Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

Abstract

Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t : C → C, i.e. a family such that T0(f) = f, T s+t (f) = Ts o Tt(f) and ρ(T(f) - T(g)) ≤ αt (f)α(f - g), where lim sup t→∞αt (f) ≤ 1 for every f ∈ C. In particular, we prove that if Lρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex. © 2013 Bin Dehaish et al.; licensee Springer.