A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities. (English) Zbl 0683.34027

Lazer, A. C.; Mckenna, P. J.

Proc. Am. Math. Soc. 106, No. 1, 119-125 (1989).

From the introduction: “If the potential in a second-order Newtonian system of differential equations is positively homogeneous of degree two and positive semidefinite, and if the unforced system has no nontrivial T-periodic solutions \((T>0)\), then for any continuous T-periodic forcing, there is at least one T-periodic solution.”

Reviewer: P.Drábek

MSC:

34C25	Periodic solutions to ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems

Keywords:

second order; differential operator; homogeneous nonlinearities; potential; second-order Newtonian system

References:

[1]	Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. · Zbl 0559.47040
[2]	Jerry L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), no. 5, 567 – 597. · Zbl 0325.35038
[3]	A. C. Lazer and P. J. McKenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues. II, Comm. Partial Differential Equations 11 (1986), no. 15, 1653 – 1676. · Zbl 0654.35082
[4]	A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 3, 243 – 274 (English, with French summary). · Zbl 0633.34037
[5]	A. C. Lazer and P. J. McKenna, A symmetry theorem and applications to nonlinear partial differential equations, J. Differential Equations 72 (1988), no. 1, 95 – 106. · Zbl 0666.47038
[6]	N. G. Lloyd, Degree theory, Cambridge University Press, Cambridge-New York-Melbourne, 1978. Cambridge Tracts in Mathematics, No. 73. · Zbl 0367.47001
[7]	P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal. 98 (1987), no. 2, 167 – 177. · Zbl 0676.35003
[8]	E. Podolak, On the range of operator equations with an asymptotically nonlinear term, Indiana Univ. Math. J. 25 (1976), no. 12, 1127 – 1137. · Zbl 0353.47035
[9]	Helmut Schaefer, Über die Methode der a priori-Schranken, Math. Ann. 129 (1955), 415 – 416 (German). · Zbl 0064.35703
[10]	Klaus Schmitt, Periodic solutions of nonlinear second order differential equations, Math. Z. 98 (1967), 200 – 207. · Zbl 0153.12501
[11]	E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 4, 283 – 300. · Zbl 0351.35037