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

Copy DOI

Abstract

LetKbe a nonempty subset of ap-uniformly convex Banach spaceE,Ga left reversible semitopological semigroup, and𝒮={Tt:t∈G}a generalized Lipschitzian semigroup ofKinto itself, that is, fors∈G,‖Tsx−Tsy‖≤as‖x−y‖+bs(‖x−Tsx‖+‖y−Tsy‖)+cs(‖x−Tsy‖+‖y−Tsx‖), forx,y∈Kwhereas,bs,cs>0such that there exists at1∈Gsuch thatbs+cs<1for alls≽t1. It is proved that if there exists a closed subsetCofKsuch that⋂sco¯{Ttx:t≽s}⊂Cfor allx∈K, then𝒮with[(α+β)p(αp⋅2p−1−1)/(cp−2p−1βp)⋅Np]1/p<1has a common fixed point, whereα=lim sups(as+bs+cs)/(1-bs-cs)andβ=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