An Iterative Method with Norm Convergence for a Class of

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 647524, 6 pages http://dx.doi.org/10.1155/2013/647524

Research Article An Iterative Method with Norm Convergence for a Class of Generalized Equilibrium Problems Haixia Zhang1 and Fenghui Wang2 1 2

Department of Mathematics, Henan Normal University, Xinxiang 453007, China Department of Mathematics, Luoyang Normal University, Luoyang 471022, China

Correspondence should be addressed to Fenghui Wang; [email protected] Received 12 January 2013; Accepted 1 July 2013 Academic Editor: Filomena Cianciaruso Copyright © 2013 H. Zhang and F. Wang. 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. Recently, Takahashi and Takahashi proposed an iterative algorithm for solving a problem for finding common solutions of generalized equilibrium problems governed by inverse strongly monotone mappings and of fixed point problems for nonexpansive mappings. In this paper, we provide a result that allows for the removal of one condition ensuring the strong convergence of the algorithm.

1. Introduction Let H be a real Hilbert space and 𝐶 a nonempty closed convex subset. A generalized equilibrium problem is formulated as a problem of finding a point 𝑥∗ ∈ 𝐶 with the property 𝐹 (𝑥∗ , 𝑦) + ⟨𝐴𝑥∗ , 𝑦 − 𝑥∗ ⟩ ≥ 0,

∀𝑦 ∈ 𝐶,

(1)

where 𝐹 : 𝐶 × 𝐶 → R is a bifunction and 𝐴 : 𝐶 → H is a nonlinear mapping. In particular, if 𝐴 is the zero mapping, then problem (1) is reduced to an equilibrium problem; find a point 𝑥∗ ∈ 𝐶 with the property 𝐹 (𝑥∗ , 𝑦) ≥ 0,

∀𝑦 ∈ 𝐶.

(2)

We will denote by EP(𝐹; 𝐴) and EP(𝐹) the solution set of problem (1) and problem (2), respectively. A fixed point problem (FPP) is to find a point 𝑥∗ with the property 𝑥∗ ∈ 𝐶,

𝑆𝑥∗ = 𝑥∗ ,

(3)

where 𝑆 : 𝐶 → 𝐶 is a nonlinear mapping. The set of fixed points of 𝑆 is denoted as Fix(𝑆). The problem under consideration in this paper is to find a common solution of problem (1) and of FPP (3). Namely, we seek a point 𝑥∗ such that 𝑥∗ ∈ Fix (𝑆) ∩ EP (𝐹; 𝐴) .

(4)

We consider problem (4) in the case whenever 𝐴 is a ]inverse strongly monotone mapping and 𝑆 is a nonexpansive mapping. To solve problem (4), Takahashi and Takahashi [1] introduced an algorithm which generates a sequence (𝑥𝑛 ) by the iterative procedure 𝐹 (𝑧𝑛 , 𝑦) + ⟨𝐴𝑥𝑛 , 𝑦 − 𝑧𝑛 ⟩ +

1 ⟨𝑦 − 𝑧𝑛 , 𝑧𝑛 − 𝑥𝑛 ⟩ ≥ 0, 𝜆𝑛 ∀𝑦 ∈ 𝐶,

(5)

𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆 [𝛼𝑛 𝑢 + (1 − 𝛼𝑛 ) 𝑧𝑛 ] , where (𝛼𝑛 ) ⊆ [0, 1], (𝛽𝑛 ) ⊆ [0, 1], and (𝜆 𝑛 ) ⊆ [0, 2]] are chosen so that 0 < 𝑎 ≤ 𝜆 𝑛 ≤ 𝑏 < 2], 0 < 𝑐 ≤ 𝛽𝑛 ≤ 𝑑 < 1, lim 𝛼𝑛 = 0,

𝑛→∞

∞

∑ 𝛼𝑛 = ∞,

(6)

𝑛=0

󵄨 󵄨󵄨 󵄨󵄨𝜆 𝑛 − 𝜆 𝑛+1 󵄨󵄨󵄨 󳨀→ 0. Under these conditions, they proved that the sequence (𝑥𝑛 ) generated by (5) can be strongly convergent to a solution of problem (4). It is the aim of this paper to continue the study of algorithm (5). We will show that problem (4) is in fact

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a special fixed point problem for a nonexpansive mapping (a composition of a nonexpansive mapping and an averaged mapping). Our approach mainly uses the properties of averaged mappings, which is different from the existing methods invented by Takahashi and Takahashi. Moreover, we shall prove that condition |𝜆 𝑛 − 𝜆 𝑛+1 | → 0 sufficient to guarantee the convergence of algorithm (5) is superfluous.

(iii) If 𝑇 : 𝐶 → H is ]-averaged, then for any 𝑧 ∈ Fix(𝑇) and for all 𝑥 ∈ 𝐶, ‖𝑇𝑥 − 𝑧‖2 ≤ ‖𝑥 − 𝑧‖2 −

1−] ‖𝑇𝑥 − 𝑥‖2 . ]

From now on, we assume that 𝐹 : 𝐶 × 𝐶 → R is a bifunction so that

2. Preliminaries and Notations

(A1) 𝐹(𝑥, 𝑥) = 0, for all 𝑥 ∈ 𝐶;

Notation 1. → strong convergence, ⇀ weak convergence and 𝜔𝑤 (𝑥𝑛 ) the set of the weak cluster points of (𝑥𝑛 ). Denote by 𝑃𝐶 the projection from H onto 𝐶; namely, for 𝑥 ∈ H, 𝑃𝐶𝑥 is the unique point in 𝐶 with the property

(A2) 𝐹 is monotone; that is, 𝐹(𝑥, 𝑦) + 𝐹(𝑦, 𝑥) 0, for all 𝑥, 𝑦 ∈ 𝐶;

󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑥 − 𝑃𝐶𝑥󵄩󵄩󵄩 = min 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 . 𝑦∈𝐶

(7)

It is well known that 𝑃𝐶𝑥 is characterized by the inequality 𝑃𝐶𝑥 ∈ 𝐶, ⟨𝑥 − 𝑃𝐶𝑥, 𝑧 − 𝑃𝐶𝑥⟩ ≤ 0,

∀𝑧 ∈ 𝐶.

(8)

We will use the following notions on nonlinear mappings 𝑇 : 𝐶 → H. (i) 𝑇 is nonexpansive if 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,

∀𝑥, 𝑦 ∈ 𝐶.

(9)

(A3) lim𝑡↓0 𝐹(𝑡𝑧 + (1 − 𝑡)𝑥, 𝑦) ≤ 𝐹(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐶; (A4) for each 𝑥 ∈ 𝐶, 𝑦 󳨃→ 𝐹(𝑥, 𝑦) is convex and lower semicontinuous. Under these assumptions, the following results hold (see [6, 7]). Lemma 3. Let 𝐹 : 𝐶×𝐶 → R satisfy (A1)–(A4). Then for any 𝜆 > 0 and 𝑥 ∈ H, there exists 𝑧 ∈ 𝐶 so that 𝐹 (𝑧, 𝑦) +

1 ⟨𝑦 − 𝑧, 𝑧 − 𝑥⟩ ≥ 0, 𝜆

∀𝑦 ∈ 𝐶.

(13)

Moreover if 𝑆𝜆 𝑥 = {𝑧 ∈ 𝐶 : 𝐹(𝑧, 𝑦) + 1/𝜆⟨𝑦 − 𝑧, 𝑧 − 𝑥⟩ ≥ 0, for all 𝑦 ∈ 𝐶}, then (i) 𝑆𝜆 is single valued and Fix(𝑆𝜆 ) = EP(𝐹); (iii) EP(𝐹) is closed and convex.

∀𝑥, 𝑦 ∈ 𝐶.

(10)

(iii) 𝑇 is 𝛼-averaged if there exist a constant 𝛼 ∈ (0, 1) and a nonexpansive mapping 𝑆 such that 𝑇 = (1−𝛼)𝐼+𝛼𝑆, where 𝐼 is the identity mapping on H. (iv) 𝑇 is ]-inverse strongly monotone if there is a constant ] > 0 such that 󵄩2 󵄩 ⟨𝑇𝑥 − 𝑇𝑦, 𝑥 − 𝑦⟩ ≥ ]󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ,

≤

(ii) 𝑆𝜆 is firmly nonexpansive;

(ii) 𝑇 is firmly nonexpansive if 󵄩2 󵄩 ⟨𝑇𝑥 − 𝑇𝑦, 𝑥 − 𝑦⟩ ≥ 󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ,

(12)

∀𝑥, 𝑦 ∈ 𝐶.

(11)

We end this section by a useful lemma (see Xu [8]). Lemma 4. Let (𝑎𝑛 ) be a nonnegative real sequence satisfying 𝑎𝑛+1 ≤ (1 − 𝛼𝑛 ) 𝑎𝑛 + 𝛼𝑛 𝑏𝑛 ,

where (𝛼𝑛 ) ⊂ (0, 1) and (𝑏𝑛 ) are real sequences. Then 𝑎𝑛 → 0 provided that (i) ∑𝑛 𝛼𝑛 = ∞, lim𝑛 𝛼𝑛 = 0;

The next lemma is referred to as the demiclosedness principle for nonexpansive mappings (see [2]).

(ii) lim sup𝑛 𝑏𝑛 ≤ 0 or ∑ 𝛼𝑛 |𝑏𝑛 | < ∞.

Lemma 1. Let 𝐶 be a nonempty closed convex subset of H and 𝑇 : 𝐶 → H a nonexpansive mapping with Fix(𝑇) ≠ 0. If (𝑥𝑛 ) is a sequence in 𝐶 such that 𝑥𝑛 ⇀ 𝑥 and (𝐼 − 𝑇)𝑥𝑛 → 0, then (𝐼 − 𝑇)𝑥 = 0; that is, 𝑥 ∈ Fix(𝑇).

3. Algorithm and Its Convergence

Averaged mappings will play important role in our convergence analysis. We therefore collect some useful properties of averaged mappings (see, e.g., [3–5]). Lemma 2. The following assertions hold. (i) 𝑇 is firmly nonexpansive if and only if 𝑇 is 1/2averaged. (ii) If 𝑇𝑖 is ]𝑖 -averaged, 𝑖 = 1, 2, then 𝑇1 𝑇2 is (]1 +]2 −]1 ]2 )averaged.

(14)

We begin with the following lemma. Lemma 5. Assume that 𝐴 : 𝐶 → H is ]-inverse strongly monotone mapping for some ] > 0. Given a real number 𝜆 such that 0 < 𝜆 < 2], set 𝑇𝜆 = 𝑆𝜆 (𝐼 − 𝜆𝐴) with 𝑆𝜆 defined as in Lemma 3. Then the following assertions hold: (a) 𝑇𝜆 is single valued and Fix(𝑇𝜆 ) = EP(𝐹; 𝐴); (b) 𝑇𝜆 is (2] + 𝜆)/4]-averaged; (c) given 𝑧 ∈ EP(𝐹; 𝐴), it follows that 2] − 𝜆 󵄩󵄩 󵄩󵄩 󵄩2 󵄩2 2 󵄩󵄩𝑇𝜆 𝑥 − 𝑥󵄩󵄩󵄩 ; 󵄩󵄩𝑇𝜆 𝑥 − 𝑧󵄩󵄩󵄩 ≤ ‖𝑥 − 𝑧‖ − 2] + 𝜆

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(d) if 0 < 𝜆 ≤ 𝜆󸀠 < 2], then for all 𝑥 ∈ 𝐶 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑇𝜆 𝑥 − 𝑥󵄩󵄩󵄩 ≤ 2 󵄩󵄩󵄩𝑇𝜆󸀠 𝑥 − 𝑥󵄩󵄩󵄩 .

(16)

Proof. (a) It is readily seen that 𝑇𝜆 is single valued because 𝑆𝜆 is single valued. The equality follows from the definition of 𝑆𝜆 . (b) It follows that 󵄩2 󵄩󵄩 󵄩󵄩(𝐼 − 2]𝐴)𝑥 − (𝐼 − 2]𝐴)𝑦󵄩󵄩󵄩 󵄩2 󵄩 = 󵄩󵄩󵄩(𝑥 − 𝑦) − 2](𝐴𝑥 − 𝐴𝑦)󵄩󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩 = 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 4]2 󵄩󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩

𝑛→∞

(17)

(18)

which implies that 𝐼 − 𝜆𝐴 is 𝜆/2]-averaged. Consequently (b) follows from part (ii) of Lemma 2 and (c) follows from part (iii) of Lemma 2. (d) Let 𝑧1 = 𝑇𝜆 𝑥 and 𝑧2 = 𝑇𝜆󸀠 𝑥. By definition of 𝑆𝜆 , 𝐹 (𝑧1 , 𝑦) + ⟨𝐴𝑥, 𝑦 − 𝑧1 ⟩ +

1 ⟨𝑦 − 𝑧1 , 𝑧1 − 𝑥⟩ ≥ 0, 𝜆

(19)

∀𝑦 ∈ 𝐶.

(26)

𝑛=0

then the sequence (𝑥𝑛 ) generated by (25) converges strongly to 𝑥∗ = 𝑃Ω 𝑢.

Lemma 7. Let the conditions in Theorem 6 be satisfied. If (𝑥𝑛 ) and (𝑦𝑛 ) are the sequences generated by (25), then both (𝑥𝑛 ) and (𝑦𝑛 ) are bounded. Proof. Let 𝑧 ∈ Ω be fixed. We have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑥𝑛+1 − 𝑧󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(1 − 𝛽𝑛 ) (𝑦𝑛 − 𝑧) + 𝛽𝑛 (𝑥𝑛 − 𝑧)󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑧󵄩󵄩󵄩 + 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 ; on the other hand, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑦𝑛 − 𝑧󵄩󵄩󵄩 = 󵄩󵄩󵄩𝛼𝑛 (𝑢 − 𝑧) + (1 − 𝛼𝑛 ) (𝑇𝑛 𝑥𝑛 − 𝑧)󵄩󵄩󵄩 󵄩 󵄩 ≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 + 𝛼𝑛 ‖𝑢 − 𝑧‖ .

(27)

(28)

Altogether

Letting 𝑦 = 𝑧2 in (19) yields 𝐹 (𝑧1 , 𝑧2 ) + ⟨𝐴𝑥, 𝑧2 − 𝑧1 ⟩ +

∞

∑ 𝛼𝑛 = ∞,

Before proving the theorem, we need some lemmas.

Since 𝐴 is ]-inverse strongly monotone, 𝐼 − 2]𝐴 is nonexpansive. Observe that 𝜆 𝜆 )𝐼 + (𝐼 − 2]𝐴) , 2] 2]

0 < 𝑎 ≤ 𝜆 𝑛 ≤ 𝑏 < 2], 0 < 𝑐 ≤ 𝛽𝑛 ≤ 𝑑 < 1, lim 𝛼𝑛 = 0,

− 4] ⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ .

𝐼 − 𝜆𝐴 = (1 −

Theorem 6. Let 𝐹 : 𝐶 × 𝐶 → R be a bifunction satisfying (A1)–(A4), 𝐴 : 𝐶 → H a ]-inverse strongly monotone mapping for some ] > 0, and 𝑆 : 𝐶 → 𝐶 a nonexpansive mapping so that the solution set Ω := Fix(𝑆) ∩ EP(𝐹; 𝐴) is nonempty. If the following conditions hold:

1 ⟨𝑧 − 𝑧1 , 𝑧1 − 𝑥⟩ ≥ 0. (20) 𝜆 2

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑥𝑛+1 − 𝑧󵄩󵄩󵄩 ≤ [1 − 𝛼𝑛 (1 − 𝛽𝑛 )] 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 + 𝛼𝑛 (1 − 𝛽𝑛 ) ‖𝑢 − 𝑧‖ .

(29)

By induction, (𝑥𝑛 ) is bounded and so is (𝑦𝑛 ).

Similarly, 𝐹 (𝑧2 , 𝑧1 ) + ⟨𝐴𝑥, 𝑧1 − 𝑧2 ⟩ +

1 ⟨𝑧 − 𝑧2 , 𝑧2 − 𝑥⟩ ≥ 0. (21) 𝜆󸀠 1

Adding up these inequalities and using the monotonicity of 𝐹, 1 1 ⟨𝑧 − 𝑧1 , 𝑧1 − 𝑥⟩ + 󸀠 ⟨𝑧1 − 𝑧2 , 𝑧2 − 𝑥⟩ ≥ 0, 𝜆 2 𝜆

(22)

(23)

Hence, ‖𝑧2 − 𝑧1 ‖ ≤ ‖𝑧2 − 𝑥‖. By the triangle inequality, 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 (24) 󵄩󵄩𝑧1 − 𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑧1 − 𝑧2 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧2 − 𝑥󵄩󵄩󵄩 ≤ 2 󵄩󵄩󵄩𝑧2 − 𝑥󵄩󵄩󵄩 , which is the result as desired. For every 𝑛 ≥ 0, if we define 𝑇𝑛 = 𝑆𝜆 𝑛 (𝐼 − 𝜆 𝑛 𝐴), where 𝑆𝜆 𝑛 is defined as in Lemma 3, then we can rewrite algorithm (5) as 𝑦𝑛 = 𝛼𝑛 𝑢 + (1 − 𝛼𝑛 ) 𝑇𝑛 𝑥𝑛 , 𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆𝑦𝑛 .

Proof. Let 𝑇𝑎 = 𝑆𝑎 (𝐼 − 𝑎𝐴). By part (d) of Lemma 5, 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑥𝑛 − 𝑇𝑎 𝑥𝑛 󵄩󵄩󵄩 ≤ 2 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑛 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.

(30)

Since 𝑇𝑎 is nonexpansive, applying the demiclosedness principle yields

or equivalently, 𝜆 󵄩2 󵄩󵄩 󵄩󵄩𝑧2 − 𝑧1 󵄩󵄩󵄩 ≤ (1 − 󸀠 ) ⟨𝑧2 − 𝑧1 , 𝑧2 − 𝑥⟩ . 𝜆

Lemma 8. Let the conditions in Theorem 6 be satisfied. If ‖𝑥𝑛 − 𝑇𝑛 𝑥𝑛 ‖ → 0 and ‖𝑥𝑛 − 𝑆𝑦𝑛 ‖ → 0, then ‖𝑥𝑛 − 𝑦𝑛 ‖ → 0 and 𝜔𝑤 (𝑥𝑛 ) ⊆ Ω.

(25)

𝜔𝑤 (𝑥𝑛 ) ⊆ Fix (𝑇𝑎 ) = EP (𝐹; 𝐴) . On the other hand, we see that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝛼𝑛 (𝑢 − 𝑥𝑛 ) + (1 − 𝛼𝑛 ) (𝑇𝑛 𝑥𝑛 − 𝑥𝑛 )󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 𝛼𝑛 󵄩󵄩󵄩𝑢 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0, which implies that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑥𝑛 − 𝑆𝑥𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑆𝑦𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑆𝑥𝑛 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑆𝑦𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.

(31)

(32)

(33)

Using again the demiclosedness principle gets the desired result.

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Proof of Theorem 6. Let 𝑥∗ = 𝑃Ω 𝑢. Using Lemma 5(c), we have 2] − 𝜆 𝑛 󵄩󵄩 󵄩2 󵄩 󵄩󵄩 ∗ 󵄩2 ∗ 󵄩2 󵄩𝑇 𝑥 − 𝑥𝑛 󵄩󵄩󵄩 . 󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 − 2] + 𝜆 𝑛 󵄩 𝑛 𝑛

(34)

By the subdifferential inequality,

Case 1. Assume that {𝑠𝑛𝑘 } is finite. Then there exists 𝑁 ∈ N so that 𝑠𝑛 > 𝑠𝑛+1 for all 𝑛 ≥ 𝑁, and therefore {𝑠𝑛 } must be convergent. It follows from (38) that 󵄩2 󵄩 󵄩2 󵄩 𝜀 (󵄩󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 ) ≤ 𝑀𝛼𝑛 + (𝑠𝑛 − 𝑠𝑛+1 ) , (39)

󵄩󵄩 󵄩 ∗ 󵄩2 ∗ ∗ 󵄩2 󵄩󵄩𝑦𝑛 − 𝑥 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝛼𝑛 (𝑢 − 𝑥 ) + (1 − 𝛼𝑛 ) (𝑇𝑛 𝑥𝑛 − 𝑥 )󵄩󵄩󵄩 󵄩2 󵄩 ≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 ⟨𝑢 − 𝑥∗ , 𝑦𝑛 − 𝑥∗ ⟩ 󵄩2 󵄩 ≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩

developed by Maing´e [9], we next consider two possible cases on (𝑠𝑛𝑘 ).

(35)

where 𝑀 > 0 is a sufficiently large real number. Consequently, both ‖𝑇𝑛 𝑥𝑛 − 𝑥𝑛 ‖ and ‖𝑆𝑦𝑛 − 𝑥𝑛 ‖ converge to zero, and by Lemma 8 we conclude that ‖𝑦𝑛 − 𝑥𝑛 ‖ → 0 and 𝜔𝑤 (𝑥𝑛 ) ⊆ Ω. Hence, lim sup ⟨𝑢 − 𝑥∗ , 𝑦𝑛 − 𝑥∗ ⟩ = lim sup ⟨𝑢 − 𝑥∗ , 𝑥𝑛 − 𝑥∗ ⟩ 𝑛→∞

+ 2𝛼𝑛 ⟨𝑢 − 𝑥∗ , 𝑦𝑛 − 𝑥∗ ⟩

𝑛→∞

= max ⟨𝑢 − 𝑥∗ , 𝑤 − 𝑥∗ ⟩ ≤ 0, 𝑤∈𝜔𝑤 (𝑥𝑛 ) (40)

(1 − 𝛼𝑛 ) (2] − 𝜆 𝑛 ) 󵄩󵄩 󵄩2 − 󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 , 2] + 𝜆 𝑛

where the inequality uses (8). It then follows from (38) that

which implies that

𝑠𝑛+1 ≤ (1 − 𝜀𝛼𝑛 ) 𝑠𝑛 + 2𝛼𝑛 (1 − 𝛽𝑛 ) ⟨𝑢 − 𝑥∗ , 𝑦𝑛 − 𝑥∗ ⟩ . (41)

󵄩󵄩 󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 ∗ 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 = 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑥 󵄩󵄩󵄩 󵄩2 󵄩 − 𝛽𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩 ≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 󵄩2 󵄩 − 𝛽𝑛 (1 − 𝛽𝑛 ) 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󵄩 󵄩2 ≤ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛 ) 󵄩2 󵄩 × (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩

We therefore apply Lemma 4 to conclude that 𝑠𝑛 → 0. Case 2. Assume now that {𝑠𝑛𝑘 } is infinite. Let 𝑛 ∈ N be fixed. Then there exists 𝑘 ∈ N so that 𝑛𝑘 ≤ 𝑛 ≤ 𝑛𝑘+1 . By the choice of {𝑠𝑛𝑘 }, we see that 𝑠𝑛𝑘 +1 is the largest one among {𝑠𝑛𝑘 , 𝑠𝑛𝑘 +1 , . . . , 𝑠𝑛𝑘+1 }; in particular (36)

󵄩 󵄩2 󵄩 󵄩2 𝜀 (󵄩󵄩󵄩󵄩𝑇𝑛𝑘 𝑥𝑛𝑘 − 𝑥𝑛𝑘 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆𝑦𝑛𝑘 − 𝑥𝑛𝑘 󵄩󵄩󵄩󵄩 ) ≤ 𝑀𝛼𝑛𝑘 󳨀→ 0.

󵄩2 󵄩 × 󵄩󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + 2𝛼𝑛 (1 − 𝛽𝑛 )

(42)

lim sup⟨𝑢 − 𝑥∗ , 𝑦𝑛𝑘 − 𝑥∗ ⟩ ≤ 0.

󵄩2 󵄩 × 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 .

𝑛→∞

By our assumption, there exists 𝜀 > 0 so that for all 𝑛 ≥ 0, (1 − 𝛼𝑛 ) (1 − 𝛽𝑛 ) (2] − 𝜆 𝑛 ) ≥ 𝜀, 2] + 𝜆 𝑛

(37)

(44)

It follows again from (38) and inequality (42) that 𝑠𝑛𝑘 ≤ 2 (1 − 𝛽𝑛𝑘 ) ⟨𝑢 − 𝑥∗ , 𝑦𝑛𝑘 − 𝑥∗ ⟩ .

(45)

Taking lim sup in (44) yields

and 1 − 𝛽𝑛 ≥ 𝛽𝑛 (1 − 𝛽𝑛 ) ≥ 𝜀. Consequently, 󵄩󵄩 󵄩 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩 ≤ (1 − 𝜀𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 − 𝜀 (󵄩󵄩󵄩𝑇𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 )

lim sup 𝑠𝑛𝑘 ≤ 0 󳨐⇒ 𝑠𝑛𝑘 󳨀→ 0. 𝑘→∞

∗󵄩 󵄩2

∗

(43)

Applying Lemma 8 yields ‖𝑦𝑛𝑘 − 𝑥𝑛𝑘 ‖ → 0 and 𝜔𝑤 (𝑥𝑛𝑘 ) ⊆ Ω. Similarly

× ⟨𝑢 − 𝑥∗ , 𝑦𝑛 − 𝑥∗ ⟩ − 𝛽𝑛 (1 − 𝛽𝑛 )

(46)

Moreover, we deduce from algorithm (25) that (38)

∗

+ 2𝛼𝑛 (1 − 𝛽𝑛 ) ⟨𝑢 − 𝑥 , 𝑦𝑛 − 𝑥 ⟩ . 2

𝑠𝑛 ≤ 𝑠𝑛𝑘 +1 .

Then we deduce from (38) that

(1 − 𝛼𝑛 ) (1 − 𝛽𝑛 ) (2] − 𝜆 𝑛 ) − 2] + 𝜆 𝑛

∗󵄩 󵄩2

𝑠𝑛𝑘 ≤ 𝑠𝑛𝑘 +1 ,

Set 𝑠𝑛 = ‖𝑥𝑛+1 − 𝑥∗ ‖ , and let (𝑠𝑛𝑘 ) be a subsequence so that it includes all elements in {𝑠𝑛 } with the property; each of them is less than or equal to the term after it. Following an idea

󵄩󵄩 󵄩󵄩 ∗ √𝑠𝑛𝑘 +1 = 󵄩󵄩󵄩(𝑥𝑛𝑘 − 𝑥 ) − (𝑥𝑛𝑘 − 𝑥𝑛𝑘 +1 )󵄩󵄩󵄩 󵄩 󵄩 (47) 󵄩 󵄩 ≤ √𝑠𝑛𝑘 + 󵄩󵄩󵄩󵄩𝑥𝑛𝑘 − 𝑥𝑛𝑘 +1 󵄩󵄩󵄩󵄩 ≤ √𝑠𝑛𝑘 + 󵄩󵄩󵄩󵄩𝑥𝑛𝑘 − 𝑆𝑦𝑛𝑘 󵄩󵄩󵄩󵄩 , which together with (43) implies that 𝑠𝑛𝑘 +1 → 0. Consequently 𝑠𝑛 → 0 immediately follows from (42).

Journal of Applied Mathematics

5

4. Applications In this section we present several applications. First we consider a problem for finding a common solution of equilibrium problem (2) and fixed point problem (3); namely, find 𝑥∗ ∈ 𝐶 so that 𝑥∗ ∈ EP (𝐹) ∩ Fix (𝑆) .

(48)

Taking 𝐴 = 0 in Theorem 6 and noting that zero mapping is ]-inverse strongly monotone for any positive number ], one can easily get the following. Corollary 9. Let 𝐹 : 𝐶 × 𝐶 → R be a bifunction satisfying (A1)–(A4) and 𝑆 : 𝐶 → 𝐶 a nonexpansive mapping so that the solution set of problem (48) is nonempty. Given 𝑢 ∈ 𝐶, let (𝑥𝑛 ) generated by the iterative algorithm: 𝐹 (𝑧𝑛 , 𝑦) +

1 ⟨𝑦 − 𝑧𝑛 , 𝑧𝑛 − 𝑥𝑛 ⟩ ≥ 0, 𝜆𝑛

∀𝑦 ∈ 𝐶, (49)

𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆 [(1 − 𝛼𝑛 ) 𝑢 + 𝛼𝑛 𝑧𝑛 ] . 0 < 𝑎 ≤ 𝜆 𝑛 ≤ 𝑏 < ∞, 0 < 𝑐 ≤ 𝛽𝑛 ≤ 𝑑 < 1, lim 𝛼 𝑛→∞ 𝑛

= 0,

∑ 𝛼𝑛 = ∞,

(50)

𝑛=0

A variational inequality problem (VIP) is formulated as a problem of finding a point 𝑥∗ with the property 𝑥 ∈ 𝐶,

∗

∗

⟨𝐴𝑥 , 𝑧 − 𝑥 ⟩ ≥ 0,

∀𝑧 ∈ 𝐶.

(51)

We will denote the solution set of VIP (51) by VI(𝐴; 𝐶). Next we consider a problem for finding a common solution of variational inequality problem (51) and of fixed point problem (3), namely; find 𝑥∗ ∈ 𝐶 so that 𝑥∗ ∈ VI (𝐴; 𝐶) ∩ Fix (𝑆) .

𝑓 (𝑥∗ ) = min𝑓 (𝑥) , 𝑥∈𝐶

(55)

where 𝑓 : H → R is a convex and differentiable function. We say that the differential ∇𝑓 is 1/]-Lipschitz continuous, if 󵄩 1󵄩 󵄩 󵄩󵄩 󵄩󵄩∇𝑓 (𝑥) − ∇𝑓 (𝑦)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , ]

∀𝑥, 𝑦 ∈ H.

(56)

Denote by Argmin(𝐶; 𝑓) the solution set of problem (55). Finally we consider a problem for finding a common solution of optimization problem (55) and of fixed point problem (3), namely; find 𝑥∗ ∈ 𝐶 so that 𝑥∗ ∈ Argmin (𝐶; 𝑓) ∩ Fix (𝑆) .

⟨∇𝑓 (𝑥∗ ) , 𝑥∗ − 𝑧⟩ ≥ 0,

then the sequence (𝑥𝑛 ) converges strongly to a solution of problem (48).

∗

Consider the optimization problem of finding a point 𝑥∗ ∈ 𝐶 with the property

(57)

By [10, Lemma 5.13], problem (55) is equivalent to the variational inequality problem

If the following conditions hold:

∞

then the sequence (𝑥𝑛 ) converges strongly to a solution of problem (52).

(52)

∀𝑧 ∈ 𝐶.

(58)

Taking 𝐴 = ∇𝑓 in Corollary 10, we have the following result. Corollary 11. Let 𝑓 : H → R be a convex and differentiable function so that ∇𝑓 is 1/]-Lipschitz continuous. Let 𝑆 : 𝐶 → 𝐶 be a nonexpansive mapping so that the solution set of problem (57) is nonempty. Given 𝑢 ∈ 𝐶, let (𝑥𝑛 ) generated by 𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 ∇𝑓 (𝑥𝑛 )) , 𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆 [(1 − 𝛼𝑛 ) 𝑢 + 𝛼𝑛 𝑧𝑛 ] .

(59)

If the following conditions hold: 0 < 𝑎 ≤ 𝜆 𝑛 ≤ 𝑏 < 2], 0 < 𝑐 ≤ 𝛽𝑛 ≤ 𝑑 < 1, lim 𝛼𝑛 = 0,

𝑛→∞

∞

∑ 𝛼𝑛 = ∞,

(60)

𝑛=0

Taking 𝐹 = 0 in (1), we note that the generalized equilibrium problem is reduced to the variational problem (51). Thus applying Theorem 6 gets the following.

then the sequence (𝑥𝑛 ) converges strongly to a solution of problem (57).

Corollary 10. Let 𝐴 : 𝐶 → H be ]-inverse strongly monotone mapping and 𝑆 : 𝐶 → 𝐶 a nonexpansive mapping so that the solution set of problem (52) is nonempty. Given 𝑢 ∈ 𝐶, let (𝑥𝑛 ) generated by the iterative algorithm:

Proof. It suffices to note that if ∇𝑓 is 1/]-Lipschitz continuous, then it is ]-inverse strongly monotone mapping (see [11, Corollary 10]). Consequently Corollary 10 applies and the result immediately follows.

𝑧𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝐴𝑥𝑛 ) , 𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑆 [(1 − 𝛼𝑛 ) 𝑢 + 𝛼𝑛 𝑧𝑛 ] .

(53)

If the following conditions hold: 0 < 𝑎 ≤ 𝜆 𝑛 ≤ 𝑏 < 2], 0 < 𝑐 ≤ 𝛽𝑛 ≤ 𝑑 < 1, lim 𝛼 𝑛→∞ 𝑛

= 0,

∞

∑ 𝛼𝑛 = ∞,

𝑛=0

Remark 12. We can further apply the previous method to find a common solution for fixed point and split feasibility problems, as well as for fixed point and convex constrained linear inverse problems (see [12]).

Acknowledgment (54)

This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227).

6

References [1] S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008. [2] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990. [3] C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. [4] P. L. Combettes, “Solving monotone inclusions via compositions of nonexpansive averaged operators,” Optimization, vol. 53, no. 5-6, pp. 475–504, 2004. [5] H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360–378, 2011. [6] E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. [7] S. D. Fl˚am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997. [8] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. [9] P.-E. Maing´e, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008. [10] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Springer, Dordrecht, The Netherlands, 1996. [11] J.-B. Baillon and G. Haddad, “Quelques propri´et´es des op´erateurs angle-born´es et n-cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977. [12] F. Wang and H.-K. Xu, “Strongly convergent iterative algorithms for solving a class of variational inequalities,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 407–421, 2010.

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