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

Abstract

We investigate the existence of solutions and positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form $x^{(4)}(t)=f(t,x(t),x^{\prime }(t),x^{\prime \prime }(t),x^{\prime \prime \prime }(t))$, $t\in [0,1]$, $x(0)=x^{\prime }(1)=0$, $x^{\prime \prime }(0)={\int }_0^{1}h(s,x(s),x^{\prime }(s),x^{\prime \prime }(s))\text{d}s$, $x^{\prime \prime \prime }(1)=0$, where $f\in C([0,1]\times {ℝ}^4)$, $h\in C([0,1]\times {ℝ}^3)$. By using a fixed point theorem due to D. O'Regan, the existence of solutions and positive solutions for the previous boundary value problems is obtained. Meanwhile, as applications, some examples are given to illustrate our results.