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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.

MSC:
39A11 Stability of difference equations (MSC2000)
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[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
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