Results on non-resonant oscillations for some nonlinear vector fourth order differential systems. (English) Zbl 1283.34043

The authors consider the following nonlinear vector differential equations of fourth order: \[ X^{(4)} + A{X}''' + B{X}'' + \frac{d}{dt}g(X) + DX = P(t) \] and \[ X^{(4)} + A{X}''' + B{X}'' + G(t,X) + DX = P(t) \] subject to the periodic boundary conditions \[ D^{(r)}X(0) = D^{(r)}X(T), (D = \frac{d}{dt}),\quad r = 0,1,2,3, \] on \([0,T]\) with \(T > 0.\)
Here, \(X \equiv (x_i )_{1 \leq i \leq n} :[0,T] \to \mathbb R^n\). They prove two existence results. They extend and improve some related results from scalar cases to vectorial cases.


34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: Link


[1] Afuwape, A. U., Omari, P., Zanolin, F.: Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems. J. Math. Anal. Appl. 143, 1 (1989), 35-56. · Zbl 0695.47044 · doi:10.1016/0022-247X(89)90027-9
[2] Afuwape, A. U., Ukpera, A. S.: Existence of solutions of periodic boundary value problems for some vector third order differential equations. J. Nig. Math. Soc. 20 (2001), 1-17.
[3] Ezeilo, J. O. C.: Non-resonant oscillations for some fourth order differential equations I. Ekhaguere, G.O.S., Ugbebor, O. (eds.): Directions in Mathematics, Associated Book Makers, Ibadan, 1999, 15-35.
[4] Ezeilo, J. O. C.: Periodic boundary value problems for some fourth order differential equations. Tejumola, H.O., Ezeilo, J.O.C., Afuwape, A.U. (eds.): Proceedings of National Mathematical Centre on Ordinary Differential Equations, National Mathematical Centre, Abuja, Nigeria, 1, 1 (2000), 1-14. · Zbl 0227.34026
[5] Ezeilo, J. O. C., Nkashama, M. N.: Resonant and nonresonant oscillations for third-order nonlinear ordinary differential equations. Nonl. Anal. T.M.A. 12, 10 (1988), 1029-1046. · Zbl 0676.34021 · doi:10.1016/0362-546X(88)90098-3
[6] Ezeilo, J. O. C., Omari, P.: Nonresonant oscillations for some third-order Differential equations II. J. Nig. Math. Soc. 8 (1989), 25-48.
[7] Ezeilo, J. O. C., Onyia, J.: Nonresonant oscillations for some third-order differential equations I. J. Nig. Math. Soc. 3 (1984), 83-96. · Zbl 0599.34055
[8] Reissig, R., Sansone, G., Conti, R.: Nonlinear Differential Equations of Higher Order. Noordhoff Pub., Groningen, 1974. · Zbl 0275.34001
[9] Tejumola, H. O.: Existence of periodic solutions for certain nonlinear fourth order differential equations. Animalu, A.O.E., Iyahen, S.O., Tejumola, H.O. (eds.): Contributions to the Development of Mathematics in Nigeria, National Mathematical Centre, Abuja, 2000, 41-49.
[10] Tejumola, H. O.: Periodic boundary value problems for some fifth, fourth and third order ordinary differential equations. J. Nig. Math. Soc. 25 (2006), 37-46.
[11] Ukpera, A. S.: Periodic solutions of certain third order differential systems with nonlinear dissipation. Acta Univ. Palacki. Olomuc., Math. 41 (2002), 113-125. · Zbl 1040.34049
[12] Ukpera, A. S.: Asymptotic conditions for solving non-symmetric problems of third order nonlinear differential systems. Electron. J. Diff. Equat. 2008, 154 (2008), 1-15. · Zbl 1176.34022
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.