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

Abstract

Abstract Let T ⊆ R $\mathbb{T}\subseteq \mathbb{R}$ 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 ) n y Δ ( 2 n ) ( t ) = f ( t , y ( t ) ) , t ∈ [ a , b ] T , y Δ ( 2 i ) ( a ) = y Δ ( 2 i ) ( σ 2 n − 2 i ( b ) ) = 0 , i = 0 , 1 , … , n − 1 . $$\begin{aligned} &(-1)^{n} y^{\Delta ^{(2n)}}(t) = f\bigl(t, y(t)\bigr), \quad \text{$t\in [a,b]_{ \mathbb{T}}$,} \\ &y^{\Delta ^{(2i)}}(a)= y^{\Delta ^{(2i)}}\bigl(\sigma ^{2n-2i}(b)\bigr)=0,\quad i=0,1,\ldots,n-1. \end{aligned}$$ Existence of multiple positive solutions is established using fixed point methods. At the end some examples are also given to illustrate our results.

Keywords

• Krasnoselskiĭ’s–Guo fixed point theorem
• Time scale
• Lidstone boundary value problem
• Positive solution