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Zbl 0939.45004
Liu, Xinzhi; Guo, Dajun
Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space.
(English)
[J] Comput. Math. Appl. 38, No. 3-4, 213-223 (1999). ISSN 0898-1221

The initial value problem is investigated for the second-order nonlinear impulsive integro-differential equations of Volterra type $$x''=f(t,x,Tx), \quad\forall t\in J,\ t\ne t_i,$$ $$\Delta x|_{t= t_i}=- \sum^i_{j=1} \alpha_{ij} x(t_j)+ \sum^i_{j=1} \beta_{ij} x'(t_j),$$ $$\Delta x'|_{t=t_i}= -\sum^i_{j=1} \gamma_{ij} x(t_j),$$ $$x(0)=x_0,\ x'(0) =x_1,$$ where $f\in C[J\times E\times E,E]$, $J=[0,a](a>0)$, $0<t_1< \cdots< t_i < \cdots< t_m<a$, $\alpha_{ij}$, $\beta_{ij}$, $\gamma_{ij}(i\ge j,\ i=1,2, \dots, m)$ are nonnegative constants $x_0,x_1\in E$, and $$(Tx)(t)= \int^t_0k (t,s) x(s)ds,\quad forall t\in J,$$ $k\in C[D,R_+]$, $D=\{(t,s)\in J\times J: t\ge s\}$, $R_+$ 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.
[I.Foltyńska (Poznań)]
MSC 2000:
*45J05 Integro-ordinary differential equations
45N05 Integral equations in abstract spaces
45G10 Nonsingular nonlinear integral equations
45L05 Theoretical approximation of solutions of integral equations

Keywords: initial value problem; second-order nonlinear impulsive integro-differential equations of Volterra type; Banach space; upper and lower solutions; maximal and minimal solutions

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