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

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′′′(t)+f(t,x(t),x′(t))=0, t∈J, x(0)=0, x′′(0)=0, and x(1)=∫01g(t)x(t)dt is considered, where f is a nonnegative continuous function, J=[0,1], and g∈L[0,1]. The emphasis here is that f depends on the first-order derivatives