Positive solutions of boundary-value problems for 2m-order differential equations

Abstract

This article concerns the existence of positive solutions to the differential equation $$ (-1)^m x^{(2m)}(t)=f(t,x(t),x'(t),dots,x^{(m)}(t)), quad 0<t<pi, $$ subject to boundary condition $$ x^{(2i)}(0)=x^{(2i)}(pi)=0, $$ or to the boundary condition $$ x^{(2i)}(0)=x^{(2i+1)}(pi)=0, $$ for $i=0,1,dots,m-1$. Sufficient conditions for the existence of at least one positive solution of each boundary-value problem are established. Motivated by references [7,17,21], the emphasis in this paper is that $f$ depends on all higher-order derivatives.

Keywords

• Higher-order differential equation
• boundary-value problem
• positive solution
• fixed point theorem