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; wfenghui@163.com
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
2
Journal of Applied Mathematics
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
(15)
Journal of Applied Mathematics
3
(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.
4
Journal of Applied Mathematics
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 Maingé [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
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[9] P.-E. Maingé, “Strong convergence of projected subgradient
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