Nonlinear differential equations with perturbed Dirichlet integral boundary conditions

Javier Iglesias,Alberto Cabada

doi:10.1186/s13661-021-01542-5

Javier Iglesias, Alberto Cabada

Open Access

https://doi.org/10.1186/s13661-021-01542-5

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Journal: Boundary Value Problems	Publication Date: Jul 29, 2021
Citations: 2	License type: open-access

Affiliation: University of Santiago de Compostela

Abstract

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green’s function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones.

Highlights

1 Introduction This paper is devoted to the study of the existence of solutions of the following family of nonlinear second order ordinary differential equations: u (t) + γ u(t) + f t, u(t) = 0, 0 < t < 1, (1)
Integral boundary conditions have been considered in much work in the literature; see, for instance, [6, 9, 12] or [1, 5, 7, 8] and the references therein
The paper is organized as follows: in Sect. 2, we study the linear part of problem (1)– (2), where we obtain the explicit expression of the related Green’s function and calculate the exact values of γ and λ for which the Green’s function has constant sign

Summary

Introduction

This paper is devoted to the study of the existence of solutions of the following family of nonlinear second order ordinary differential equations:. We will analyze each of them and give optimal sufficient conditions on γ , λ and f that allow us to ensure the existence of a solution of the considered problem. We prove the existence of positive solutions for the nonlinear problem (1)–(2) Such solutions are given as the fixed points of a related integral operator defined on a suitable cone. Theorem 2 (Krasnosels’kiı) Let X be a Banach space and K ⊂ X a cone in X. T has a fixed point at K ∩ ( 2 \ 1)

Linear part

The result is immediately deduced from the following equality: