Abstract
In this paper, we prove the existence of a solution of the mixed equilibrium problem by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we introduce a hybrid iterative scheme for finding a common element of the set of solutions of and the set of common fixed points of a countable family of quasi-ϕ-asymptotically nonexpansive multivalued mappings. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of and the set of common fixed points of a countable family of quasi-ϕ-asymptotically nonexpansive multivalued mappings.
Highlights
Let E be a Banach space with the norm · and the dual E∗
Motivated and inspired by the above results, we investigate the problem of finding a common element of the set of solutions of MEP(f, φ, C) and the set of common fixed points of a countable family of quasi-φ-asymptotically nonexpansive multivalued mappings in Banach spaces
In Section, we prove that our proposed hybrid iterative scheme converges strongly to a common element of the set of MEP(f, φ, C) and the set of common fixed points of a countable family of quasi-φ-asymptotically nonexpansive multivalued mappings
Summary
Let E be a Banach space with the norm · and the dual E∗. We denote by N and R the sets of positive integers and real numbers, respectively. Let C be a nonempty, closed and convex subset of a real smooth, strictly convex and reflexive Banach space E, let f be a bifunction from C × C to R satisfying (A )-(A ), and let φ be a lower semicontinuous and convex function from C to R. Let C be a nonempty, bounded, closed and convex subset of a real smooth, strictly convex and reflexive Banach space E, let f be a bifunction from C × C to R satisfying (A ) and (A ), and let φ be a lower semicontinuous and convex function from C to R. Let C be a closed, bounded and convex subset of a uniformly smooth, strictly convex Banach space E, let f be a bifunction from C × C to R satisfying (A )-(A ), and let φ be a lower semicontinuous and convex function from C to R.
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