## 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.

### MSC:

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

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