Solvability of a Fourth-Order Boundary Value Problem with Integral Boundary Conditions

We investigate the existence of solutions and positive solutionsfor a nonlinear fourth-order differential equation with integral boundary conditions of theform x(4)(t)=f(t,x(t),x′(t),x′′(t),x′′′(t)), t∈[0,1], x(0)=x′(1)=0, x′′(0)=∫01h(s,x(s),x′(s),x′′(s))ds, x′′′(1)=0, where f∈C([0,1]×ℝ4), h∈C([0,1]×ℝ3). By using a fixed point theorem due to D. O'Regan, the existence of solutions and positive solutions for the previous boundary valueproblems is obtained. Meanwhile, as applications, some examples are given to illustrateour results