# Article

 Title: Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations (English) Author: Parhi, N. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 136 Issue: 3 Year: 2011 Pages: 241-258 Summary lang: English . Category: math . Summary: In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $\Delta (p_{n-1}\Delta y_{n-1}) + q y_{n} =0 , \quad n\geq 1,$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $\Delta (p_{n-1}\Delta y_{n-1}) + q_{n}g( y_{n}) = f_{n-1}, \quad n\geq 1,$ where, unlike earlier works, $f_{n}\geq 0$ or $\leq 0$ (but $\not \equiv 0)$ for large $n$. Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form $y_{n+2}+ a_{n}y_{n+1}+ b_{n}y_{n}+ c_{n}y_{n-1}= g_{n-1}, \quad n\geq 1.$ (English) Keyword: oscillation Keyword: non-oscillation Keyword: second order difference equation Keyword: third order difference equation Keyword: generalized zero MSC: 39A06 MSC: 39A10 MSC: 39A12 MSC: 39A21 idZBL: Zbl 1249.39015 idMR: MR2893974 DOI: 10.21136/MB.2011.141647 . Date available: 2011-09-22T14:55:17Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/141647 . Reference: [1] Chen, S.: Disconjugacy, disfocality and oscillation of second order difference equations.J. Diff. Eqs. 107 (1994), 383-394. Zbl 0791.39001, MR 1264528, 10.1006/jdeq.1994.1018 Reference: [2] Elaydi, S. N.: An Introduction to Difference Equations.Springer, New York (2005). Zbl 1071.39001, MR 2128146 Reference: [3] Hooker, J. W., Patula, W. T.: Riccati type transformations for second order linear difference equations.J. Math. Anal. Appl. 82 (1981), 451-462. Zbl 0471.39007, MR 0629769, 10.1016/0022-247X(81)90208-0 Reference: [4] Hooker, J. W., Kwong, M. K., Patula, W. T.: Oscillatory second order linear difference equations and Riccati equations.SIAM J. Math. Anal. 18 (1987), 54-63. Zbl 0619.39005, MR 0871820, 10.1137/0518004 Reference: [5] Kelley, W. G., Peterson, A. C.: Difference Equations: An Introduction with Applications.Harcourt/Academic Press, San Diego (2001). Zbl 0970.39001, MR 1765695 Reference: [6] Parhi, N.: On disconjugacy and conjugacy of second order linear difference equations.J. Indian Math. Soc. 68 (2001), 221-232. Zbl 1141.39300, MR 1929838 Reference: [7] Parhi, N.: Oscillation of forced nonlinear second order self-adjoint difference equations.Indian J. Pure Appl. Math. 34 (2003), 1611-1624. Zbl 1044.39010, MR 2020679 Reference: [8] Parhi, N., Panda, A.: Oscillation of solutions of forced nonlinear second order difference equations.Proc. Eighth Ramanujan Symposium on Recent Developments in Nonlinear Systems R. Sahadevan, M. Lakshmanan Narosa Pub. House, New Delhi (2002), 221-238. Zbl 1056.39010, MR 2010625 Reference: [9] Parhi, N., Panda, A.: Oscillatory and non-oscillatory behaviour of solutions of difference equations of the third order.Math. Bohem. 133 (2008), 99-112. MR 2400154 Reference: [10] Parhi, N., Tripathy, A. K.: Oscillatory behaviour of second order difference equations.Commun. Appl. Nonlin. Anal. 6 (1999), 79-100. MR 1665966 Reference: [11] Parhi, N., Tripathy, A. K.: On oscillatory third-order difference equations.J. Difference Eq. Appl. 6 (2000), 53-74. Zbl 0963.39009, MR 1752155, 10.1080/10236190008808213 Reference: [12] Parhi, N., Tripathy, A. K.: On the behaviour of solutions of a class of third order difference equations.J. Difference Eq. Appl. 8 (2002), 415-426. Zbl 1037.39001, MR 1897066, 10.1080/10236190290017423 Reference: [13] Patula, W.: Growth and oscillation properties of second order linear difference equations.SIAM J. Math. Anal. 10 (1979), 55-61. Zbl 0397.39001, MR 0516749, 10.1137/0510006 Reference: [14] Patula, W.: Growth, oscillation and comparison theorems for second order difference equations.SIAM J. Math. Anal. 10 (1979), 1272-1279. MR 0547812, 10.1137/0510114 .

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