Existence of periodic solutions of second order differential equations with delay

Author:
Gerhard Metzen

Journal:
Proc. Amer. Math. Soc. **103** (1988), 765-772

MSC:
Primary 34K15; Secondary 34C25, 34K10

DOI:
https://doi.org/10.1090/S0002-9939-1988-0947655-5

MathSciNet review:
947655

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of -periodic solutions of second order ordinary differential equations with delay. In particular we study the effect the delay has with respect to the solvability of such problems. Resonance and nonresonance problems are considered.

**[1]**D. G. de Figueiredo,*Semilinear elliptic equations at resonance: Higher eigenvalues and unbounded nonlinearities*, Advances in Differential Equations (Roberto Conti, ed.), Academic Press, New York, 1981, pp. 89-99. MR**643128 (83i:35074)****[2]**D. G. de Figueiredo and W. M. Ni,*Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition*, Nonlinear Anal. TMA**3**(1979), 629-634. MR**541873 (81k:35052)****[3]**E. de Pascale and R. Iannacci,*Periodic solutions of generalized Lienard equations with delay*, Equadiff 82, Würzburg, Lecture Notes in Math., vol. 1017, Springer-Verlag, Berlin and New York, 1983 pp. 148-156. MR**726579 (85m:34118)****[4]**P. Habets and G. Metzen,*Periodic solutions of Duffing equations*J. Differential Equations (to appear). MR**986151 (90c:34040)****[5]**R. Iannacci and M. N. Nkashama,*Nonresonance conditions for periodic solutions of forced Lienard and Duffing equations with delay*, Ann. Soc. Sci. Bruxelles**99**(1985), 29-43. MR**844400 (87i:34088)****[6]**-,*On periodic solutions of forced second order differential equations with deviating argument*, Ordinary and Partial Differential Equations, Proc. (Dundee, U. K. 1984), Lecture Notes in Math., vol. 1151, Springer-Verlag, Berlin, 1985.**[7]**-,*Nonlinear two point boundary value problems as resonance without Landesman-Lazer's condition*(preprint).**[8]**-,*Unbounded perturbations of forced second order ordinary differential equations at resonance*J. Differential Equations**69**(1987), 289-309. MR**903389 (88i:34105)****[9]**W. Layton,*Periodic solutions of nonlinear delay equations*, J. Math. Anal. Appl.**77**(1980), 198-204. MR**591270 (82a:34093)****[10]**J. R. Ward,*A note on the Dirichlet problem for some semilinear elliptic equations*(preprint).**[11]**T. Weidman,*Lineare Operatoren in Hilberträumen*, Teubner, Stuttgart, 1976. MR**0634110 (58:30345)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
34K15,
34C25,
34K10

Retrieve articles in all journals with MSC: 34K15, 34C25, 34K10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0947655-5

Keywords:
Delay,
normal operator,
periodic solution,
resonance,
nonresonance

Article copyright:
© Copyright 1988
American Mathematical Society