In this paper, we introduce some new iterative algorithms for the split common solution problems for equilibrium problems and fixed point problems of nonlinear mappings. Some examples illustrating our results are also given.
MSC: 47J25, 47H09, 65K10.
Keywords:fixed point problem; iterative algorithm; equilibrium problem; split common solution problem
Throughout this paper, we assume that H is a real Hilbert space with zero vector θ, whose inner product and norm are denoted by and , respectively. Let K be a nonempty subset of H and T be a mapping from K into itself. The set of fixed points of T is denoted by . The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively.
Let C and K be nonempty subsets of real Banach spaces and , respectively. Let be a bounded linear mapping, T a mapping from C into itself with and f a bi-function from into R. The classical equilibrium problem is to find such that
The equilibrium problem contains optimization problems, variational inequalities problems, saddle point problems, the Nash equilibrium problems, fixed point problems, complementary problems, bilevel problems, and semi-infinite problems as special cases and have many applications in mathematical program with equilibrium constraint; for detail, one can refer to [1-4] and references therein.
In this paper, we study the following split common solution problem (SCSP) for equilibrium problems and fixed point problems of nonlinear mappings A, T and f:
Many authors had proposed some methods to find the solution of the equilibrium problem (1.1). As a generalization of the equilibrium problem (1.1), finding a common solution for some equilibrium problems and fixed point problems of nonlinear operators, it has been considered in the same subset of the same space; see [5-15]. However, some equilibrium problems and fixed point problems of nonlinear mappings always belong to different subsets of spaces in general. So the split common solution is very important for the research on generalized equilibriums problems and fixed point problems.
Example 1.2 Let with the norm for and with the standard norm . Let and . Let for and for all . Then and A is a bounded and linear operator from into with . Now define a bi-function f as for all . Then f is a bi-function from into ℝ with .
Remark 1.1 It is worth to mention that the split common solution problem in Example 1.1 lies in two different subsets of the same space and the split common solution problem in Example 1.2 lies in two different subsets of the different space. So, Examples 1.1 and 1.2 also show that the split common solution problem is meaningful.
In this paper, we introduce a weak convergence algorithm and a strong convergence algorithm for the split common solution problem when the nonlinear operator T is a quasi-nonexpansive mapping. Some strong and weak convergence theorems are established. We also give some examples to illustrate our results.
It is well known that any Hilbert space satisfies Opial’s condition.
Hence, T is a continuous quasi-nonexpansive mapping but not nonexpansive.
Definition 2.1 (see )
Let K be a nonempty closed convex subset of a real Hilbert space H and T a mapping from K into K. The mapping T is said to be demiclosed if, for any sequence which weakly converges to y, and if the sequence strongly converges to z, then .
Remark 2.1 In Definition 2.1, the particular case of demiclosedness at zero is frequently used in some iterative convergence algorithms, which is the particular case when , the zero vector of H; for more detail, one can refer to .
The following concept of zero-demiclosedness was introduced in .
Definition 2.2 (see )
The following result was essentially proved in , but we give the proof for the sake of completeness.
Proposition 2.1LetKbe a nonempty, closed, and convex subset of a real Hilbert space with zero vectorθandTa mapping fromKintoK. Then the following statements hold.
Then T is a discontinuous quasi-nonexpansive mapping but not zero-demiclosed.
Let and be two Hilbert spaces. Let and be two bounded linear operators. B is called the adjoint operator (or adjoint) of A, if for all , , B satisfies . It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that if B is an adjoint operator of A, then .
Example 2.4 Let with the norm for and with the norm for . Let and denote the inner product of and , respectively, where , , , . Let for . Then A is a bounded linear operator from into with . For , let . Then B is a bounded linear operator from into with . Moreover, for any and , , so B is an adjoint operator of A.
Let K be a closed and convex subset of a real Hilbert space H. For each point , there exists a unique nearest point in K, denoted by , such that , . The mapping is called the metric projection from H onto K. It is well known that has the following characterizations:
The following lemmas are crucial in our proofs.
Lemma 2.1 (see )
Lemma 2.2 (see )
Lemma 2.3 (see, e.g., )
LetHbe a real Hilbert space. Then the following hold.
The following result is simple, but it is very useful in this paper; see also .
So, combining (2.2), (2.3), and (A2), we get
3 Main results
In this section, we first introduce a weak convergence iterative algorithms for the split common solution problem.
Theorem 3.1Letandbe two real Hilbert spaces andandbe two nonempty closed convex sets. Letbe zero-demiclosed quasi-nonexpansive mappings andbe bi-functions with. Letbe a bounded linear operator with its adjointB.
so it follows from (3.1), (3.3), and (3.4) that
From (3.1) and (3.10), we have
Corollary 3.1Letandbe two real Hilbert spaces. Letbe a zero-demiclosed quasi-nonexpansive mapping withandbe a bi-function with. Letbe a bounded linear operator with its adjointB. Given. Letandbe sequences generated by
Next, we introduce a strong convergence algorithm for the split common solution problem.
Theorem 3.2Letandbe two nonempty, closed, and convex sets, zero-demiclosed quasi-nonexpansive mappings anda bi-function with. Letbe a bounded linear operator with the adjointB. Given, and. Letandbe sequences generated by
By (3.13), (3.14), and (3.15), we get
So , which say that . On the other hand, since by (3.19) and , we have . Notice that T is zero-demiclosed quasi-nonexpansive mappings, by (3.20), , namely, . So . From (3.21), we also have converges strongly to . The proof is completed. □
Corollary 3.2Letandbe two real Hilbert spaces. Letbe a zero-demiclosed quasi-nonexpansive mappings withandbe a bi-function with. Letbe a bounded linear operator with the adjoint B. Given, , and. Letandbe sequences generated by
Because (3.24) is equivalent with
The authors declare that they have no competing interests.
Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
The authors would like to express their sincere thanks to the anonymous referee for their valuable comments and useful suggestions in improving the paper. The first author was supported by the Natural Science Foundation of Yunnan Province (2010ZC152). The second author was supported partially by Grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.
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