# A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations

Archivum Mathematicum (2003)

- Volume: 039, Issue: 4, page 335-345
- ISSN: 0044-8753

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topDošlý, Ondřej, and Jaroš, Jaroslav. "A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations." Archivum Mathematicum 039.4 (2003): 335-345. <http://eudml.org/doc/249136>.

@article{Došlý2003,

abstract = {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.},

author = {Došlý, Ondřej, Jaroš, Jaroslav},

journal = {Archivum Mathematicum},

keywords = {Picone’s identity; forced quasilinear equation; principal solution; Picone's identity; principal solution},

language = {eng},

number = {4},

pages = {335-345},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {A singular version of Leighton's comparison theorem for forced quasilinear second order differential equations},

url = {http://eudml.org/doc/249136},

volume = {039},

year = {2003},

}

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

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 039

IS - 4

SP - 335

EP - 345

AB - 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.

LA - eng

KW - Picone’s identity; forced quasilinear equation; principal solution; Picone's identity; principal solution

UR - http://eudml.org/doc/249136

ER -

## References

top- Methods of oscillation theory of half–linear second order differential equations, Czech. Math. J. 125 (2000), 657–671. MR1777486
- A remark on conjugacy of half-linear second order differential equations, Math. Slovaca, 50 (2000), 67–79. MR1764346
- Half-linear oscillation theory, Stud. Univ. Žilina, Ser. Math. Phys. 13 (2001), 65–73. (2001) Zbl1040.34040MR1874005
- Integral characterization of principal solution of half-linear second order differential equations, Studia Sci. Math. Hungar. 36 (2000), 455-469. (2000) MR1798750
- Regular half-linear second order differential equations, Arch. Math. (Brno) 39 (2003), 233–245. MR2010724
- A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), 153–180. (1979) MR0680591
- 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) MR1711081
- Forced superlinear oscillations via Picone’s identity, Acta Math. Univ. Comenianae LXIX (2000), 107–113. (2000) MR1796791
- Generalized Picone’s formula and forced oscillation in quasilinear differential equations of the second order, Arch. Math. (Brno) 38 (2002), 53–59. MR1899568
- A generalization of Leighton’s variational theorem, Appl. Anal. 2 (1973), 377–383. (1973) MR0414994
- Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603–610. (1962) Zbl0118.08202MR0140759
- On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418–425. (1976) Zbl0327.34027MR0402184
- Principal and nonprincipal solutions of a nonoscillatory system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. (1988) MR1001343
- Comparison theorems for Sturm-Liouville equations, Arch. Math. 22 (1986), 65–73. (1986) MR0868121
- Sturm comparison theorems for non-selfadjoint differential equations on non-compact intervals, Math. Nachr. 159 (1992), 291–298. (1992) MR1237116
- [unknown], Comparison and Oscillation Theory of Linear Differential Equation, Acad. Press, New York, 1968. Zbl1168.92026MR0463570

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