Hakl, Robert; Torres, Pedro J. Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign. (English) Zbl 1235.34064 Appl. Math. Comput. 217, No. 19, 7599-7611 (2011). 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. Reviewer: Alfonso Montes-Rodriguez (Sevilla) Cited in 10 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:maximum principle; anti-maximum principle; differential operators PDF BibTeX XML Cite \textit{R. Hakl} and \textit{P. J. Torres}, Appl. Math. Comput. 217, No. 19, 7599--7611 (2011; Zbl 1235.34064) Full Text: DOI References: [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 [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 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.