Multiple symmetric positive solutions for two-point even order boundary value problems on time scales

Abstract

This paper is concerned with symmetric positive solutions of the even order dynamic equation on a time scale,$$(-1)^n y^{{(\Delta \nabla)}^{n}}(t)= f(y(t),y^{\Delta \nabla}(t),...,y^{(\Delta \nabla)^{i}}(t),...,y^{(\Delta\nabla)^{n-1}}(t)),~~t\in[a, b]$$ subject to thetwo-point boundary conditions$$y^{{(\Delta\nabla)}^{i}}(a)=0=y^{{(\Delta\nabla)}^{i}}(b),~~~0\leq i\leq n-1,$$ where $n\in \N$, $[a,b]$ is a time scale interval and$f:\R^{n}\rightarrow[0,\infty)$ is continuous. We establish theexistence of at least three symmetric positive solutions by usingthe well-known Avery generalization of the Leggett-Williams fixed point theorem. And then, we establish at least $2m-1$symmetric positive solutions for an arbitrary positive integer $m$.