Geodesic
ASYMPTOTIC INTEGRATION OF SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS
Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 223-243

Voir la notice de l'article provenant de la source Cambridge University Press

In this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems—fixed initial data and fixed asymptote, respectively—we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform convergence for both the solution and its derivative, while in the other case the convergence is somewhat weaker. Two different techniques are utilized, and for each one has to employ ad-hoc methods for the unbounded interval. Of particular importance is the exact form of the operators and metric spaces formulated in the earlier sections.
DOI : 10.1017/S0017089510000650
Mots-clés : 39A12, 34D05 
EHRNSTRÖM, MATS; TISDELL, CHRISTOPHER C.; WAHLÉN, ERIK. ASYMPTOTIC INTEGRATION OF SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS. Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 223-243. doi: 10.1017/S0017089510000650
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[1] 1.Agarwal, R. P., Difference equations, in Theory, methods and applications, 2nd edn (Monographs and textbooks in pure and applied mathematics) (Marcel Dekker, New York, 2000), 228. Google Scholar

[2] 2.Agarwal, R. P., Bohner, M., Grace, S. R. and O'Regan, D., Discrete oscillation theory (Hindawi Publishing Corporation, New York, 2005). Google Scholar | DOI

[3] 3.Agarwal, R. P. and Ravi, P., On multipoint boundary value for discrete equations, J. Math. Anal. Appl. 96 (2) (1983), 520–534. Google Scholar | DOI

[4] 4.Atkinson, F. V., On second order nonlinear oscillation, Pac. J. Math. 5 (1955), 643–647. Google Scholar | DOI

[5] 5.Bielecki, A., Une remarque sur la méthode de Banach–Cacciopoli–Tikhonov dans la théorie des équations differéntielles ordinaires, Bull. Acad. Polon. Sci. 4 (1956), 261–264. Google Scholar

[6] 6.Dubé, S. G. and Mingarelli, A. B., Note on a non-oscillation theorem of Atkinson, Electron. J. Differ. Equ. 22 (2004), 1–6. Google Scholar

[7] 7.Dugundji, J. and Granas, A., Fixed point theory: I, Monografie Matematyczne (Mathematical Monographs) vol 61 (Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982). Google Scholar

[8] 8.Ehrnström, M., Positive solutions for second-order nonlinear differential equations, Nonlinear Anal. 64 (2006), 1608–1620. Google Scholar | DOI

[9] 9.Ehrnström, M., Linear asymptotic behaviour of second order ordinary differential equations, Glasgow Math. J. 49 (2007), 105–120. Google Scholar | DOI

[10] 10.Ehrnström, M. and Mustafa, O. G., On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. 67 (2007), 1147–1154. Google Scholar | DOI

[11] 11.Kelley, W. G. and Peterson, A. C., Difference equations: An introduction with applications 2nd edn (Harcourt/Academic Press, San Diego, CA, 2001). Google Scholar

[12] 12.Mingarelli, A. B. and Sadarangani, K., Asymptotic solutions of forced nonlinear second order differential equations and their extensions, Electron. J. Differ. Equ. 2007, no. 40 (2007), 1–40. Google Scholar

[13] 13.Mustafa, O. G. and Rogovchenko, Y. V., Asymptotic integration of nonlinear differential equations, Nonlinear Anal. 63 (2005), 2135–2143. Google Scholar | DOI

[14] 14.Rachůnková, I. and Tisdell, C. C., Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions, Nonlinear Anal. 67 (2007), 1236–1245. Google Scholar | DOI

[15] 15.Wahlén, E., Positive solutions of second-order differential equations. Nonlinear Anal. 58 (2004), 359–366. Google Scholar | DOI

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