Positive Solutions for Third-Order Boundary-Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

Abstract

By using a fixed point theorem in a cone and the nonlocal third-order BVP's Green function, the existence of at least one positive solution for the third-order boundary-value problem with the integral boundary conditions $x^{\mathrm{\prime \prime \prime }}$ $(t)+f(t,x(t),x^{\prime }(t))=0$, $t\in J$, $x(0)=0$, $x^{\mathrm{\prime \prime }}(0)=0$, and $x(1)={\displaystyle {\int }_0^{1}g(t)x(t)dt}$ is considered, where $f$ is a nonnegative continuous function, $J=[0,1]$, and $g\in L[0,1].$ The emphasis here is that $f$ depends on the first-order derivatives.