Inequalities of Green’s functions and positive solutions to nonlocal boundary value problems

Abstract

In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in C 2 [0, 1], and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients. MSC: Primary 34B18; 34B10; secondary 34B15 Keywords: Positive solution; Fixed point index; Cone; Spectral radius u(t) dB i (t) and α i [u] = 1 0 u(t) dA i (t) © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.