Opuscula Mathematica
Opuscula Math. 37, no. 3 (2017), 389-402

General solutions of second-order linear difference equations of Euler type

Akane Hongyo
Naoto Yamaoka
Abstract. The purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation \(y^{\prime\prime}+(\lambda/t^2)y=0\) or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations.
Keywords: Euler-Cauchy equations, oscillation, conditionally oscillatory.
Mathematics Subject Classification: 39A06, 39A12, 39A21.
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References ▼
  1. [1] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An introduction with applications, Birkhäuser, Boston, 2001.
  2. [2] O. Došlý, R. Hilscher, A class of Sturm-Liouville difference equations: (non)oscillation constants and property BD, Comput. Math. Appl. 45 (2003), 961-981.
  3. [3] S. Elaydi, An Introduction to Difference Equations, Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.
  4. [4] L. Erbe, A. Peterson, Recent results concerning dynamic equations on time scales, Electron. Trans. Numer. Anal. 27 (2007), 51-70.
  5. [5] S. Fišnarová, Oscillation of two-term Sturm-Liouville difference equations, Int. J. Difference Equ. 1 (2006), 81-99.
  6. [6] P. Hartman, Ordinary differential equations, John Wiley & Sons, New York, London, Sydney, 1964.
  7. [7] P. Hasil, M. Veselý, Oscillation constants for half-linear difference equations with coefficients having mean values, Adv. Difference Equ. 2015, 2015:210.
  8. [8] E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234-252.
  9. [9] W. Kelley, A. Peterson, Difference Equations: An introduction with applications, 2nd ed., Harcourt/Academic Press, San Diego, 2001.
  10. [10] F. Luef, G. Teschl, On the finiteness of the number of eigenvalues of Jacobi operators below the essential spectrum, J. Difference Equ. Appl. 10 (2004), 299-307.
  11. [11] P. Řehák, How the constants in Hille-Nehari theorems depend on time scales, Adv. Difference Equ. 2006, Art. ID 64534.
  12. [12] P. Řehák, A critical oscillation constant as a variable of time scales for half-linear dynamic equations, Math. Slovaca 60 (2010), 237-256.
  13. [13] J. Sugie, N. Yamaoka, An infinite sequence of nonoscillation theorems for second-order nonlinear differential equations of Euler type, Nonlinear Anal. 50 (2002), 373-388.
  14. [14] C.A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968.
  15. [15] N. Yamaoka, Oscillation criteria for second-order nonlinear difference equations of Euler type, Adv. Difference Equ. 2012, 2012:218.
  16. [16] N. Yamaoka, J. Sugie, Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations, Ukraïn. Mat. Zh. 58 (2006), 1704-1714.
  17. [17] G. Zhang, S.S. Cheng, A necessary and sufficient oscillation condition for the discrete Euler equation, Panamer. Math. J. 9 (1999), 29-34.
About the authors ▼
Akane Hongyo
Osaka Prefecture University, Department of Mathematical Sciences, Sakai 599-8531, Japan
Naoto Yamaoka
Osaka Prefecture University, Department of Mathematical Sciences, Sakai 599-8531, Japan
About the article ▼
Communicated by P.A. Cojuhari.
Received: 2016-08-30.
Revised: 2016-09-30.
Accepted: 2016-09-30.
Published online: 2017-01-30.
Cite this article as ▼
Akane Hongyo, Naoto Yamaoka, General solutions of second-order linear difference equations of Euler type, Opuscula Math. 37, no. 3 (2017), 389-402, http://dx.doi.org/10.7494/OpMath.2017.37.3.389.
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