×

Existence theorems for second-order discrete boundary value problems. (English) Zbl 1113.39019

The authors consider the following second order difference boundary value problem:
\[ \Delta (p_n\,(\Delta x_{n-1})^\delta)+q_n\, x_n^\delta=f(n,x_n), \; n \in \{1, \dots,k\}; \quad \Delta\, x_0=A, \quad x_{k+1}=B. \]
Here \(\delta >0\), \(\{p_n\}\) and \(\{q_n\}\) are real sequences, \(p_n \neq 0\) for all \(n \in \{1,\dots,k+1\}\) and \(A\) and \(B\) are two given constants.
Under different suitable assumptions on functions \(f\), \(p_n\) and \(q_n\), the authors prove some existence results of at least one (or at least two) solution of the considered problem. The proofs follow from the Linking Theorem and the Mountain Pass Lemma in the critical point theory.

MSC:

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P., Difference equations and applications: theory, methods and applications, (2000), Dekker New York · Zbl 0952.39001
[2] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Dordrecht · Zbl 0923.39002
[3] Agarwal, R.P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems, Nonlinear anal., 58, 69-73, (2004) · Zbl 1070.39005
[4] Agarwal, R.P.; Stanek, S., Existence of positive solutions to singular semi-positone boundary value problems, Nonlinear anal., 51, 821-842, (2002) · Zbl 1018.34023
[5] Ahlbrandt, C.D.; Peterson, A.C., Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations, (1996), Kluwer Academic · Zbl 0860.39001
[6] Atici, F.M.; Guseinov, G.S., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. math. anal. appl., 232, 166-182, (1999) · Zbl 0923.39010
[7] Cabada, A.; Espinar, V.O., Existence and comparison results for difference φ-Laplacian boundary value problems with lower and upper solutions in reverse order, J. math. anal. appl., 267, 501-521, (2002) · Zbl 0995.39003
[8] Castro, A.; Shivaji, R., Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. partial differential equations, 14, 1091-1100, (1989) · Zbl 0688.35025
[9] Cecchi, M.; Marini, M.; Villari, G., On the monotonicity property for a certain class of second order differential equations, J. differential equations, 82, 15-27, (1998) · Zbl 0694.34035
[10] Deimling, K., Nonlinear functional analysis, (1985), Springer Berlin · Zbl 0559.47040
[11] Esteban, J.R.; Vazguez, J.L., On the equation of turbulent filtration in one-dimensional porous media, Nonlinear anal., 10, 1303-1325, (1986) · Zbl 0613.76102
[12] Herrero, M.A.; Vazguez, J.L., On the propagation properties of a nonlinear degenerate parabolic equation, Comm. partial differential equations, 7, 1381-1402, (1982) · Zbl 0516.35041
[13] Kaper, H.G.; Knapp, M.; Kwong, M.K., Existence theorems for second order boundary value problems, Differential integral equations, 4, 543-554, (1991) · Zbl 0732.34019
[14] Li, W.T., Oscillation of certain second-order nonlinear differential equations, J. math. anal. appl., 217, 1-14, (1998) · Zbl 0893.34023
[15] Liu, Y.; Ge, W., Twin positive solutions for boundary value problems for finite difference equations with p-Laplacian operator, J. math. anal. appl., 278, 551-561, (2003) · Zbl 1019.39002
[16] Marini, M., On nonoscillatory solutions of a second order nonlinear differential equation, Boll. un. mat. ital. ser. VI, 3-C, 189-202, (1984) · Zbl 0574.34022
[17] Marini, M.; Zezza, P., On the asymptotic behavior of the solutions of a class of second order linear differential equations, J. differential equations, 28, 1-17, (1978) · Zbl 0371.34032
[18] 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. Providence, RI · Zbl 0609.58002
[19] Wen, L.Z., Asymptotic and oscillation of second-order functional differential equation, Sci. China ser. A, 2, 149-161, (1986), (in Chinese)
[20] Wong, P.J.Y.; Agarwal, R.P., Oscillatory behavior of solutions certain second order nonlinear differential equations, J. math. anal. appl., 198, 337-354, (1996) · Zbl 0855.34039
[21] Zhuang, W.; Chen, Y.; Cheng, S.S., Monotone methods for a discrete boundary value problem, Comput. math. appl., 32, 41-49, (1996) · Zbl 0872.39005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.