×

Existence of multiple solutions for a second-order difference equation with a parameter. (English) Zbl 1205.39002

The author studies the existence of solutions of the second order difference boundary value problem with parameter dependence \(\Delta(p_{k-1}\, \Delta\, x_{k-1})+q_k\, x_k +\lambda\, f(k,x_k)=0\), \(k \in \{1, \dots,N\}\), \(x_0=x_N\), \(p_0\, \Delta\, x_0=p_N\, \Delta\, x_N\). Here \(N>2\) is an integer, \(\Delta\) denotes the usual forward difference operator, \(p_k,q_k \in \mathbb R\) and \(\lambda >0\).
Under suitable assumptions on the function \(f\), the author proves the existence of at least \(2\, N\) distinct solutions for \(\lambda\) belonging to an interval related with the eigenvalues of a given matrix that depends on \(p_k\) and \(q_k\).
The proofs follow from the application of a critical point theorem of P. H. Rabinowitz [CBMS Reg. Conf. Ser. Math. vol. 65, Am. Math. Soc., Providence, RI (1986; Zbl 0609.58002)].

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 0609.58002
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Yu, J. S.; Guo, Z. M., On boundary value problems for a discrete generalized Emden-Fowler equation, J. Differ. Equ., 231, 18-31 (2006) · Zbl 1112.39011
[2] Wong, J. S., On the generalized Emden-Fowler equation, SIAM Rev., 2, 17, 339-360 (1975) · Zbl 0295.34026
[3] Atici, F. M.; Guseinov, G. Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl., 232, 166-182 (1999) · Zbl 0923.39010
[4] Atici, F. M.; Cabada, A., Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. Math. Appl., 45, 1417-1427 (2003) · Zbl 1057.39008
[5] Agarwal, R. P.; O’Regan, D., Nonpositone discrete boundary value problems, Nonlinear Anal., 39, 207-215 (2000) · Zbl 0937.39001
[6] Agarwal, R. P.; O’Regan, D., A fixed-point approach for nonlinear discrete boundary value problems, Comput. Math. Appl., 36, 115-121 (1998) · Zbl 0933.39004
[7] Agarwal, R. P.; O’Regan, D., Boundary value problems for discrete equations, Appl. Math. Lett., 10, 83-89 (1997) · Zbl 0890.39001
[8] Henderson, J.; Thompson, H. B., Existence of multiple solutions for second-order discrete boundary value problems, Comput. Math. Appl., 43, 1239-1248 (2002) · Zbl 1005.39014
[9] Thompson, H. B.; Tisdell, C., Systems of difference equations associated with boundary value problems for second-order systems of ordinary differential equations, J. Math. Anal. Appl., 248, 333-347 (2000) · Zbl 0963.65081
[10] Wan, Z.; Chen, Y. B.; Cheng, S. S., Monotone methods for discrete boundary value problem, Comput. Math. Appl., 32, 41-49 (1996) · Zbl 0872.39005
[11] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0609.58002
[12] Bai, D. Y.; Xu, Y. T., Nontrivial solutions of boundary value problems of second-order difference equations, J. Math. Anal. Appl., 326, 297-302 (2007) · Zbl 1113.39018
[13] Liang, H. H.; Weng, P. X., Existence and multiple solutions for a second-order difference boundary value problem via critical point theory, J. Math. Anal. Appl., 326, 511-520 (2007) · Zbl 1112.39008
[14] He, X.; Wu, X., Existence and multiplicity of solutions for nonlinear second order difference boundary problems, Comput. Math. Appl., 57, 1-8 (2009) · Zbl 1165.39303
[15] Guo, D. J., Nonlinear Functional Analysis (2001), Shandong Publishing House of Science and Technology: Shandong Publishing House of Science and Technology Jinan, (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.