Fixed point theorems for generalized Lipschitzian semigroups

Jong Soo Jung,Balwant Singh Thakur

doi:10.1155/s0161171201007426

Jong Soo Jung, Balwant Singh Thakur

Open Access

https://doi.org/10.1155/s0161171201007426

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Abstract

Let K be a nonempty subset of a p‐uniformly convex Banach space E, G a left reversible semitopological semigroup, and 𝒮 = {Tt : t ∈ G} a generalized Lipschitzian semigroup of K into itself, that is, for s ∈ G, ‖Tsx − Tsy‖ ≤ as‖x − y‖ + bs(‖x − Tsx‖ + ‖y − Tsy‖) + cs(‖x − Tsy‖ + ‖y − Tsx‖), for x, y ∈ K where as, bs, cs > 0 such that there exists a t1 ∈ G such that bs + cs < 1 for all s≽t1. It is proved that if there exists a closed subset C of K such that for all x ∈ K, then 𝒮 with has a common fixed point, where α = lim sups(as + bs + cs)/(1 − bs − cs) and β = lim sups(2bs + 2cs)/(1 − bs − cs).

Highlights

Let K be a nonempty subset of a Banach space E and T a mapping of K into itself
We prove a fixed point theorem for generalized Lipschitzian semigroups in a p-uniformly convex Banach space
Let K be a nonempty subset of a Hilbert space H, G a left reversible semitopological semigroup, and ᏿ = {Tt : t ∈ G} a generalized Lipschitzian semigroup on K with (α + β)2 2α2 − 1 1/2

Summary

Introduction

Let K be a nonempty subset of a Banach space E and T a mapping of K into itself. The mapping T is said to be Lipschitzian mapping if for each n ≥ 1, there exists a positive real number kn such thatT nx − T ny ≤ kn x − y (1.1)for all x, y in K. Let K be a nonempty subset of a Banach space E and T a mapping of K into itself.

Results

Conclusion