Existence of solutions of first-order differential equations via a fixed point theorem for discontinuous operators

Rubén Figueroa Sestelo,Rodrigo López Pouso

doi:10.1186/s13663-015-0472-5

Abstract

AbstractWe use a recent Schauder-type result for discontinuous operators in order to look for solutions for first-order differential equations subject to initial functional conditions. We show how this abstract fixed-point result allows us to consider a nonlinearity which can be strongly discontinuous. Some examples of applications and comparison with recent literature are included.

Highlights

1 Introduction and preliminaries In this paper we are concerned with the existence of absolutely continuous solutions of the initial value problem x = f (t, x) for a.a. t ∈ I = [t – L, t + L], x(t ) = F(x)
Our goal is to show that the following general version of Schauder’s theorem proven in [ ] can be employed to prove the existence of solutions of ( . ) under very general conditions
We lose no generality if we assume that all admissible discontinuity curves are viable and m(Jn) > for all n ∈ N, where

Summary

Introduction

An admissible discontinuity curve for the differential equation x = f (t, x) is an absolutely continuous function γ : [a, b] ⊂ I −→ R satisfying one of the following conditions: either γ (t) = f (t, γ (t)) for a.a. t ∈ [a, b] (and we say that γ is viable for the differential equation), or there exist ε > and ψ ∈ L (a, b), ψ(t) > for a.a. t ∈ [a, b], such that either γ (t) + ψ(t) < f (t, y) for a.a. t ∈ I and all y ∈ γ (t) – ε, γ (t) + ε , ) has at least one absolutely continuous solution x : I −→ R such that x ∞ ≤ R, provided that F : C(I) −→ R is continuous and the following conditions are satisfied: (H ) There exist N ≥ and M ∈ L (I, [ , ∞)) such that N + M ≤ R, |F(x)| ≤ N if x ∞ ≤ R, and for a.a. t ∈ I and all x ∈ [–R, R] we have |f (t, x)| ≤ M(t).

Objectives

Results

Conclusion