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New results for the periodic boundary value problem for impulsive integro-differential equations. (English) Zbl 1166.45002

Consider $J=\left[0,T\right]$, $T>0$, the continuous function $f:J×{ℝ}^{3}\to ℝ$, the continuous functions ${I}_{k}:ℝ\to ℝ$, $1\le k\le m$, $0={t}_{0}<{t}_{1}<\cdots <{t}_{m}<{t}_{m+1}=T$, the set $D=\left\{\left(t,s\right)\in J×J$; $t\ge s\right\}$, the functions $K\in C\left(D,\left[0,+\infty \right)\right)$, $H\in C\left(J×J,\left[0,+\infty \right)\right)$ and the functions

$\left[𝒯u\right]\left(t\right)={\int }_{0}^{t}K\left(t,s\right)u\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,\phantom{\rule{1.em}{0ex}}t\in J,\phantom{\rule{2.em}{0ex}}\left[𝒮u\right]\left(t\right)={\int }_{0}^{T}H\left(t,s\right)u\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,\phantom{\rule{1.em}{0ex}}t\in J,$

where $u:J\to ℝ$.

Suppose that there exist the limits

$u\left({t}_{k}^{+}\right)=\underset{\begin{array}{c}t\to {t}_{k}\\ t<{t}_{k}\end{array}}{lim}u\left(t\right),\phantom{\rule{1.em}{0ex}}u\left({t}_{k}^{-}\right)=\underset{\begin{array}{c}t\to {t}_{k}\\ t>{t}_{k}\end{array}}{lim}u\left(t\right),\phantom{\rule{1.em}{0ex}}1\le k\le m,$

and denote ${\Delta }u\left({t}_{k}\right)=u\left({t}_{k}^{+}\right)-u\left({t}_{k}^{-}\right)$, $1\le k\le m$.

The authors consider the first-order impulsive integrodifferential equation

${u}^{\text{'}}\left(t\right)=f\left(t,u\left(t\right),\left[𝒯u\right]\left(t\right),\left[𝒮u\right]\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in J\setminus \left\{{t}_{1},\cdots ,{t}_{m}\right\}\phantom{\rule{2.em}{0ex}}\left(1\right)$

with periodic boundary value conditions

$\left\{\begin{array}{cc}{\Delta }u\left({t}_{k}\right)={I}_{k}\left(u\left({t}_{k}\right)\right),\phantom{\rule{1.em}{0ex}}\hfill & 1\le k\le m,\hfill \\ u\left(0\right)=u\left(T\right)\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(2\right)$

and prove some comparison principles and establish existence results for extremal solutions $u$ of the problem $\left(1\right)\wedge \left(2\right)$ using these principles and the monotone iterative technique.

For example, they consider the Banach spaces $\left(PC\left(J\right),\parallel ·{\parallel }_{PC}\right)$ and $\left(P{C}^{1}\left(J\right),\parallel ·{\parallel }_{P{C}^{1}}\right)$, where

$\begin{array}{c}{PC\left(J\right)=\left\{u:J\to ℝ;u|}_{\left({t}_{k},{t}_{k+1}\right]}\in C\left(\left({t}_{k},{t}_{k+1}\left[,ℝ\right),\phantom{\rule{4pt}{0ex}}0\le k\le m,\phantom{\rule{4pt}{0ex}}\exists u\left({t}_{k}^{+}\right),\hfill \\ \hfill \exists u\left({t}_{k}^{-}\right)=u\left({t}_{k}\right),\phantom{\rule{4pt}{0ex}}1\le k\le m\right\},\end{array}$ $\begin{array}{c}P{C}^{1}\left(J\right)={\left\{u\in PC\left(J\right);u|}_{\left({t}_{k},{t}_{k+1}\right)}\in {C}^{1}\left(\left({t}_{k},{t}_{k+1}\right],ℝ\right),\phantom{\rule{4pt}{0ex}}0\le k\le m,\phantom{\rule{4pt}{0ex}}\exists {u}^{\text{'}}\left({0}^{+}\right),\hfill \\ \hfill \exists {u}^{\text{'}}\left({T}^{-}\right),\phantom{\rule{4pt}{0ex}}\exists {u}^{\text{'}}\left({t}_{k}^{+}\right),\phantom{\rule{4pt}{0ex}}\exists {u}^{\text{'}}\left({t}_{k}^{-}\right),\phantom{\rule{4pt}{0ex}}1\le k\le m\right\}\end{array}$

with the norms ${\parallel u\parallel }_{PC}=sup\left\{|u\left(t\right)|;t\in J\right\}$, respectively, ${\parallel u\parallel }_{P{C}^{1}}={\parallel u\parallel }_{PC}+{\parallel {u}^{\text{'}}\parallel }_{PC}$ and if there exist the functions $\alpha$ and $\beta$ in $P{C}^{1}\left(J\right)$, $\alpha \le \beta$, satisfying some hypotheses, then there exist monotone sequences ${\left({\alpha }_{n}\right)}_{n}$, ${\left({\beta }_{n}\right)}_{n}$ of functions with

$\alpha ={\alpha }_{0}\le {\alpha }_{n}\le {\beta }_{n}\le {\beta }_{0}=\beta ,\phantom{\rule{1.em}{0ex}}n\in ℕ,$

which converge uniformly on $J$ to the extremal solutions $u$ of the problem $\left(1\right)\wedge \left(2\right)$ in

$\left[\alpha ,\beta \right]=\left\{u\in PC\left(J\right);\phantom{\rule{0.166667em}{0ex}}\alpha \left(t\right)\le u\left(t\right)\le \beta \left(t\right),\phantom{\rule{0.166667em}{0ex}}t\in J\right\}·$

##### MSC:
 45J05 Integro-ordinary differential equations 45L05 Theoretical approximation of solutions of integral equations