De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation

Báňa, Libor   ; Došlý, Ondřej

Archivum mathematicum, Tome 50 (2014) no. 4, pp. 193-203

Abstract (VO)

We study the generalized half-linear second order differential equation via the associated Riccati type differential equation and Prüfer transformation. We establish a de la Vallée Poussin type inequality for the distance of consecutive zeros of a nontrivial solution and this result we apply to the “classical” half-linear differential equation regarded as a perturbation of the half-linear Euler differential equation with the so-called critical oscillation constant. In the second part of the paper we study a Dirichlet eigenvalue problem associated with the investigated half-linear equation.

Détail

DOI : 10.5817/AM2014-4-193

Classification : 34C10
Keywords: generalized half-linear differential equation; de la Vallée Poussin inequality; half-linear Euler differential equation; Dirichlet eigenvalue problem

Bibliographie

[1] Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory of Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers,Dordrecht-Boston-London, 2002. | MR

[2] Andres, J.: On the criterion of de la Vallée Poussin. Publ. Math. Debrecen 45 (1994), 145–152. | MR

[3] Bihari, I.: On the second order half-linear differential equation. Studia Sci. Math. Hungar. 3 (1968), 411–437. | MR | Zbl

[4] Bihari, I.: Notes on eigenvalues and zeros of the solutions of half-linear second order ordinary differential equation. Period. Math. Hungar. (1976), 117–125. | DOI | MR

[5] Bognár, G., Došlý, O.: Conditional oscillation and principal solution of generalized half-linear differential equation. Publ. Math. Debrecen 82 (2013), 451–459. | DOI | MR | Zbl

[6] Cohn, J.H.E.: On an oscillation criterion of de la Vallée Poussin. Quart. J. Math. Oxford Ser. (2) 39 (159) (1988), 173–174. | DOI | MR | Zbl

[7] de Vallée Poussin, Ch.: Sur l’équation différentielle linéqire du second order. Détermination d’une intégrale par deux valuers assignés. Extension aux équasions d’ordre $n$. J. Math. Pures Appl. (8) (1929), 125–144.

[8] Došlý, O., Fišnarová, S.: Half-linear oscillation criteria: perturbation in term involving derivative. Nonlinear Anal. 73 (2010), 3756–3766. | DOI | MR | Zbl

[9] Došlý, O., Funková, H.: Perturbations of half-linear Euler differential equation and transformations of modified Riccati equation. Abstr. Appl. Anal. 2012 (2012), 19pp. | MR | Zbl

[10] Došlý, O., Lomtatidze, A.: Disconjugacy and disfocality criteria for singular half-linearsecond order differential equations. Ann. Polon. Math. 72 (1999), 273–284. | MR

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

[12] Došlý, O., Řezníčková, J.: Conjugacy and principal solution of generalized half-linear second order differential equations. Electron. J. Qual. Theory Differ. Equ., Proc.9th Coll. QTDE 2012 (5) (2012), 1–13. | MR

[13] Došlý, O., Ünal, M.: Conditionally oscillatory half-linear differential equations. Acta Math. Hungar. 120 (2008), 147–163. | DOI | MR

[14] Došlý, O., Yamaoka, N.: Oscillation constants for second-order ordinary differential equations related to elliptic equations with $p$-Laplacian. Nonlinear Anal. 113 (2015), 115–136. | MR

[15] Elber, Á., Kusano, T.: On differential equation $y^{\prime \prime }+p(t)\vert y^{\prime }\vert \,{\rm sgn}\, y+q(t)y=0$. Hiroshima Math. J. 22 (1992), 203–218. | MR

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

[17] Elbert, Á.: Generalized Riccati equation for half-linear second order differentia lequations. Colloq. Math. Soc. János Bolyai 47 (1984), 227–249. | MR

[18] Elbert, Á., Schneider, A.: Perturbations of the half-linear Euler differential equation. Result. Math. 37 (2000), 56–83. | DOI | MR | Zbl

[19] Fišnarová, S., Mařík, R.: Half-linear ODE and modified Riccati equation: comparison theorems, integral characterization of principal solution. Nonlinear Anal. 74 (2011), 6427–6433. | DOI | MR | Zbl

[20] Fišnarová, S., Mařík, R.: Local estimates for modified Riccati equation in theory of half-linear differential equation. Electron. J. Qual. Theory Differ. Equ. 2012 (63) (2012), 15 pp. | MR

[21] Harris, B.J.: On the oscillation criterion of Cohn. Quart. J. Math. Oxford Ser. (2) 42 (167) (1988), 309–9313. | MR

[22] Hartman, P., Wintner, A.: On an oscillation criterion of De la Vallée Poussin. Quart. Appl. Math. 13 (1955), 330–332. | MR | Zbl

[23] Jaroš, J., Kusano, T., Tanigawa, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions. Nonlinear Anal. 64 (2006), 762–787. | DOI | MR

[24] Kiguradze, I., Půža, B.: On the Vallée-Poussin problem for singular differential equations with deviating arguments. Arch. Math. (Brno) 33 (1–2) (1997), 127–138. | MR | Zbl

[25] Kusano, T., Manojlović, J., Tanigawa, T.: Existence of regularly varying solutions with nonzero indices of half-linear differential equations with retarded arguments. Comput. Math. Appl. 59 (2010), 411–425. | DOI | MR | Zbl

[26] Willet, D.: Generalized de la Vallée Pousin disconjugacy test for linear differential equations. Canad. Math. Bull 14 (1971), 419–430. | DOI