International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 1, Pages 41-50
doi:10.1155/S0161171201007426

Fixed point theorems for generalized Lipschitzian semigroups

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).