Existence of positive periodic solutions for a periodic logistic equation
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 \(\sigma,r\in C(\mathbb{R},(0,\infty))\) are \(\omega\)-periodic functions with \(\sigma(t)<r(t)\), \(f\), \(g\), \(\tau\), \(\mu\) \(\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})\) 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 [\textit{R. E. Gaines} and \textit{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.
scientific article; zbMATH DE number 1978042