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.


34K13 Periodic solutions to functional-differential equations
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