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

Luo, Zhiguo; Nieto, Juan J.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 6, 2248-2260 (2009).

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
\[ [\alpha,\beta]= \{u\in PC(J);\, \alpha(t)\leq u(t)\leq \beta(t),\,t\in J\}. \]

Reviewer: Dan-Mircea Borş (Iaşi)

MSC:

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

Keywords:

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

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