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Existence of positive periodic solutions for neutral functional differential equations. (English) Zbl 1099.34063

Consider the following two types of nonlinear neutral functional-differential equations with delay \[ \frac d{dt} \big[x(t)-cx(t-\tau(t))\big] = -a(t) x(t)+g\big(t,x(t-\tau(t))\big),\tag{1} \]
\[ \frac d{dt} \Big[x(t)-c\int_{-\infty}^0 K(r) x(t+r)dr\Big] = -a(t) x(t) + b(t) \int_{-\infty}^0 K(r) g\big(t,x(t+r)\big)dr,\tag{2} \]
where \(a,\tau\in C(\mathbb R,\mathbb R)\), \(\int_0^\omega a(t)dt >0\), \(b\in C\big(\mathbb R,(0,\infty)\big)\), \(g\in C\big(\mathbb R\times [0,\infty),[0,\infty)\big)\), \(a,b,\tau,g(\cdot,*)\) are \(\omega\)-periodic functions, \(\omega>0\) and \(c\in [0,1)\) are constants, and \(K\in C\big((-\infty,0],[0,\infty)\big)\) with \(\int_{-\infty}^0 K(r) dr=1\). Using Krasnoselskii’s fixed-point theorem in cones, sufficient conditions for the existence of positive periodic solutions of equations (1) and (2) are obtained.

MSC:

34K13 Periodic solutions to functional-differential equations
34K40 Neutral functional-differential equations
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