Weak and strong convergence theorems for $m$-generalized hybrid mappings in Hilbert spaces

Abstract

In this paper, we prove a weak convergence theorem of Ishikawa's type for m-generalized hybrid mappings in a Hilbert space. Further, by using a new modification of Ishikawa iteration, we prove a strong convergence theorem for m-generalized hybrid mappings in a Hilbert space.

Weak Convergence Theorems for 2-Generalized Hybrid Mappings and Equilibrium Problems

“…Recently, Alizadeh and Moradlou [2] have considered the class of m-generalized hybrid mappings in Hilbert spaces and they proved weak and strong convergence theorems for this class of nonlinear mappings.…”

Strong Convergence to Common Fixed Points Using Ishikawa and Hybrid Methods for Mean-Demiclosed Mappings in Hilbert Spaces

“…If λ n = 1 for all n ∈ N in (1.2), then, Ishikawa iteration coincides with Mann's iteration (1.1). Various convergence theorems basing on the Ishikawa iteration have been studied by many researchers; see [1,11,13].…”

“…where x 1 ∈ C is provided and a n ∈ [0, 1]. They demonstrated weak convergence to a common fixed point of nonexpansive mappings S and T that satisfy ST = T S. A mapping T is nonexpansive if ∥T x − T y∥ ≤ ∥x − y∥ for all x, y ∈ C. For successive studies of the mean-valued iterative method, see [1,6,13,14,16,17,18]. For a nonempty, closed, and convex subset D of H, we use P D to represent a metric projection from H onto D. In 2003, Nakajo and Takahashi [21] proved a strong convergence theorem for finding a fixed point of a nonexpansive mapping: Theorem 1 ( [21]).…”

Weak convergence theorems for symmetric generalized hybrid mappings in uniformly convex Banach spaces

“…In recent years, many authors have proved weak or strong convergence theorems for some nonlinear mappings by using various iteration processes in the framework of Hilbert spaces and Banach spaces, see, [2,9,11,13].…”