Volume 2009, Article ID 719360, 18 pagesdoi:10.1155/2009/719360 Research Article A Hybrid Iterative Scheme for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Prob
Trang 1Volume 2009, Article ID 719360, 18 pages
doi:10.1155/2009/719360
Research Article
A Hybrid Iterative Scheme for Equilibrium
Problems, Variational Inequality Problems, and
Fixed Point Problems in Banach Spaces
Prasit Cholamjiak
School of Science and Technology, Naresuan University at Phayao, Phayao 56000, Thailand
Correspondence should be addressed to Prasit Cholamjiak,prasitch2008@yahoo.com
Received 5 February 2009; Accepted 10 April 2009
Recommended by Simeon Reich
The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space Then we show a strong convergence theorem Using this result, we obtain some applications in a Banach space
Copyrightq 2009 Prasit Cholamjiak 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
Let E be a real Banach space and let E∗be the dual of E Let C be a closed convex subset of E Let A : C → E∗be an operator The classical variational inequality problem for A is to find
x ∈ C such that
The set of solutions of1.1 is denoted by V IA, C Such a problem is connected with the
convex minimization problem, the complementarity, the problem of finding a point x ∈ E
satisfying 0 Ax, and so on First, we recall that
1 an operator A is called monotone if
Trang 22 an operator A is called α-inverse-strongly monotone if there exists a constant α > 0
with
Assume that
C1 A is α-inverse-strongly monotone,
C2 V IA, C / ∅,
C3 Ay ≤ Ay − Au for all y ∈ C and u ∈ V IA, C.
Iiduka and Takahashi1 introduced the following algorithm for finding a solution
of the variational inequality for an operator A that satisfies conditions C1–C3 in a
2-uniformly convex and 2-uniformly smooth Banach space E For an initial point x1 x ∈ C,
define a sequence{x n} by
where J is the duality mapping on E, and ΠC is the generalized projection from E onto
C Assume that λ n ∈ a, b for some a, b with 0 < a < b < c2α/2 where 1/c is the
p-uniformly convexity constant of E They proved that if J is weakly sequentially continuous,
then the sequence {x n } converges weakly to some element z in V IA, C where z
limn → ∞ΠV IA,C x n
The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance,2 4 and the references cited therein
Let f : C × C → R be a bifunction The equilibrium problem for f is to find x ∈ C such
that
f
The set of solutions of1.5 is denoted by EPf.
For solving the equilibrium problem, let us assume that a bifunction f satisfies the
following conditions:
A1 fx, x 0 for all x ∈ C;
A2 f is monotone, that is, fx, y fy, x ≤ 0 for all x, y ∈ C;
A3 for all x, y, z ∈ C,
lim sup
t↓0 f
tz 1 − tx, y≤ fx, y
A4 for all x ∈ C, fx, · is convex and lower semicontinuous.
Trang 3Recently, Takahashi and Zembayashi 5, introduced the following iterative scheme which is called the shrinking projection method:
x0 x ∈ C, C0 C,
y n J−1α n Jx n 1 − α n JTx n ,
u n ∈ C such that fu n , y
r1
n y − u n , Ju n − Jy n ≥ 0, ∀y ∈ C,
C n1z ∈ C n : φz, u n ≤ φz, x n,
x n1 ΠC n1 x0, ∀n ≥ 0,
1.7
where J is the duality mapping on E andΠC is the generalized projection from E onto C They
proved that the sequence {x n } converges strongly to q Π FT∩EPf x0 under appropriate conditions
Very recently, Qin et al.6 extend the iteration process 1.7 from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings:
x0∈ E, chosen arbitrarily,
C1 C, x1 ΠC1x0,
y n J−1
α n Jx n β n JTx n γ n JSx n
,
u n ∈ C such that fu n , y
r1
n y − u n , Ju n − Jy n ≥ 0, ∀y ∈ C,
C n1z ∈ C n : φz, u n ≤ φz, x n,
x n1 ΠC n1 x0.
1.8
Under suitable conditions over {α n }, {β n }, and {γ n }, they obtain that the sequence {x n} generated by1.8 converges strongly to q Π FT∩FS∩EPf x0
The problem of finding a common element of the set of fixed points and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces has been studied by many authors; see5,7 16
Motivated by Iiduka and Takahashi 1, Takahashi and Zembayashi 5, and Qin
et al 6, we introduce a new general process for finding common elements of the set of the equilibrium problem and the set of the variational inequality problem for an inverse-strongly monotone operator and the set of the fixed points for relatively quasi-nonexpansive mappings
Trang 42 Preliminaries
Let E be a real Banach space and let U {x ∈ E : x 1} be the unit sphere of E A Banach space E is said to be strictly convex if for any x, y ∈ U,
x / y implies
x y2 < 1. 2.1
It is also said to be uniformly convex if for each ε ∈ 0, 2, there exists δ > 0 such that for any
x, y ∈ U,
x − y ≥ ε impliesx y
2
It is known that a uniformly convex Banach space is reflexive and strictly convex; and we
define a function δ : 0, 2 → 0, 1 called the modulus of convexity of E as follows:
δε inf
1−
x y2 : x,y ∈ E, x y 1, x − y ≥ ε . 2.3
Then E is uniformly convex if and only if δε > 0 for all ε ∈ 0, 2 Let p be a fixed real number with p ≥ 2 A Banach space E is said to be p-uniformly convex if there exists a constant
c > 0 such that δε ≥ cε p for all ε ∈ 0, 2; see 17–19 for more details A Banach space E is said to be smooth if the limit
lim
t → 0
x ty − x
exists for all x, y ∈ U It is also said to be uniformly smooth if the limit 2.4 is attained
uniformly for x, y ∈ U One should note that no Banach space is p-uniformly convex for
1 < p < 2; see19 It is well known that a Hilbert space is 2-uniformly convex, uniformly
smooth For each p > 1, the generalized duality mapping J p : E → 2E∗
is defined by
J p x x∗∈ E∗:x, x∗ x p , x∗ x p−1 2.5
for all x ∈ E In particular, J J2is called the normalized duality mapping If E is a Hilbert space, then J I, where I is the identity mapping It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E See20,21 for more details
Banach space Then, for all x, y ∈ E, j x ∈ J p x and j y ∈ J p y,
x − y, j x − j y ≥ c p
where J p is the generalized duality mapping of E and 1/c is the p-uniformly convexity constant of E.
Trang 5Let E be a smooth Banach space The function φ : E × E → R is defined by
φ
x, y
for all x, y ∈ E In a Hilbert space H, we have φx, y x − y 2for all x, y ∈ H.
Recall that a mapping T : C → C is called nonexpansive if Tx − Ty ≤ x − y for all
x, y ∈ C and relatively nonexpansive if T satisfies the following conditions:
1 FT / ∅, where FT is the set of fixed points of T;
2 φp, Tx ≤ φp, x for all p ∈ FT and x ∈ C;
3 F T FT, where F T is the set of all asymptotic fixed points of T;
see10,23,24 for more details
T is said to be relatively quasi-nonexpansive if T satisfies the conditions 1 and 2 It
is easy to see that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings9,25,26
We give some examples which are closed relatively quasi-nonexpansive; see6
Example 2.2 Let E be a uniformly smooth and strictly convex Banach space and A ⊂ E × E∗
be a maximal monotone mapping such that its zero set A−10/ ∅ Then, J r J rA−1J is a
closed relatively quasi-nonexpansive mapping from E onto DA and FJ r A−10
Example 2.3 Let ΠC be the generalized projection from a smooth, strictly convex, and
reflexive Banach space E onto a nonempty closed convex subset C of E Then,ΠCis a closed
relatively quasi-nonexpansive mapping with FΠ C C.
space and let {x n }, {y n } be two sequences of E If φx n , y n → 0 and either {x n } or {y n } is bounded,
then x n − y n → 0 as n → ∞.
Let C be a nonempty closed convex subset of E If E is reflexive, strictly convex and smooth, then there exists x0 ∈ C such that φx0, x minφy, x for x ∈ E and y ∈ C The
generalized projectionΠC : E → C defined by Π C x x0 The existence and uniqueness of the operatorΠC follows from the properties of the functional φ and strict monotonicity of the duality mapping J; for instance, see20,27–30 In a Hilbert space, ΠCis coincident with the metric projection
x ∈ E Then x0 ΠC x if and only if x0− y, Jx − Jx0 ≥ 0 for all y ∈ C.
and smooth Banach space E and let x ∈ E Then
φ
y, Π C x
φΠ C x, x ≤ φy, x
, ∀y ∈ C. 2.8
convex subset of E, let T be a closed and relatively quasi-nonexpansive mapping from C into itself Then FT is a closed convex subset of C.
Trang 6Lemma 2.8 Cho et al 31 Let E be a uniformly convex Banach space and let B r 0 be a closed
ball of E Then there exists a continuous strictly increasing convex function g : 0, ∞ → 0, ∞ with g0 0 such that
αx βy γz2≤ α x 2 βy2 γ z 2− αβgx − y, 2.9
for all x, y, z ∈ B r 0, and α, β, γ ∈ 0, 1 with α β γ 1.
Lemma 2.9 Blum and Oettli 7 Let C be a closed convex subset of a smooth, strictly convex, and
reflexive Banach space E, let f be a bifunction from C × C to R satisfying (A1)–(A4), and let r > 0 and x ∈ E Then, there exists z ∈ C such that
f
z, y
Lemma 2.10 Qin et al 6 Let C be a closed convex subset of a uniformly smooth, strictly convex,
and reflexive Banach space E, and let f be a bifunction from C × C to R satisfying (A1)–(A4) For all
r > 0 and x ∈ E, define a mapping T r : E → C as follows:
T r x
z ∈ C : fz, y
1r y − z, Jz − Jx ≥ 0, ∀y ∈ C
Then, the following hold:
1 T r is single-valued;
2 T r is a firmly nonexpansive-type mapping [ 32 ], that is, for all x, y ∈ E,
T r x − T r y, JT r x − JT r y ≤ T r x − T r y, Jx − Jy; 2.12
3 FT r EPf;
4 EPf is closed and convex.
strictly, and reflexive Banach space E, let f be a bifucntion from C × C to R satisfying (A1)–(A4), let
r > 0 Then, for all x ∈ E and q ∈ FT r ,
φ
q, T r x
φT r x, x ≤ φq, x
We make use of the following mapping V studied in Alber28:
V x, x∗ x 2− 2x, x∗ x∗ 2 2.14
for all x ∈ E and x∗∈ E∗, that is, V x, x∗ φx, J−1x∗
Trang 7Lemma 2.12 Alber 28 Let E be a reflexive, strictly convex, smooth Banach space and let V be as
in2.14 Then
V x, x∗ 2J−1x∗ − x, y∗ ≤ Vx, x∗ y∗
2.15
for all x ∈ E and x∗, y∗∈ E∗.
An operator A of C into E∗is said to be hemicontinuous if for all x, y ∈ C, the mapping
F of 0, 1 into E∗defined by Ft Atx 1 − ty is continuous with respect to the weak∗ topology of E∗ We define by N C v the normal cone for C at a point v ∈ C, that is,
N C v x∗∈ E∗:v − y, x∗ ≥ 0, ∀y ∈ C. 2.16
and A a monotone, hemicontinuous operator of C into E∗ Let T e ⊂ E × E∗be an operator defined as follows:
T e v
⎧
⎨
⎩
Av N C v, v ∈ C;
∅, otherwise. 2.17
Then T e is maximal monotone and T−1
e 0 V IA, C.
3 Strong Convergence Theorems
Theorem 3.1 Let E be a 2-uniformly convex, uniformly smooth Banach space, let C be a nonempty
closed convex subset of E Let f be a bifunction from C × C to R satisfying (A1)–(A4), let A be an operator of C into E∗ satisfying (C1)–(C3), and let T, S be two closed relatively quasi-nonexpansive mappings from C into itself such that F : FT ∩ FS ∩ EPf ∩ V IA, C / ∅ For an initial point
x0∈ E with x1 ΠC1x0and C1 C, define a sequence {x n } as follows:
z n ΠC J−1Jx n − λ n Ax n ,
y n J−1
α n Jx n β n JTx n γ n JSz n
,
u n ∈ C such that fu n , y
1
r n y − u n , Ju n − Jy n ≥ 0, ∀y ∈ C,
C n1z ∈ C n : φz, u n ≤ φz, x n,
x n1 ΠC n1 x0, ∀n ≥ 1,
3.1
where J is the duality mapping on E Assume that {α n }, {β n }, and {γ n } are sequences in 0, 1
satisfying the restrictions:
B1 α n β n γ n 1;
B2 lim infn → ∞ α n β n > 0, lim inf n → ∞ α n γ n > 0;
Trang 8B3 {r n } ⊂ s, ∞ for some s > 0;
B4 {λ n } ⊂ a, b for some a, b with 0 < a < b < c2α/2, where 1/c is the 2-uniformly convexity constant of E.
Then, {x n } and {u n } converge strongly to q Π F x0.
Proof We divide the proof into eight steps.
Step 1 Show thatΠF x0andΠC n1 x0are well defined
It is obvious that V IA, C is a closed convex subset of C ByLemma 2.7, we know that
FT ∩ FS is closed and convex FromLemma 2.104, we also have EPf is closed and convex Hence F : FT ∩ FS ∩ EPf ∩ V IA, C is a nonempty, closed, and convex subset
of C; consequently,ΠF x0is well defined
Clearly, C1 C is closed and convex Suppose that C k is closed and convex for k ∈ N For all z ∈ C k , we know φz, y k ≤ φz, x k is equivalent to
So, C k1 is closed and convex By induction, C n is closed and convex for all n≥ 1 This shows thatΠC n1 x0is well-defined
Step 2 Show that F ⊂ C n for all n ∈ N.
Put v n J−1Jx n − λ n Ax n First, we observe that u n T r n y n for all n ≥ 1 and F ⊂ C1
C Suppose F ⊂ C k for k ∈ N Then, for all u ∈ F, we know fromLemma 2.6andLemma 2.12
that
φu, z k φu, Π C v k
≤ φu, v k
φu, J−1Jx k − λ k Ax k
V u, Jx k − λ k Ax k
≤ V u, Jx k − λ k Ax k λ k Ax k − 2J−1Jx k − λ k Ax k − u, λ k Ax k
V u, Jx k − 2λ k v k − u, Ax k
φu, x k − 2λ k x k − u, Ax k 2v k − x k , −λ k Ax k .
3.3
Since u ∈ V IA, C and from C1, we have
−2λ k x k − u, Ax k −2λ k x k − u, Ax k − Au − 2λ k x k − u, Au
Trang 9FromLemma 2.1andC3, we obtain
2vk − x k , −λ k Ax k 2J−1Jx k − λ k Ax k − J−1Jx k , −λ k Ax k
≤ 2J−1Jx k − λ k Ax k − J−1Jx k λ
k Ax k
≤ c42JJ−1Jx k − λ k Ax k − JJ−1Jx k λ
k Ax k
c42 Jx k − λ k Ax k − Jx k λ k Ax k
c42λ2
k Ax k 2
≤ c42λ2
k Ax k − Au 2.
3.5
Replacing3.4 and 3.5 into 3.3, we get
φu, z k ≤ φu, x k 2λ k
2
c2λ k − α
Ax k − Au 2≤ φu, x k . 3.6
By the convexity of · 2, for each u ∈ F ⊂ C k, we obtain
φu, u k φu, T r k y k
≤ φu, y k
φu, J−1
α k Jx k β k JTx k γ k JSz k
u 2− 2α k u, Jx k − 2β k u, JTx k − 2γ k u, JSz k
α k Jx k β k JTx k γ k JSz k2
≤ u 2− 2α k u, Jx k − 2β k u, JTx k − 2γ k u, JSz k
α k Jx k 2 β k JTx k 2 γ k JSz k 2
α k φu, x k β k φu, Tx k γ k φu, Sz k
≤ α k φu, x k β k φu, x k γ k φu, z k
≤ φu, x k .
3.7
This shows that u ∈ C k1 ; consequently, F ⊂ C k1 Hence F ⊂ C n for all n≥ 1
Step 3 Show that lim n → ∞ φx n , x0 exists
From x n ΠC n x0and x n1 ΠC n1 x0 ∈ C n1 ⊂ C n, we have
Trang 10FromLemma 2.6, we have
φx n , x0 φΠ C n x0, x0 ≤ φu, x0 − φu, x n ≤ φu, x0. 3.9
Combining3.8 and 3.9, we obtain that limn → ∞ φx n , x0 exists
Step 4 Show that {x n } is a Cauchy sequence in C.
Since x m ΠC m x0∈ C m ⊂ C n for m > n, byLemma 2.6, we also have
φx m , x n φx m , Π C n x0
≤ φx m , x0 − φΠ C n x0, x0
φx m , x0 − φx n , x0.
3.10
Taking m, n → ∞, we obtain that φx m , x n → 0 FromLemma 2.4, we have x m − x n → 0 Hence{x n } is a Cauchy sequence By the completeness of E and the closedness of C, one can assume that x n → q ∈ C as n → ∞ Further, we obtain
lim
Since x n1 ΠC n1 x0∈ C n1, we have
as n → ∞ ApplyingLemma 2.4to3.11 and 3.12, we get
lim
This implies that u n → q as n → ∞ Since J is uniformly norm-to-norm continuous on bounded subsets of E, we also obtain
lim
Step 5 Show that x n → q ∈ FT ∩ FS.