Abstract
In this article, a Krasnoselskii-type and a Halpern-type algorithm for approximating a common fixed point of a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps and a solution of a system of generalized mixed equilibrium problem are constructed. Strong convergence of the sequences generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property. Several applications of our theorems are also presented. Finally, our theorems are a significant improvement of several important recent results.
Highlights
1 Introduction In what follows, we assume that X is a real Banach space with dual space X∗, K is a nonempty, closed, and convex subset of X, and → and will, respectively, denote strong and weak convergence
A subset K of X is said to be a retract of X, if there exists a continuous map P : X → K such that Pu = u, for all u ∈ X
The results of Zhao and Chang [ ] are proved in uniformly smooth and uniformly convex real Banach spaces, while Theorem . is proved in the more general uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property
Summary
We assume that X is a real Banach space with dual space X∗, K is a nonempty, closed, and convex subset of X, and → and will, respectively, denote strong and weak convergence. (C ∗) F is a nonempty bounded subset of K , where {Gi}∞ i= is a countable family of uniformly L-Lipschitz continuous, closed and uniformly totally quasi-φ-asymptotically nonexpansive nonself maps. Later in the same year, Yi [ ] established the results in the paper of Zhao and Chang [ ], when {Gi}∞ i= is a countable family of uniformly L-Lipschitz continuous and uniformly totally quasi-φ-asymptotically nonexpansive nonself maps, under conditions (C ), (C ), and the following condition:. GMEP(hi, Ai, ζi)), where countable family of continuous and totally quasi-φ-asymptotically nonexpansive nonself multi-valued maps; {Ai}∞ i= , Ai : K → X∗ is a sequence of continuous and monotone maps; {hi}∞ i= , hi : K × K → R is a sequence of bifunctions satisfying appropriate conditions and {ζi}∞ i= , ζi : K → R is a sequence of lower-semicontinuous and convex functions.
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