Anti-periodic boundary value problems for nonlinear higher order impulsive differential equations
The authors consider the impulsive boundary value problem for higher order differential equation \[ \begin{aligned} &x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t)),\quad t \in [0,1]\setminus\{t_1,\ldots,t_p\}, \\ &\triangle x^{(i)}(t_k) = I_{i,k}(x(t_k),x'(t_k),\ldots,x^{(n-1)}(t_k)), \quad k = 1,\ldots,p,\;i = 0,\ldots,n-1,\\ &x^{(i)}(0) = -x^{(i)}(T),\quad i = 0,\ldots,n-1, \end{aligned} \] where \(0 < t_1 < \ldots < t_p < 1\), \(f\) is an impulsive Carathéodory function, \(I_{i,k}\) are continuous functions. Sufficient conditions for the existence of at least one solution of the problem are obtained. The proofs of the main results are based on the Fredholm operator theory.
higher order impulsive differential equation
scientific article; zbMATH DE number 5304995