On the existence of solutions of some second order nonlinear difference equations
Migda, Małgorzata ; Schmeidel, Ewa ; Zbąszyniak, Małgorzata
Archivum Mathematicum, Tome 041 (2005), p. 379-388 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

We consider a second order nonlinear difference equation \[ \Delta ^2 y_n = a_n y_{n+1} + f(n,y_n,y_{n+1})\,,\quad n\in N\,. \qquad \mathrm {(\mbox{E})}\] The necessary conditions under which there exists a solution of equation (E) which can be written in the form \[ y_{n+1} = \alpha _{n}{u_n} + \beta _{n}{v_n}\,,\quad \mbox{are given.} \] Here $u$ and $v$ are two linearly independent solutions of equation \[ \Delta ^2 y_n = a_{n+1} y_{n+1}\,, \quad ({\lim \limits _{n \rightarrow \infty } \alpha _{n} = \alpha <\infty } \quad {\rm and} \quad {\lim \limits _{n \rightarrow \infty } \beta _{n} = \beta <\infty })\,. \] A special case of equation (E) is also considered.

Publié le : 2005-01-01
Classification:  39A10,  39A11 
@article{107967,
     author = {Ma\l gorzata Migda and Ewa Schmeidel and Ma\l gorzata Zb\k aszyniak},
     title = {On the existence of solutions of some second order nonlinear difference equations},
     journal = {Archivum Mathematicum},
     volume = {041},
     year = {2005},
     pages = {379-388},
     zbl = {1122.39001},
     mrnumber = {2195491},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107967}
}

Migda, Małgorzata; Schmeidel, Ewa; Zbąszyniak, Małgorzata. On the existence of solutions of some second order nonlinear difference equations. Archivum Mathematicum, Tome 041 (2005) pp. 379-388. http://gdmltest.u-ga.fr/item/107967/

Agarwal R. P. Difference equations and inequalities. Theory, methods and applications, Marcel Dekker, Inc., New York 1992. (1992) | MR 1155840 | Zbl 0925.39001

Cheng S. S.; Li H. J.; Patula W. T. Bounded and zero convergent solutions of second order difference equations, J. Math. Anal. Appl. 141 (1989), 463–483. (1989) | MR 1009057

Drozdowicz A. On the asymptotic behavior of solutions of the second order difference equations, Glas. Mat. 22 (1987), 327–333. (1987)

Elaydi S. N. An introduction to difference equation, Springer-Verlag, New York 1996. (1996) | MR 1410259

Kelly W. G.; Peterson A. C. Difference equations, Academic Press, Inc., Boston-San Diego 1991. (1991) | MR 1142573

Medina R.; Pinto M. Asymptotic behavior of solutions of second order nonlinear difference equations, Nonlinear Anal. 19 (1992), 187–195. (1992) | MR 1174467 | Zbl 0773.39003

Migda J.; Migda M. Asymptotic properties of the solutions of second order difference equation, Arch. Math. (Brno) 34 (1998), 467–476. (1998) | MR 1679641

Migda M. Asymptotic behavior of solutions of nonlinear delay difference equations, Fasc. Math. 31 (2001), 57–62. | MR 1860548

Migda M.; Schmeidel E.; Zbąszyniak M. Some properties of solutions of second order nonlinear difference equations, Funct. Differ. Equ. 11 (2004), 147–152. | MR 2056707

Popenda J.; Werbowski J. On the asymptotic behavior of the solutions of difference equations of second order, Ann. Polon. Math. 22 (1980), 135–142. (1980) | MR 0610341

Schmeidel E. Asymptotic behaviour of solutions of the second order difference equations, Demonstratio Math. 25 (1993), 811–819. (1993) | MR 1265844 | Zbl 0799.39001

Thandapani E.; Arul R.; Graef J. R.; Spikes P. W. Asymptotic behavior of solutions of second order difference equations with summable coefficients, Bull. Inst. Math. Acad. Sinica 27 (1999), 1–22. (1999) | MR 1681601 | Zbl 0920.39001

Thandapani E.; Manuel M. M. S.; Graef J. R.; Spikes P. W. Monotone properties of certain classes of solutions of second order difference equations, Advances in difference equations II, Comput. Math. Appl. 36 (1998), 291–297. (1998) | MR 1666147