Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Picone’s identity; forced quasilinear equation; principal solution
Picone’s identity; forced quasilinear equation; principal solution
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.
References:
[1] Došlý, O.: Methods of oscillation theory of half–linear second order differential equations. Czech. Math. J. 125 (2000), 657–671. MR 1777486
[2] Došlý, O.: A remark on conjugacy of half-linear second order differential equations. Math. Slovaca, 50 (2000), 67–79. MR 1764346
[3] Došlý, O.: Half-linear oscillation theory. Stud. Univ. Žilina, Ser. Math. Phys. 13 (2001), 65–73. MR 1874005 | Zbl 1040.34040
[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
[5] Došlý, O., Řezíčková, J.: Regular half-linear second order differential equations. Arch. Math. (Brno) 39 (2003), 233–245. MR 2010724
[6] Elbert, Á.: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. MR 0680591
[7] Elbert, Á. and Kusano, T.: Principal solutions of nonoscillatory half-linear differential equations. Advances in Math. Sci. Appl. 18 (1998), 745–759.
[8] Jaroš, J., Kusano, T.: A Picone type identity for half-linear differential equations. Acta Math. Univ. Comenianae 68 (1999), 127–151. MR 1711081
[9] Jaroš, J., Kusano, T., Yoshida, N.: Forced superlinear oscillations via Picone’s identity. Acta Math. Univ. Comenianae LXIX (2000), 107–113. MR 1796791
[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
[11] Komkov, V.: A generalization of Leighton’s variational theorem. Appl. Anal. 2 (1973), 377–383. MR 0414994
[12] Leighton, W.: Comparison theorems for linear differential equations of second order. Proc. Amer. Math. Soc. 13 (1962), 603–610. MR 0140759 | Zbl 0118.08202
[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. MR 0402184 | Zbl 0327.34027
[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
[15] Müller-Pfeiffer, E.: Comparison theorems for Sturm-Liouville equations. Arch. Math. 22 (1986), 65–73. MR 0868121
[16] Müller-Pfeiffer, E.: Sturm comparison theorems for non-selfadjoint differential equations on non-compact intervals. Math. Nachr. 159 (1992), 291–298. MR 1237116
[17] Swanson, C. A.: Comparison and Oscillation Theory of Linear Differential Equation, Acad. Press, New York, 1968. MR 0463570 | Zbl 1168.92026
Search
Partner of
