Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems
© Ma and Wang; licensee Springer. 2014
Received: 19 December 2013
Accepted: 8 April 2014
Published: 6 May 2014
In this paper, the demiclosed principle for a k-asymptotically strictly pseudononspreading mapping is shown. Meanwhile, an iterative scheme is introduced to approximate a common element of the set of common fixed points of k-asymptotically strictly pseudononspreading mappings and the set of solutions of mixed equilibrium problems in Hilbert spaces, and some weak and strong convergence theorems are proved. The results presented in this paper improve and extend some recent corresponding results.
The set of solutions of EP is denoted by . Given a mapping , let for all . Then if and only if is a solution of the variational inequality for all , i.e., is a solution of the variational inequality.
The set of solutions of is denoted by .
If , then mixed equilibrium problem (1.2) reduces to (1.1).
The set of solutions of (1.3) is denoted by .
The mixed equilibrium problem (MEP) includes several important problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of abstract equilibrium problems (e.g., [1, 2]). Many authors have proposed some useful methods to solve the EP, ; see, for instance, [1–8] and the references therein.
In 1967, Browder and Petryshyn  introduced the concept of k-strictly pseudononspreading mapping.
Definition 1.1 
Clearly, every nonspreading mapping is k-strictly pseudononspreading.
In 2012, Osilike  introduced a class of nonspreading type mappings, which is more general than the mappings studied in  in Hilbert spaces, and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, Chang  studied the multiple-set split feasibility problem for asymptotically strict pseudocontraction in the framework of infinite-dimensional Hilbert spaces.
Definition 1.2 
holds for all .
It is easy to see that the class of k-asymptotically strictly pseudononspreading mappings is more general than the classes of k-strictly pseudononspreading mappings and k-asymptotically strict pseudocontractions.
Next we prove that T is a k-asymptotically strictly pseudononspreading mapping.
In fact, for any .
Case 1. If and , then we have , , and so inequality (1.4) holds for any .
Therefore inequality (1.4) holds for any .
So, inequality (1.4) holds for any .
Thus inequality (1.4) still holds for any . Therefore the mapping defined by (1.5) is a k-asymptotically strictly pseudononspreading mapping.
A Banach space E is said to satisfy Opial’s condition if, for any sequence in E, implies that for all with . It is well known that every Hilbert space satisfies Opial’s condition.
A mapping T with domain and range in E is said to be demiclosed at p if whenever is a sequence in such that converges weakly to and converges strongly to p, then .
T is said to be semi-compact if for any bounded sequence with , there exists a subsequence of such that converges strongly to a point .
where , . Under some suitable conditions, they proved that the sequences , weakly and strongly converge to a solution of the problem .
where γ is a constant and , λ is the spectral of the operator , , and is a sequence in with . Under some suitable conditions, they proved that weakly and strongly converges to a split fixed point .
Inspired and motivated by the recent works of Zhao and Chang , Quan and Chang , etc., in this paper, we propose an iterative scheme to approximate a common element of the set of solutions of k-asymptotically strictly pseudononspreading mappings and mixed equilibrium problem in infinite-dimensional Hilbert spaces. Some weak and strong convergence theorems are proved. At the same time, the demiclosed principle of a k-asymptotically strictly pseudononspreading mapping is shown. The results presented in this paper improve and extend some recent corresponding results.
Throughout this paper, we denote the strong convergence and weak convergence of a sequence to a point by , , respectively.
For solving mixed equilibrium problems, we assume that the bifunction satisfies the following conditions:
(A1) , ;
(A2) , ;
(A3) For all , ;
(A4) For each , the function is convex and lower semi-continuous.
Lemma 2.1 
For each , ;
- (3)is firmly nonexpansive, that is, ,
is closed and convex.
Lemma 2.2 
- (i)For all and for all ,
- (iii)If is a sequence in H which converges weakly to , then
The demiclosed principle and the closeness and convexity of the set of fixed points of a nonlinear mapping play very important roles in investigating many nonlinear problems. We now show the demiclosed principle of k-asymptotically strictly pseudononspreading mapping and the closeness and convexity of the set of fixed points of such a mapping, respectively.
Lemma 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a continuous k-asymptotically strictly pseudononspreading mapping. If , then it is closed and convex.
Since and as , we get that . Since T is continuous, which implies that . Hence, .
Now, we show that is convex.
Since as , we obtain that , which implies that , . Hence, , which means that is convex. □
Lemma 2.4 Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a k-asymptotically strictly pseudononspreading and uniformly L-Lipschitzian mapping. Then, for any sequence in C converging weakly to a point p and converging strongly to 0, we have .
In fact, it is obvious that the conclusion is true for . Suppose that the conclusion holds for , now we prove that the conclusion is also true for .
This is , as desired. The proof is completed. □
Lemma 2.5 
if , then .
3 Main results
where is a sequence in with , is a sequence in with , , and the sequence satisfies that and . If , then the sequence converges weakly to a point .
Proof The proof is divided into four steps.
Step 1. Firstly, we prove that exists for any .
Using Lemma 2.5, we show that exists. Further, it follows from (3.2) and (3.12) that and are bounded.
Step 2. Now, we prove that , and .
Since exists for any and , it follows from (3.36) that holds. Similarly, holds for any .
Step 3. We show that .
Firstly, we show that .
In fact, since is bounded, there exists a subsequence such that . Hence, for any positive integer , there exists a subsequence with such that . Again, by (3.34) we know that as , therefore we have that .
Now, we show that .
This implies that . Hence .
Step 4. Finally, we prove that and , .
This is a contradiction. Therefore . By (3.1) and (3.22), we have . Therefore, the conclusion follows.
This completes the proof of Theorem 3.1. □
Taking , in Theorem 3.1, we have the following result.
where , , is a sequence in with , is a sequence in with and the sequence with and . If , then the sequence converges weakly to a point .
where , , is a sequence in with , is a sequence in with and the sequence with and . If , and there exists a positive integer j such that is semi-compact, then the sequence converges strongly to a point .
That is, , and converge strongly to the point . This completes the proof. □
4.1 Application to a convex minimization problem
It is well known that mixed equilibrium problem (1.2) reduces to the convex minimization problem as . Therefore, Theorem 3.1 can be used to solve convex minimization problem (1.3), and the following result can be directly deduced from Theorem 3.1.
where , , is a sequence in with , is a sequence in with , and the sequence with and . If , then the sequence converges weakly to a point .
4.2 Application to a convex feasibility problem
The so-called convex feasibility problem for a family of mappings (where ω may be a finite positive integer or +∞) is to find a point of the nonempty intersection , where is the fixed point set of mapping , .
In Theorem 3.1 if , , then the condition ‘ such that , ’ is equivalent to . Therefore, the following result can be directly obtained from Theorem 3.1.
where is a sequence in with and is a sequence in with , . If , then the sequence converges weakly to a point , which is a solution of the convex feasibility problem for mappings and .
4.3 Application to the mixed variational inequality problem of Browder type
A variational inequality problem (VIP) is formulated as a problem of finding a point with property , , . We will denote the solution set of VIP by . We know that given a mapping , let for all . Then if and only if is a solution of the variational inequality for all , i.e., is a solution of the variational inequality.
We will denote the solution set of a mixed variational inequality of Browder type by .
A mapping is said to be an α-inverse-strongly monotone mapping if there exists a constant such that for any . Setting , it is easy to show that F satisfies conditions (A1)-(A4) as A is an α-inverse-strongly monotone mapping. Then it follows from Theorem 3.1 that the following result holds.
where is a sequence in with , is a sequence in with , , and the sequence satisfies and . If , then the sequence converges weakly to a point .
The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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