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Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space
- Jia-wei Chen^{1},
- Yeol Je Cho^{2}Email author and
- Zhongping Wan^{1}
https://doi.org/10.1186/1687-1812-2011-91
© Chen et al; licensee Springer. 2011
Received: 16 July 2011
Accepted: 30 November 2011
Published: 30 November 2011
Abstract
Shrinking projection algorithms for finding a solution of an equilibrium problem with a bifunction defined on the dual space of a Banach space, in this paper, are introduced and studied. Under some suitable assumptions, strong and weak convergence results of the shrinking projection algorithms are established, respectively. Finally, we give an example to illustrate the algorithms proposed in this paper.
2000 Mathematics Subject Classification: 47H09; 65J15; 90C99.
Keywords
- equilibrium problem
- strong and weak convergence
- shrinking projection algorithm
- sunny generalized nonexpansive retraction
- fixed point
1 Introduction
Many problems in structural analysis, optimization, management sciences, economics, variational inequalities and complementary problems coincide to find a solution of the equilibrium problem. Various methods have been proposed to solve some kinds of equilibrium problems in Hilbert and Banach spaces (see [1–8]).
In [9], Takahashi and Zembayashi proved strong and weak convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in Banach spaces. Ibaraki and Takahashi [10] introduced a new resolvent of a maximal monotone operator in Banach spaces and the concept of the generalized nonexpansive mapping in Banach spaces. Honda et al. [11], Kohsaka and Takahashi [12] also studied some properties for the generalized nonexpansive retractions in Banach spaces. Takahashi et al. [13] proved a strong convergence theorem for nonexpansive mapping by hybrid method. In 2009, Ceng et al. [2] proved strong and weak convergence theorems for equilibrium problems and dealt maximal monotone operators by hybrid proximal-point methods. Motivated by Ibaraki and Takahashi [10] and Takahashi et al. [13], Takahashi and Zembayashi [14] considered the following equilibrium problem:
Then they proved a strong convergence theorem for finding a solution of the equilibrium problem (1.1) in Banach spaces. Forward, we denote the set of solutions of the problem (1.1) by EP(f):
Inspired and motivated by Ceng et al. [2], Takahashi and Zembayashi [14], Takahashi and Zembayashi [9], the main aim of this paper is to introduce and investigate a new iterative method for finding a solution of the equilibrium problem (1.1). Under some appropriate assumptions, strong and weak convergence results of the iterative algorithms are established, respectively. Furthermore, we also give an example to illustrate the algorithms proposed in this paper.
2 Preliminaries
Throughout this paper, we denote the sets of nonnegative integers and real numbers by Z_{+} and R, respectively.
J is said to be weakly sequentially continuous if the strong convergence of a sequence {x_{ n }} to x in E implies the weak* convergence of {J(x_{ n })} to J(x) in E*.
Many properties of the normalized duality mapping J can be found in [15–17] and, now, we list the following properties:
(p^{1}) J(x) is nonempty for any x ∈ E;
(p^{2}) J is a monotone and bounded operator in Banach spaces;
(p^{3}) J is a strictly monotone operator in strictly convex Banach spaces;
(p^{4}) J is the identity operator in Hilbert spaces;
(p^{5}) If E is a reflexive, smooth and strictly convex Banach space and J*: E* → 2^{ E } is the normalized duality mapping on E*, then J^{-1} = J*; JJ* = I_{ E }* and J*J = I_{ E }; where I_{ E }* and IE*are the identity mappings on E and E*, respectively.
(p^{7}) If E is smooth, then J is single-valued;
(p^{8}) E is a uniformly convex Banach space if and only if E* is uniformly smooth;
(p^{9}) If E is uniformly convex and uniformly smooth Banach space, then J is uniformly norm-to-norm continuous on bounded subsets of E and J^{-1} = J* is also uniformly norm-to-norm continuous on bounded subsets of E*:
- (1)
If E is a reflexive, strictly convex and smooth Banach space, then, for all x, y ∈ E, ϕ(x; y) = 0 if and only if x = y;
- (2)
If E is a Hilbert space, then ϕ(x, y) = ║x - y║^{2} for all x; y ∈ E;
- (3)
For all x, y ∈ E, (║x║ - ║y║)^{2} ≤ ϕ(x, y) ≤ (║x║ + ║y║)^{2}.
For solving the equilibrium problem (1.1), we assume that f: J(C) × J(C) → R satisfies the following conditions (A 1) - (A 4) [9]:
(A1) f(x*, x*) = 0 for all x* ∈ J(C);
(A2) f is monotone, that is, f(x*; y*) + f(y*, x*) ≤ 0 for all x*, y* ∈ J(C);
(A4) For all x* ∈ J(C), f(x*, ·) is convex and lower semicontinuous.
In the sequel, we recall some concepts and results.
where F(T) denotes the set of fixed points of T, that is, F(T) = {x ∈ C: Tx = x}.
- (1)
a retraction if R^{2} = R;
- (2)
sunny if R(Rx + t(x - Rx)) = Rx for all x ∈ E and t > 0.
Definition 2.3. (see [11]) A nonempty closed subset C of a smooth Banach space E is called a sunny generalized nonexpansive retract of E if there exists a sunny generalized nonexpansive retraction R from E onto C.
Lemma 2.1. (see [19]) Let E be a uniformly convex and smooth Banach space, and let {x_{ n }} and {y_{ n }} be two sequences of E. If ϕ(x_{ n }, y_{ n }) → 0 and either {x_{ n }} or {y_{ n }} is bounded, then x_{n} - y_{ n } → 0.
where B_{ r } = {z ∈ E: ║z║ ≤ r}.
- (1)
T_{ r } is single-valued;
- (2)For all x, y ∈ E,$\u3008{T}_{r}\left(x\right)-{T}_{r}\left(y\right),J\left({T}_{r}\left(x\right)\right)-J\left({T}_{r}\left(y\right)\right)\u3009\le \u3008x-y,J\left({T}_{r}\left(x\right)\right)-J\left({T}_{r}\left(y\right)\right)\u3009;$
- (3)
F(T_{ r }) = EP(f) and J(EP(f)) is closed and convex;
- (4)
ϕ(x, T_{ r }(x)) + ϕ(T_{ r }(x), p) ≤ ϕ(x, p) for all x ∈ E and p ∈ F(T_{ r }).
- (1)
R is sunny generalized nonexpansive;
- (2)
〈x - R x, J(y) - J(Rx)〉 ≤ 0 for all (x, y) ∈ E × C.
Lemma 2.6. (see [20]) Let C be a nonempty closed sunny generalized nonexpansive retract of a smooth and strictly convex Banach space E. Then the sunny generalized nonexpansive retraction from E onto C is uniquely determined.
- (1)
z = Rx if and only if 〈x - z, J(y) ≤ J(z)〉 ≤ 0 for all y ∈ C;
- (2)
ϕ(x, Rx) + ϕ(Rx, z) ≤ ϕ(x, z).
- (1)
C is a sunny generalized nonexpansive retract of E;
- (2)
J(C) is closed and convex.
Remark 2.2. If E is a Hilbert space, then, from Lemmas 2.6 and 2.8, a sunny generalized nonexpansive retraction from E onto C reduces to a metric projection operator P from E onto C.
If ${\sum}_{n=0}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$, then lim_{n →∞}a_{ n } exists.
3 Main results
In this section, we propose iterative algorithms for finding a solution of the equilibrium problem (1.1) and prove the strong and weak convergence for the algorithms in a Banach space under some suitable conditions.
Then the sequence {R_{EP(f)}x_{ n }} converges strongly to a point ω ∈ EP(f), where R_{EP(f)}is the sunny generalized nonexpansive retraction from E onto EP(f).
Proof. For the sake of simplicity, let ${u}_{n}={T}_{{r}_{n}}{x}_{n}$ and y_{ n } = β_{ n }x_{ n } + (1 - β_{ n })u_{ n }. Then x_{n+1}= α_{ n } x_{0} + (1 - α_{ n })y_{ n }. From Lemma 2.4, it follows that EP(f) is a nonempty closed and convex subset of E.
which shows that {z_{ n }} is a Cauchy sequence. Since EP(f) is closed, there exists ω ∈ EP(f) such that z_{ n } → ω. Therefore, the sequence {R_{EP(f)}x_{ n }} converges strongly to the ω ∈ EP(f). This completes the proof. □
If J is weakly sequentially continuous, then the sequence {x_{ n }} converges weakly to a point ω ∈ EP(f), where ω = lim_{n→∞}R_{EP(f)}x_{ n }and R_{EP(f)}is the sunny generalized nonexpansive retraction from E onto EP(f).
we get $f\left({y}_{t}^{*},J\left(y\right)\right)\ge 0$. By (A 3), one has f (p*, J(y)) ≥ 0.Therefore, p* ∈ J(EP(f)).
this together with the strictly monotonicity of J yields that J^{-1}(p*) = ω. Therefore, the sequence {x_{ n }} converges weakly to the point ω ∈ EP(f), where ω = lim_{n→∞}R_{EP(f)}x_{ n }. This completes the proof. □
4 Numerical test
In this section, we give an example of numerical test to illustrate the algorithms given in Theorems 3.1 and 3.2.
First, we verify that f satisfies the conditions (A 1)-(A 4) as follows:
(A1) f(x, x) = - 5x^{2} + x^{2} + 4x^{2} = 0 for all x ∈ [-1000, 1000];
(A2) f(x, y) + f(y, x) = -(x - y)^{2} ≤ 0 for all x, y ∈ [-1000, 1000];
(A4) For all x ∈ [-1000, 1000], F(y) = f(x, y) = -5x^{2} + xy + 4y^{2} is convex and lower semicontinuous.
Therefore, by Theorem 3.1, the sequence {P_{EP(f)}x_{ n }} must converge strongly to a solution of the problem (4.1). In fact, P_{EP(f)}x_{ n }= 0 for all n ∈ Z_{+}. Also, according to Theorem 3.2, the sequence {x_{ n }} converges weakly to a solution of the problem (4.1). For a number ε = 10^{-3}, if we use MATLAB, then we generate a sequence {x_{ n }} as follows:
Selected values of {u_{ n }}
u _{ n } | u _{ n } | u _{ n } | u _{ n } | u _{ n } | u _{ n } |
---|---|---|---|---|---|
0.1818 | 0.0607 | 0.0026 | 0.0012 | 0.0007 | 0.0003 |
0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0000 | 0.0000 |
Selected values of {x_{ n }}
x _{ n } | x _{ n } | x _{ n } | x _{ n } | x _{ n } | x _{ n } |
---|---|---|---|---|---|
0.4247 | 0.0204 | 0.0100 | 0.0059 | 0.0028 | 0.0021 |
0.0016 | 0.0013 | 0.0010 | 0.0008 | 0.0007 | 0.0006 |
0.0005 | 0.0004 | 0.0003 | 0.0003 | 0.0002 | 0.0002 |
0.0002 | 0.0002 | 0.0002 | 0.0001 | 0.0001 | 0.0001 |
0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |
0.0001 | 0.0001 | 0.0001 | 0.0000 | 0.0000 | 0.0000 |
From Table 1, we can see that the sequence {u_{ n }} converges to 0. Moreover, F(T_{ r }) = EP(f) = {0}. Table 2 shows that the iterative sequence {x_{ n }} converges to 0, which is indeed a solution of the problem (4.1). Moreover, ${lim}_{n\to \mathrm{\infty}}{P}_{EP\left(f\right)}{x}_{n}=0$.
Declarations
Acknowledgements
The authors would like to thank three anonymous referees for their invaluable comments and suggestions, which led to an improved presentation of the results. This work was supported by the Natural Science Foundation of China (71171150, 70771080), the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050), the Academic Award for Excellent Ph.D. Candidates Funded by Wuhan University and the Fundamental Research Fund for the Central Universities (201120102020004).
Authors’ Affiliations
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