Positive periodic solutions of first-order singular systems. (English) Zbl 1300.34090

This paper deals with existence and the number of periodic solutions of the first order system \[ u'_i(t)=-a_i(t)u_i(t)+\lambda b_i(t)f_i(\vec u(t)),\quad i=1,\,\dots,\,n\eqno(1) \] where \(\vec u=(u_1,\,\dots,\,u_n)\in{\mathbb R}^n\), \(a_i\) and \(b_i\) are continuous \(T\)-periodic functions with positive mean value in a period, \(f_i: {\mathbb R}^n\setminus0\to]0,\infty[\) are continuous functions and \(\lambda\) is a positive parameter.
In addition it is assumed that \(f_i(\vec u)\to +\infty\) as \(\|\vec u\|\to0\) and \(\frac{f_i(\vec u)}{\|\vec u\|}\to+\infty\) as \(\|\vec u\|\to\infty\).
The main result (showing that (1) mimics the behaviour of the autonomous single equation, which can be reduced to that of a real function) is the following: there exists \(\lambda^*>0\) such that (1) has at least 2, at least 1, or no \(T\)-periodic solutions according to whether \(0<\lambda<\lambda^*\), \(\lambda=\lambda^*\) or \(\lambda>\lambda^*\) respectively.
The proof makes use of lower and upper solutions and the usual degree-theoretic argument.
Related results had been given by H. Wang [J. Differ. Equations 249, No. 12, 2986–3002 (2010; Zbl 1364.34032); Appl. Math. Comput. 218, No. 5, 1605–1610 (2011; Zbl 1239.34043)].


34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] Gurney, W. S.; Blythe, S. P.; Nisbet, R. N., Nicholson’s blowflies revisited, Nature, 287, 17-21 (1980)
[2] Mackey, M. C.; Glass, L., Oscillations and chaos in physiological control systems, Science, 197, 287-289 (1997) · Zbl 1383.92036
[3] Wazewska-Czyzewska, M.; Lasota, A., Mathematical problems of the dynamics of a system of red blood cells, Mat. Stosow., 6, 23-40 (1976), (in Polish)
[4] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[5] Freedman, H. I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal., 23, 689-701 (1992) · Zbl 0764.92016
[6] Chow, S. N., Existence of periodic solutions of autonomous functional differential equations, J. Differ. Equat., 15, 350-378 (1974) · Zbl 0295.34055
[7] Cheng, S.; Zhang, G., Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. Differ. Equat., 59, 1-8 (2001)
[8] Wan, A.; Jiang, D., Existence of positive periodic solutions for functional differential equations, Kyushu J. Math., 56, 193-202 (2002) · Zbl 1012.34068
[9] Graef, J. R.; Kong, L., Existence of multiple periodic solutions for first order functional differential equations, Math. Comput. Modell., 54, 11-12, 2962-2968 (2011) · Zbl 1235.34190
[11] Ma, R.; Chen, R.; Chen, T., Existence of positive periodic solutions of nonlinear first-order delayed differential equations, J. Math. Anal. Appl., 384, 527-535 (2011) · Zbl 1229.34109
[12] Wang, H., Positive periodic solutions of functional differential systems, J. Differ. Equat., 202, 354-366 (2004) · Zbl 1064.34052
[13] Jin, Z.; Wang, H., A note on positive periodic solutions of delayed differential equations, Appl. Math. Lett., 23, 5, 581-584 (2010) · Zbl 1194.34130
[14] Chu, J.; Torres, P. J.; Zhang, M., Periodic solutions of second order non-autonomous singular dynamical systems, J. Differ. Equat., 239, 196-212 (2007) · Zbl 1127.34023
[15] Franco, D.; Webb, J. R.L., Collisionless orbits of singular and nonsingular dynamical systems, Discrete Contin. Dynam. Syst., 15, 747-757 (2006) · Zbl 1120.34029
[16] Wang, H., Positive periodic solutions of singular systems with a parameter, J. Differ. Equat., 249, 2986-3002 (2010) · Zbl 1364.34032
[17] Wang, H., Positive periodic solutions of singular systems of first order ordinary differential equations, Appl. Math. Comput., 218, 1605-1610 (2011) · Zbl 1239.34043
[18] Ma, R., Multiplicity of positive solutions for second-order three-point boundary value problems, Comput. Math. Appl., 40, 193-204 (2000) · Zbl 0958.34019
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