Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces
Advances in Difference Equations volume 2007, Article number: 086925 (2007)
We consider the difference equation , , in the context of a Hilbert space. In this setting, we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory, as well as conditions which guarantee the existence of directionally positive monotone increasing solutions.
Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.
Agarwal RP, Bohner M, Grace SR, O'Regan D: Discrete Oscillation Theory. Hindawi, New York, NY, USA; 2005:xiv+961.
Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:viii+507.
Agarwal RP, O'Regan D: Difference equations in abstract spaces. Journal of the Australian Mathematical Society 1998,64(2):277–284. 10.1017/S1446788700001762
Franco D, O'Regan D, Peran J: The antipodal mapping theorem and difference equations in Banach spaces. Journal of Difference Equations and Applications 2005,11(12):1037–1047. 10.1080/10236190500331305
González C, Jiménez-Melado A: Set-contractive mappings and difference equations in Banach spaces. Computers & Mathematics with Applications 2003,45(6–9):1235–1243.
González C, Jiménez-Melado A, Lorente M: Existence and estimate of solutions of some nonlinear Volterra difference equations in Hilbert spaces. Journal of Mathematical Analysis and Applications 2005,305(1):63–71. 10.1016/j.jmaa.2004.10.015
Medina R: Delay difference equations in infinite-dimensional spaces. Journal of Difference Equations and Applications 2006,12(8):799–809. 10.1080/10236190600734192
Medina R, Gil' MI: Delay difference equations in Banach spaces. Journal of Difference Equations and Applications 2005,11(10):889–895. 10.1080/10236190512331333860
Medina R, Gil' MI: The freezing method for abstract nonlinear difference equations. Journal of Mathematical Analysis and Applications 2007,330(1):195–206. 10.1016/j.jmaa.2006.07.074
Jiang J, Li X: Oscillation and nonoscillation of two-dimensional difference systems. Journal of Computational and Applied Mathematics 2006,188(1):77–88. 10.1016/j.cam.2005.03.054
Dubé SG, Mingarelli AB: Note on a non-oscillation theorem of Atkinson. Electronic Journal of Differential Equations 2004,2004(22):1–6.
Ehrnström M: Positive solutions for second-order nonlinear differential equations. Nonlinear Analysis: Theory, Methods & Applications 2006,64(7):1608–1620. 10.1016/j.na.2005.07.010
Wahlén E: Positive solutions of second-order differential equations. Nonlinear Analysis: Theory, Methods & Applications 2004,58(3–4):359–366. 10.1016/j.na.2004.05.008
Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.
Grace SR, El-Morshedy HA: Oscillation criteria for certain second order nonlinear difference equations. Bulletin of the Australian Mathematical Society 1999,60(1):95–108. 10.1017/S0004972700033360
About this article
Cite this article
González, C., Jiménez-Melado, A. Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces. Adv Differ Equ 2007, 086925 (2007). https://doi.org/10.1155/2007/86925