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A relationship between the periodic and the Dirichlet BVPs of singular differential equations*

Published online by Cambridge University Press:  14 November 2011

Meirong Zhang
Meirong Zhang
Affiliation:
Department of Applied Mathematics, Tsinghua University, Beijing 100084, People's Republic of China; Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A., e-mail: [email protected]
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Abstract

In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 5 , 1998 , pp. 1099 - 1114
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Ambrosetti, A.. Critical points and nonlinear variational problems. Mem. Soc. Math. France 49, in Bull Soc. Math. France 120 (1992), 5–132.Google Scholar
2Ambrosetti, A. and Coti Zelati, V.. Critical points with lack of compactness and singular dynamical systems. Ann. Mat. Pura Appl. (4) 149 (1987), 237–59.CrossRefGoogle Scholar
3Bahri, A. and Rabinowitz, P.. A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 82 (1989), 412–28.CrossRefGoogle Scholar
4Bevc, V., Palmer, J. L. and Süsskind, C.. On the design of the transition region of axi-symmetric magnetically focused beam valves. J. British Inst. Radio Engineers 18 (1958), 696–708.CrossRefGoogle Scholar
5Capietto, A., Mawhin, J. and Zanolin, F.. Continuation theorems for periodic perturbations of autonomous systems. Trans. Amer. Math. Soc. 329 (1992), 41–72.CrossRefGoogle Scholar
6Coti Zelati, V.. Dynamical systems with effective-like potentials. Nonlinear Anal. 12 (1988), 209–22.CrossRefGoogle Scholar
7del Pino, M. A. and Manasevich, R. F.. Infinitely many T–periodic solutions for a problem arising in nonlinear elasticity. J. Differential Equations 103 (1993), 260–77.CrossRefGoogle Scholar
8del Pino, M. A., Manásevich, R. F. and Montero, A.. T-periodic solutions for some second-order differential equations with singularities. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 231–43.CrossRefGoogle Scholar
9Ding, T.. A boundary value problem for the periodic Brillouin focusing system. Acta Sci. Natur. Univ. Pekinensis 11 (1965), 31–8 [in Chinese].Google Scholar
10Ding, T., Iannacci, R. and Zanolin, F.. Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differential Equations 105 (1993), 364–409.CrossRefGoogle Scholar
11Fonda, A., Manasevich, R. and Zanolin, F.. Subharmonic solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24 (1993), 1294–311.CrossRefGoogle Scholar
12Gordon, W.. Conservative dynamical systems involving strong forces. Trans. Amer. Math. Soc. 204 (1975), 113–35.CrossRefGoogle Scholar
13Habets, P. and Sanchez, L.. Periodic solutions of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990), 1035–44.Google Scholar
14Habets, P. and Sanchez, L.. Periodic solutions of dissipative dynamical systems with singular potentials. Differential Integral Equations 3 (1990), 1139–49.CrossRefGoogle Scholar
15Lazer, A. C. and Solimini, S.. On periodic solutions of nonlinear differential equations with singularities. Proc. Amer. Math. Soc. 99 (1987), 109–14.CrossRefGoogle Scholar
16Majer, P.. Ljusternik-Schnirelmann theory with local Palais-Smale conditions and singular dynamical systems. Ann. Inst. H. Poincaré Anal. Non Lineaire 8 (1991), 459–76.CrossRefGoogle Scholar
17Majer, P. and Terracini, S.. Periodic solutions to some problems of n–body type. Arch. Rational Mech. Anal. 124 (1993), 381–404.CrossRefGoogle Scholar
18Mawhin, J.. Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS—Regional Conference Series in Mathematics 40 (Providence, RI: American Mathematical Society, 1979).CrossRefGoogle Scholar
19Mawhin, J.. Topological degree and boundary value problems for nonlinear differential equations. In Topological Methods for Ordinary Differential Equations, eds Furi, M. and Zecca, P., pp. 74–142, Lecture Notes in Mathematics 1537 (New York/Berlin: Springer, 1993).CrossRefGoogle Scholar
20Mawhin, J. and Ward, J. R.. Nonuniform non-resonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations. Rocky Mountain J. Math. 12 (1982), 643–54.CrossRefGoogle Scholar
21Solimini, S.. On forced dynamical systems with a singularity of repulsive type. Nonlinear Anal. 14 (1990), 489–500.CrossRefGoogle Scholar
22Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110 (1976), 353–72.CrossRefGoogle Scholar
23Wang, C.. Multiplicity of periodic solutions for Duffing equations under nonuniform non-resonance conditions (Preprint).Google Scholar
24Ye, Y. and Wang, X.. Nonlinear differential equations in electron beam focusing theory. Acta Math. Appl. Sinica 1 (1978), 13–41 [in Chinese].Google Scholar
25Zhang, M.. Periodic solutions of Lienard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203 (1996), 254–69.CrossRefGoogle Scholar
26Zhang, M.. Periodic solutions of damped differential systems with repulsive singular forces (Preprint).Google Scholar
27Zhang, S.. Multiple closed orbits of fixed energy for N-body-type problems with gravitational potentials. J. Math. Anal. Appl. 208 (1997), 462–75.Google Scholar