A New Iterative Method for Solving a System of Generalized Mixed Equilibrium Problems for a Countable Family of Generalized Quasi-ϕ-Asymptotically Nonexpansive Mappings
By using a specific way of choosing the indexes, we introduce an up-to-date iterative algorithm for approximating common fixed points of a countable family of generalized quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. As application, an iterative solution to a system of generalized mixed equilibrium problems is studied. The results extend those of other authors, in which the involved mappings consist of just finite families.
Throughout this paper, we assume that is a real Banach space with its dual , is a nonempty closed convex subset of , and is the normalized duality mapping defined by In the sequel, we use to denote the set of fixed points of a mapping .
Definition 1. (1)  A mapping is said to be generalized quasi-ϕ-asymptotically nonexpansive in the light of , if , and there exist nonnegative real sequences and with (as ) such that
where denotes the Lyapunov functional defined by
It is obvious from the definition of that
(2) A mapping is said to be uniformly L-Lipschitz continuous, if there exists a constant such that
Example 2. Let be a unit ball in a real Hilbert space , and let be a mapping defined by
where is a sequence in satisfying . It is shown by Goebel and Kirk  that
where , , for all , and is a nonnegative real sequence with as . This shows that the mapping defined earlier is a generalized quasi--asymptotically nonexpansive mapping.
Let be a bifunction, a real valued function, and a nonlinear mapping. The so-called generalized mixed equilibrium problem (GMEP) is to find a such that whose set of solutions is denoted by .
The equilibrium problem is a unifying model for several 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 the abstract equilibrium problems. Many authors have proposed some useful methods to solve the (equilibrium problem) EP, (generalized equilibrium problem) GEP, (mixed equilibrium problem) MEP, and GMEP. Concerning the weak and strong convergence of iterative sequences to approximate a common element of the set of solutions for the GMEP, the set of solutions to variational inequality problems and the set of common fixed points for relatively nonexpansive mappings, quasi--nonexpansive mappings, and quasi--asymptotically nonexpansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (e.g., see [3–16] and the references therein).
Inspired and motivated by the study mentioned earlier, in this paper, by using a specific way of choosing the indexes, we propose an up-to-date iteration scheme for approximating common fixed points of a countable family of generalized quasi--asymptotically nonexpansive mappings and obtain a strong convergence theorem for solving a system of generalized mixed equilibrium problems. The results extend those of the authors, in which the involved mappings consist of just finite families.
A Banach space is strictly convex if the following implication holds for : It is also said to be uniformly convex if for any , there exists a such that It is known that if is uniformly convex Banach space, then is reflexive and strictly convex. A Banach space is said to be smooth if exists for each . In this case, the norm of is said to be Gâteaux differentiable. The space is said to have uniformly Gâteaux differentiable norm if for each , the limit (11) is attained uniformly for . The norm of is said to be Fréchet differentiable if for each , the limit (11) is attained uniformly for . The norm of is said to be uniformly Fréchet differentiable (and is said to be uniformly smooth) if the limit (11) is attained uniformly for . Note that (, resp.) is uniformly convex (, resp.) is uniformly smooth.
Following Alber , the generalized projection is defined by
Lemma 3 (see ). Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then, the following conclusions hold:(1) for all and ;(2)if and , then , for all ;(3)for , if and only if .
Remark 4. The following basic properties for a Banach space can be found in Cioranescu .(i)If is uniformly smooth, then is uniformly continuous on each bounded subset of .(ii)If is reflexive and strictly convex, then is norm-weak continuous.(iii)If is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping is single valued, one-to-one, and onto.(iv)A Banach space is uniformly smooth if and only if is uniformly convex.(v)Each uniformly convex Banach space has the Kadec-Klee property; that is, for any sequence , if and , then , where denotes that converges weakly to .
Lemma 5 (see ). Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let and be two sequences in C such that and , where is the function defined by (3); then, .
Lemma 6 (see ). Let E and C be the same as those in Lemma 5. Let be a closed and generalized quasi--asymptotically nonexpansive mapping with nonnegative real sequences and ; then, the fixed point set F(T) of T is a closed and convex subset of C.
Lemma 7 (see ). Let be a real uniformly convex Banach space, and let be the closed ball of with center at the origin and radius . Then, there exists a continuous strictly increasing convex function with such that for all , and with .
3. Main Results
Theorem 8. Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, C a nonempty closed convex subset of E, and , a countable family of closed and generalized quasi--asymptotically nonexpansive mappings with nonnegative real sequences and satisfying and (as and for each ), and each is uniformly -Lipschitz continuous. Let be a sequence in for some , and let be a sequence in satisfying . Let be the sequence generated by where , is the generalized projection of onto , and and satisfy the positive integer equation: , (, ); that is, for each , there exist unique and such that If is nonempty and bounded, then converges strongly to .
Proof. We divide the proof into several steps.(I) and (for all ) both are closed and convex subsets in .
In fact, it follows from Lemma 6 that each is a closed and convex subset of , so is . In addition, with (=) being closed and convex, we may assume that is closed and convex for some . In view of the definition of , we have that where , and . This shows that is closed and convex.(II) is a subset of .
It is obvious that . Suppose that for some . Since is uniformly smooth, is uniformly convex. Then, for any , we have that Furthermore, it follows from Lemma 7 that for any , we have that Substituting (18) into (17) and simplifying it, we have that This implies that , and so .(III) as .
In fact, since , from Lemma 3 (2), we have that , for all . Again, since , we have that , for all . It follows from Lemma 3 (1) that for each and for each , which implies that is bounded, so is . Since for all , and , we have . This implies that is nondecreasing; hence, the limit Since is reflexive, there exists a subsequence of such that as . Since is closed and convex and , this implies that is weakly, closed and for each . In view of , we have that Since the norm is weakly lower semicontinuous, we have that and so This implies that , and so as . Since , by virtue of Kadec-Klee property of , we obtain that Since is convergent, this, together with , shows that . If there exists some subsequence of such that as , then, from Lemma 3 (1), we have that that is, , and so (IV) is some member of .
Set for each . Note that and whenever for each . For example, by the definition of , we have that , and . Then, we have that Note that ; that is, as . It follows from (27) and (28) that Since , it follows from (14), (27), and (29) that as . Since as , it follows from (30) and Lemma 5 that Note that whenever for each . From (17) and (18), for any , we have that that is, This, together with assumption conditions imposed on the sequences and , shows that . In view of property of , we have that In addition, implies that . From Remark 4 (ii), it yields that, as , Again, since for each , as , This, together with (35) and the Kadec-Klee property of , shows that On the other hand, by the assumptions that for each , is uniformly -Lipschitz continuous, and noting again that , that is, for all , we then have From (37) and , we have that , and ; that is, . It then follows that, for each , In view of the closeness of , it follows from (37) that , namely, for each , , and, hence, . (V), and so as .
Put . Since and , we have that , for all . Then, which implies that since , and, hence, .
This completes the proof.
Let be a smooth, strictly convex, and reflexive Banach space, and let be a nonempty closed convex subset of . Let be a sequence of -inverse strongly monotone mappings, a sequence of lower semicontinuous and convex functions, and a sequence of bifunctions satisfying the following conditions:(A1);(A2) is monotone; that is, ;(A3);(A4) the mapping is convex and lower semicontinuous.
A system of generalized mixed equilibrium problems (GMEPs), for , , and is to find an such that whose set of common solutions is denoted by , where denotes the set of solutions to generalized mixed equilibrium problem for , , and .
Define a countable family of mappings with as follows: where , for all , . It has been shown by Zhang  that (1) is a sequence of single-valued mappings;(2) is a sequence of closed quasi--nonexpansive mappings;(3).
Now, we have the following result.
Theorem 9. Let be the same as that in Theorem 8, and let C be a nonempty closed convex subset of E. Let be a sequence of mappings defined by (42) with . Let be a sequence in for some , and let be a sequence in satisfying . Let be the sequence generated by where satisfies the positive integer equation: , and (, ). Then, converges strongly to which is some solution to the system of generalized mixed equilibrium problems for .
Proof. Note that are quasi--nonexpansive mappings; so, they are obviously generalized quasi--asymptotically nonexpansive. Therefore, this conclusion can be obtained immediately from Theorem 8.
The author is greatly grateful to the referees for their useful suggestions by which the contents of this paper are improved. This work was supported by the National Natural Science Foundation of China (Grant no. 11061037).
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