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Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space. (English) Zbl 0939.45004

The initial value problem is investigated for the second-order nonlinear impulsive integro-differential equations of Volterra type

${x}^{\text{'}\text{'}}=f\left(t,x,Tx\right),\phantom{\rule{1.em}{0ex}}\forall t\in J,\phantom{\rule{4pt}{0ex}}t\ne {t}_{i},$
${{\Delta }x|}_{t={t}_{i}}=-\sum _{j=1}^{i}{\alpha }_{ij}x\left({t}_{j}\right)+\sum _{j=1}^{i}{\beta }_{ij}{x}^{\text{'}}\left({t}_{j}\right),$
${\Delta }{x}^{\text{'}}{|}_{t={t}_{i}}=-\sum _{j=1}^{i}{\gamma }_{ij}x\left({t}_{j}\right),$
$x\left(0\right)={x}_{0},\phantom{\rule{4pt}{0ex}}{x}^{\text{'}}\left(0\right)={x}_{1},$

where $f\in C\left[J×E×E,E\right]$, $J=\left[0,a\right]\left(a>0\right)$, $0<{t}_{1}<\cdots <{t}_{i}<\cdots <{t}_{m}, ${\alpha }_{ij}$, ${\beta }_{ij}$, ${\gamma }_{ij}\left(i\ge j,\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,m\right)$ are nonnegative constants ${x}_{0},{x}_{1}\in E$, and

$\left(Tx\right)\left(t\right)={\int }_{0}^{t}k\left(t,s\right)x\left(s\right)ds,\phantom{\rule{1.em}{0ex}}forallt\in J,$

$k\in C\left[D,{R}_{+}\right]$, $D=\left\{\left(t,s\right)\in J×J:t\ge s\right\}$, ${R}_{+}$ 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.

##### MSC:
 45J05 Integro-ordinary differential equations 45N05 Abstract integral equations, integral equations in abstract spaces 45G10 Nonsingular nonlinear integral equations 45L05 Theoretical approximation of solutions of integral equations