Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 147915, 16 pages
doi:10.1155/2012/147915
Research Article
Strong Convergence Theorems for a Generalized
Mixed Equilibrium Problem and a Family of
Total Quasi-φ-Asymptotically Nonexpansive
Multivalued Mappings in Banach Spaces
J. F. Tan1 and S. S. Chang2
1
2
Department of Mathematics, Yibin University, Yibin 644007, China
College of Statistics and Mathematics, Yunnan University of Finance and Economics,
Kunming 650221, China
Correspondence should be addressed to S. S. Chang, changss@yahoo.cn
Received 30 December 2011; Accepted 2 February 2012
Academic Editor: Khalida Inayat Noor
Copyright q 2012 J. F. Tan and S. S. Chang. 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.
The main purpose of this paper is by using a hybrid algorithm to find a common element of the
set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational
inequality problems, and the set of common fixed points for a infinite family of total quasi-φasymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex
Banach space with Kadec-Klee property. The results presented in this paper improve and extend
some recent results announced by some authors.
1. Introduction
Throughout this paper, we always assume that X is a real Banach space with the dual X ∗ , C
is a nonempty closed convex subset of X, and J : X → 2X is the normalized duality mapping
defined by
2 Jx f ∗ ∈ X ∗ : x, f ∗ x2 f ∗ ,
∀x ∈ E.
1.1
In the sequel, we use FT to denote the set of fixed points of a mapping T and use R and R
to denote the set of all real numbers and the set of all nonnegative real numbers, respectively.
We denote by xn → x and xn x the strong convergence and weak convergence of a
sequence {xn }, respectively.
2
Abstract and Applied Analysis
Let Θ : C × C → R be a bifunction, ψ : C → R a real valued function, and A : C → X ∗
a nonlinear mapping. The so-called generalized mixed equilibrium problem is to find u ∈ C such
that
Θ u, y Au, y − u ψ y − ψu ≥ 0,
∀y ∈ C.
1.2
The set of solutions to 1.2 is denoted by Ω, that is,
Ω u ∈ C : Θ u, y Au, y − u ψ y − ψu ≥ 0, ∀y ∈ C .
1.3
Special examples are follows.
i If A ≡ 0, the problem 1.2 is equivalent to finding u ∈ C such that
Θ u, y ψ y − ψu ≥ 0,
∀y ∈ C,
1.4
which is called the mixed equilibrium problem MEP 1.
ii If Θ ≡ 0, the problem 1.2 is equivalent to finding u ∈ C such that
Au, y − u ψ y − ψu ≥ 0,
∀y ∈ C,
1.5
which is called the mixed variational inequality of Browder type VI 2.
A Banach space X is said to be strictly convex if x y/2 < 1 for all x, y ∈ U {z ∈
y. X is said to be uniformly convex if, for each ∈ 0, 2, there exists δ > 0
X : z 1} with x /
such that x y/2 < 1 − δ for all x, y ∈ U with x − y ≥ . X is said to be smooth if the limit
lim
t→0
x ty − x
t
1.6
exists for all x, y ∈ U. X is said to be uniformly smooth if the above limit is attained uniformly
in x, y ∈ U.
Remark 1.1. The following basic properties of a Banach space X can be found in Cioranescu
3.
i If X is uniformly smooth, then J is uniformly continuous on each bounded subset
of X.
ii If X is a reflexive and strictly convex Banach space, then J −1 is norm-weakcontinuous.
iii If X is a smooth, strictly convex, and reflexive Banach space, then J is single-valued,
one-to-one and onto.
iv A Banach space X is uniformly smooth if and only if X ∗ is uniformly convex.
v Each uniformly convex Banach space X has the Kadec-Klee property, that is, for any
sequence {xn } ⊂ X, if xn x ∈ X and xn → x, then xn → x.
Abstract and Applied Analysis
3
Let X be a smooth Banach space. We always use φ : X × X → R to denote the
Lyapunov functional defined by
2
φ x, y x2 − 2 x, Jy y ,
∀x, y ∈ X.
1.7
It is obvious from the definition of the function φ that
2
2
x − y ≤ φ x, y ≤ x y ,
∀x, y ∈ X.
1.8
Following Alber 4, the generalized projection ΠC : X → C is defined by
ΠC x arg inf φ y, x ,
y∈C
∀x ∈ X.
1.9
Lemma 1.2 see 4. Let X be a smooth, strictly convex, and reflexive Banach space and C a
nonempty closed convex subset of X. Then, the following conclusions hold:
a φx, ΠC y φΠC y, y ≤ φx, y for all x ∈ C and y ∈ X,
b if x ∈ X and z ∈ C, then
z ΠC x
iff z − y, Jx − Jz ≥ 0,
∀y ∈ C,
1.10
c for x, y ∈ X, φx, y 0 if and only if x y.
Let X be a smooth, strictly convex, and reflexive Banach space, C a nonempty closed convex
subset of X, and T : C → C a mapping. A point p ∈ C is said to be an asymptotic fixed point of T if
there exists a sequence {xn } ⊂ C such that xn p and xn − T xn → 0. We denoted the set of all
.
asymptotic fixed points of T by FT
Definition 1.3. 1 A mapping T : C → C is said to be relatively nonexpansive 5 if
FT /
∅, FT FT and
φ p, T x ≤ φ p, x ,
∀x ∈ C, p ∈ FT .
1.11
2 A mapping T : C → C is said to be closed if, for any sequence {xn } ⊂ C with
xn → x and T xn → y, T x y.
Let C be a nonempty closed convex subset of a Banach space X. Let NC be the family
of nonempty subsets of C.
4
Abstract and Applied Analysis
Definition 1.4. 1 Let T : C → NC be a multivalued mapping and q a point in C. The
definitions of T q, T 2 q, T 3 q, . . . , T n q, n ≥ 1 are as follows:
T q : q1 : q1 ∈ T q ,
T q1 ,
T 2 q T T q :
q1 ∈T q
T 3 q T T 2 q :
T q2 ,
1.12
q2 ∈T 2 q
..
.
T n q T T n−1 q :
T qn−1 ,
n ≥ 1.
qn−1 ∈T n−1 q
2 Let T : C → NC be a multivalued mapping. A point p ∈ C is said to be
an asymptotic fixed point of T if there exists a sequence {xn } ⊂ C such that xn p and
.
limn → ∞ dxn , T xn 0. We denoted the set of all asymptotic fixed points of T by FT
3 A multivalued mapping T : C → NC is said to be relatively nonexpansive 5 if
and
FT /
∅, FT FT
φ p, w ≤ φ p, x ,
∀x ∈ C, w ∈ T x, p ∈ FT .
1.13
4 A multivalued mapping T : C → NC is said to be closed if, for any sequence
{xn } ⊂ C with xn → x and wn ∈ T xn with wn → y, then y ∈ T x.
Definition 1.5. 1 A multivalued mapping T : C → NC is said to be quasi-φ-nonexpansive if
FT /
∅ and
φ p, w ≤ φ p, x ,
∀x ∈ C, w ∈ T x, p ∈ FT .
1.14
2 A multivalued mapping T : C → NC is said to be quasi-φ-asymptotically
nonexpansive if FT /
∅ and there exists a real sequence {kn } ⊂ 1, ∞ with kn → 1 such
that
φ p, wn ≤ kn φ p, x ,
∀n ≥ 1, x ∈ C, wn ∈ T n x, p ∈ FT .
1.15
3 A multivalued mapping T : C → NC is said to be total quasi-φ-asymptotically
nonexpansive if FT /
∅ and there exist nonnegative real sequences {νn }, {μn } with νn →
0, μn → 0 as n → ∞ and a strictly increasing continuous function ζ : R → R with
ζ0 0 such that for all x ∈ C, p ∈ FT φ p, wn ≤ φ p, x νn ζ φ p, x μn ,
∀n ≥ 1, wn ∈ T n x.
1.16
Abstract and Applied Analysis
5
∞
Definition 1.6. 1 Let {Ti }∞
i1 : C → NC be a sequence of mappings. {Ti }i1 is said to be a
∅
family of uniformly total quasi-φ-asymptotically nonexpansive multivalued mappings if ∩∞
i1 FTi /
and there exist nonnegative real sequences {νn }, {μn } with νn → 0, μn → 0 as n → ∞
and a strictly increasing continuous function ζ : R → R with ζ0 0 such that for all
i ≥ 1, x ∈ C, p ∈ ∩∞
i1 FTi φ p, wn,i ≤ φ p, x νn ζ φ p, x μn ,
∀wn,i ∈ Tin x, ∀n ≥ 1.
1.17
2 A total quasi-φ-asymptotically nonexpansive multivalued mapping T : C → NC
is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that
wn − sn ≤ Lx − y,
∀x, y ∈ C, wn ∈ T n x, sn ∈ T n y, n ≥ 1.
1.18
In 2005, Matsushita and Takahashi 5 proved weak and strong convergence theorems
to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly
convex and uniformly smooth Banach space X. In 2008, Plubtieng and Ungchittrakool 6
proved the strong convergence theorems to approximate a fixed point of two relatively
nonexpansive mappings in a uniformly convex and uniformly smooth Banach space X.
In 2010, Chang et al. 7 obtained the strong convergence theorem for an infinite family
of quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly
convex Banach space X with Kadec-Klee property. In 2011, Chang et al. 8 proved some
approximation theorems of common fixed points for countable families of total quasi-φasymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach
space X with Kadec-Klee property. In 2011, Homaeipour and Razani 9 proved weak and
strong convergence theorems for a single relatively nonexpansive multivalued mapping
in a uniformly convex and uniformly smooth Banach space X. On the other hand, In
2009, Zhang 10 proved the strong convergence theorem for finding a common element
of the set of solutions of a generalized mixed equilibrium problem, the set of solutions
for variational inequality problems, and the set of fixed points of a finite family of quasiφ-asymptotically nonexpansive mappings in a uniformly smooth and uniformly convex
Banach space. Recently, Tang 11, Cho et al. 12–21, and Noor et al. 22–26 extended the
finite family of quasi-φ-asymptotically nonexpansive mappings to infinite family of quasi-φasymptotically nonexpansive mappings.
Motivated and inspired by the researches going on in this direction, the purpose
of this paper is by using the hybrid iterative algorithm to find a common element of
the set of solutions of a generalized mixed equilibrium problem, the set of solutions for
variational inequality problems, and the set of fixed points of a infinite family of total quasiφ-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly
convex Banach space with Kadec-Klee property. In order to get the strong convergence
theorems, the hybrid algorithms are presented and used to approximate the fixed point. The
results presented in the paper improve and extend some recent results announced by some
authors.
6
Abstract and Applied Analysis
2. Preliminaries
Lemma 2.1 see 8. Let X be a real uniformly smooth and strictly convex Banach space with KadecKlee property and C a nonempty closed convex set of X. Let {xn } and {yn } be two sequences in C such
that xn → p and φxn , yn → 0, where φ is the function defined by 1.7, and then yn → p.
Lemma 2.2. Let X and C be as in Lemma 2.1. Let T : C → NC be a closed and total quasi-φasymptotically nonexpansive multivalued mapping with nonnegative real sequences {νn }, {μn } and a
strictly increasing continuous function ζ : R → R such that νn → 0, μn → 0 (as n → ∞), and
ζ0 0. If μ1 0, then the fixed point set FT is a closed and convex subset of C.
Proof. Letting {xn } be a sequence in FT with xn → p as n → ∞, we prove that p ∈ FT .
In fact, by the assumption that T is a total quasi-φ-asymptotically nonexpansive multivalued
mapping and μ1 0, we have
φxn , u ≤ φ xn , p ν1 ζ φ xn , p ,
∀u ∈ T p.
2.1
Furthermore, we have
φ p, u lim φxn , u
n→∞
≤ lim φ xn , p ν1 ζ φ xn , p
0,
n→∞
∀u ∈ T p.
2.2
By Lemma 1.2c, p u. Hence, p ∈ T p. This implies that p ∈ FT , that is, FT is closed.
Next, we prove that FT is convex. For any x, y ∈ FT , t ∈ 0, 1, putting q tx 1 − ty, we prove that q ∈ FT . Indeed, let {un } be a sequence generated by
u1 ∈ T q,
u2 ∈ T u1 ⊂ T 2 q,
u3 ∈ T u2 ⊂ T 3 q,
..
.
2.3
un ∈ T un−1 ⊂ T n q,
..
.
In view of the definition of φx, y, for all un ∈ T un−1 ⊂ T n q, we have
2
φ q, un q − 2 q, Jun un 2
2
q − 2tx, Jun − 21 − t y, Jun un 2
2
2
q tφx, un 1 − tφ y, un − tx2 − 1 − ty
2.4
Abstract and Applied Analysis
7
since
tφx, un 1 − tφ y, un
≤ t φ x, q νn ζ φ x, q μn 1 − t φ y, q νn ζ φ y, q μn
2
t x2 − 2 x, Jq q νn ζ φ x, q μn
2
2
1 − t y − 2 y, Jq q νn ζ φ y, q μn
2.5
2 2
tx2 1 − ty − q tνn ζ φ x, q 1 − tνn ζ φ y, q μn .
Substituting 2.5 into 2.4 and simplifying it, we have
φ q, un ≤ tνn ζ φ x, q 1 − tνn ζ φ y, q μn −→ 0 n −→ ∞.
2.6
By Lemma 2.1, we have un → q as n → ∞. This implies that un1 → q as n → ∞. Since
T is closed, we have q ∈ T q, that is, q ∈ FT .
This completes the proof of Lemma 2.2.
Lemma 2.3 see 7. Let X be a uniformly convex Banach space, r > 0, a positive number, and Br 0
∞
a closed ball of X. Then, for any given sequence {xn }∞
n1 ⊂ Br 0 and for any given sequence {λn }n1 of
∞
positive numbers with Σn1 λn 1, there exists a continuous, strictly increasing, and convex function
g : 0, 2r → 0, ∞ with g0 0 such that for any positive integers i, j with i < j,
2
∞
∞
λn xn 2 − λi λj g xi − xj .
λn xn ≤
n1
n1
2.7
For solving the generalized mixed equilibrium problem, let us assume that the function ψ :
C → R is convex and lower semicontinuous, the nonlinear mapping A : C → X ∗ is continuous and
monotone, and the bifunction Θ : C × C → R satisfies the following conditions:
A1 Θx, x 0, for all x ∈ C,
A2 Θ is monotone, that is, Θx, y Θy, x ≤ 0, for all x, y ∈ C,
A3 lim supt↓0 Θx tz − x, y ≤ Θx, y, for all x, y, z ∈ C,
A4 the function y → Θx, y is convex and lower semicontinuous.
Lemma 2.4. Let X be a smooth, strictly convex and reflexive Banach space and C a nonempty closed
convex subset of X. Let Θ : C × C → R be a bifunction satisfying conditions (A1 )–(A4 ). Let r > 0
and x ∈ X. Then, the following hold.
i [27] There exists z ∈ C such that
1
Θ z, y y − z, Jz − Jx ≥ 0,
r
∀y ∈ C.
2.8
8
Abstract and Applied Analysis
ii [28] Define a mapping Tr : X → C by
Tr x 1
z ∈ C : Θ z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C ,
r
x ∈ X.
2.9
Then, the following conclusions hold:
a Tr is single-valued,
b Tr is a firmly nonexpansive-type mapping, that is, for all z, y ∈ X,
Tr z − Tr y , JTr z − JTr y ≤ Tr z − Tr y , Jz − Jy ,
2.10
c FTr EP Θ FT
r ,
d EP Θ is closed and convex,
e φq, Tr x φTr x, x ≤ φq, x, for all q ∈ FTr .
Lemma 2.5 see 10. Let X be a smooth, strictly convex, and reflexive Banach space and C a
nonempty closed convex subset of X. Let A : C → X ∗ be a continuous and monotone mapping,
ψ : C → R a lower semicontinuous and convex function, and Θ : C × C → R a bifunction satisfying
conditions (A1 )–(A4 ). Let r > 0 be any given number and x ∈ X any given point. Then, the following
hold.
i There exists u ∈ C such that for all y ∈ C
1
Θ u, y Au, y − u ψ y − ψu y − u, Ju − Jx ≥ 0.
r
2.11
ii If one defines a mapping Kr : C → C by
Kr x u ∈ C : Θ u, y Au, y − u ψ y − ψu
1
y − u, Ju − Jx ≥ 0, ∀y ∈ C ,
r
2.12
x ∈ C,
then, the mapping Kr has the following properties:
a Kr is single-valued,
b Kr is a firmly nonexpansive-type mapping, that is, for all z, y ∈ X
Kr z − Kr y , JKr z − JKr y ≤ Kr z − Kr y , Jz − Jy ,
c FKr Ω FK
r ,
d Ω is a closed convex set of C,
e φp, Kr z φKr z, z ≤ φp, z, for all p ∈ FKr , z ∈ X.
2.13
Abstract and Applied Analysis
9
Remark 2.6. It follows from Lemma 2.4 that the mapping Kr : C → C defined by 2.12 is a
relatively nonexpansive mapping. Thus, it is quasi-φ-nonexpansive.
3. Main Results
In this section, we will use the hybrid iterative algorithm to find a common element of
the set of solutions of a generalized mixed equilibrium problem, the set of solutions for
variational inequality problems, and the set of fixed points of a infinite family of total quasiφ-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly
convex Banach space with Kadec-Klee property.
Theorem 3.1. Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee
property and C a nonempty closed and convex subset of X. Let Θ : C × C → R be a bifunction
satisfying conditions (A1 )–(A4 ), A : C → X ∗ a continuous and monotone mapping, and ψ : C → R
a lower semicontinuous and convex function. Let {Ti }∞
i1 : C → NC be an infinite family of closed
and uniformly total quasi-φ-asymptotically nonexpansive multivalued mappings with nonnegative
real sequences {νn }, {μn } and a strictly increasing continuous function ζ : R → R such that
μ1 0, νn → 0, μn → 0 (as n → ∞) and ζ0 0 and for each i ≥ 1, Ti is uniformly Li -Lipschitz
continuous. Let x0 ∈ C, C0 C, and let {xn } be a sequence generated by
xn1 x0 ,
Cn1 ν ∈ Cn : φν, un ≤ φν, xn ξn ,
∀n ≥ 0,
Cn1
yn J −1 αn Jxn 1 − αn Jzn ,
∞
−1
βn,0 Jxn βn,i Jwn,i ,
zn J
3.1
i1
un ∈ C such that, ∀y ∈ C,
1
Θ un , y Aun , y − un ψ y − ψun y − un , Jun − Jyn ≥ 0,
rn
where wn,i ∈ Tin xn , for all n ≥ 1, i ≥ 1, ξn νn supp∈F ζφp, xn μn , ΠCn1 is the generalized
projection of X onto Cn1 , and {αn } and {βn,0 , βn,i } are sequences in 0, 1 satisfying the following
conditions:
a for each n ≥ 0, βn,0 Σ∞
i1 βn,i 1,
b lim infn → ∞ βn,0 βni > 0 for any i ≥ 1,
c 0 ≤ αn ≤ α < 1 for some α ∈ 0, 1.
If G : F ∩ Ω ∩∞
i1 FTi ∩ Ω is a nonempty and bounded subset of C, then the sequence {xn }
converges strongly to ΠG x0 .
Proof. First, we define two functions H : C × C → R and Kr : C → C by
H x, y Θ x, y Ax, y − x ψ y − ψx, ∀x, y ∈ C,
1
Kr x u ∈ C : H u, y y − u, Ju − Jx ≥ 0, ∀y ∈ C , x ∈ C.
r
3.2
10
Abstract and Applied Analysis
By Lemma 2.5, we know that the function H satisfies conditions A1 –A4 and Kr has
properties a–e. Therefore, 3.1 is equivalent to
xn1 x0 ,
Cn1 ν ∈ Cn : φν, un ≤ φν, xn ξn ,
∀n ≥ 0,
Cn1
yn J −1 αn Jxn 1 − αn Jzn ,
∞
−1
βn,0 Jxn βn,i Jwn,i ,
zn J
3.3
i1
un ∈ C such that, ∀y ∈ C,
1
H un , y y − un , Jun − Jyn ≥ 0.
rn
Now we divide the proof of Theorem 3.1 into six steps.
i F and Cn are closed and convex for each n ≥ 0.
In fact, it follows from Lemma 2.2 that FTi , i ≥ 1, is a closed and convex subset of C.
Therefore, F is a closed and convex subset C.
Again by the assumption, C0 C is closed and convex. Suppose that Cn is closed and
convex for some n ≥ 1. Since the condition φν, yn ≤ φν, xn ξn is equivalent to
2
2 ν, Jxn − Jyn ≤ xn 2 − yn ξn ,
n 1, 2, . . . ,
3.4
the set
2
Cn1 ν ∈ Cn : 2 ν, Jxn − Jyn ≤ xn 2 − yn ξn
3.5
is closed and convex. Therefore, Cn is closed and convex for each n ≥ 0.
ii {xn } is bounded and {φxn , x0 } is a convergent sequence.
Indeed, it follows from 3.1 and Lemma 1.2a that for all n ≥ 0, u ∈ FT φxn , x0 φ
x0 , x0
≤ φu, x0 − φ u,
Cn
x0
≤ φu, x0 .
3.6
Cn
This implies that {φxn , x0 } is bounded. By virtue of 1.3, we know that {xn } is bounded.
In view of the structure of {Cn }, we have Cn1 ⊂ Cn , xn ΠCn x0 and xn1 ΠCn1 x0 .
This implies that xn1 ∈ Cn and
φxn , x0 ≤ φxn1 , x0 ,
Therefore, {φxn , x0 } is a convergent sequence.
iii G : F ∩ Ω ⊂ Cn for all n ≥ 0.
∀n ≥ 0.
3.7
Abstract and Applied Analysis
11
Indeed, it is obvious that G ⊂ C0 C. Suppose that G ⊂ Cn for some n ∈ N. Since
un Krn yn , by Lemma 2.5 and Remark 2.6, Krn is quasi-φ-nonexpansive. Hence, for any given
u ∈ G ⊂ Cn and n ≥ 1 we have
φu, un φ u, Krn yn ≤ φ u, yn
φ u, J −1 αn Jxn 1 − αn Jzn u2 − 2u, αn Jxn 1 − αn Jzn αn Jxn 1 − αn Jzn 2
2
3.8
2
≤ u − 2αn u, Jxn − 21 − αn u, Jzn αn xn 1 − αn zn 2
αn φu, xn 1 − αn φu, zn .
Furthermore, it follows from Lemma 2.3 that for any u ∈ G ⊂ Cn , wn,i ∈ Tin xn , and i ≥ 1 we
have
φu, zn φ u, J
−1
βn,0 Jxn ∞
βn,i Jwn,i
i1
2
u − 2 u, βn,0 Jxn ∞
βn,i Jwn,i
i1
2
∞
βn,0 Jxn βn,i Jwn,i i1
∞
≤ u2 − 2βn,0 u, Jxn − 2 βn,i u, Jwn,i βn,0 xn 2
i1
∞
βn,i wn,i 2 − βn,0 βn,l gJxn − Jwn,l i1
βn,0 φu, xn ∞
3.9
βn,i φu, wn,i − βn,0 βn,l gJxn − Jwn,l i1
≤ βn,0 φu, xn ∞
βn,i φu, xn νn ζ φu, xn μn
i1
− βn,0 βn,l gJxn − Jwn,l ≤ φu, xn νn supζ φ p, xn μn − βn,0 βn,l gJxn − Jwn,l p∈F
φu, xn ξn − βn,0 βn,l gJxn − Jwn,l .
Substituting 3.9 into 3.8 and simplifying it, we have for all u ∈ G
φu, un ≤ φ u, yn
≤ φu, xn 1 − αn ξn − 1 − αn βn,0 βn,l gJxn − Jwn,l 12
Abstract and Applied Analysis
≤ φu, xn ξn − 1 − αn βn,0 βn,l gJxn − Jwn,l ≤ φu, xn ξn ,
3.10
that is, u ∈ Cn1 and so G ⊂ Cn1 for all n ≥ 0.
By the way, in view of the assumption on {νn }, {μn } we have
ξn νn sup ζ φ p, xn μn −→ 0
n −→ ∞.
3.11
p∈F
iv {xn } converges strongly to some point p∗ ∈ C.
In fact, since {xn } is bounded and X is reflexive, there exists a subsequence {xni } ⊂
{xn } such that xni p∗ some point in C. Since Cn is closed and convex and Cn1 ⊂ Cn , this
implies that Cn is weakly closed and p∗ ∈ Cn for each n ≥ 0. In view of xni ΠCni x0 , we have
φxni , x0 ≤ φ p∗ , x0 ,
∀ni ≥ 0.
3.12
Since the norm · is weakly lower semicontinuous, we have
lim inf φxni , x0 lim inf xni 2 − 2xni , Jx0 x0 2
ni → ∞
ni → ∞
2
≥ p∗ − 2 p∗ , Jx0 x0 2 φ p∗ , x0 ,
3.13
and so
φ p∗ , x0 ≤ lim inf φxni , x0 ≤ lim sup φxni , x0 ≤ φ p∗ , x0 .
ni → ∞
ni → ∞
3.14
This implies that limni → ∞ φxni , x0 φp∗ , x0 , and so xni → p∗ . Since xni p∗ , by virtue
of Kadec-Klee property of X, we obtain that
lim xni p∗ .
3.15
ni → ∞
Since {φxn , x0 } is convergent, this together with limni → ∞ φxni , x0 φp∗ , x0 , shows that
limn → ∞ φxn , x0 φp∗ , x0 . If there exists some sequence {xnj } ⊂ {xn } such that xnj → q,
then from Lemma 1.2a we have that
φ p∗ , q lim φ xni , xnj
ni ,nj → ∞
⎛
≤
⎛
lim φ⎝xni ,
ni ,nj → ∞
⎛
lim ⎝φxni , x0 − φ⎝
ni ,nj → ∞
⎞
x0 ⎠
Cnj
⎞⎞
x0 , x0 ⎠⎠
Cnj
Abstract and Applied Analysis
13
lim
ni ,nj → ∞
φxni , x0 − φ xnj , x0
φ p∗ , x0 − φ p∗ , x0 0.
3.16
This implies that p∗ q and
lim xn p∗ .
3.17
n→∞
v Now we prove that p∗ ∈ G F ∩ Ω.
First, we prove that p∗ ∈ F. In fact, since xn1 ∈ Cn1 ⊂ Cn , it follows from 3.1 and
3.17 that
φ xn1 , yn ≤ φxn1 , xn ξn −→ 0 n −→ ∞.
3.18
By the virtue of Lemma 2.1, we have
lim yn p∗ .
3.19
n→∞
From 3.10, for any u ∈ F and wn,i ∈ Tin xn , we have
φ u, yn ≤ φu, xn ξn − 1 − αn βn,0 βn,l gJxn − Jwn,l ,
3.20
that is,
1 − αn βn,0 βn,l gJxn − Jwn,l ≤ φu, xn ξn − φ u, yn −→ 0
n −→ ∞.
3.21
By conditions b and c it is shown that limn → ∞ gJxn − Jwn,l 0. In view of property of
g, we have
Jxn − Jwn,l −→ 0 n −→ ∞.
3.22
Since Jxn → Jp∗ , this implies that Jwn,l → Jp∗ . From Remark 1.1ii it yields
wn,l p∗ n −→ ∞,
∀l ≥ 1.
3.23
Again since
wn,l − p∗ Jwn,l − Jp∗ ≤ Jwn,l − Jp∗ −→ 0 n −→ ∞,
3.24
this together with 3.23 and the Kadec-Klee property of X shows that
lim wn,l p∗ ,
n→∞
∀l ≥ 1.
3.25
14
Abstract and Applied Analysis
Let {sn,l } be a sequence generated by
s2,l ∈ Tl w1,l ⊂ Tl2 x1 ,
s3,l ∈ Tl w2,l ⊂ Tl3 x2 ,
..
.
3.26
sn1,l ∈ Tl wn,l ⊂ Tln1 xn ,
..
.
By the assumption that each Ti is uniformly Li -Lipschitz continuous, for any wn,l ∈ Tln xn and
sn1,l ∈ Tl wn ⊂ Tln1 xn we have
sn1,l − wn,l ≤ sn1,l − wn1,l wn1,l − xn1 xn1 − xn xn − wn,l ≤ Ll 1xn1 − xn wn1,l − xn1 xn − wn,l .
3.27
This together with 3.17 and 3.27 shows that limn → ∞ sn1,l − wn,l 0 and limn → ∞ sn1,l p∗ . In view of the closeness of Tl , it yields that p∗ ∈ T p∗ , that is, p∗ ∈ FTl . By the arbitrariness
of l ≥ 1, we have
p∗ ∈ F ∞
FTi .
3.28
i1
Next, we prove that p∗ ∈ Ω. Since xn1 ΠCn1 x0 ∈ Cn , it follows from 3.1 and 3.17
that
φxn1 , un ≤ φxn1 , xn ξn −→ 0
n −→ ∞.
3.29
Since xn → p∗ , by virtue of Lemma 2.1 we have
lim un p∗ .
3.30
n→∞
This together with 3.19 shows that un − yn → 0 and limn → ∞ Jun − Jyn → 0. By the
assumption that rn ≥ a, for all n ≥ 0, we have
Jun − Jyn 3.31
0.
lim
n→∞
rn
Since Hun , y 1/rn y − un , Jun − Jyn ≥ 0, for all y ∈ C, by condition A1 , we have
1
y − un , Jun − Jyn ≥ −H un , y ≥ H y, un ,
rn
∀y ∈ C.
3.32
By the assumption that y → Hx, y is convex and lower semicontinuous, letting n → ∞ in
3.32, from 3.30 and 3.31, we have Hy, p∗ ≤ 0, for all y ∈ C.
Abstract and Applied Analysis
15
For t ∈ 0, 1 and y ∈ C, letting yt ty 1 − tp∗ , there are yt ∈ C and Hyt , p∗ ≤ 0.
By conditions A1 and A4 , we have
0 H yt , yt ≤ tH yt , y 1 − tH yt , p∗ ≤ tH yt , y .
3.33
Dividing both sides of the above equation by t, we have Hyt , y ≤ 0, for all y ∈ C. Letting
t ↓ 0, from condition A3 , we have Hp∗ , y ≤ 0, for all y ∈ C, that is, p∗ ∈ Ω, and p∗ ∈ G F ∩ Ω.
vi We prove that xn → p∗ ΠG x0 .
Let q ΠG x0 . Since q ∈ G ⊂ Cn and xn ΠCn x0 , we have
φxn , x0 ≤ φ q, x0 ,
∀n ≥ 0.
3.34
This implies that
φ p∗ , x0 lim φxn , x0 ≤ φ q, x0 .
n→∞
3.35
In view of the definition of ΠG x0 , from 3.35 we have p∗ q. Therefore, xn → p∗ ΠG x0 .
This completes the proof of Theorem 3.1.
4. Conclusions
Recently the extended general variational inequalities have been introduced and studied in Noor
24, 25. We would like to point out that the results and the methods presented in this
paper will be used to study this kind of extended general variational inequalities and its
multivalued version.
Acknowledgment
The authors would like to express their thanks to the referees for their helpful suggestions
and comments.
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