Some results on continuous pseudo-contractions in a reflexive Banach space
© Lv and Hao; licensee Springer. 2013
Received: 17 April 2013
Accepted: 1 October 2013
Published: 13 November 2013
In this paper, we investigate fixed point problems of a continuous pseudo-contraction based on a viscosity iterative scheme. Strong convergence theorems are established in a reflexive Banach space which also enjoys a weakly continuous duality mapping.
MSC:47H05, 47H09, 47H10, 47J05, 47J25.
1 Introduction and preliminaries
Fixed point problems of nonlinear operators, which include many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, and saddle point problems, recently have been studied as an effective and powerful tool for studying many real world problems which arise in economics, finance, medicine, image reconstruction, ecology, transportation, and network; see [1–20] and the references therein. Interest in pseudo-contractive operators, an important class of nonlinear operators, stems mainly from their firm connection with equations of evolution. It is known that many physically significant problems can be modeled by initial value problems of the form , , where A is a pseudo-contractive operator in the framework of Banach spaces. Typical examples where such evolution equations occur can be found in the heat, wave or Schrödinger equations. Fixed points of a pseudo-contractive operator have been investigated by many authors; see [21–32] and the references therein. For the existence of fixed points of pseudo-contractive operators, we refer readers to  and  and the references therein. In this paper, we consider the mean regularization viscosity method for treating fixed points of pseudo-contractive operators in a Banach space.
where denotes the generalized duality pairing. In the case that , we write J for and call J the normalized duality mapping.
Recall that a Banach space E is said to have a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping is single-valued and weak-to-weak∗ sequentially continuous (i.e., if is a sequence in E weakly convergent to a point x, then the sequence converges weakly∗ to ). It is known that has a weakly continuous duality mapping with a gauge function for all . Set , . Then , , where ∂ denotes the sub-differential in the sense of convex analysis.
whenever for and . A mapping Q of C into itself is called a retraction if . If a mapping Q of C into itself is a retraction, then for all , where is the range of Q. A nonempty subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.
The mapping T is said to be nonexpansive if the contractive coefficient .
where is a fixed point. Banach’s contraction mapping principle guarantees that has a unique fixed point in C. In the case of T having a fixed point, Browder  proved that if E is a Hilbert space, then converges strongly to a fixed point of T that is nearest to u. Reich  extended Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth Banach space, then converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from C onto .
Xu  proved that Reich’s results hold in reflexive Banach spaces which have a weakly continuous duality map. To be more precise, he proved the following theorem.
Then T has a fixed point if and only if remains bounded as , and in this case, converges as strongly to a fixed point of T.
The mapping T is said to be pseudo-contractive if the coefficient .
Recently, Zegeye and Shahzad  improved Theorem X about the mapping T from nonexpansive mappings to pseudo-contractions by the viscosity approximation method which was first introduced by Moudafi . More precisely, they proved the following results.
Suppose further that is bounded or and E is a reflexive Banach space having a weakly continuous duality mapping for some gauge φ. Then converges strongly to a fixed point of T.
We give some remarks about Moudafi’s viscosity approximation method which was recently studied by many authors. From Suzuki , we know that Moudafi’s viscosity approximation with a contraction is trivial. Since the mapping , where is a metric projection from H onto its nonempty closed and convex subset C, is a contraction, we can get the conclusion if Browder’s property is satisfied; see  for more details.
Next, we propose the following question.
What happens if the mapping f is a strong pseudo-contraction instead of a contraction? Does Theorem ZS still hold?
Since we do not know whether the mapping , where f is a strong pseudo-contraction, has a unique fixed point or not, we cannot answer the above question easily based on Suzuki’s results.
It is our purpose in this paper to consider the convergence of paths for a continuous pseudo-contraction by Moudafi’s viscosity approximation with continuous strong pseudo-contractions instead of contractions, which gives an answer to the above question.
To prove our main results, we need the following lemma and definitions.
Let C be a nonempty subset of a Banach space E. For , the inward set of x, , is defined by . A mapping is called weakly inward if for all , where denotes the closure of the inward set. Every self-map is trivially weakly inward.
The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in .
- (i)For all , the following inequality holds:
- (ii)Assume that a sequence in E converges weakly to a point . Then the following identity holds:
Lemma 1.2 
Let E be a Banach space, C be a nonempty closed and convex subset of E and be a continuous and strong pseudo-contraction. Then T has a unique fixed point in C.
2 Main results
(2.1) has a unique solution for every ;
If is bounded, then as ;
- (iii)If , where denotes the fixed point set of T, then is bounded and satisfies
This shows that is a continuous and strong pseudo-contraction for each . By Lemma 1.2, we obtain that has a unique fixed point in C for each . Therefore, one obtains that (i) holds.
This completes the proof. □
On the other hand, from (2.6), we obtain that . It follows from (2.8) that . Hence, .
It follows that .
Adding up the above inequalities, we obtain that . This completes the proof. □
Remark 2.3 In the case that T and f are non-self mappings, we remark that the results of Lemma 2.1 and Theorem 2.2 still hold under the assumption that both T and f satisfy the weak inward condition. We give an affirmative answer to the question purposed in Section 1.
Remark 2.4 It is clear that every contraction is a continuous, bounded and strong pseudo-contraction. The main results in this paper develop the so-called viscosity approximation method which was first introduced by Moudafi  from the class of contractions to the class of strong pseudo-contractions.
If for all in Theorem 2.2, we have the following result.
Corollary 2.5 Let E be a reflexive Banach space which has a weakly continuous duality mapping for some gauge φ, and let C be a nonempty closed and convex subset of E. Let be a continuous pseudo-contraction. Fix and . Let be the unique solution of the equation . If is bounded, then converges strongly to a fixed point p of T as .
Remark 2.6 Corollary 2.5 improves Theorem X (Theorem 3.1 of Xu ). To be more precise, it improves the mapping T from a nonexpansive mapping to continuous pseudo-contractions.
Remark 2.7 In the case that T is a non-self mapping, we remark that Corollary 2.5 still holds under the assumption that T satisfies the weak inward condition.
The authors are grateful to the reviewers for useful suggestions which improved the contents of the article. The first author was supported by the Natural Science Foundation of Henan Province (122300410420). The second author was supported by the Natural Science Foundation of Zhejiang Province (Y6110270).
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