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

Let T⊆R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}\subseteq \mathbb{R}$\end{document} 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} $$\begin{aligned} &(-1)^{n} y^{\Delta ^{(2n)}}(t) = f\bigl(t, y(t)\bigr), \quad \text{$t\in [a,b]_{ \mathbb{T}}$,} \\ &y^{\Delta ^{(2i)}}(a)= y^{\Delta ^{(2i)}}\bigl(\sigma ^{2n-2i}(b)\bigr)=0,\quad i=0,1,\ldots,n-1. \end{aligned}$$ \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.