Regular half-linear second order differential equations
Došlý, Ondřej ; Řezníčková, Jana
Archivum Mathematicum, Tome 039 (2003), p. 233-245 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation \[ \left(r(t)\Phi (x^{\prime })\right)^{\prime }+c(t)\Phi (x)=0\,,\quad \Phi (x):=|x|^{p-2}x\,,\quad p>1 \qquad \mathrm {{(*)}}\] and we show that if (*) is regular, a solution $x$ of this equation such that $x^{\prime }(t)\ne 0$ for large $t$ is principal if and only if \[ \int ^\infty \frac{dt}{r(t)x^2(t)|x^{\prime }(t)|^{p-2}}=\infty \,. \] Conditions on the functions $r,c$ are given which guarantee that (*) is regular.

Publié le : 2003-01-01
Classification:  34C10 
@article{107870,
     author = {Ond\v rej Do\v sl\'y and Jana \v Rezn\'\i \v ckov\'a},
     title = {Regular half-linear second order differential equations},
     journal = {Archivum Mathematicum},
     volume = {039},
     year = {2003},
     pages = {233-245},
     zbl = {1119.34029},
     mrnumber = {2010724},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107870}
}

Došlý, Ondřej; Řezníčková, Jana. Regular half-linear second order differential equations. Archivum Mathematicum, Tome 039 (2003) pp. 233-245. http://gdmltest.u-ga.fr/item/107870/

Allegretto W.; Huang Y. X. A Picone’s identity for the $p$-Laplacian and applications, Nonlin. Anal. 32 (1998), 819–830. (1998) | MR 1618334 | Zbl 0930.35053

Cecchi M.; Došlá Z.; Marini M. Principal solutions and minimal set for quasilinear differential equations, to appear in Dynam. Syst. Appl. | MR 2140874

Došlý O.; Elbert Á. Integral characterization of the principal solution of half-linear differential equations, Studia Sci. Math. Hungar. 36 (2000), No. 3-4, 455–469. | MR 1798750

Došlý O.; Lomtatidze A. Oscillation and nonoscillation criteria for half-linear second order differential equations, submitted. | Zbl 1123.34028

Elbert Á. A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), 158–180. (1979)

Elbert Á. Asymptotic behaviour of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar. 19 (1984), 447–464. (1984) | MR 0874513 | Zbl 0629.34066

Elbert Á. The Wronskian and the half-linear differential equations, Studia Sci. Math. Hungar. 15 (1980), 101–105. (1980) | MR 0681431 | Zbl 0522.34034

Elbert Á.; Kusano T. Principal solutions of nonoscillatory half-linear differential equations, Advances in Math. Sci. Appl. 18 (1998), 745–759. (1998) | MR 1657164

Elbert Á.; Schneider A. Perturbation of the half-linear Euler differential equations, Result. Math. 37 (2000), 56–83. | MR 1742294

Hartman P. Ordinary Differential Equations, John Wiley, New York, 1964. (1964) | MR 0171038 | Zbl 0125.32102

Jaroš J.; Kusano T. A Picone type identity for half-linear differential equations, Acta Math. Univ. Comenianea 68 (1999), 137–151. (1999) | MR 1711081

Leighton W.; Morse M. Singular quadratic functionals, Trans. Amer. Math. Soc. 40 (1936) 252–286. (1936) | MR 1501873 | Zbl 0015.02701

Lorch L.; Newman J. D. A supplement to the Sturm separation theorem, with applications, Amer. Math. Monthly 72 (1965), 359–366, 390. (1965) | MR 0176147 | Zbl 0135.29702

Mirzov J. D. On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418–426. (1976) | MR 0402184 | Zbl 0327.34027

Mirzov J. D. Principal and nonprincipal solutions of a nonoscillatory system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. (1988) | MR 1001343