Strong Convergence to Common Fixed Points of a Countable Family of Asymptotically Strictly Quasi--Pseudocontractions
Based on an original idea, namely, a specific way of choosing the indexes of the involved mappings, we propose a new hybrid shrinking iteration scheme for approximating some common fixed points of a countable family of asymptotically strictly quasi--pseudocontractions and obtain a strong convergence theorem in the framework of Banach space. Our result extends other authors, related results existing in the current literature. As application, an iterative solution to a system of equilibrium problems is provided.
Fixed point theory as an important branch of nonlinear analysis theory has been applied in many disciplines. From the viewpoint of applications in real world, it is not only to show the existence of fixed points of nonlinear mappings but also to construct iterative algorithms to approximate their fixed points, which explains the popularity of iteration methods for finding the fixed points of various nonlinear mappings. For instance, iteration approximation of common fixed points of asymptotically nonexpansive mappings, an important class of nonlinear mappings introduced by Goebel and Kirk , has been studied by many authors (see, e.g., [2–7]).
In this paper, we discuss the approximation of common fixed points of a family of asymptotically strictly quasi--pseudocontractions.
Throughout this paper we assume that is a real Banach space with 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 (see ). A mapping is said to be an asymptotically strictly quasi- -pseudocontraction, if , and there exists a nonnegative real sequence with (as ) and a constant such that where denotes the Lyapunov functional defined by where stands for the duality product. It is obvious from the definition of 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 in  that where for all . This shows that the mapping defined above is an asymptotically strictly quasi--pseudocontraction.
Definition 3 (see ). A mapping is said to be asymptotically regular on if, for any bounded subset of ,
In 2012, Zhang  used the following hybrid projection algorithm for a single asymptotically strictly quasi--pseudocontraction to obtain strong convergence under a limit condition only in the framework of reflexive, strictly convex, and smooth Banach spaces such that both and have the Kadec-Klee property where and is the generalized projection (see (10)) of onto . Their results improved the corresponding results of Zhou and Gao  and Qin et al. .
Inspired and motivated by the studies mentioned above, we introduce an up-to-date iteration scheme for approximating common fixed points of a countable family of asymptotically strictly quasi--pseudocontractions and obtain a strong convergence theorem. The result extends the corresponding one for one map in .
Following Alber , the generalized projection is defined by
Lemma 4 (see ). Let be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of . Then the following conclusions hold:(1) for all and ;(2)if and , then , for all ;(3)for , if and only if .
Remark 5. The following basic properties for a Banach space can be found in :(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 as .
Lemma 6 (see ). Let be a strictly convex and smooth Banach space, let be a nonempty and closed convex subset of , and let be a quasi--nonexpansive mapping from into . Then is closed and convex.
Lemma 7 (see ). The unique solutions to the positive integer equation are where denotes the maximal integer that is not larger than .
3. Main Results
Recall that an operator in a Banach space is closed if and as ; then .
Theorem 8. Let be a reflexive, strictly convex, and smooth Banach space such that both and have Kadec-Klee property, let C be a nonempty closed and convex subset of , and be a countable family of closed and asymptotically strictly quasi--pseudocontractions with a nonnegative real sequence satisfying (as for each ) and a sequence . Let be the sequence generated by where ; is the generalized projection of onto ; and are the solutions to the positive integer equation ; that is, for each , there exist unique and such that If are asymptotically regular on C and is nonempty and bounded, then converges strongly to .
Proof. We divide the proof into several steps.
(i) Both and (for all ) are closed and convex subsets in .
For each , the closedness of follows from that of , and hence is closed. In addition, using the same argument as presented in the proof of Theorem 2.1 in , we have that each is convex and so is .
Next, we show that is closed and convex for all . 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 . It follows from (4) that for any . Then, from (16) we have This implies that , and so .
(iii) Consider as .
In fact, since , from Lemma 4 (2) we have , for all . Again since , we have , for all . It follows from Lemma 4 (1) that for each and for each , which implies that is bounded and hence 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 Since the norm is weakly lower semicontinuous, we have 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 4 (1) we have that That is, and so
(iv) is a member of .
Set for each . Note that , and whenever for each . For example, by Lemma 7 and the definition of , we have and . By virtue of , we have for all . Note that ; that is, as . It then follows from (25) that In view of (5), we have This, in view of (25), implies that which yields that This shows that is bounded for each . Note that both and are reflexive. We may assume that as . In view of the reflexivity of , we know that there exists an element such that . Then we have Taking on both sides of the above equation yields that That is, , which in turn implies that . It follows that as . In the light of the Kadec-Klee property of , we obtain from (30) that . Since is norm-weak-continuous, as . This implies, from (29) and the Kadec-Klee property of , that Again note that ; that is, whenever for each . We then have It follows from (33) and the asymptotic regularity of each that That is, In view of the closedness of , it follows from (33) that , namely, for each , and hence .
(v) Consider , and so as .
Put . Since and , we have , for all . Then which implies that since , and hence . This completes the proof.
4. Applications to a System of Equilibrium Problems
The kind of equilibrium problems 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) and generalized equilibrium problem (GEP) (see, e.g., [15–19]).
Let be a smooth, strictly convex, and reflexive Banach space, a nonempty closed and convex subset of , and a countable family of bifunctions satisfying the conditions: for each ,(A1);(A2) is monotone; that is, ;(A3);(A4) the mapping is convex and lower semicontinuous. A system of equilibrium problems for is to find an such that whose set of common solutions is denoted by , where denotes the set of solutions to the equilibrium problem for . It will be shown in Theorem 9 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonlinear mappings.
Let . Define a countable family of mappings as follows: It is shown in  that (1) is a sequence of single-valued mappings;(2) is a sequence of closed quasi--nonexpansive mappings; that is, (3).
Now, we have the following result.
Theorem 9. Let be a reflexive, strictly convex, and smooth Banach space such that both and have Kadec-Klee property, C a nonempty closed and convex subset of , and be a countable family of mappings defined by (39). Let be the sequence generated by where is the generalized projection of onto ; is the solution to the positive integer equation . If , then strongly converges to which is a common solution of the system of equilibrium problems for .
Proof. Firstly, it follows from Lemma 6 that is closed and convex. Secondly, are obviously asymptotically strictly quasi--pseudocontractions with and for all , since they are quasi--nonexpansive. Finally, (13) clearly reduces to (41) with for all . Therefore conclusion can be obtained immediately from Theorem 8.
The author is very grateful to the referees for their useful suggestions, by which the content of this paper has been improved. This work is supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2013A02).
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