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Hybrid algorithms of nonexpansive semigroups for mixed equilibrium problems, variational inequalities, and fixed point problems
Journal of Inequalities and Applications volume 2014, Article number: 174 (2014)
Abstract
The purpose of this paper is to introduce two hybrid algorithms for the variational inequalities and mixed equilibrium problems over the common fixed points set of nonexpansive semigroups in Hilbert space. Under suitable conditions some strong convergence theorems for these two hybrid algorithms are proved. The results presented in the paper extend and improve some recent results.
1 Introduction
Throughout this paper, we always assume that H is a real Hilbert space with inner product and norm , C is a nonempty closed convex subset of H and is the metric projection of H onto C. In the sequel, we denote by → and ⇀ the strong convergence and weak convergence, respectively. Let be a real-valued function and be an equilibrium bifunctions, i.e., for each . We consider the mixed equilibrium problem (MEP) which is to find such that
In particular, if , this problem reduces to the equilibrium problem (EP), which is to find such that
Denote the set of solutions of MEP by Ω. The MEP includes fixed point problems, optimization problems, variational inequality problems, Nash EPS and the EP as special cases.
Recall that a mapping is said to be nonexpansive, if , .
Let C be a closed convex subset of a Hilbert space H. A family of mappings is said to be a nonexpansive semigroup, if it satisfies the following conditions:
-
(i)
, and ;
-
(ii)
, , ;
-
(iii)
the mapping is continuous for each .
We denote by the set of common fixed points of , i.e., . It is well known that is closed and convex.
Now let be a nonlinear operator. The variational inequality problem is formulated as finding a point such that
It is well known that the is equivalent to the fixed point equation
where is an arbitrarily fixed constant. So, fixed point methods can be implemented to find a solution of the provided F satisfies some conditions and is chosen appropriately. The fixed point formulation (1.1) involves the projection , which may not be easy to compute, due to the complexity of the convex set C. In order to reduce the complexity probably caused by the projection , Yamada [1] recently introduced a hybrid steepest-descent method for solving the . Assume that F is an η-strongly monotone and κ-Lipschitzian mapping with , on C. An equally important problem is how to find an approximate solution of the if any. A great deal of effort has been done in this problem.
In 2007, Ceng and Yao [2] investigate the problem of finding a common element of the set of solutions of a mixed equilibrium problem (MEP) and the set of common fixed points of finitely many nonexpansive mappings in a real Hilbert space. Very recently, Yang et al. [3] introduce two hybrid algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert space.
Motivated and inspired by Ceng and Yao [2] and Yang et al. [3], the purpose of this paper is to introduce two hybrid algorithms for the variational inequalities and mixed equilibrium problems over the common fixed points set of nonexpansive semigroups in Hilbert space. Under suitable conditions some strong convergence theorem for these two hybrid algorithms are proved. The results presented in the paper extend and improve some recent results.
2 Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. For solving mixed equilibrium problems, let us assume that the function satisfies the following conditions:
(H1) Θ is monotone, i.e., , ;
(H2) for each fixed , the mapping is concave and upper semicontinuous;
(H3) for each fixed , the mapping is convex.
A mapping is said to be:
-
(i)
κ-Lipschitz continuous, if there exists a constant such that
-
(ii)
η-strongly monotone, if there exists a constant such that
A differentiable function is said to be:
-
(i)
ξ-convex [4], if
where is the Fréchet derivative of K at x;
-
(ii)
ξ-strongly convex [5], if there exists a constant such that
The following lemmas will be needed in proving our main results.
Lemma 2.1 [2]
Let C be a nonempty closed convex subset of a real Hilbert space H and be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying the conditions (H1)-(H3). Assume that
-
(i)
is λ-Lipschitz continuous such that
-
(a)
, ;
-
(b)
is affine in the first variable;
-
(c)
for each fixed , the mapping is sequentially continuous from the weak topology to the weak topology;
-
(ii)
is ξ-strongly convex with constant , and its derivative is sequentially continuous from the weak topology to the strong topology;
-
(iii)
for each there exist a bounded subset and a point such that, for any ,
For given , let be the mapping defined by
Then
-
(i)
is single-valued;
-
(ii)
is nonexpansive if is Lipschitz continuous with constant and
where for ;
-
(iii)
, where is the set of solutions of the following mixed equilibrium problem:
-
(iv)
is closed and convex.
Lemma 2.2 [6]
Let and be bounded sequences in a Banach space E and let be a sequence in with . Suppose that for all integers and . Then .
Lemma 2.3 [7]
Let be a sequence of nonnegative real numbers such that
where is some nonnegative integer, , , and are sequences satisfying
-
(i)
and ,
-
(ii)
or ,
-
(iii)
(), .
Then .
Lemma 2.4 [8]
Let C be a bounded closed convex subset of H and be a nonexpansive semigroup on C, then for any
Lemma 2.5 [9]
Let C be a nonempty bounded closed convex subset of H, be a sequence in C and be a nonexpansive semigroup on C. If the following conditions are satisfied:
-
(i)
;
-
(ii)
,
then .
Lemma 2.6 [10]
Let F be an η-strongly monotone and κ-Lipschitzian operator on a Hilbert space H with and . Then is a contraction with contraction coefficient .
Lemma 2.7 In a real Hilbert space H, we have the inequality
for all and .
Recall that a Banach space E is said to satisfy the Opial condition, if for any sequence in E with , then for every with we have
It is well known that each Hilbert space satisfies the Opial condition.
3 Main results
Now we will show our main results.
Theorem 3.1 Let H be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium functions satisfying conditions (H1)-(H3). Let be a nonexpansive semigroup on H. Let F be an η-strongly monotone and κ-Lipschitzian operator on H. Let be a continuous net of positive real numbers such that . Putting , for each , let the net be defined by the following implicit scheme:
where is the mapping defined by (2.1). Suppose the following conditions are satisfied:
-
(i)
is λ-Lipschitz continuous such that
-
(a)
, ;
-
(b)
is affine;
-
(c)
is sequentially continuous from the weak topology to the weak topology;
-
(ii)
is ξ-strongly convex with constant , and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant and ;
-
(iii)
for each there exist a bounded subset and a point such that, for any ,
and if . Then, as , the net converges strongly to an element of Γ provided is firmly nonexpansive which is the unique solution of the following variational inequality:
Proof We divide the proof into several steps.
Step 1. First, we note that the net defined by (3.1) is well defined. In fact, we define a mapping
Because is nonexpansive. It follows from Lemma 2.6 that
Hence, the is a contraction, and so it has a unique fixed point. Therefore, the net defined by (3.1) is well defined.
Step 2. We prove that is bounded. Taking and using Lemma 2.6, we have
It follows that
Observe that
Thus, (3.4) and (3.5) imply that the net is bounded for small enough t. Without loss of generality, we may assume that the net is bounded for all . Consequently, we deduce that and are also bounded.
Step 3. On the other hand, from (3.1) and (3.4), we have
In fact, we have
so
observe that
then
This together with Lemma 2.4 and (3.6) implies that
Let be a sequence such that as . Put and . Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . Next we prove that
-
(a)
In fact, we have
With (3.7) and (3.8), we have
Indeed, from Lemma 2.5 and (3.9) we know that , i.e., , .
-
(b)
Now we prove that . In fact, since , we have
From the monotonicity of Θ, we have
and hence
Since (3.7), then and , from the weak lower semicontinuity of φ and in the second variable y, we have for all . For and , let . Since and , we have and hence . From the convexity of equilibrium bifunction in the second variable y, we have
and hence . Then we have for all . So .
We can obtain and .
Step 4. Finally, from (3.1), we have
Therefore,
It follows that
Thus, implies that .
Again, from (3.11), we obtain
It is clear that , , and . We deduce immediately from (3.12) that , which is equivalent to its dual variational inequality . That is, is a solution of the variational inequality (3.2).
Suppose that and both are solutions to the variational inequality (3.2); then
Adding up (3.13) and the last inequality yields . The strong monotonicity of F implies that and the uniqueness is proved. Later, we will use to denote the unique solution of (3.2). This completes the proof. □
Next we introduce an explicit algorithm for finding an element of Γ.
Theorem 3.2 Let H be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium functions satisfying conditions (H1)-(H3). Let be a nonexpansive semigroup on H such that . Let F be an η-strongly monotone and κ-Lipschitzian operator on H with . For given arbitrarily, define a sequence iteratively by
where , are sequences in , is a sequence in , and is the mapping defined by (2.1). Suppose the following conditions are satisfied:
-
(i)
is λ-Lipschitz continuous such that
-
(a)
, ;
-
(b)
is affine;
-
(c)
is sequentially continuous from the weak topology to the weak topology;
-
(ii)
is ξ-strongly convex with constant , and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant and ;
-
(iii)
For each there exist a bounded subset and a point such that, for any ,
-
(iv)
and ;
-
(vi)
and ;
-
(vii)
, for some .
If , then the sequences and converge strongly to an element of Γ provided is firmly nonexpansive if and only if , where is the unique solution of the following variational inequality:
Proof The necessity is obvious. We only need to prove the sufficiency. Suppose that .
Step 1. First, we show that , , , , and are bounded. In fact, letting , we have .
Then
From condition (iv), without loss of generality, we can assume that , . By (3.14) and Lemma 2.6, we have
where .
Then, from (3.15) and (3.16), we obtain
Observe that , we have by induction
where . Hence is bounded. Consequently, we deduce that , , and are also bounded.
Step 2. Define , , then .
Observe that
Next, we estimate
where . From (3.18) and (3.19), we have
This together with condition (vii) implies that
Namely,
Since and condition (vi), we get
Consequently, by Lemma 2.2, we deduce . Therefore,
Step 3. Next, we claim that . Observe that
Note that
It follows that
From (3.15), we have
then
This together with condition (vii), , and (3.21), we have
By Lemma 2.4, (3.21)-(3.24), and , we derive
Step 4. Next, we show that , where and is defined by . Since is bounded, there exists a subsequence of that converges weakly to w. Similarly be able to prove like Theorem 3.1. Hence, by Theorem 3.1, we have
Step 5. Finally, we prove that converges strongly to . From (3.14), we have
where and . Obviously, and . Hence, all conditions of Lemma 2.3 are satisfied. Therefore, we immediately deduce that the sequence converges strongly to .
Observe that
It is clear that converges strongly to . From and Theorem 3.1, we see that is the unique solution of the variational inequality (3.2). This completes the proof. □
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Acknowledgements
The research was supported by Fujian Nature Science Foundation.
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Jiang, Q., Wang, J. Hybrid algorithms of nonexpansive semigroups for mixed equilibrium problems, variational inequalities, and fixed point problems. J Inequal Appl 2014, 174 (2014). https://doi.org/10.1186/1029-242X-2014-174
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DOI: https://doi.org/10.1186/1029-242X-2014-174