Abstract
In this paper, we consider the class of monotone ρ-nonexpansive semigroups and give existence and convergence results for common fixed points. First, we prove that the set of common fixed points is nonempty in uniformly convex modular spaces and modular spaces. Then we introduce an iteration algorithm to approximate a common fixed point for the same class of semigroups.
Highlights
We prove the existence and convergence to a common fixed point of monotone ρnonexpansive semigroups in modular spaces
Motivated by the results cited, we begin by generalizing Theorem 1.1 for the monotone ρ-nonexpansive semigroups in uniformly convex and uniformly convex in every direction modular spaces
We show under some assumptions that the sequencen ρ-converges to a common fixed point of a monotone ρ-nonexpansive semigroup
Summary
We prove the existence and convergence to a common fixed point of monotone ρnonexpansive semigroups in modular spaces. Kozlowski [9] has demonstrated the existence of common fixed points for semigroups of monotone contractions and monotone ρ-nonexpansive mappings in Banach spaces. Under the frame of modular function spaces, Kozlowski [7] has shown that the set of common fixed points of any ρ-nonexpansive semigroups, acting on a ρ-closed convex and ρ-bounded subset of a uniformly convex modular function space Lρ , is nonempty ρ-closed and convex (see Theorem 6.5 in [8]). For finding a common fixed point of a nonexpansive mapping, Halpern [5] has introduced in Hilbert spaces H the following explicit iteration scheme for elements u ∈ H and x0 ∈ H: xn+1 = αnu + (1 – αn)T(xn) for all n ≥ 0,. Many mathematicians paid their attention to studying the convergence of Halpern iteration for semigroups of various nonlinear mappings in different spaces and under different conditions
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