# Article

 Title: A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations (English) Author: Došlý, Ondřej Author: Jaroš, Jaroslav Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 39 Issue: 4 Year: 2003 Pages: 335-345 Summary lang: English . Category: math . Summary: We extend the classical Leighton comparison theorem to a class of quasilinear forced second order differential equations $(r(t)|x^{\prime }|^{\alpha -2}x^{\prime })^{\prime }+c(t)|x|^{\beta -2}x=f(t)\,,\quad 1<\alpha \le \beta ,\ t\in I=(a,b)\,, \qquad \mathrm {(*)}$ where the endpoints $a$, $b$ of the interval $I$ are allowed to be singular. Some applications of this statement in the oscillation theory of (*) are suggested. (English) Keyword: Picone’s identity Keyword: forced quasilinear equation Keyword: principal solution MSC: 34C10 idZBL: Zbl 1116.34316 idMR: MR2032106 . Date available: 2008-06-06T22:42:34Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107881 . Reference: [1] Došlý, O.: Methods of oscillation theory of half–linear second order differential equations.Czech. Math. J. 125 (2000), 657–671. MR 1777486 Reference: [2] Došlý, O.: A remark on conjugacy of half-linear second order differential equations.Math. Slovaca, 50 (2000), 67–79. MR 1764346 Reference: [3] Došlý, O.: Half-linear oscillation theory.Stud. Univ. Žilina, Ser. Math. Phys. 13 (2001), 65–73. Zbl 1040.34040, MR 1874005 Reference: [4] Došlý, O., Elbert, Á.: Integral characterization of principal solution of half-linear second order differential equations.Studia Sci. Math. Hungar. 36 (2000), 455-469. MR 1798750 Reference: [5] Došlý, O., Řezíčková, J.: Regular half-linear second order differential equations.Arch. Math. (Brno) 39 (2003), 233–245. MR 2010724 Reference: [6] Elbert, Á.: A half-linear second order differential equation.Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. MR 0680591 Reference: [7] Elbert, Á. and Kusano, T.: Principal solutions of nonoscillatory half-linear differential equations.Advances in Math. Sci. Appl. 18 (1998), 745–759. Reference: [8] Jaroš, J., Kusano, T.: A Picone type identity for half-linear differential equations.Acta Math. Univ. Comenianae 68 (1999), 127–151. MR 1711081 Reference: [9] Jaroš, J., Kusano, T., Yoshida, N.: Forced superlinear oscillations via Picone’s identity.Acta Math. Univ. Comenianae LXIX (2000), 107–113. MR 1796791 Reference: [10] Jaroš, J., Kusano, T., Yoshida, N.: Generalized Picone’s formula and forced oscillation in quasilinear differential equations of the second order.Arch. Math. (Brno) 38 (2002), 53–59. MR 1899568 Reference: [11] Komkov, V.: A generalization of Leighton’s variational theorem.Appl. Anal. 2 (1973), 377–383. MR 0414994 Reference: [12] Leighton, W.: Comparison theorems for linear differential equations of second order.Proc. Amer. Math. Soc. 13 (1962), 603–610. Zbl 0118.08202, MR 0140759 Reference: [13] Mirzov, J. D.: On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems.J. Math. Anal. Appl. 53 (1976), 418–425. Zbl 0327.34027, MR 0402184 Reference: [14] Mirzov, J. D.: Principal and nonprincipal solutions of a nonoscillatory system.Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. MR 1001343 Reference: [15] Müller-Pfeiffer, E.: Comparison theorems for Sturm-Liouville equations.Arch. Math. 22 (1986), 65–73. MR 0868121 Reference: [16] Müller-Pfeiffer, E.: Sturm comparison theorems for non-selfadjoint differential equations on non-compact intervals.Math. Nachr. 159 (1992), 291–298. MR 1237116 Reference: [17] Swanson, C. A.: .Comparison and Oscillation Theory of Linear Differential Equation, Acad. Press, New York, 1968. Zbl 1168.92026, MR 0463570 .

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