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


45J05 Integro-ordinary differential equations
45L05 Theoretical approximation of solutions to integral equations
Full Text: DOI


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