Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions

Misir J. Mardanov,Carlo Cattani,Yusif S. Gasimov,Yagub A. Sharifov

doi:10.3390/fractalfract5010015

Abstract

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique.

Highlights

IntroductionBoundary value problems for ODEs, with special initial-boundary conditions, are intensively investigated for their many applications in physics and mathematics [1,2] in a wide range of problems from vibrations to the theory of elasticity [3]
Boundary value problems for ODEs, with special initial-boundary conditions, are intensively investigated for their many applications in physics and mathematics [1,2] in a wide range of problems from vibrations to the theory of elasticity [3]. These problems are often described by the multipoint boundary value problems [1]
An original approach based on the construction of a suitable Green function is proposed for the analysis of the multipoint BPV, so that a problem on a differential system is converted into an equivalent integral equation

Summary

Introduction

Boundary value problems for ODEs, with special initial-boundary conditions, are intensively investigated for their many applications in physics and mathematics [1,2] in a wide range of problems from vibrations to the theory of elasticity [3]. In mathematical terms, these problems are often described by the multipoint boundary value problems [1]. Many authors have studied various aspects of boundary value problems with multipoint boundary conditions for the differential equations having broad applications in several branches of physics and applied mathematics [5–11].

Preliminary Remarks

Uniqueness of the Solution

Existence of the Solution

4: Let us show now that the set

Example

Conclusions