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Existence of positive periodic solutions for a periodic logistic equation. (English) Zbl 1031.45005

The paper deals with existence of ω-periodic solutions for the following generalized logistic equations:

x ' (t)=±x(t)ft, -r(t) -σ(t) x(t+s)dμ(t,s)-g(t,x(t-τ(t,x(t)))),

where σ,rC(,(0,)) are ω-periodic functions with σ(t)<r(t), f, g, τ, μ C(×,) are ω-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 [R. E. Gaines and 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 ω-periodic solution for each of the above equations.

45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
45M15Periodic solutions of integral equations