Geodesic
A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations
Archivum mathematicum, Tome 39 (2003) no. 4, pp. 335-345
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 34C10
Keywords: Picone’s identity; forced quasilinear equation; principal solution 
@article{ARM_2003_39_4_a8,
     author = {Do\v{s}l\'y, Ond\v{r}ej and Jaro\v{s}, Jaroslav},
     title = {A singular version of {Leighton's} comparison theorem for forced quasilinear second order differential equations},
     journal = {Archivum mathematicum},
     pages = {335--345},
     year = {2003},
     volume = {39},
     number = {4},
     mrnumber = {2032106},
     zbl = {1116.34316},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2003_39_4_a8/}
}
TY  - JOUR
AU  - Došlý, Ondřej
AU  - Jaroš, Jaroslav
TI  - A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations
JO  - Archivum mathematicum
PY  - 2003
SP  - 335
EP  - 345
VL  - 39
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2003_39_4_a8/
LA  - en
ID  - ARM_2003_39_4_a8
ER  - 
%0 Journal Article
%A Došlý, Ondřej
%A Jaroš, Jaroslav
%T A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations
%J Archivum mathematicum
%D 2003
%P 335-345
%V 39
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2003_39_4_a8/
%G en
%F ARM_2003_39_4_a8
Došlý, Ondřej; Jaroš, Jaroslav. A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations. Archivum mathematicum, Tome 39 (2003) no. 4, pp. 335-345. http://geodesic.mathdoc.fr/item/ARM_2003_39_4_a8/

[1] Došlý, O.: Methods of oscillation theory of half–linear second order differential equations. Czech. Math. J. 125 (2000), 657–671. | MR

[2] Došlý, O.: A remark on conjugacy of half-linear second order differential equations. Math. Slovaca, 50 (2000), 67–79. | MR

[3] Došlý, O.: Half-linear oscillation theory. Stud. Univ. Žilina, Ser. Math. Phys. 13 (2001), 65–73. | MR | Zbl

[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

[5] Došlý, O., Řezíčková, J.: Regular half-linear second order differential equations. Arch. Math. (Brno) 39 (2003), 233–245. | MR

[6] Elbert, Á.: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. | MR

[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

[9] Jaroš, J., Kusano, T., Yoshida, N.: Forced superlinear oscillations via Picone’s identity. Acta Math. Univ. Comenianae LXIX (2000), 107–113. | MR

[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

[11] Komkov, V.: A generalization of Leighton’s variational theorem. Appl. Anal. 2 (1973), 377–383. | MR

[12] Leighton, W.: Comparison theorems for linear differential equations of second order. Proc. Amer. Math. Soc. 13 (1962), 603–610. | MR | Zbl

[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 | Zbl

[14] Mirzov, J. D.: Principal and nonprincipal solutions of a nonoscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. | MR

[15] Müller-Pfeiffer, E.: Comparison theorems for Sturm-Liouville equations. Arch. Math. 22 (1986), 65–73. | MR

[16] Müller-Pfeiffer, E.: Sturm comparison theorems for non-selfadjoint differential equations on non-compact intervals. Math. Nachr. 159 (1992), 291–298. | MR

[17] Swanson, C. A.: Comparison and Oscillation Theory of Linear Differential Equation, Acad. Press, New York, 1968. | MR | Zbl