Optimal conditions for maximum and antimaximum principles of the periodic solution problem. (English) Zbl 1200.34001

The aim of this paper is to give several optimal criteria for the existence of maximum or antimaximum principles for the second order operator \(L_q x=x''+q x\), with periodic conditions, where \(q\) is a given periodic, integrable potential. This problem can be fully characterized in terms of periodic and antiperiodic eigenvalues. Alternatively, the maximum or antimaximum principle is closely related with the associated Green’s function and its sign. The author presents an illuminating and unifying review of this topic.


34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B27 Green’s functions for ordinary differential equations
Full Text: DOI EuDML


[1] Barteneva, IV; Cabada, A; Ignatyev, AO, Maximum and anti-maximum principles for the general operator of second order with variable coefficients, Applied Mathematics and Computation, 134, 173-184, (2003) · Zbl 1037.34014
[2] Grunau, H-C; Sweers, G, Optimal conditions for anti-maximum principles, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 30, 499-513, (2001) · Zbl 1072.35066
[3] Mawhin, J; Ortega, R; Robles-Pérez, AM, Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications, Journal of Differential Equations, 208, 42-63, (2005) · Zbl 1082.35040
[4] Pinchover, Y, Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Mathematische Annalen, 314, 555-590, (1999) · Zbl 0928.35010
[5] Takác, P, An abstract form of maximum and anti-maximum principles of Hopf’s type, Journal of Mathematical Analysis and Applications, 201, 339-364, (1996) · Zbl 0855.35016
[6] Campos, J; Mawhin, J; Ortega, R, Maximum principles around an eigenvalue with constant eigenfunctions, Communications in Contemporary Mathematics, 10, 1243-1259, (2008) · Zbl 1170.35029
[7] Cabada, A; Cid, JÁ, On the sign of the Green’s function associated to Hill’s equation with an indefinite potential, Applied Mathematics and Computation, 205, 303-308, (2008) · Zbl 1161.34014
[8] Torres, PJ, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, Journal of Differential Equations, 190, 643-662, (2003) · Zbl 1032.34040
[9] Torres, PJ; Zhang, M, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Mathematische Nachrichten, 251, 101-107, (2003) · Zbl 1024.34030
[10] Magnus W, Winkler S: Hill’s Equation. Dover, New York, NY, USA; 1979:viii+129. · Zbl 1141.34002
[11] Zhang, M, The rotation number approach to eigenvalues of the one-dimensional [inlineequation not available: see fulltext.]-Laplacian with periodic potentials, Journal of the London Mathematical Society. Second Series, 64, 125-143, (2001) · Zbl 1109.35372
[12] Pöschel J, Trubowitz E: The Inverse Spectrum Theory. Academic Press, New York, NY, USA; 1987. · Zbl 0623.34001
[13] Zettl A: Sturm-Liouville Theory, Mathematical Surveys and Monographs. Volume 121. American Mathematical Society, Providence, RI, USA; 2005:xii+328.
[14] Ortega, R, The twist coefficient of periodic solutions of a time-dependent Newton’s equation, Journal of Dynamics and Differential Equations, 4, 651-665, (1992) · Zbl 0761.34036
[15] Ortega, R, Periodic solutions of a Newtonian equation: stability by the third approximation, Journal of Differential Equations, 128, 491-518, (1996) · Zbl 0855.34058
[16] Meng G, Zhang M:Measure differential equations, II, Continuity of eigenvalues in measures with[InlineEquation not available: see fulltext.]topology. preprint · Zbl 1170.35029
[17] Feng, H; Zhang, M, Optimal estimates on rotation number of almost periodic systems, Zeitschrift für Angewandte Mathematik und Physik, 57, 183-204, (2006) · Zbl 1099.34044
[18] Johnson, R; Moser, J, The rotation number for almost periodic potentials, Communications in Mathematical Physics, 84, 403-438, (1982) · Zbl 0497.35026
[19] Johnson, R; Moser, J, Erratum: “The rotation number for almost periodic potentials” [comm. math. phys. vol. 84 (1982), no. 3, 403-438], Communications in Mathematical Physics, 90, 317-318, (1983)
[20] Zhang, M, Continuity in weak topology: higher order linear systems of ODE, Science in China. Series A, 51, 1036-1058, (2008) · Zbl 1151.34007
[21] Chu, J; Lei, J; Zhang, M, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, Journal of Differential Equations, 247, 530-542, (2009) · Zbl 1175.34053
[22] Zhang, M; Li, W, A Lyapunov-type stability criterion using [inlineequation not available: see fulltext.] norms, Proceedings of the American Mathematical Society, 130, 3325-3333, (2002) · Zbl 1007.34053
[23] Chu, J; Nieto, JJ, Recent existence results for second-order singular periodic differential equations, 20, (2009) · Zbl 1182.34057
[24] Cabada, A; Cid, JÁ; Tvrdý, M, A generalized anti-maximum principle for the periodic one-dimensional [inlineequation not available: see fulltext.]-Laplacian with sign-changing potential, Nonlinear Analysis: Theory, Methods & Applications, 72, 3436-3446, (2010) · Zbl 1192.34025
[25] Cabada, A; Lomtatidze, A; Tvrdý, M, Periodic problem involving quasilinear differential operator and weak singularity, Advanced Nonlinear Studies, 7, 629-649, (2007) · Zbl 1146.34016
[26] Zhang, M, Certain classes of potentials for [inlineequation not available: see fulltext.]-Laplacian to be non-degenerate, Mathematische Nachrichten, 278, 1823-1836, (2005) · Zbl 1092.34044
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