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Positive periodic solutions of functional differential equations. (English) Zbl 1064.34052

The author considers the existence, multiplicity and nonexistence of positive \(\omega\)-periodic solutions for the periodic equation \[ x'(t)=a(t)g(x(t))x(t)-\lambda b(t)f(x(t-\tau(t))), \] where \(\lambda>0\) is a positive parameter, \(a,b\in C(\mathbb{R}\to [0,\infty))\) are \(\omega\)-periodic, \(\int_0^\omega a(t)\,dt>0\), \(\int_0^\omega b(t)\,dt>0\), \(f,g\in C([0,\infty),[0,\infty))\), and \(f(u)>0\) for \(u>0\), \(g(x)\) is bounded and \(\tau\in C(\mathbb{R}\to \mathbb{R})\) is an \(\omega\)-periodic function. Define \[ f_0:=\lim_{u\to 0+}\frac{f(u)}{u}, \qquad f_\infty:=\lim_{u\to\infty}\frac{f(u)}{u}, \] \(i_0:=\) number of zeros in the set \(\{f_0,f_\infty\}\) and \(i_\infty=\) number of infinities in the set \(\{f_0,f_\infty\}\). The author shows that the equation has \(i_0\) or \(i_\infty\) positive \(\omega\)-periodic solutions for sufficiently large or small \(\lambda>0\), respectively. The proof is based on the fixed-point index theorem.

MSC:

34K13 Periodic solutions to functional-differential equations
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