Green’s Function and Positive Solutions for a Second-Order Singular Boundary Value Problem with Integral Boundary Conditions and a Delayed Argument

Abstract

This paper investigates the expression and properties of Green’s function for a second-order singular boundary value problem with integral boundary conditions and delayed argument $-x^{\prime \prime }\left(t\right)-a\left(t\right)x^{\prime }\left(t\right)+b\left(t\right)x\left(t\right)=\omega \left(t\right)f\left(t, x\left(\alpha \left(t\right)\right)\right),\hspace{0.17em} t\in \left(0, 1\right);\hspace{0.17em} x^{\prime }\left(0\right)=0,\hspace{0.17em} x\left(1\right)-{\int }_0^{1}h\left(t\right)x\left(t\right)dt=0$, where $a\in \left(\left[0, 1\right], \left[0, +\infty \right)\right), b\in C\left(\left[0, 1\right], \left(0, +\infty \right)\right)$ and, $\omega$ may be singular at $t=0$ or/and at $t=1$. Furthermore, several new and more general results are obtained for the existence of positive solutions for the above problem by using Krasnosel’skii’s fixed point theorem. We discuss our problems with a delayed argument, which may concern optimization issues of some technical problems. Moreover, the approach to express the integral equation of the above problem is different from earlier approaches. Our results cover a second-order boundary value problem without deviating arguments and are compared with some recent results.