Positive solutions of higher-order Sturm–Liouville boundary value problems with fully nonlinear terms

Abstract

Abstract In this paper we consider the existence of positive solutions of nth-order Sturm–Liouville boundary value problems with fully nonlinear terms, in which the nonlinear term f involves all of the derivatives u′,…,u(n−1) $u',\ldots, u^{(n-1)}$ of the unknown function u. Such cases are seldom investigated in the literature. We present some inequality conditions guaranteeing the existence of positive solutions. Our inequality conditions allow that f(t,x0,x1,…,xn−1) $f(t, x_{0}, x_{1},\ldots, x_{n-1})$ is superlinear or sublinear growth on x0,x1,…,xn−1 $x_{0}, x_{1},\ldots, x_{n-1}$. Our discussion is based on the fixed point index theory in cones.

Keywords

• Sturm–Liouville boundary value problem
• Fully nonlinear term
• Positive solution
• Cone
• Fixed point index