×

On periodic solutions of sublinear Duffing equations. (English) Zbl 0727.34030

The authors are concerned with the scalar Duffing equation \(x''+g(x)=p(t),\) where g: \({\mathbb{R}}\to {\mathbb{R}}\) is continuous and p: \({\mathbb{R}}\to {\mathbb{R}}\) is continuous and \(2\pi\)-periodic. They prove that the equation has at least one \(2\pi\)-periodic solution if the following assumptions are satisfied: (i) there exist two constants E, \(d>0\), such that \(g(x) sgn(x)\geq E>\sup \{| p(t)| :\;t\in {\mathbb{R}}\}\) for \(| x| \geq d\); \((ii)\quad \liminf_{x\to +\infty}g(x)/x=0;\) (iii) g is continuously differentiable on \([d,+\infty]\) and there is a constant \(M>0\) such that \(xg'(x)/g(x)\leq M\) for \(x>d\). The result is achieved by a continuation lemma based on coincidence degree. An example is presented which shows that this result is not contained in the results of the (about 30) articles quoted in the bibliography.
Reviewer: W.Müller (Berlin)

MSC:

34C25 Periodic solutions to ordinary differential equations
47H11 Degree theory for nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Capozzi, A., Remarks on subquadratic non-autonomous Hamiltonian system, Boll. Un. Mat. Ital. B (6), 4, 113-124 (1985) · Zbl 0565.34038
[2] Cesari, L.; Kannan, R., Periodic solutions in the large of Lienard systems with forcing terms, Boll. Un. Mat. Ital. A (6), 1, 217-224 (1982) · Zbl 0492.34030
[3] De Pascale, L.; Iannacci, R., Periodic solutions of generalized Liénard equations with delay, (Equadiff ’82 (Würzburg 1982). Equadiff ’82 (Würzburg 1982), Lecture Notes in Mathematics, Vol. 1017 (1983), Springer-Verlag: Springer-Verlag Berlin/New York), 148-156 · Zbl 0522.34064
[4] Ding, T., An infinite class of periodic solutions of periodically perturbed Duffing’s equations at resonance, (Proc. Amer. Math. Soc., 86 (1982)), 47-54 · Zbl 0511.34031
[5] Ding, T., Boundedness of solutions of Duffing’s equations, J. Differential Equations, 61, 178-207 (1986) · Zbl 0619.34038
[6] T. Ding, R. Iannacci, and F. Zanolin; T. Ding, R. Iannacci, and F. Zanolin · Zbl 0785.34033
[7] Ding, T.; Ding, W., Resonance problems for a class of Duffing’s equations, Chinese Ann. Math. Ser. B, 6, 427-432 (1985) · Zbl 0593.34042
[8] Ding, W., Fixed points of twist mappings and periodic solutions of ordinary differential equations, Acta Math. Sinica, 25, 227-235 (1982) · Zbl 0501.34037
[9] Fernandes, L.; Zanolin, F., Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term, Arch. Math., 51, 151-163 (1988) · Zbl 0665.34045
[10] Fonda, A.; Lupo, D., Periodic solutions of second order ordinary differential equations, Boll. Uni. Mat. Ital. A (7), 3, 291-299 (1989) · Zbl 0706.34038
[11] Fučik, S., Solvability of Nonlinear Equations and Boundary Value Problems (1980), Reidel: Reidel Dordrecht · Zbl 0453.47035
[12] Gossez, J.-P; Omari, P., Nonresonance with respect to the Fučik spectrum for periodic solutions of second order ordinary differential equations, Nonlinear Anal., 14, 1079-1104 (1990) · Zbl 0724.34048
[13] Iannacci, R.; Nkashama, M. N.; Omari, P.; Zanolin, F., Periodic solutions of forced Liénard equations with jumping nonlinearities, under nonuniform conditions, (Proc. Roy. Soc. Edinburgh Sect. A, 110 (1988)), 183-198 · Zbl 0693.34046
[14] Lazer, A. C., On Schauder’s fixed point and forced second order nonlinear oscillations, J. Math. Anal. Appl., 21, 421-425 (1968) · Zbl 0155.14001
[15] Martelli, M., On forced non-linear oscillations, J. Math. Anal. Appl., 69, 496-504 (1979) · Zbl 0413.34040
[16] Mawhin, J., An extension of a theorem of A. C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl., 40, 20-29 (1972) · Zbl 0245.34035
[17] Mawhin, J., Topological Degree Methods in Nonlinear Boundary Value Problems, (CBMS Conf. in Math., Vol. 40 (1979), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI) · Zbl 0414.34025
[18] Mawhin, J., Remarks on the preceding paper of Ahmad and Lazer on periodic solutions, Boll. Un. Mat. Ital. A (6), 3, 229-236 (1984) · Zbl 0547.34032
[19] Mawhin, J.; Ward, J. R., Periodic solutions of some forced Liénard differential equations at resonance, Arch. Math., 41, 337-351 (1983) · Zbl 0537.34037
[20] Mawhin, J.; Willem, M., Critical points of convex perturbations of some indefinite quadratic forms and semi-linear boundary value problems at resonance, Ann. Inst. H. Poincaré, Anal. Non linéare, 3, 431-453 (1986) · Zbl 0678.35091
[21] Omari, P.; Villari, G.; Zanolin, F., Periodic solutions of the Liénard equation with one-sided growth restrictions, J. Differential Equations, 67, 278-293 (1987) · Zbl 0615.34037
[22] Omari, P.; Zanolin, F., On the existence of periodic solutions of forced Liénard differential equations, Nonlinear Anal., 11, 275-284 (1987) · Zbl 0644.34030
[23] Opial, Z., Sur les solutions périodiques de l’équation différentielle \(x\)″ + \(g(x) = p(t)\), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 8, 151-156 (1960) · Zbl 0134.07202
[24] Reissig, R., Schwingungssätze für die verallgemeinerte Liénardsche Differentialgleichung, (Abh. Math. Sem. Univ. Hamburg, 44 (1975)), 45-51 · Zbl 0323.34033
[25] Reissig, R., Periodic solutions of a second order differential equation including a one-sided restoring term, Arch. Math, 33, 85-90 (1979) · Zbl 0412.34032
[26] Reíssig, R.; Sansone, G.; Conti, R., Qualitative Theorie nichtlinearer Differentialgleichungen (1963), Cremonese: Cremonese Roma · Zbl 0114.04302
[27] Schmitt, K., Periodic solutions of a forced nonlinear oscillator involving a one-sided restoring force, Arch. Math., 31, 70-73 (1978) · Zbl 0399.34035
[28] Seifert, G., A note on periodic solutions of second order differential equations without damping, (Proc. Amer. Math. Soc., 10 (1959)), 396-398 · Zbl 0091.26601
[29] Ward, J., Periodic solutions for systems of second order ordinary differential equations, J. Math. Anal. Appl., 81, 92-98 (1981) · Zbl 0462.34023
[30] Willem, M., Subharmonic oscillations of convex Hamiltonian systems, Nonlinear Anal., 9, 1303-1311 (1985) · Zbl 0579.34030
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.