Existence of solutions for odd-order multi-point impulsive boundary value problems on time scales (Q2161986)

The paper deals with \(2q+1\)st order (\(q\geq1\)) impulsive dynamic equations on time scales with periodic multi-point boundary conditions (PBVP). The authors first obtain the integral representation of the solutions for the impulsive PBVP of the form \[ \begin{cases} x^{\Delta^{2q+1}}(t)&= f(t, x(\sigma(t))),\quad t\in J_0,\\ x^{\Delta^p}(0)&= 0,\quad p\in \{2,3, \ldots, 2q\},\\ x^{\Delta^2}(t_k^+)&= x^{\Delta^2}(t_k)+I_k(x(t_k)),\quad k\in \{1, \ldots, m\},\\ x^{\Delta}(0)&= \sum\limits_{j=1}^m \alpha_j x^{\Delta}(\xi_j),\\ x(t_k^+)&= x(t_k)+J_k(x(t_k)),\quad k\in \{1, \ldots, m\},\\ x(0)&= x(\sigma(T)), \end{cases}\tag{1} \] where \[ z(t_k^\pm)=\lim_{t\to t_k^\pm}z(t),\quad k\in \{1, \ldots, m\} \] with \(z(t_k^-)=z(t_k)\), and \(t_k\)'s are right dense for \(k\in \{1, \ldots, m\}\), and \(J_0=[0,T]_{\mathbb{T}}\backslash \{t_k\}_{k=1}^m\). Then, by setting some reasonable conditions on the nonlinear terms \(f(t, x(\sigma(t)))\), \( I_k(x(t_k))\) and \(J_k(x(t_k))\), the authors prove that the impulsive PBVP (1) has at least one solution. The main tool for the proof of existence of solutions is the Schaefer fixed point theorem. An example supporting the results is also given.