Existence of Solutions for Singular Second-Order Ordinary Differential Equations with Periodic and Deviated Nonlocal Multipoint Boundary Conditions

Ahmed M A El-Sayed,Fatma M Gaafar

doi:10.1155/2018/9726475

Abstract

This paper studies the existence of continuous solutions for a class of nonlinear singular second-order ordinary differential equations subject to one of the following boundary conditions: periodic-deviated multipoint boundary conditions, periodic-integral boundary conditions, and periodic-nonlocal integral conditions in the Riemann-Stieltjes sense. An existence result based on the Schauder fixed point theorem and Leray-Schauder continuation principle is used to obtain at least one continuous solution for the singular second-order ordinary differential problems. Two examples are given to show the application of our results.

Highlights

This paper studies the existence of continuous solutions for a class of nonlinear singular second-order ordinary differential equations subject to one of the following boundary conditions: periodic-deviated multipoint boundary conditions, periodicintegral boundary conditions, and periodic-nonlocal integral conditions in the Riemann-Stieltjes sense
We study the existence of at least one continuous solution for the second-order differential equation x󸀠󸀠 = f (t, x (t), x󸀠 (t)), a.e. t ∈ (0, 1), (1)
The existence of solutions of ordinary differential equation with integral and periodic boundary conditions has been widely considered in recent years; they constitute very important and interesting problems because they have various applications in thermoelasticity, chemical engineering, population dynamics, and underground water flow and include nonlocal and multipoint boundary conditions

Summary

Introduction

The existence of solutions of ordinary differential equation with integral and periodic boundary conditions has been widely considered in recent years; they constitute very important and interesting problems because they have various applications in thermoelasticity, chemical engineering, population dynamics, and underground water flow and include nonlocal and multipoint boundary conditions (see, for example, Feng et al [36], Feng and Cong [37], Hua et al [38], Jiang et al [18], Nanware and Dhaigude [39], Song et al [40], Stanek [41], Webb and Infante [42], Yan et al [31], Yang [43], Yao [44], Zhang and Ge [45], and Zhang and Xu [46], and the references therein)

Preliminary Lemmas

Existence Results

Problems with Periodic and Integral Boundary Conditions