Jong Soo Jung¹ and Balwant Singh Thakur²

Copyright © 2001 Jong Soo Jung and Balwant Singh Thakur. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ⋂sco¯{Ttx:t≽s}⊂C for all x∈K, then 𝒮 with [(α+β)p(αp⋅2p−1−1)/(cp−2p−1βp)⋅Np]1/p<1 has a common fixed point, where α=lim sups(as+bs+cs)/(1-bs-cs) and β=lim sups(2bs+2cs)/(1-bs-cs).