Existence of positive solutions for Lidstone boundary value problems on time scales

Abstract

Let mathbb{T}subseteq mathbb{R} be a time scale. The purpose of this paper is to present sufficient conditions for the existence of multiple positive solutions of the following Lidstone boundary value problem on time scales: (−1)nyΔ(2n)(t)=f(t,y(t)),t∈[a,b]T,yΔ(2i)(a)=yΔ(2i)(σ2n−2i(b))=0,i=0,1,…,n−1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &(-1)^{n} y^{\\Delta ^{(2n)}}(t) = f\\bigl(t, y(t)\\bigr), \\quad \ ext{$t\\in [a,b]_{ \\mathbb{T}}$,} \\\\ &y^{\\Delta ^{(2i)}}(a)= y^{\\Delta ^{(2i)}}\\bigl(\\sigma ^{2n-2i}(b)\\bigr)=0,\\quad i=0,1,\\ldots,n-1. \\end{aligned}$$ \\end{document} Existence of multiple positive solutions is established using fixed point methods. At the end some examples are also given to illustrate our results.