Positive Solutions for Second-Order Singular Semipositone Differential Equations Involving Stieltjes Integral Conditions

Abstract

By means of the fixed point theory in cones, we investigate the existence of positive solutions for the following second-order singular differential equations with a negatively perturbed term: $-u^{\prime \prime }\left(t\right)=\lambda \left[f\left(t,u\left(t\right)\right)-q\left(t\right)\right]$, , $\alpha u\left(0\right)-\beta u^{\prime }\left(0\right)={\displaystyle {\int }_0^{1}u\left(s\right)d\xi \left(s\right)}$, $\gamma u\left(1\right)+\delta u^{\prime }\left(1\right)={\displaystyle {\int }_0^{1}u\left(s\right)d\eta (s),}$where $\lambda >0$ is a parameter; $f:\left(0,1\right)\times \left(0,\infty \right)\to \left[0,\infty \right)$ is continuous; $f\left(t,x\right)$ may be singular at $t=0$, $t=\mathrm{1,}$ and $x=0$, and the perturbed term $q:\left(0,1\right)\to \left[0,+\infty \right)$ is Lebesgue integrable and may have finitely many singularities in $\left(0,1\right)$, which implies that the nonlinear term may change sign.