Existence of Solutions for a Class of Fourth-Order Multipoint Boundary-Value Problems on Time Scales

Abstract

The paper deals with the existence of solutions for the dynamic equation on time scales $$ \begin{array}{cc}\hfill {u}^{\varDelta \varDelta \varDelta \varDelta}(t)= f\left( t, u\left(\sigma (t)\right),{u}^{\varDelta \varDelta}(t)\right),\hfill & \hfill t\in {\left[0,1\right]}_T,\hfill \end{array} $$ with the multipoint boundary conditions $$ \begin{array}{cccc}\hfill u(0)=0,\hfill & \hfill u\left(\sigma (1)\right)={\displaystyle \sum_{i=1}^{m-2}{a}_i u\left({\xi}_i\right),}\hfill & \hfill {u}^{\varDelta \varDelta}(0)=0,\hfill & \hfill {u}^{\varDelta \varDelta}\left(\sigma (1)\right)={\displaystyle \sum_{j=1}^{n-2}{b}_j{u}^{\varDelta \varDelta}\left({\eta}_j\right),}\hfill \end{array} $$ where T is a time scale [0, 1] T = {t ∈ T | 0 ≤ t ≤ 1}, a i > 0, i = 1, 2, …, m − 2, b j > 0, j = 1, 2, …, n − 2, 0 < ξ1 < ξ2 < … < ξ m−2 < ρ(1), and 0 < η 1 < η 2 < … < η n−2 < ρ(1). The existence result is given by using Green’s function, the method of upper and lower solutions, and the monotone iterative technique. We also give an example to illustrate our result.