Geodesic
Limit and integral properties of principal solutions for half-linear differential equations
Archivum mathematicum, Tome 43 (2007) no. 1, pp. 75-86
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Some asymptotic properties of principal solutions of the half-linear differential equation \[ (a(t)\Phi (x^{\prime }))^{\prime }+b(t)\Phi (x)=0\,, \qquad \mathrm {(*)}\] $\Phi (u)=|u|^{p-2}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.
Some asymptotic properties of principal solutions of the half-linear differential equation \[ (a(t)\Phi (x^{\prime }))^{\prime }+b(t)\Phi (x)=0\,, \qquad \mathrm {(*)}\] $\Phi (u)=|u|^{p-2}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.
Classification : 34C10, 34C11
Keywords: half-linear equation; principal solution; limit characterization; integral characterization 
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Cecchi, Mariella; Došlá, Zuzana; Marini, Mauro. Limit and integral properties of principal solutions for half-linear differential equations. Archivum mathematicum, Tome 43 (2007) no. 1, pp. 75-86. http://geodesic.mathdoc.fr/item/ARM_2007_43_1_a7/

[1] Agarwal R. P., Grace S. R., O’Regan D.: Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations. Kluwer Acad. Publ., Dordrecht, The Netherlands, 2002. | MR | Zbl

[2] Cecchi M., Došlá Z., Marini M.: On nonoscillatory solutions of differential equations with $p$-Laplacian. Adv. Math. Sci. Appl. 11 1 (2001), 419–436. | MR | Zbl

[3] Cecchi M., Došlá Z., Marini M.: Half-linear equations and characteristic properties of the principal solution. J. Differential Equ. 208, 2005, 494-507; Corrigendum, J. Differential Equations 221 (2006), 272–274. | MR

[4] Cecchi M., Došlá Z., Marini M.: Half-linear differential equations with oscillating coefficient. Differential Integral Equations 18 11 (2005), 1243–1256. | MR | Zbl

[5] Cecchi M., Došlá Z., Marini M., Vrkoč I.: Integral conditions for nonoscillation of second order nonlinear differential equations. Nonlinear Anal. 64 (2006), 1278–1289. | MR | Zbl

[6] Došlá Z., Vrkoč I.: On extension of the Fubini theorem and its application to the second order differential equations. Nonlinear Anal. 57 (2004), 531–548. | MR

[7] Došlý O., Elbert Á.: Integral characterization of the principal solution of half-linear second order differential equations. Studia Sci. Math. Hungar. 36 (2000), 455–469. | MR

[8] Došlý O., Řehák P.: Half-linear Differential Equations. North-Holland, Mathematics Studies 202, Elsevier, Amsterdam, 2005. | MR | Zbl

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

[10] Elbert Á.: On the half-linear second order differential equations. Acta Math. Hungar. 49 (1987), 487–508. | MR | Zbl

[11] Elbert Á., Kusano T.: Principal solutions of non-oscillatory half-linear differential equations. Adv. Math. Sci. Appl. 8 2 (1998), 745–759. | MR | Zbl

[12] Fan X., Li W. T., Zhong C.: A classification scheme for positive solutions of second order iterative differential equations. Electron. J. Differential Equations 25 (2000), 1–14.

[13] Hartman P.: Ordinary Differential Equations. 2nd ed., Birkhäuser, Boston–Basel–Stuttgart, 1982. | MR | Zbl

[14] Hoshino H., Imabayashi R., Kusano T., Tanigawa T.: On second-order half-linear oscillations. Adv. Math. Sci. Appl. 8 1 (1998), 199–216. | MR | Zbl

[15] Jaroš J., Kusano T., Tanigawa T.: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation. Results Math. 43 (2003), 129–149. | MR | Zbl

[16] Jingfa W.: On second order quasilinear oscillations. Funkcial. Ekvac. 41 (1998), 25–54. | MR | Zbl

[17] Mirzov J. D.: Asymptotic Properties of Solutions of the Systems of Nonlinear Nonautonomous Ordinary Differential Equations. (Russian), Maikop, Adygeja Publ., 1993; the english version: Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. 14 2004. | MR