A monotone projection algorithm for fixed points of nonlinear operators
© Wu and Sun; licensee Springer. 2013
Received: 23 August 2013
Accepted: 7 November 2013
Published: 25 November 2013
In this paper, a monotone projection algorithm is investigated for equilibrium and fixed point problems. Strong convergence theorems for common solutions of the two problems are established in the framework of reflexive Banach spaces.
MSC:47H09, 47J25, 90C33.
1 Introduction and preliminaries
Recently, Alber  introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Recall that the generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem . Existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping J. If E is a reflexive, strictly convex and smooth Banach space, then if and only if . In Hilbert spaces, . It is obvious from the definition of function ϕ that , .
Next, we give some special cases:
which is called the mixed equilibrium problem.
which is called the mixed variational inequality of Browder type.
which is called the generalized equilibrium problem.
which is called the equilibrium problem.
F is monotone, i.e., , ;
for each , is convex and weakly lower semi-continuous.
Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [2–27] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real world problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models forms the major part of numerical mathematics.
where is a sequence such that as .
Remark 1.1 The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in .
Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings, which is an extension of the class of quasi-ϕ-nonexpansive mappings, was considered in [29–31]. The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction .
Remark 1.3 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings  is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings in the framework of Banach spaces which was studied by Agarwal et al. .
In this paper, we consider a projection algorithm for a common solution of a family of generalized asymptotically quasi-ϕ-nonexpansive mappings and generalized mixed equilibrium problems. A strong convergence theorem is established in a Banach space. In order to prove our main results, we need the following lemmas.
Lemma 1.4 
for all and such that .
Lemma 1.5 
Lemma 1.6 
Lemma 1.7 
Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let be a generalized asymptotically quasi-ϕ-nonexpansive mapping. Then is closed and convex.
Lemma 1.8 
is a single-valued firmly nonexpansive-type mapping, i.e., for all , ;
is closed and convex;
2 Main results
where is a real number sequence in for every , is a real number sequence in , where r is some positive real number, and . Assume that and for every . Then the sequence converges strongly to , where is the generalized projection from E onto Ψ.
Proof The proof is split into five steps.
Step 1. Show that the common solution set Ψ is convex and closed.
This step is clear in view of Lemma 1.7 and Lemma 1.8.
Step 2. Show that the set is convex and closed.
This proves that is convex. This proves that is closed and convex. This completes Step 2.
Step 3. Show that .
which proves that . This completes Step 3.
Step 4. Show that , where .
This gives that . Hence, we have . Since the space E enjoys the Kadec-Klee property, we find that as .
Taking the limit as , we find that , . For and , define . It follows that , which yields that . It follows from conditions (A1) and (A4) that . This yields that . Letting , we find from condition (A3) that , . This implies that for every .
On the other hand, we have . Combining (2.5) with (2.6), one obtains that . Since is demicontinuous, one sees that . Notice that . This yields that . Since the space E enjoys the Kadec-Klee property, we obtain that . Note that . Since T is asymptotically regular, we find that . That is, as . It follows from the closedness of that for every . This completes Step 4.
Step 5. Show that .
From Lemma 1.6, we can immediately obtain that . This completes the proof. □
Remark 2.3 The framework of the space in Theorem 2.1 can be applicable to , .
If and , then Theorem 2.1 is reduced to the following.
where , , and are real number sequences in , is a real number sequence in , where r is some positive real number, and . Assume that and . Then the sequence converges strongly to , where is the generalized projection from E onto .
Remark 2.5 Corollary 2.4 mainly improves the corresponding results in Qin et al. . To be more clear, the mapping is extended from quasi-ϕ-nonexpansive mappings to generalized asymptotically quasi-ϕ-nonexpansive mappings and the framework of spaces is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space.
The authors are grateful to the reviewers’ useful suggestions which improved the contents of the article.
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