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

Consider $$J= [0,T]$$, $$T> 0$$, the continuous function $$f: J\times\mathbb R^3\to\mathbb R$$, the continuous functions $$I_k:\mathbb R\to\mathbb R$$, $$1\leq k\leq m$$, $$0= t_0< t_1<\cdots< t_m< t_{m+1}= T$$, the set $$D= \{(t,s)\in J\times J$$; $$t\geq s\}$$, the functions $$K\in C(D,[0,+\infty))$$, $$H\in C(J\times J,[0,+\infty))$$ and the functions
$[{\mathcal T}u](t)= \int^t_0 K(t,s)u(s)\,ds,\quad t\in J,\qquad [{\mathcal S}u](t)= \int^T_0 H(t,s)u(s)\,ds,\quad t\in J,$
where $$u: J\to\mathbb R$$.
Suppose that there exist the limits
$u(t^+_k)= \lim_{\substack{ t\to t_k\\ t< t_k}} u(t),\quad u(t^-_k)= \lim_{\substack{ t\to t_k\\ t> t_k}} u(t),\quad 1\leq k\leq m,$
and denote $$\Delta u(t_k)= u(t^+_k)- u(t^-_k)$$, $$1\leq k\leq m$$.
The authors consider the first-order impulsive integrodifferential equation
$u'(t)= f(t,u(t), [{\mathcal T}u](t), [{\mathcal S}u](t)),\quad t\in J\setminus\{t_1,\dots, t_m\}\tag{1}$
with periodic boundary value conditions
$\begin{cases} \Delta u(t_k)= I_k(u(t_k)),\quad & 1\leq k\leq m,\\ u(0)= u(T)\end{cases}\tag{2}$
and prove some comparison principles and establish existence results for extremal solutions $$u$$ of the problem $$(1)\wedge (2)$$ using these principles and the monotone iterative technique.
For example, they consider the Banach spaces $$(PC(J),\|.\|_{PC})$$ and $$(PC^1(J),\|.\|_{PC^1})$$, where
$\begin{split} PC(J)= \{u: J\to\mathbb R; u|_{(t_k,t_{k+1}]}\in C((t_k, t_{k+1}[,\mathbb R),\;0\leq k\leq m,\;\exists u(t^+_k),\\ \exists u(t^-_k)= u(t_k),\;1\leq k\leq m\},\end{split}$
$\begin{split} PC^1(J)= \{u\in PC(J); u|_{(t_k, t_{k+1})}\in C^1((t_k, t_{k+1}],\mathbb R),\;0\leq k\leq m,\;\exists u'(0^+),\\ \exists u'(T^-),\;\exists u'(t^+_k),\;\exists u'(t^-_k),\;1\leq k\leq m\}\end{split}$
with the norms $$\| u\|_{PC}= \sup\{|u(t)|; t\in J\}$$, respectively, $$\| u\|_{PC^1}= \| u\|_{PC}+\| u'\|_{PC}$$ and if there exist the functions $$\alpha$$ and $$\beta$$ in $$PC^1(J)$$, $$\alpha\leq \beta$$, satisfying some hypotheses, then there exist monotone sequences $$(\alpha_n)_n$$, $$(\beta_n)_n$$ of functions with $\alpha= \alpha_0\leq \alpha_n\leq \beta_n\leq \beta_0= \beta,\quad n\in\mathbb{N},$ which converge uniformly on $$J$$ to the extremal solutions $$u$$ of the problem $$(1)\wedge (2)$$ in
$[\alpha,\beta]= \{u\in PC(J);\, \alpha(t)\leq u(t)\leq \beta(t),\,t\in J\}.$

### MSC:

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

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