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

For a measure space (Ω,Σ,μ) with μ(Ω)⩽1, under some general conditions on a bijective function φ:[0,∞)→[0,∞), a family of μ-integrable functions x:Ω→R with the functional pφ defined bypφ(x):=φ−1(∫Ωφ∘|x|dμ), forms a paranormed uniformly convex space (Sφ(Ω,Σ,μ),pφ) (an extension of Lp 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∈Sφ(Ω,Σ,μ) of a nonlinear functional equation. Moreover some new fixed results are proved.