Twin nonnegative solutions for higher-order boundary value problems
The authors prove the existence of twin nonnegative solutions to the \((p,n-p)\) conjugate problem: \[ (-1)^{n-p} y^{(n)} = \varphi(t) f(t,y), \quad 0< t< 1, \] \[ y^{(i)}(0)= 0, \quad 0\leq i\leq p-1, \qquad y^{(i)}(1)= 0, \quad 0\leq i\leq n-p-1, \] with \(n\geq 2\), \(1\leq p\leq n-1\), and to the \((n,p)\) problem: \[ y^{(n)}+ \varphi(t) f(t,y)= 0, \quad 0< t< 1, \] \[ y^{(i)}(0)= 0, \quad 0\leq i\leq p-2, \qquad y^{(p)}(1)= 0, \quad 1\leq p\leq n-1, \] is fixed with \(n\geq 2\). To do so, they develop a new technique which involves combining existence principles from the literature with Krasnoselskii's fixed-point theorem in a cone. The existence theory given in this paper is very general and extends many results in the literature. Also, the ideas here could be extended to discuss other higher-order problems.
scientific article; zbMATH DE number 1539291