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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[1] Iannacci, R.; Nkashama, M.N., On periodic solutions of forced second order differential equations with a deviating argument, (), 224-232 · Zbl 0568.34056
[2] Omari, P.; Zanolin, F., Boundary value problems for forced nonlinear equations at resonance, (), 285-294
[3] Huang, X.K.; Xiang, Z.G., The \(2 \pi\)-periodic solution of Duffing equation \(x'' + g(x(t - \tau)) = p(t)\) with delay, Chinese sci. bull., 39, 3, 201-203, (1994), (in Chinese)
[4] Huang, X.K.; Chen, W.D., The \(2 \pi\)-periodic solution of delay Duffing equation \(x'' + c x^\prime + g(x(t - \tau)) = p(t)\), Progr. natur. sci., 8, 1, 118-121, (1998), (in Chinese)
[5] Zhang, Z.Q.; Yu, J.S., Periodic solution of a kind of Duffing equation, Appl. math. - JCU (gaoxiao yingyong shuxue xuebao), 13A, 4, 389-392, (1998), (in Chinese) · Zbl 0921.34066
[6] Wang, G.Q.; Cheng, S.S., A priori bounds for periodic solutions of a delay Rayleigh equation, Appl. math. lett., 12, 41-44, (1999) · Zbl 0980.34068
[7] Ma, S.W.; Wang, Z.C.; Yu, J.S., Periodic solutions of Duffing equations with delay, Differential equations dynam. systems, 8, 3-4, 243-255, (2000) · Zbl 0981.34063
[8] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, () · Zbl 0326.34020
[9] Ding, T.R., Nonlinear oscillations at a point of resonance, Sci. sinica ser. A., 25, 9, 918-931, (1982) · Zbl 0509.34043
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