zbMATH — the first resource for mathematics

Periodic solutions of the third order functional differential equations. (English) Zbl 1057.34098
The authors deal with the existence of \(2\pi\)-periodic solutions of third-order differential equations of the type \[ x'''(t)+ a(x'')^{2k-1}(t)+ b(x')^{2k-1}(t)+ cx^{2k-1}(t)+ g(t,x(t-\tau_1), x'(t-\tau_2))= p(t), \tag{1} \] where \(p(t+2\pi)= p(t)\), \(a\), \(b\), \(c\), \(\tau_1\) and \(\tau_2\) are real numbers, \(k\) is a positive integer, \(g: \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is continuous and \(2\pi\)-periodic with respect to the first variable \(t\). The authors derive sufficient conditions for the existence of periodic solutions of (1). To this end, the authors use the continuation theorem from the theory of the coincidence degree, and a priori estimates.

34K13 Periodic solutions to functional-differential equations
Full Text: DOI
[1] Afuwape, A.U., Conditions on the behaviour of solutions for a certain class of third order nonlinear differential equations, An. ştiinţ. univ. al. I. cuza iaşi. mat., 30, 31-34, (1984) · Zbl 0625.34049
[2] Afuwape, A.U., Frequency-domain approach to nonlinear oscillations of some third order differential equations, Nonlinear anal., 10, 1459-1470, (1986) · Zbl 0612.34050
[3] Afuwape, A.U., On the applications of the frequency-domain method for uniformly dissipative nonlinear systems, Math. nachr., 130, 217-224, (1987) · Zbl 0689.34009
[4] Hale, J.K., Ordinary differential equations, (1969), Wiley-Interscience New York · Zbl 0186.40901
[5] Amann, H., Existence theorems for equations of Hammerstein type, Appl. anal., 2, 385-397, (1973) · Zbl 0244.47047
[6] Barbalat, I.; Halanay, A., Conditions de comportement « presque linéaire » dans la théorie des oscillations, Rev. roumaine sci. tech. Sér. electrotech. energet., 29, 321-341, (1974)
[7] Popov, V.M., Hyperstability of control systems, (1973), Springer-Verlag Berlin · Zbl 0276.93033
[8] Cronin, J., Some mathematics of biological oscillations, SIAM rev., 19, 100-137, (1977) · Zbl 0366.92001
[9] Levinson, N., Transformation theory of nonlinear differential equations of the second order, Ann. of math., 45, 723-735, (1944)
[10] Ezeilo, J.O.C., On the existence of periodic solutions of a certain third-order differential equation, Proc. Cambridge philos. soc., 56, 381-389, (1960) · Zbl 0097.29404
[11] Lasalle, J.P.; Lefschetz, S., Stability, by Liapunov’s direct method, with applications, (1961), Academic Press New York
[12] Yacubovich, V.A., Frequency conditions for the absolute stability and dissipativity of control systems with a single differentiable nonlinearity, Soviet math., 6, 98-101, (1965) · Zbl 0142.36301
[13] Li, B., Uniqueness and stability of a limit cycle for a third-order dynamical system arising in neuron modelling, Nonlinear anal., 5, 13-19, (1981) · Zbl 0448.34046
[14] Friedrichs, K.O., On nonlinear vibrations of the third-order, (), 65-103
[15] Pliss, V.A., Nonlocal problems in the theory of oscillations, (1966), Academic Press New York · Zbl 0151.12104
[16] Voraćek, J., Einige bemerkungen über eine nicht lineare differential gleichungen dritter ordnung, Abh. Deutsch. akad. wiss. Berlin kl. math. phys. tech., 1, 373-378, (1965) · Zbl 0178.43401
[17] Rauch, L.L., Oscillations of a third order nonlinear autonomous system, (), 39-88
[18] Mulholland, R.S.; Keener, M.S., Analysis of linear compartment models for ecosystems, J. theoret. biol., 44, 106-116, (1974)
[19] Villari, G., Soluzioni periodiche di una classa di equazioni differenziali de terz’ordine quasi lineari, Ann. mat. pura appl. (4), 73, 103-110, (1966) · Zbl 0144.11401
[20] Reissig, R., Periodic solutions of a third-order nonlinear differential equation, Ann. mat. pura appl. (4), 92, 193-198, (1972) · Zbl 0257.34043
[21] Reissig, R., An extension of Ezeilo’s result, Ann. mat. pura appl. (4), 92, 199-210, (1972) · Zbl 0268.34045
[22] Sedziwy, S., On periodic solutions of a certain third-order nonlinear differential equation, Ann. polon. math., 17, 147-154, (1965) · Zbl 0134.30802
[23] Ezeilo, J.O.C.; Onyla, J., Nonresonant oscillations for some third-order differential equations, I, J. Nigerian math. soc., 3, 83-96, (1984) · Zbl 0599.34055
[24] Mawhin, J., Degré topologique et solutions périodiques des systèmes différentiels non linéaires, Bull. soc. roy. sci. liège, 38, 308-398, (1969) · Zbl 0186.41704
[25] Afuwape, A.U., Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems, J. math. anal. appl., 143, 35-56, (1989) · Zbl 0695.47044
[26] Omari, P.; Zanolin, F., Sharp nonresonance conditions for periodic boundary value problems, (), 73-88
[27] Mawhin, J., Topological degree methods in nonlinear boundary value problems, Regional conference series in mathematics, vol. 40, (1978), American Mathematical Society Providence, RI · Zbl 0414.34025
[28] Mawhin, J., Periodic solutions of nonlinear functional differential equations, J. differential equations, 10, 240-261, (1971) · Zbl 0223.34055
[29] Manásevich, R.; Mawhin, J.; Zanolin, F., Periodic solutions of some complex-valued Liénard and Rayleigh equations, Nonlinear anal., 36, 997-1014, (1999) · Zbl 0980.34041
[30] Gaines, R.E.; Mawhin, J.L., Coincidence degree, and non-linear differential equations, (1977), Springer-Verlag Berlin
[31] Yoshixawa, T., Stability theory by Liapunov’s method, (1966), Mathematical Society of Japan Tokyo
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.