Fixed point theorems for generalized Lipschitzian semigroups
Jong Soo Jung, Balwant Singh Thakur · International Journal of Mathematics and Mathematical Sciences · 2001
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).