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.
@article{107881,
author = {Ond\v rej Do\v sl\'y and Jaroslav Jaro\v s},
title = {A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations},
journal = {Archivum Mathematicum},
volume = {039},
year = {2003},
pages = {335-345},
zbl = {1116.34316},
mrnumber = {2032106},
language = {en},
url = {http://dml.mathdoc.fr/item/107881}
}
Došlý, Ondřej; Jaroš, Jaroslav. A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations. Archivum Mathematicum, Tome 039 (2003) pp. 335-345. http://gdmltest.u-ga.fr/item/107881/
Methods of oscillation theory of half–linear second order differential equations, Czech. Math. J. 125 (2000), 657–671. | MR 1777486
A remark on conjugacy of half-linear second order differential equations, Math. Slovaca, 50 (2000), 67–79. | MR 1764346
Half-linear oscillation theory, Stud. Univ. Žilina, Ser. Math. Phys. 13 (2001), 65–73. (2001) | MR 1874005 | Zbl 1040.34040
Integral characterization of principal solution of half-linear second order differential equations, Studia Sci. Math. Hungar. 36 (2000), 455-469. (2000) | MR 1798750
Regular half-linear second order differential equations, Arch. Math. (Brno) 39 (2003), 233–245. | MR 2010724
A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. (1979) | MR 0680591
Principal solutions of nonoscillatory half-linear differential equations, Advances in Math. Sci. Appl. 18 (1998), 745–759. (1998)
A Picone type identity for half-linear differential equations, Acta Math. Univ. Comenianae 68 (1999), 127–151. (1999) | MR 1711081
Forced superlinear oscillations via Picone’s identity, Acta Math. Univ. Comenianae LXIX (2000), 107–113. (2000) | MR 1796791
Generalized Picone’s formula and forced oscillation in quasilinear differential equations of the second order, Arch. Math. (Brno) 38 (2002), 53–59. | MR 1899568
A generalization of Leighton’s variational theorem, Appl. Anal. 2 (1973), 377–383. (1973) | MR 0414994
Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603–610. (1962) | MR 0140759 | Zbl 0118.08202
On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418–425. (1976) | MR 0402184 | Zbl 0327.34027
Principal and nonprincipal solutions of a nonoscillatory system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. (1988) | MR 1001343
Comparison theorems for Sturm-Liouville equations, Arch. Math. 22 (1986), 65–73. (1986) | MR 0868121
Sturm comparison theorems for non-selfadjoint differential equations on non-compact intervals, Math. Nachr. 159 (1992), 291–298. (1992) | MR 1237116
, Comparison and Oscillation Theory of Linear Differential Equation, Acad. Press, New York, 1968. | MR 0463570 | Zbl 1168.92026