New iterative scheme with strict pseudo-contractions and multivalued nonexpansive mappings for fixed point problems and variational inequality problems
© Vahidi et al.; licensee Springer. 2013
Received: 7 May 2013
Accepted: 25 July 2013
Published: 9 August 2013
In this paper, we introduce an iterative scheme for finding a common element of the sets of fixed points for multivalued nonexpansive mappings, strict pseudo-contractive mappings and the set of solutions of an equilibrium problem for a pseudomonotone, Lipschitz-type continuous bifunctions. We prove the strong convergence of the sequence, generated by the proposed scheme, to the solution of the variational inequality. Our results generalize and improve some known results.
MSC:47H10, 65K10, 65K15, 90C25.
for all (see ). Strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems, see Scherzer . In the literature, many interesting and important results have been appeared to approximate the fixed points of pseudo-contractive mappings. For example, see [4–6] and the references therein.
for all .
Such problems arise frequently in mathematics, physics, engineering, game theory, transportation, economics and network. Due to importance of the solutions of such problems, many researchers are working in this area and studying on the existence of the solutions of such problems, see, e.g., [15–20]. Further, in the recent years, iterative algorithms for finding a common element of the set of solutions of equilibrium problem and the set of fixed points of nonexpansive mappings in a real Hilbert space have been studied by many authors (see, e.g., [21–33]).
- (i)strongly monotone on C with if
- (ii)monotone on C if
- (iii)pseudomonotone on C if
- (iv)Lipschitz-type continuous on C with constants and (in the sense of Mastroeni ) if
Recently, Anh [35, 36] introduced some methods for finding a common element of the set of solutions of monotone Lipschitz-type continuous equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H. In , he proved the following theorem.
, , where ,
In this paper, we introduce an iterative algorithm for finding a common element of the sets of fixed points for multivalued nonexpansive mappings, strict pseudo-contractive mappings and the set of solutions of an equilibrium problem for a pseudomonotone, Lipschitz-type continuous bifunctions. We prove the strong convergence of the sequence generated by the proposed algorithm to the solution of the variational inequality. Our results generalize and improve a number of known results including the results of Anh .
The following lemmas are crucial for the proofs of our results.
Lemma 2.4 
Lemma 2.5 
Lemma 2.6 
Lemma 2.7 
Lemma 2.8 
Let C be nonempty closed convex subset of a real Hilbert space H, and let be β-pseudo-contraction mapping. Then is demiclosed at 0. That is, if is a sequence in C such that and , then .
Lemma 2.9 
Let C be a closed convex subset of a Hilbert space H, and let be a β-strict pseudo-contraction on C and the fixed-point set of T is nonempty, then is closed and convex.
Lemma 2.10 
Let C be a closed convex subset of a real Hilbert space H. Let be a nonexpansive multivalued mapping. Assume that for all . Then is closed and convex.
Lemma 2.11 
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive multivalued mapping. If and , then .
3 Main results
Now, we are in a position to give our main results.
, , ,
, where ,
, and .
In order to prove that as , we consider the following two cases.
From Lemma 2.4, we conclude that the sequence converges strongly to q.
Therefore, converges strongly to . This completes the proof. □
Then, we have . Indeed, if , then , hence . On the other hand, if , since , we have . Now, using the similar arguments as in the proof of Theorem 3.1, we obtain the following result by replacing T by , and removing the strict condition for all .
, , ,
, where ,
, and .
As a consequence, we obtain the following result for single-valued mappings.
, , ,
, where ,
, and .
4 Application to variational inequalities
and, hence, f satisfies Lipschitz-type continuous condition with . Now, using Theorem 3.1, we obtain the following convergence theorem for finding a common element of the set of common fixed points of a strict pseudo-contractive mapping and a multivalued nonexpansive mapping and the solution set of the variational inequality problem.
, , ,
, and .
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second author acknowledges with thanks DSR, KAU for financial support. The authors thank the referees for their valuable comments and suggestions.
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