Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator. (English) Zbl 1421.34046

In this paper, existence of positive periodic solutions is considered for differential equations of the form \[[x(t)-cx(t-\tau(t))]''+a(t)x(t)=f(t, x(t-\tau(t)))+e(t),\] where \(c\) is a constant, \(a(t) \geq 0\) and \(\tau, a, e\) are real valued \(\omega\)-periodic functions, \(f(t, u)\) is \(\omega\)-periodic in \(t\) and with a singularity at \(u=0\): \(\lim_{u\to 0^+}f(t, u)=+\infty\) (or \(\lim_{u\to 0^+}f(t, u)=-\infty\)) uniformly in \(t\). By using a fixed-point theorem of Krasnoselskii, the existence of positive \(\omega\)-periodic solutions is proved under various sufficient conditions. There are many results in literature on existence of periodic solutions for delay differential equations, neutral differential equations, equations with a difference operator, and differential equations with a singularity term. But it seems that this is the first paper on periodic solutions of differential equations combining a difference operator with a singularity term.


34K13 Periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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