Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space
The initial value problem is investigated for the second-order nonlinear impulsive integro-differential equations of Volterra type \[ x''=f(t,x,Tx), \quad\forall t\in J,\;t\neq t_i, \] \[ \Delta x|_{t= t_i}=- \sum^i_{j=1} \alpha_{ij} x(t_j)+ \sum^i_{j=1} \beta_{ij} x'(t_j), \] \[ \Delta x'|_{t=t_i}= -\sum^i_{j=1} \gamma_{ij} x(t_j), \] \[ x(0)=x_0,\;x'(0) =x_1, \] where \(f\in C[J\times E\times E,E]\), \(J=[0,a](a>0)\), \(0<t_1< \cdots< t_i < \cdots< t_m<a\), \(\alpha_{ij}\), \(\beta_{ij}\), \(\gamma_{ij}(i\geq j,\;i=1,2, \dots, m)\) are nonnegative constants \(x_0,x_1\in E\), and \[ (Tx)(t)= \int^t_0k (t,s) x(s)ds,\quad forall t\in J, \] \(k\in C[D,R_+]\), \(D=\{(t,s)\in J\times J: t\geq s\}\), \(R_+\) is the set of all nonnegative numbers, in a real Banach space by means of upper and lower solutions. Conditions for the existence of maximal and minimal solutions are established.
second-order nonlinear impulsive integro-differential equations of Volterra type
scientific article; zbMATH DE number 1391631