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Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign. (English) Zbl 1235.34064

The authors study criteria for the existence of a maximum or anti-maximum principle of a general second order operator with periodic conditions and conditions for non-resonance.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Barteneva, I.V.; Cabada, A.; Ignatyev, A.O., Maximum and anti-maximum principles for the general operator of second order with variable coefficients, Appl. math. comput., 134, 1, 173-184, (2003) · Zbl 1037.34014
[2] Cabada, A.; Cid, J.A., On the sign of the green’s function associated to hill’s equation with an indefinite potential, Appl. math. comput., 205, 303-308, (2008) · Zbl 1161.34014
[3] Chu, J.; Torres, P.J., Applications of schauder’s fixed point theorem to singular differential equations, Bull. London math. soc., 39, 653-660, (2007) · Zbl 1128.34027
[4] Hakl, R.; Mukhigulashvili, S., A periodic boundary value problem for functional differential equations of higher order, Georgian math. J., 16, 4, 651-665, (2009) · Zbl 1187.34085
[5] I.T. Kiguradze, Boundary value problems for systems of ordinary differential equations, Itogi Nauki i Tekhniki, Current Problems in Mathematics, Newest results, vol. 30, No. 204, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, pp. 3-103. Translated in J. Soviet Math. 43(2) (1988) 2259-2339. · Zbl 0631.34020
[6] Kiguradze, I., On periodic solutions of nth order ordinary differential equations, Nonlinear anal., 40, 1-8, 309-321, (2000) · Zbl 0953.34028
[7] Kiguradze, I.; Lomtatidze, A., Periodic solutions of nonautonomous ordinary differential equations, Monatsh. math., 159, 3, 235-252, (2010) · Zbl 1194.34076
[8] Lasota, A.; Opial, Z., Sur LES solutions périodiques des équations différentielles ordinaires, Ann. polon. math., 16, 69-94, (1964) · Zbl 0142.35303
[9] Li, X.; Zhang, Z., Periodic solutions for damped differential equations with a weak repulsive singularity, Nonlinear anal., 70, 2395-2399, (2009) · Zbl 1165.34349
[10] Omari, P.; Trombetta, M., Remarks on the lower and upper solutions method for second and third-order periodic boundary value problems, Appl. math. comput., 50, 1-21, (1992) · Zbl 0760.65078
[11] Torres, P.J., Existence of one – signed periodic solutions of some second – order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040
[12] Torres, P.J., Weak singularities may help periodic solutions to exist, J. differential equations, 232, 277-284, (2007) · Zbl 1116.34036
[13] Torres, P.J., Existence and stability of periodic solutions of a Duffing equation by using a new maximum principle, Mediterr. J. math., 1, 4, 479-486, (2004) · Zbl 1115.34037
[14] Torres, P.J.; Zhang, M., A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. nachr., 251, 101-107, (2003) · Zbl 1024.34030
[15] Wang, H.; Li, Y., Existence and uniqueness of periodic solutions for Duffing equations across many points of resonance, J. differential equations, 108, 1, 152-169, (1994) · Zbl 0799.34038
[16] Wang, H.; Li, Y., Periodic solutions for Duffing equations, Nonlinear anal., 24, 7, 961-979, (1995) · Zbl 0828.34030
[17] Wang, Y.; Lian, H.; Ge, W., Periodic solutions for a second order nonlinear functional differential equation, Appl. math. lett., 20, 110-115, (2007) · Zbl 1151.34056
[18] Zhang, M., Optimal criteria for maximum and antimaximum principles of the periodic solution problem, Bound. value probl., 2010, 26, (2010), Article ID 410986 · Zbl 1200.34001
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