Fixed point theorem in a uniformly convex paranormed space and its application

For a measure space (Omega, Sigma, mu) with mu(Omega) <= 1, under some general conditions on a bijective function phi:[0, infinity) -> [0, infinity), a family of mu-integrable functions x:Omega -> R with the functional p(phi) defined by p(phi)(x) := phi(-1) (integral(Omega) phi o vertical bar x vertical bar d mu), forms a paranormed uniformly convex space (S-phi(Omega, Sigma, mu), p(phi)) (an extension of L-P space). Applying a generalization of the Browder-Goehde-Kirk-type fixed point theorem due to Pasicki, we present sufficient conditions for existence of a solution x is an element of S-phi(Omega, Sigma, mu) of a nonlinear functional equation. Moreover some new fixed results are proved. (C) 2013 Elsevier B.V. All rights reserved.