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A new existence theory for positive periodic solutions to functional differential equations with impulse effects. (English) Zbl 1156.34053

The authors study the existence of positive periodic solutions for the nonautonomous functional differential equation

$\stackrel{˙}{y}\left(t\right)=-a\left(t\right)y\left(t\right)+g\left(t,y\left(t-\tau \left(t\right)\right)\right),t\ne {t}_{j},j\in ℤ,y\left({t}_{j}^{+}\right)=y\left({t}_{j}^{-}\right)+{I}_{j}\left(y\left({t}_{j}\right)\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $a\in C\left(ℝ,\left(0,\infty \right)\right)$, $\tau \in C\left(ℝ,ℝ\right)$, $g\in C\left(ℝ×\left[0,\infty \right),\left[0,\infty \right)\right)$, ${I}_{j}\in C\left(\left[0,\infty \right),\left[0,\infty \right)\right)$, $a\left(t\right)$, $\tau \left(t\right)$, $g\left(t,y\right)$ are $w$-periodic functions, and $w>0$ is a constant. Here, $y\left({t}_{j}^{+}\right)$, $y\left({t}_{j}^{-}\right)=y\left({t}_{j}\right)$ denote, respectively, the right and the left-hand side limits of $y$ at ${t}_{j}$, and the following hypotheses are assumed ${t}_{j+p}={t}_{j}+w$, ${I}_{j+p}={I}_{j}$, $j\in ℤ$, for a certain positive integer $p$ in such a way that $\left[0,w\right)\cap \left\{{t}_{j}:j\in ℤ\right\}=\left\{{t}_{1},{t}_{2},\cdots ,{t}_{p}\right\}$.

The results included in this paper are connected with those given in the papers of A. Wan, D. Jiang and X. Xu [Comput. Math. Appl. 47, 1257–1262 (2004; Zbl 1073.34082)], and A. Wan and D. Jiang [Kyushu J. Math. 56, 193–202 (2002; Zbl 1012.34068)] for the nonimpulsive case.

The new results extend those in [Comput. Math. Appl. 47, 1257–1262 (2004; Zbl 1073.34082)], and their proof is based on the application of fixed-point theorems in cones [see K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, New York (1985; Zbl 0559.47040), and K. Lan and J. R. L. Webb, J. Differ. Equations 148, 407–421 (1998; Zbl 0909.34013)]. The calculus of the Green’s function associated with problem (1) is also essential in their procedure.

To this purpose, the authors consider an impulsive ‘linear’ problem related to (1) which is written on its equivalent integral form. This expression is useful to write problem (1) as an integral equation. As a particular case of the main result, it is analyzed the case of sublinear and superlinear behavior. Finally, the authors include some examples which illustrate the applicability of the new results to some problems with biological meaning.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34K45 Functional-differential equations with impulses 47N20 Applications of operator theory to differential and integral equations