Solutions for second order nonlocal BVPs via the generalized Miranda theorem

Abstract

In this paper, the generalized Miranda theorem is applied for second-order systems of differential equations with one boundary condition given by Riemann-Stieltjes integral \[ x'' = f(t,x,x'), \quad x(0) = 0, \ x'(1) = \int _0^1 x(s) \, dg(s),\] where $f : [0,1]\times \mathbb{R} ^k\times \mathbb{R} ^k \to \mathbb{R} ^k$ is continuous and $g : [0,1] \to \mathbb{R} ^k$ has bounded variation. Under suitable assumptions upon $f$ and $g$ we prove the existence of solutions to such posed problem.