A Fixed Point Theorem and Its Application to Integral Equations in Modular Function Spaces

In this paper we present a fixed point theorem of Banach type in modular space. We give an application of this result to a nonlinear integral equation in Musielak-Orlicz space. 0. Introduction It is well known that one of the standard proofs of Banach’s fixed point theorem is based on Cantor’s theorem in complete metric spaces [3, 4]. To this end, using some convenient constants in the contraction assumption, we present a generalization of Banach’s fixed point theorem in some classes of modular spaces, where the modular is s-convex, having the Fatou property and satisfying the ∆2-condition. As an application we study the existence of a solution for an integral equation of Lipschitz type in a Musielak-Orlicz space. We begin by recalling some basic concepts of modular spaces; for more information, we refer to the books by Musielak [8] and Kozlowski [7]. Definition 0-1. Let X be an arbitrary vector space over K (= R or C). a) A functionnal ρ : X → [0, +∞] is called modular if: i) ρ(x) = 0 ⇐⇒ x = 0. ii) ρ(αx) = ρ(x) for α ∈ K with |α| = 1, ∀x ∈ X . iii) ρ(αx + βy) ≤ ρ(x) + ρ(y) if α, β ≥ 0,α + β = 1,∀x, y ∈ X . b) If iii) is replaced by: iii′) ρ(αx + βy) ≤ αρ(x) + βρ(y) for α, β ≥ 0,α + β = 1 with an s ∈]0, 1], then the modular ρ is called an s-convex modular; and if s = 1, ρ is called convex modular. c) A modular ρ defines a corresponding modular space, i.e. the space Xρ given by Xρ = {x ∈ X |ρ(λx) → 0 as λ → 0}. Remarks. 1) Note that in general there is no reason to expect the subadditivity of a modular ρ. Nevertheless, in view of iii) from Definition 0-1 the inequality ρ(x + y) ≤ ρ(2x) + ρ(2y) holds. Received by the editors January 5, 1996 and, in revised form, October 30, 1997. 1991 Mathematics Subject Classification. Primary 46A80, 47H10, 45G10, 46E30.