possesses a unique solution in H1 × H2 .
Proof. For any x, y ∈ H1 × H2 , define
G ,η1
Fρ1 x, y x − f1 − g1 x RM11 y,·,ρ
x G1 f1 − g1 x − ρ1 N1 A1 x, B1 y, C1 x ,
1
G ,η2
Fρ2 x, y y − f2 − g2 y RM22 x,·,ρ
y G2 f2 − g2 y − ρ2 N2 A2 y, B2 x, C2 y .
2
3.6
For each u1 , v1 , u2 , v2 ∈ H1 × H2 , it follows from Lemma 2.6 that
Fρ u1 , v1 − Fρ u2 , v2 1
1
1
≤ u1 − u2 − f1 − g1 u1 − f1 − g1 u2 1 σ1
d1 − ρ1 s1
× u1 − u2 G1 f1 − g1 u1 − G1 f1 − g1 u2 1
3.7
ρ1 N1 A1 u1 , B1 v1 , C1 u1 − N1 A1 u2 , B1 v2 , C1 u2 1 rv1 − v2 2 .
Because f1 −g1 is δf1 ,g1 -strongly monotone, f1 , g1 and G1 are Lipschitz continuous, and G1 f1 −
g1 is ζ1 -relaxed Lipschitz, we deduce that
u1 − u2 − f1 − g1 u1 − f1 − g1 u2 2
≤ 1 − 2δf1 ,g1
1
ϑf21 2 ζ1 ϑf1 ϕ1 ϑg1 − 1 ϑg21 u1 − u2 21 ,
u1 − u2 G1 f1 − g1 u1 − G1 f1 − g1 u2 2
≤ 1 − 2ζ1 t21 ϑf1 ϑg1
1
2 u1 − u2 21 .
3.8
3.9
Fixed Point Theory and Applications
7
Since A1 , B1 , C1 are all Lipschitz continuous, N1 is λ1 , ξ1 -relaxed cocoercive with respect to
A1 , τ1 -relaxed Lipschitz with respect to C1 , and is Lipschitz continuous in the first, second
and third arguments, respectively, we infer that
N1 A1 u1 , B1 v1 , C1 u1 − N1 A1 u2 , B1 v1 , C1 u1 − u1 − u2 21
≤ μ21 α21 2λ1 α1 − ξ1 1 u1 − u2 21 ,
3.10
N1 A1 u2 , B1 v2 , C1 u1 − N1 A1 u2 , B1 v2 , C1 u2 u1 − u2 21
≤ ω12 γ12 − 2τ1 1 u1 − u2 21 ,
3.11
N1 A1 u2 , B1 v1 , C1 u1 − N1 A1 u2 , B1 v2 , C1 u1 ≤ ν1 β1 v1 − v2 2 .
3.12
In terms of 3.7–3.12, we obtain that
Fρ u1 − v1 − Fρ u2 , v2 1
1
≤ m1 u1 − u2 1 σ1
c1 ρ1 l1 u1 − u2 1 χ1 v1 − v2 2 rv1 − v2 2 .
d1 − ρ1 s1
3.13
Similarly, we deduce that
Fρ u1 , v1 − Fρ u2 , v2 2
2
≤ m2 v1 − v2 2 σ2
c2 ρ2 l2 v1 − v2 2 χ2 u1 − u2 1 ru1 − u2 1 .
d2 − ρ2 s2
3.14
Define · ∗ on H1 × H2 by u, v∗ u1 v1 for any u, v ∈ H1 × H2 . It is easy to see
that H1 × H2 , · ∗ is a Banach space. Define Lρ1 ,ρ2 : H1 × H2 → H1 × H2 by
Lρ1 ,ρ2 u, v Fρ1 u, v, Fρ2 u, v ,
∀u, v ∈ H1 × H2 .
3.15
By virtue of 3.3,3.4,3.13 and 3.14, we achieve that 0 < k < 1 and
Lρ ,ρ u1 , v1 − Lρ ,ρ u2 , v2 ≤ ku1 , v1 − u2 , v2 ,
1 2
1 2
∗
∗
3.16
which means that Lρ1 ,ρ2 : H1 × H2 → H1 × H2 is a contractive mapping. Hence, there exists a
unique x, y ∈ H1 × H2 such that Lρ1 ,ρ2 x, y x, y. That is,
G ,η1
f1 x g1 x RM11 y,·,ρ
x G1 f1 − g1 x − ρ1 N1 A1 x, B1 y, C1 x ,
1
G ,η2
f2 y g2 y RM22 x,·,ρ
y G2 f2 − g2 y − ρ2 N2 A2 y, B2 x, C2 y .
2
3.17
8
Fixed Point Theory and Applications
By Lemma 3.1, we derive that x, y is a unique solution of the problem 2.13. This completes
the proof.
Theorem 3.3. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, let ηi , Ai , Ci , Mi , fi , gi , fi − gi , Gi be all the same as in
Theorem 3.2, Bi : Hj → Hi be ri -expanding, Ni : Hj × Hi × Hj → Hi be Lipschitz continuous in
the first, second and third arguments with constants μi , νi , ωi respectively, and Ni be λi , ξi -relaxed
cocoercive with respect to Ai in the first argument, be θi -cocoercive with respect to Bi in the second
argument, be τi -relaxed Lipschtz with respect to Ci in the third argument. If there exist constants
ρ1 , ρ2 and k such that 3.3 and 3.4, but
ci ti ϑf2i 2 ζi ϑfi ϕi ϑgi − i ϑg2i ,
χi ρi2 νi2 βi2 − 2ρi θi ri 1,
i ∈ {1, 2},
3.18
then the problem 2.13 possesses a unique solution in H1 × H2 .
Theorem 3.4. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, let ηi , Ai , Bi , Ci , Mi , fi , gi , fi − gi , Gi , Gi fi − gi be all
the same as in Theorem 3.2, Ni : Hj × Hi × Hj → Hi be Lipschitz continuous in the first, second
and third arguments with constants μi , νi , ωi respectively, and Ni be λi , ξi -relaxed cocoercive with
respect to Ai in the first argument, be θi -relaxed Lipschitz with respect to Bi in the second argument,
be τi -relaxed monotone with respect to Ci in the third argument. If there exist constants ρ1 , ρ2 and k
such that 3.3 and 3.4, but
li μi αi ωi γi
2
2λi αi − ξi τi 1,
χi ρi νi2 βi2 − 2θi 1,
i ∈ {1, 2},
3.19
then the problem 2.13 possesses a unique solution in H1 × H2 .
Remark 3.5. In this paper, there are three aspects which are worth of being mentioned as
follows:
1 Theorem 3.2 extends and improves in 4, Theorem 3.1 and in 19, Theorem 4.1;
2 the class of ζ, ϕ, -g-relaxed cocoercive operators includes the class of α, ξrelaxed cocoercive operators in 8 as a special case;
3 the class of s-G, η-maximal monotone operators is a generalization of the classes
of η-subdifferential operators in 3, maximal η-monotone operators in 6, Hmonotone operators in 5 and H, η-monotone operators in 4.
Acknowledgments
This work was supported by the Science Research Foundation of Educational Department
of Liaoning Province 2009A419 and the Korea Research Foundation Grant funded by the
Korean Government KRF-2008-313-C00042.
Fixed Point Theory and Applications
9
References
1 Q. H. Ansari and J.-C. Yao, “A fixed point theorem and its applications to a system of variational
inequalities,” Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp. 433–442, 1999.
2 Y. J. Cho and X. Qin, “Systems of generalized nonlinear variational inequalities and its projection
methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4443–4451, 2008.
3 X. P. Ding and C. L. Luo, “Perturbed proximal point algorithms for general quasi-variational-like
inclusions,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 153–165, 2000.
4 Y.-P. Fang, N.-J. Huang, and H. B. Thompson, “A new system of variational inclusions with H, ηmonotone operators in Hilbert spaces,” Computers & Mathematics with Applications, vol. 49, no. 2-3, pp.
365–374, 2005.
5 Y.-P. Fang and N.-J. Huang, “H-monotone operator and resolvent operator technique for variational
inclusions,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 795–803, 2003.
6 N.-J. Huang and Y.-P. Fang, “A new class of general variational inclusions involving maximal ηmonotone mappings,” Publicationes Mathematicae Debrecen, vol. 62, no. 1-2, pp. 83–98, 2003.
7 N.-J. Huang and Y.-P. Fang, “Fixed point theorems and a new system of multivalued generalized
order complementarity problems,” Positivity, vol. 7, no. 3, pp. 257–265, 2003.
8 H.-Y. Lan, “A, η-accretive mappings and set-valued variational inclusions with relaxed cocoercive
mappings in Banach spaces,” Applied Mathematics Letters, vol. 20, no. 5, pp. 571–577, 2007.
9 H.-Y. Lan, “New proximal algorithms for a class of A, η-accretive variational inclusion problems
with non-accretive set-valued mappings,” Journal of Applied Mathematics & Computing, vol. 25, no. 1-2,
pp. 255–267, 2007.
10 H.-Y. Lan, “Stability of iterative processes with errors for a system of nonlinear A, η-accretive
variational inclusions in Banach spaces,” Computers & Mathematics with Applications, vol. 56, no. 1,
pp. 290–303, 2008.
11 H.-Y. Lan, “Nonlinear parametric multi-valued variational inclusion systems involving A, ηaccretive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no.
5-6, pp. 1757–1767, 2008.
12 H.-Y. Lan, “A stable iteration procedure for relaxed cocoercive variational inclusion systems based
on A, η-monotone operators,” Journal of Computational Analysis and Applications, vol. 9, no. 2, pp.
147–157, 2007.
13 H.-Y. Lan, “Nonlinear random multi-valued variational inclusion systems involving A, η-accretive
mappings in Banach spaces,” Journal of Computational Analysis and Applications, vol. 10, no. 4, pp. 415–
430, 2008.
14 H.-Y. Lan, J. I. Kang, and Y. J. Cho, “Nonlinear A, η-monotone operator inclusion systems involving
non-monotone set-valued mappings,” Taiwanese Journal of Mathematics, vol. 11, no. 3, pp. 683–701,
2007.
15 H.-Y. Lan and R. U. Verma, “Iterative algorithms for nonlinear fuzzy variational inclusion systems
with A, η-accretive mappings in Banach spaces,” Advances in Nonlinear Variational Inequalities, vol.
11, no. 1, pp. 15–30, 2008.
16 Z. Liu, J. S. Ume, and S. M. Kang, “On existence and iterative algorithms of solutions for mixed
nonlinear variational-like inequalities in reflexive Banach spaces,” Dynamics of Continuous, Discrete &
Impulsive Systems. Series B, vol. 14, no. 1, pp. 27–45, 2007.
17 Z. Liu, S. M. Kang, and J. S. Ume, “The solvability of a class of general nonlinear implicit variational
inequalities based on perturbed three-step iterative processes with errors,” Fixed Point Theory and
Applications, vol. 2008, Article ID 634921, 13 pages, 2008.
18 X. Qin, M. Shang, and Y. Su, “A general iterative method for equilibrium problems and fixed point
problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp.
3897–3909, 2008.
19 S. H. Shim, S. M. Kang, N. J. Huang, and Y. J. Cho, “Perturbed iterative algorithms with errors for
completely generalized strongly nonlinear implicit quasivariational inclusions,” Journal of Inequalities
and Applications, vol. 5, no. 4, pp. 381–395, 2000.
20 L.-C. Zeng, Q. H. Ansari, and J.-C. Yao, “General iterative algorithms for solving mixed quasivariational-like inclusions,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2455–2467,
2008.