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 p φ defined by p φ ( x ) : = φ − 1 ( ∫ Ω φ ∘ | x | d μ ) , forms a paranormed uniformly convex space ( S φ ( Ω , Σ , μ ) , p φ ) (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 ∈ S φ ( Ω , Σ , μ ) of a nonlinear functional equation. Moreover some new fixed results are proved.