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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 $$ax''(t)+bx(t)+q\bigl (x(t-\tau)\bigr)=p(t),\quad t\in \Bbb R,\tag *$$ where $a,b$ and $\tau>0$ are real constants, the forcing function $p:\Bbb R\to \Bbb R$ is a $2\pi$-periodic continuous function and $g:\Bbb R\to \Bbb R$ 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 $|g(t)|\le\beta|x|^\alpha$ 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 $x\le-\rho$, or $g(x)\le\beta|x|$ for $x\ge\rho$, and $xg(x)> 0$ for $|x|\ge \rho$, 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 $g(x)\ge-\beta|x|^\alpha$ for $x\le-\rho$, or $g(x) \le\beta|x|$ for $x\ge\rho$, and $xg(x)>0$ for $|x|\ge\rho$, then (*) has a $2\pi$-periodic solution.

MSC:
34K13Periodic solutions of functional differential equations
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References:
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