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Zbl 1166.45002
Luo, Zhiguo; Nieto, Juan J.
New results for the periodic boundary value problem for impulsive integro-differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 6, A, 2248-2260 (2009). ISSN 0362-546X

Consider $J= [0,T]$, $T> 0$, the continuous function $f: J\times\Bbb R^3\to\Bbb R$, the continuous functions $I_k:\Bbb R\to\Bbb R$, $1\le k\le m$, $0= t_0< t_1<\cdots< t_m< t_{m+1}= T$, the set $D= \{(t,s)\in J\times J$; $t\ge s\}$, the functions $K\in C(D,[0,+\infty))$, $H\in C(J\times J,[0,+\infty))$ and the functions $$[{\Cal T}u](t)= \int^t_0 K(t,s)u(s)\,ds,\quad t\in J,\qquad [{\Cal S}u](t)= \int^T_0 H(t,s)u(s)\,ds,\quad t\in J,$$ where $u: J\to\Bbb R$.\par Suppose that there exist the limits $$u(t^+_k)= \lim\Sb t\to t_k\\ t< t_k\endSb u(t),\quad u(t^-_k)= \lim\Sb t\to t_k\\ t> t_k\endSb u(t),\quad 1\le k\le m,$$ and denote $\Delta u(t_k)= u(t^+_k)- u(t^-_k)$, $1\le k\le m$. The authors consider the first-order impulsive integrodifferential equation $$u'(t)= f(t,u(t), [{\Cal T}u](t), [{\Cal S}u](t)),\quad t\in J\setminus\{t_1,\dots, t_m\}\tag 1$$ with periodic boundary value conditions $$\cases \Delta u(t_k)= I_k(u(t_k)),\quad & 1\le k\le m,\\ u(0)= u(T)\endcases\tag2$$ 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 $(PC(J),\Vert.\Vert_{PC})$ and $(PC^1(J),\Vert.\Vert_{PC^1})$, where $$\multline PC(J)= \{u: J\to\Bbb R; u|_{(t_k,t_{k+1}]}\in C((t_k, t_{k+1}[,\Bbb R),\ 0\le k\le m,\ \exists u(t^+_k),\\ \exists u(t^-_k)= u(t_k),\ 1\le k\le m\},\endmultline$$ $$\multline PC^1(J)= \{u\in PC(J); u|_{(t_k, t_{k+1})}\in C^1((t_k, t_{k+1}],\Bbb R),\ 0\le k\le m,\ \exists u'(0^+),\\ \exists u'(T^-),\ \exists u'(t^+_k),\ \exists u'(t^-_k),\ 1\le k\le m\}\endmultline$$ with the norms $\Vert u\Vert_{PC}= \sup\{|u(t)|; t\in J\}$, respectively, $\Vert u\Vert_{PC^1}= \Vert u\Vert_{PC}+\Vert u'\Vert_{PC}$ and if there exist the functions $\alpha$ and $\beta$ in $PC^1(J)$, $\alpha\le \beta$, satisfying some hypotheses, then there exist monotone sequences $(\alpha_n)_n$, $(\beta_n)_n$ of functions with $$\alpha= \alpha_0\le \alpha_n\le \beta_n\le \beta_0= \beta,\quad n\in\bbfN,$$ which converge uniformly on $J$ to the extremal solutions $u$ of the problem $(1)\wedge (2)$ in $$[\alpha,\beta]= \{u\in PC(J);\, \alpha(t)\le u(t)\le \beta(t),\,t\in J\}.$$
[Dan-Mircea Borş (Iaşi)]
MSC 2000:
*45J05 Integro-ordinary differential equations
45L05 Theoretical approximation of solutions of integral equations

Keywords: periodic boundary value problem; impulsive integro-differential equations; comparison principle; monotone iterative technique; extremal solutions

Cited in: Zbl 1190.45005

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