## Periodic solutions of a second order forced sublinear differential equation with delay.(English)Zbl 1097.34050

The authors consider the existence of $$2\pi$$-periodic solutions to the second-order sublinear differential equation with delay $ax''(t)+bx(t)+q\bigl (x(t-\tau)\bigr)=p(t),\quad t\in \mathbb R,\tag{*}$ where $$a,b$$ and $$\tau>0$$ are real constants, the forcing function $$p:\mathbb R\to \mathbb R$$ is a $$2\pi$$-periodic continuous function and $$g:\mathbb R\to \mathbb R$$ is a continuous function. By means of a priori estimation and continuation theorems, the authors obtain criteria for the existence of $$2\pi$$-periodic solutions of equation (*) under a sublinear condition on the function $$g$$. The main results of this paper are the following new criteria:
(1) If $$0<|b|<|a|/ \pi^2$$ and if there are constants $$\rho>0$$, $$\beta> 0$$ and $$\alpha\in [0,1)$$ such that $$|g(t)|\leq\beta|x|^\alpha$$ for $$|x|>\rho$$, then (*) has a $$2\pi$$-periodic solution.
(2) If $$b=0$$, $$a=1$$ and if there are constants $$\rho>0$$ and $$\beta\in(0,1/2\pi^2)$$ such that $$g(x)=-\beta |x|$$ for $$x\leq-\rho$$, or $$g(x)\leq\beta|x|$$ for $$x\geq\rho$$, and $$xg(x)> 0$$ for $$|x|\geq \rho$$, then (*) has a $$2\pi$$-periodic solution.
(3) If $$b=0$$, $$a=1$$ and there are constants $$\rho>0$$, $$\beta>0$$ and $$\alpha \in[0,1)$$ such that $$g(x)\geq-\beta|x|^\alpha$$ for $$x\leq-\rho$$, or $$g(x) \leq\beta|x|$$ for $$x\geq\rho$$, and $$xg(x)>0$$ for $$|x|\geq\rho$$, then (*) has a $$2\pi$$-periodic solution.

### MSC:

 34K13 Periodic solutions to functional-differential equations
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### References:

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