Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

Abstract

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation $x^{\mathrm{\prime \prime }}\left(t\right)=f(t,x(\varphi \left(t\right)\left)\right)$, $t\in \left(\mathrm{0,1}\right)$, with the nonlocal condition ${\sum }_{k=1}^m{a}_{k}x\left({\tau }_k\right)=x_0$, $x^{\prime }\left(0\right)+{\sum }_{j=1}^n{b}_j{x}^{\prime }\left({\eta }_j\right)=x_1$, where ${\tau }_k\in (a,d)\subset \left(\mathrm{0,1}\right)$, ${\eta }_j\in (c,e)\subset \left(\mathrm{0,1}\right)$, and $x_0,x_1>0$. As an application the integral and the nonlocal conditions ${\int }_a^{d}x\left(t\right)dt=x_0$, $x^{\prime }\left(0\right)+x\left(e\right)-x\left(c\right)=x_1$ will be considered.