Global solutions of nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces.
This paper is concerned with the existence of global solutions of the initial value problem for nonlinear second order impulsive integro-differential equation of mixed type in a Banach space \((E,\| \cdot\| )\) \[ u''=f(t,u,u',Tu,Su), \;\;t\in J:=[0,a], \;\;t\not=t_{k}, \;k=1,\dots,m, \] \[ \Delta u| _{t=t_{k}}=I_{k}(u(t_{k}),u'(t_{k})), \] \[ \Delta u'| _{t=t_{k}}=\overline I_{k}(u(t_{k}),u'(t_{k})), \] \[ u(0)=u_{0}, \;\;u'(0)=u_{1}, \] where \(f\in C(J\times E\times E\times E\times E,E), \;0<t_{1}<t_{2}<\dots<t_{m}<a, \;I_{k}, \;\overline I_{k}\in C(E\times E,E), k=1,2,\dots, m, \;u_{0}, \;u_{1}\in E,\) and \[ (Tu)(t)=\int_{0}^{t}k(t,s)u(s)ds, \;\;(Su)(t)=\int_{0}^{a}h(t,s)u(s)ds, \;\;t\in J, \] where \(k\in C(D,\mathbb{R}), \;h\in C(J\times J, \mathbb{R}), \;D=\{(t,s)\in J\times J: t\geq s\}.\) \(\Delta u| _{t=t_{k}}\) and \(\Delta u'| _{t=t_{k}}\) denote the jump of \(u(t)\) and \(u'(t)\) at \(t=t_{k}\), respectively. The proof is based on a comparison result and the \textit{H. Mönch}'s fixed-point theorem [Nonlinear Anal., Theory Methods Appl. 4, 985--999 (1980; Zbl 0462.34041)]. As applications, two classes of mixed boundary value problems for fourth-order impulsive differential equations are considered.
impulsive integro-differential equations of mixed type
scientific article; zbMATH DE number 2076390