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Zbl 1031.45005
Fan, Guihong; Li, Yongkun
Existence of positive periodic solutions for a periodic logistic equation.
(English)
[J] Appl. Math. Comput. 139, No.2-3, 311-321 (2003). ISSN 0096-3003

The paper deals with existence of $\omega$-periodic solutions for the following generalized logistic equations: $$x'(t)=\pm x(t)\left[f\left(t, \int_{-r(t)}^{-\sigma(t)}x(t+s) d\mu(t,s)\right)-g(t, x(t-\tau(t, x(t))))\right],$$ where $\sigma,r\in C(\bbfR,(0,\infty))$ are $\omega$-periodic functions with $\sigma(t)<r(t)$, $f$, $g$, $\tau$, $\mu$ $\in C(\bbfR\times\bbfR,\bbfR)$ are $\omega$-periodic functions with respect to their first variable and nondecreasing with respect to their second variable. Using the well known Mawhin's coincidence degree theorem [{\it R. E. Gaines} and {\it J. L. Mawhin}, Coincidence degree and nonlinear differential equations, Springer, Berlin (1977; Zbl 0339.47031), p. 40], the authors prove the existence of at least one positive $\omega$-periodic solution for each of the above equations.
[Panagiotis Ch.Tsamatos (Ioannina)]
MSC 2000:
*45J05 Integro-ordinary differential equations
45G10 Nonsingular nonlinear integral equations
45M15 Periodic solutions of integral equations

Keywords: positive periodic solutions; logistic equation; coincidence degree

Citations: Zbl 0339.47031

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