Some notes on a second-order random boundary value problem

Fairouz Tchier,Calogero Vetro

doi:10.15388/na.2017.6.6

Abstract

We consider a two-point boundary value problem of second-order random differential equation. Using a variant of the α-ψ-contractive type mapping theorem in metric spaces, we show the existence of at least one solution.

Highlights

In this paper, we consider the following two-point boundary value problem of secondorder random differential equation: d2u− dt2 (ω, t) = f ω, t, u(ω, t), t ∈ [0, 1], (1)u(ω, 0) = u(ω, 1) = 0 for all ω ∈ Ω, where f : Ω × [0, 1] × R → R has certain regularities and Ω is a nonempty set.By a random solution of system (1), we mean a measurable mapping u : Ω → C([0, 1], R) satisfying (1), where C([0, 1], R) denote the space of all continuous functions defined on [0, 1]
Using a variant of the α-ψ-contractive type mapping theorem in metric spaces, we show the existence of at least one solution
The interest for the random version of well-known ordinary differential equations is motivated by the necessity to model and understand certain nonspecific dynamic processes of natural phenomena arising in the applied sciences; see the books of Bharucha-Reid [2] and Skorohod [13]

Summary

Introduction

We consider the following two-point boundary value problem of secondorder random differential equation: d2u. In this setting, Samet et al [11] investigated the solvability of system (2) by using a new concept of α-ψ-contractive type mapping, which generalizes the Banach contraction in [1] and many others fixed-point theorems in the literature (see, for example, Nieto and Rodríguez-López [7] and Ran and Reurings [10]). The interesting feature of our work is that we do not impose contractive conditions to all points of the involved space, but just to the ones satisfying a specific inequality relation (defined by using a given function α; see Definition 2 below). This means that we enlarge the class of operators such that our results apply. By appropriate choices of the function α, we are able to control the whole process (as shown by the proofs of the results)

Mathematical background

Random fixed point results