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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 \[ ax''(t)+bx(t)+q\bigl (x(t-\tau)\bigr)=p(t),\quad t\in \mathbb R,\tag{*} \] 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:

[1] Iannacci, R.; Nkashama, M. N., On periodic solutions of forced second order differential equations with a deviating argument, (Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Dundee, 1984. Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Dundee, 1984, Lecture Notes in Math., vol. 1151 (1984), Springer: Springer Berlin), 224-232 · Zbl 0568.34056
[2] Omari, P.; Zanolin, F., Boundary value problems for forced nonlinear equations at resonance, (Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Dundee, 1984. Ordinary and Partial Differential Equations. Ordinary and Partial Differential Equations, Dundee, 1984, Lecture Notes in Math., vol. 1151 (1984), Springer: Springer Berlin), 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, (Lecture Notes in Math., No. 568 (1977), Springer-Verlag) · 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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