zbMATH — the first resource for mathematics

Oscillatory properties of second order half-linear difference equations. (English) Zbl 0982.39004
Summary: We study oscillatory properties of the second order half-linear difference equation \[ \Delta(r_k|\Delta y_k|^{\alpha -2}\Delta y_k)-p_k|y_{k+1}|^{\text{ad}}y_{k+1}=0, \quad \alpha >1. \tag{HL} \] It is shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation \[ \Delta(r_k\Delta y_k)-p_ky_{k+1}=0. \] We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.

39A11 Stability of difference equations (MSC2000)
Full Text: DOI EuDML
[1] R. P. Agarwal: Difference equations and inequalities, theory, methods, and applications, the second edition. Pure and Appl. Math, M. Dekker, New York-Basel-Hong Kong, 2000.
[2] C. D. Ahlbrandt and A. C. Peterson: Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Kluwer Academic Publishers, Boston, 1996. · Zbl 0860.39001
[3] O. Došlý: Oscillation criteria for higher order Sturm-Liouville difference equations. J. Differ. Equations Appl. 4 (1998), 425-450. · Zbl 0921.39005
[4] O. Došlý: Oscillation criteria for half-linear second order differential equations. Hiroshima Math. J. 28 (1998), 507-521. · Zbl 0920.34042
[5] O. Došlý: A remark on conjugacy of half-linear second order differential equations. Math. Slovaca 50 (2000), 67-79. · Zbl 0959.34025
[6] O. Došlý and P. Řehák: Nonoscillation criteria for second order half-linear difference equations. Comput. Math. Appl, In press. · Zbl 1006.39012
[7] Á. Elbert: A half-linear second order differential equations. Colloq. Math. Soc. János Bolayi 30 (1979), 158-180.
[8] Á. Elbert and T. Kusano: Principal solutions of nonoscillatory half-linear differential equations. Adv. Math. Sci. Appl. (Tokyo) 8 (1998), 745-759. · Zbl 0914.34031
[9] J. Jaroš and T. Kusano: A Picone type identity for second order half-linear differential equations. Acta Math. Univ. Comenian. (N. S.) 68 (1999), 137-151. · Zbl 0926.34023
[10] W. G. Kelley and A. Peterson: Difference Equations: An Introduction with Applications. Acad. Press, San Diego, 1991. · Zbl 0733.39001
[11] J. D. Mirzov: On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems. J. Math. Anal. Appl. 3 (1976), 418-425. · Zbl 0327.34027
[12] J. D. Mirzov: Principial and nonprincipial solutions of a nonoscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100-117.
[13] P. Řehák: Half-linear discrete oscillation theory. Proceedings of 6th Colloquium on the qualitative theory of DE, Szeged 1999, , EJQTDE, Szeged, 2000, pp. 1-14. · Zbl 0967.39005
[14] P. Řehák: Half-linear dynamic equations on time scales: IVP and oscillatory properties. Submitted. · Zbl 1037.34002
[15] P. Řehák: Hartman-Wintner type lemma, oscillation and conjugacy criteria for half-linear difference equations. J. Math. Anal. Appl. 252 (2000), 813-827. · Zbl 0969.39009
[16] P. Řehák: Oscillation criteria for second order half-linear difference equations. J. Differ. Equations Appl, In press. · Zbl 1009.39012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.