Fixed Point Theorems for Generalized Lipschitzian Semigroups

Let K be a nonempty subset of a p-uniformly convex Banach space E, G a left reversible semitopological semigroup, and = {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 ⋂ s co{Ttx : t s} ⊂ C for all x ∈ K, then with [(α+β)p(αp · 2p−1−1)/(cp−2p−1βp)·Np]1/p < 1 has a common fixed point, where α= limsups(as + bs +cs)/(1−bs −cs) and β= limsups(2bs +2cs)/(1−bs −cs). 2000 Mathematics Subject Classification. 47H10.