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Impulsive periodic boundary value problems of first-order differential equations. (English) Zbl 1110.34019
The authors deal with a periodic boundary value problem for a nonlinear differential equation of first order and impulse effects of the forn \[ \begin{aligned} u'(t)+\lambda u(t)&=f(t,u(t)),\qquad t\in [0,T],\;t\neq t_{k},\;k=1,\dots, p; \\ u(t_{k}^{+})&=u(t_{k}^{-})+I_j(u(t_{k})),\qquad k=1,\dots, p,\\ u(0)&=u(T), \end{aligned} \] where \(\lambda\neq 0\), \(0=t_0<t_1<\dots<t_p<t_{p+1}=T,\) \(f(t,x)\) is continuous for \(t\neq t_ k\) and \(I_ k\) are continuous functions for \(k=1,\dots , p\). First, the case when \(f(t,u(t))=\sigma (t)\) is considered, the so-called \`\` linear problem\'\'.
The authors obtain an integral representation for the solutions of this problem, which is used to convert the problem into an equivalent abstract operator equation. Then, an existence and uniqueness result is proved by using Banach’s contraction principle. An example is presented to show that such a problem is not always solvable. Afterwards, the general situation, the “nonlinear problem” is studied. Some existence results are obtained by using Schaefer’s fixed-point theorem. The results of this paper complement and improve some results in [J. J. Nieto, J. Math. Anal. Appl. 205, 423–433 (1997; Zbl 0870.34009)].

MSC:
34B37 Boundary value problems with impulses for ordinary differential equations
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