Unbounded solutions of second order discrete BVPs on infinite intervals

Lian, Hairong¹ ; Li, Jingwu¹ ; Agarwal, Ravi P²

¹ School of Science, China University of Geosciences, Beijing 100083, P. R. China
² Department of Mathematics, Texas A&M University-Kingsville, Kingsville, Texas 78363, USA

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 2, p. 357-369.

Résumé

In this paper, we study Sturm-Liouville boundary value problems for second order difference equations on a half line. By using the discrete upper and lower solutions, the Schäuder fixed point theorem, and the degree theory, the existence of one and three solutions are investigated. An interesting feature of our existence theory is that the solutions may be unbounded.

Détail

DOI : 10.22436/jnsa.009.02.02

Classification : 47H10, 54H25, 54C60
Keywords: Coincidence point, discrete boundary value problem, infinite interval, upper solution, lower solution, degree theory common fixed point.

Bibliographie

[1] Agarwal, R. P.; Bohner, M.; O'Regan, D. Time scale boundary value problems on infinite intervals, J. Comput. Appl. Math., Volume 141 (2002), pp. 27-34

[2] Agarwal, R. P.; Grace, S. R.; O'Regan, D. Nonoscillatory solutions for discrete equation, Comput. Math. Appl., Volume 45 (2003), pp. 1297-1302

[3] Agarwal, R. P.; O'Regan, D. Boundary value problems for discrete equations, Appl. Math. Lett., Volume 10 (1997), pp. 83-89

[4] Agarwal, R. P.; D. O'Regan Boundary value problems for general discrete systems on infinite intervals , Comput. Math. Appl., Volume 33 (1997), pp. 85-99

[5] Agarwal, R. P.; O'Regan, D. Discrete systems on infinite intervals, Comput. Math. Appl., Volume 35 (1998), pp. 97-105

[6] Agarwal, R. P.; D. O'Regan Existence and approximation of solutions of nonlinear discrete systems on infinite intervals, Math. Meth. Appl. Sci., Volume 22 (1999), pp. 91-99

[7] Agarwal, R. P.; O'Regan, D. Continuous and discrete boundary value problems on the infinite interval: existence theory, Mathematika, Volume 48 (2001), pp. 273-292

[8] Agarwal, R. P.; O'Regan, D. Nonlinear Urysohn discrete equations on the infinite interval: a fixed point approach , Comput. Math. Appl., Volume 42 (2001), pp. 273-281

[9] Apreutesei, N. C. On a class of difference equations of monotone type , J. Math. Anal. Appl., Volume 288 (2003), pp. 833-851

[10] Benchohra, M.; Ntouyas, S. K.; Ouahab, A. Upper and lower solutions method for discrete inclusions with nonlinear boundary conditions, J. Pure Appl. Math., Volume 7 (2006), pp. 1-7

[11] Bereanu, C.; J. Mawhin Existence and multiplicity results for periodic solutions of nonlinear difference equations , J. Difference Equ. Appl., Volume 12 (2006), pp. 677-695

[12] Guseinov, G. Sh. A boundary value problem for second order nonlinear difference equations on the semi-infinite interval , J. Difference Equ. Appl., Volume 8 (2002), pp. 1019-1032

[13] Henderson, J.; H. B. Thompson Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations, J. Difference Equ. Appl., Volume 7 (2001), pp. 297-321

[14] Henderson, J.; H. B. Thompson Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl., Volume 43 (2002), pp. 1239-1248

[15] Jiang, D.; O'Regan, D.; Agarwal, R. P. A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensional p-Laplacian , J. Appl. Anal., Volume 11 (2005), pp. 35-47

[16] Kubiaczyk, I.; Majcher, P. On some continuous and discrete equations in Banach spaces on unbounded intervals, Appl. Math. Comput., Volume 136 (2003), pp. 463-473

[17] Liu, Z.; Hou, X.; Ume, T. Sh.; Kang, Sh. M. Unbounded positive solutions and Mann iterative schemes of a second order nonlinear neutral delay difference equation, Abstr. Appl. Anal., Volume 2013 (2013), pp. 1-12

[18] Mohamed, M.; Thompson, H. B.; M. S. Jusoh Solvability of discrete two-point boundary value problems, J. Math. Res., Volume 3 (2011), pp. 15-26

[19] Rachůnek, L.; I. Rachůnková Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics, Nonlinear Anal. Real World Appl., Volume 12 (2011), pp. 14-23

[20] Rachůnková, I.; Rachůnek, L. Solvability of discrete Dirichlet problem via lower and upper functions method, J. Difference Equ. Appl., Volume 13 (2007), pp. 423-429

[21] Rachůnková, I.; Tisdell, C. C. Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions , Nonlinear Anal., Volume 67 (2007), pp. 1236-1245

[22] Rodriguez, J. Nonlinear discrete systems with global boundary conditions, J. Math. Anal. Appl., Volume 286 (2003), pp. 782-794

[23] Ryaben'kiim, V. S.; S. V. Tsynkov An effective numerical technique for solving a special class of ordinary difference equations, Appl. Numer. Math., Volume 18 (1995), pp. 489-501

[24] Thompson, H. B.; Tisdell, C. C. Systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations , J. Math. Anal. Appl., Volume 248 (2000), pp. 333-347

[25] Thompson, H. B. Topological methods for some boundary value problems, Comput. Math. Appl., Volume 42 (2001), pp. 487-495

[26] Tian, Y.; Ge, W. Multiple positive solutions of boundary value problems for second order discrete equations on the half line, J. Difference Equ. Appl., Volume 12 (2006), pp. 191-208

[27] Tian, Y.; Tisdell, C. C.; Ge, W. The method of upper and lower solutions for discrete BVP on infinite intervals, J. Difference Equ. Appl., Volume 17 (2011), pp. 267-278

[28] Wang, Y. M. Monotone methods of a boundary value problem of second order discrete equation , Comput. Math. Appl., Volume 36 (1998), pp. 77-92

[29] Y. M. Wang Accelerated monotone iterative methods for a boundary value problem of second order discrete equations, Comput. Math. Appl., Volume 39 (2000), pp. 85-94