Positive solutions for higher order differential equations with integral boundary conditions

Abstract

In this paper, we consider the existence of at least three positive solutions for the 2nth order differential equations with integral boundary conditions $$ \left \{ \textstyle\begin{array}{l} x^{(2n)}(t)=f(t, x(t), x''(t),\ldots,x^{(2(n-1))}(t)), \quad 0\leq t\leq 1, x^{(2i)}(0)=\int_{0}^{1}k_{i}(s)x^{(2i)}(s) \,\mathrm{d}s,\qquad x^{(2i)}(1)=0, \quad 0\leq i\leq n-1, \end{array}\displaystyle \right . $$ where $(-1)^{n}f>0$ is continuous, and $k_{i}(t)\in L^{1}[0,1]$ ( $i=0,1,\ldots,n-1$ ) are nonnegative. The associated Green’s function for the higher order differential equations with integral boundary conditions is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett-Williams fixed point theorem.