Journal Menu
- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 471435, 19 pages
doi:10.1155/2012/471435
Research Article
Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients
1Department of Mathematics, Mendel University in Brno, Zemědělská 1, 613 00 Brno, Czech Republic
2Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
Received 28 May 2012; Revised 10 September 2012; Accepted 13 September 2012
Academic Editor: Elena Braverman
Copyright © 2012 Petr Hasil and Michal Veselý. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- S. Z. Chen and L. H. Erbe, “Riccati techniques and discrete oscillations,” Journal of Mathematical Analysis and Applications, vol. 142, no. 2, pp. 468–487, 1989. View at Publisher · View at Google Scholar
- J. W. Hooker and W. T. Patula, “Riccati type transformations for second-order linear difference equations,” Journal of Mathematical Analysis and Applications, vol. 82, no. 2, pp. 451–462, 1981. View at Publisher · View at Google Scholar
- M. K. Kwong, J. W. Hooker, and W. T. Patula, “Riccati type transformations for second-order linear difference equations, II,” Journal of Mathematical Analysis and Applications, vol. 107, no. 1, pp. 182–196, 1985. View at Publisher · View at Google Scholar
- Z. Došlá and Š. Pechancová, “Conjugacy and phases for second order linear difference equation,” Computers & Mathematics with Applications, vol. 53, no. 7, pp. 1129–1139, 2007. View at Publisher · View at Google Scholar
- O. Došlý and Š. Pechancová, “Generalized zeros of symplectic difference system and of its reciprocal system,” Advances in Difference Equations, vol. 2011, Article ID 571935, 23 pages, 2011. View at Publisher · View at Google Scholar
- S. Z. Chen and L. H. Erbe, “Oscillation and nonoscillation for systems of self-adjoint second-order difference equations,” SIAM Journal on Mathematical Analysis, vol. 20, no. 4, pp. 939–949, 1989. View at Publisher · View at Google Scholar
- R. Koplatadze, G. Kvinikadze, and I. P. Stavroulakis, “Oscillation of second-order linear difference equations with deviating arguments,” Advances in Mathematical Sciences and Applications, vol. 12, no. 1, pp. 217–226, 2002.
- S. H. Saker and S. S. Cheng, “Kamenev type oscillation criteria for nonlinear difference equations,” Czechoslovak Mathematical Journal, vol. 54, no. 4, pp. 955–967, 2004. View at Publisher · View at Google Scholar
- O. Došlý and R. Hilscher, “A class of Sturm-Liouville difference equations: (non)oscillation constants and property BD,” Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 961–981, 2003. View at Publisher · View at Google Scholar
- S. Fišnarová, “Oscillation of two-term Sturm-Liouville difference equations,” International Journal of Difference Equations, vol. 1, no. 1, pp. 81–99, 2006.
- P. Hasil, “Conjugacy of self-adjoint difference equations of even order,” Abstract and Applied Analysis, vol. 2011, Article ID 814962, 16 pages, 2011. View at Publisher · View at Google Scholar
- O. Došlý, “Constants in the oscillation theory of higher order Sturm-Liouville differential equations,” Electronic Journal of Differential Equations, vol. 2002, no. 34, pp. 1–12, 2002.
- L. Erbe and S. Hilger, “Sturmian theory on measure chains,” Differential Equations and Dynamical Systems, vol. 1, no. 3, pp. 223–244, 1993.
- L. Erbe, “Oscillation results for second-order linear equations on a time scale,” Journal of Difference Equations and Applications, vol. 8, no. 11, pp. 1061–1071, 2002. View at Publisher · View at Google Scholar
- B. Jia, “Wong's comparison theorem for second order linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 349, no. 2, pp. 556–567, 2009. View at Publisher · View at Google Scholar
- L. Erbe, J. Baoguo, and A. Peterson, “Oscillation and nonoscillation of solutions of second order linear dynamic equations with integrable coefficients on time scales,” Applied Mathematics and Computation, vol. 215, no. 5, pp. 1868–1885, 2009. View at Publisher · View at Google Scholar
- D. Willett, “On the oscillatory behavior of the solutions of second order linear differential equations,” Annales Polonici Mathematici, vol. 21, pp. 175–194, 1969.
- J. S. W. Wong, “Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients,” Transactions of the American Mathematical Society, vol. 144, pp. 197–215, 1969.
- B. G. Zhang and Y. Zhou, “Comparison theorems and oscillation criteria for difference equations,” Journal of Mathematical Analysis and Applications, vol. 247, no. 2, pp. 397–409, 2000. View at Publisher · View at Google Scholar
- L. H. Erbe and B. G. Zhang, “Oscillation of discrete analogues of delay equations,” Differential and Integral Equations, vol. 2, no. 3, pp. 300–309, 1989.
- A. Kneser, “Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen,” Mathematische Annalen, vol. 42, no. 3, pp. 409–435, 1893. View at Publisher · View at Google Scholar
- F. Gesztesy and M. Ünal, “Perturbative oscillation criteria and Hardy-type inequalities,” Mathematische Nachrichten, vol. 189, pp. 121–144, 1998. View at Publisher · View at Google Scholar
- K. M. Schmidt, “Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators,” Communications in Mathematical Physics, vol. 211, no. 2, pp. 465–485, 2000. View at Publisher · View at Google Scholar
- O. Došlý and P. Hasil, “Critical oscillation constant for half-linear differential equations with periodic coefficients,” Annali di Matematica Pura ed Applicata, vol. 190, no. 3, pp. 395–408, 2011. View at Publisher · View at Google Scholar
- P. Hasil, “Conditional oscillation of half-linear differential equations with periodic coefficients,” Archivum Mathematicum, vol. 44, no. 2, pp. 119–131, 2008.
- H. Krüger, “On perturbations of quasiperiodic Schrödinger operators,” Journal of Differential Equations, vol. 249, no. 6, pp. 1305–1321, 2010. View at Publisher · View at Google Scholar
- H. Krüger and G. Teschl, “Effective Prüfer angles and relative oscillation criteria,” Journal of Differential Equations, vol. 245, no. 12, pp. 3823–3848, 2008. View at Publisher · View at Google Scholar
- H. Krüger and G. Teschl, “Relative oscillation theory for Sturm-Liouville operators extended,” Journal of Functional Analysis, vol. 254, no. 6, pp. 1702–1720, 2008. View at Publisher · View at Google Scholar
- H. Krüger and G. Teschl, “Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function,” Communications in Mathematical Physics, vol. 287, no. 2, pp. 613–640, 2009. View at Publisher · View at Google Scholar
- K. Ammann and G. Teschl, “Relative oscillation theory for Jacobi matrices,” in Difference Equations and Applications, pp. 105–115, Uğur-Bahçeşehir University Publications, Istanbul, Turkey, 2009.
- P. B. Naĭman, “The set of isolated points of increase of the spectral function pertaining to a limit-constant Jacobi matrix,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 1959, no. 1 (8), pp. 129–135, 1959 (Russian).
- F. Luef and G. Teschl, “On the finiteness of the number of eigenvalues of Jacobi operators below the essential spectrum,” Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 299–307, 2004. View at Publisher · View at Google Scholar
- J. Yu, Z. Guo, and X. Zou, “Periodic solutions of second order self-adjoint difference equations,” Journal of the London Mathematical Society, vol. 71, no. 1, pp. 146–160, 2005. View at Publisher · View at Google Scholar
- Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China A, vol. 46, no. 4, pp. 506–515, 2003. View at Publisher · View at Google Scholar
- Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations,” Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419–430, 2003. View at Publisher · View at Google Scholar
- H. J. Li and C. C. Yeh, “An oscillation criterion of almost-periodic Sturm-Liouville equations,” The Rocky Mountain Journal of Mathematics, vol. 25, no. 4, pp. 1417–1429, 1995. View at Publisher · View at Google Scholar
- P. Řehák, “Comparison theorems and strong oscillation in the half-linear discrete oscillation theory,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 333–352, 2003. View at Publisher · View at Google Scholar
- O. Došlý and P. Řehák, “Recessive solution of half-linear second order difference equations,” Journal of Difference Equations and Applications, vol. 9, no. 1, pp. 49–61, 2003. View at Publisher · View at Google Scholar
- H. A. El-Morshedy, “Oscillation and nonoscillation criteria for half-linear second order difference equations,” Dynamic Systems and Applications, vol. 15, no. 3-4, pp. 429–450, 2006.
- P. Řehák, “A critical oscillation constant as a variable of time scales for half-linear dynamic equations,” Mathematica Slovaca, vol. 60, no. 2, pp. 237–256, 2010. View at Publisher · View at Google Scholar
- P. Řehák, “On certain comparison theorems for half-linear dynamic equations on time scales,” Abstract and Applied Analysis, vol. 2004, no. 7, pp. 551–565, 2004. View at Publisher · View at Google Scholar
- P. Řehák, “New results on critical oscillation constants depending on a graininess,” Dynamic Systems and Applications, vol. 19, no. 2, pp. 271–287, 2010.
- R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, New York, NY, USA, 2000.
- W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, Calif, USA, 2001.
- C. Corduneanu, Almost Periodic Functions, John Wiley & Sons, New York, NY, USA, 1968.
- C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, NY, USA, 2009. View at Publisher · View at Google Scholar
- A. M. Fink, Almost Periodic Differential Equations, Springer, Berlin, Germany, 1974.
- M. Veselý, “Almost periodic sequences and functions with given values,” Archivum Mathematicum, vol. 47, no. 1, pp. 1–16, 2011.
- K. M. Schmidt, “Oscillation of the perturbed Hill equation and the lower spectrum of radially periodic Schrödinger operators in the plane,” Proceedings of the American Mathematical Society, vol. 127, no. 8, pp. 2367–2374, 1999. View at Publisher · View at Google Scholar
- R. Jajte, “On almost-periodic sequences,” Colloquium Mathematicum, vol. 13, pp. 265–267, 1964-1965.
- W. M. Ruess and W. H. Summers, “Minimal sets of almost periodic motions,” Mathematische Annalen, vol. 276, no. 1, pp. 145–158, 1986. View at Publisher · View at Google Scholar
- H. L. Yao, “Some results for asymptotically almost periodic functions and asymptotically almost periodic sequences,” Heilongjiang Daxue Ziran Kexue Xuebao, vol. 23, no. 6, pp. 794–796, 2006.