×

Periodic solutions of certain third order differential systems with nonlinear dissipation. (English) Zbl 1040.34049

Summary: We present \(n\)-dimensional analogues to some results obtained by Ezeilo and Omari by studying the existence of \(T\)-periodic solutions for certain third-order nonlinear differential systems of the form \[ X'''+ AX''+ G(t, X')+ CX = P(t), \] where the dissipative term \(G\) and forcing term \(P\) are vector-valued functions, and \(A\) and \(C\) are nonsingular constant matrices. We demonstrate in this study that a transition from the scalar to the vector field is by no means trivial.

MSC:

34C25 Periodic solutions to ordinary differential equations
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] Aftabizadeh A. R., Xu J. M., Gupta C. P.: Periodic boundary value problems for third-order ordinary differential equations. Nonlinear Analysis, T.M.A. 14, 1 (1990), 1-10. · Zbl 0706.34018 · doi:10.1016/0362-546X(90)90130-9
[2] 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
[3] Afuwape A. U., Ukpera A. S.: Existence of solutions of periodic boundary value problems for some vector third order differential equations. J. Nigerian Math. Soc. 20 (2001), 1-17.
[4] Andres J.: Recent results on third order nonlinear ODEs. J. Nigerian Math. Soc. 14/15 (1995/96), 41-66.
[5] Andres J., Vlček V.: Periodic solutions of the third order parametric differential equations involving large nonlinearities. Math. Slovaca 41, 4 (1991), 337-349. · Zbl 0753.34025
[6] Conti G., Iannacci R., Nkashama M. N.: Periodic solutions of Liénard systems at resonance. Ann. Math. Pura Appl. 141, 4 (1985), 313-327. · Zbl 0577.34035 · doi:10.1007/BF01766859
[7] Ezeilo J. O. C., Nkashama M. N.: Resonant and nonresonant oscillations for third-order nonlinear ordinary differential equations. Nonlinear Analysis, T.M.A. 12, 10 (1988), 1029-1046. · Zbl 0676.34021 · doi:10.1016/0362-546X(88)90098-3
[8] Ezeilo J. O. C., Omari P.: Nonresonant oscillations for some third-order Differential equations II. J. Nigerian Math. Soc. 8 (1989), 25-48.
[9] Ezeilo J. O. C., Onyia J.: Nonresonant oscillations for some third-order differential equations I. J. Nigerian Math. Soc. 3 (1984), 83-96. · Zbl 0599.34055
[10] Mawhin J.: Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS Regional Conference Series in Mathematics 40, American Math. Soc, Providence, R.I., 1979. · Zbl 0414.34025
[11] Minhós F.: Periodic solutions for a third order differential equation under conditions on the potential. Portugaliae Math. 55, 4 (1998), 475-484. · Zbl 0923.34045
[12] Omari P., Zanolin F.: Existence results for forced nonlinear periodic BVPs at resonance. Ann. Mat. Pura Appl. 141, 4 (1985), 127-158. · Zbl 0589.34005 · doi:10.1007/BF01763171
[13] Rachůnková I.: Periodic boundary value problems for third-order differential equations. Math. Slovaca 41, 3 (1991), 241-248. · Zbl 0753.34013
[14] Tejumola H. O., Afuwape A. U.: Periodic solutions of certain third-order nonlinear differential systems with delay. I.C.T.P. Trieste, Preprint IC/90/418)
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.