Multiple solutions for discrete boundary value problems. (English) Zbl 1169.39008

Consider the following boundary value problem with homogeneous Dirichlet conditions:
\[ -\Delta _{p}x(k-1)=\lambda f(k,x(k)) ,\quad k\in [ 1,T];\;x(0)=0=x(T+1),\tag{\(*\)} \]
where \(T\geq 2\) is a positive integer, \([1,T]=\{1,2,\dots,T\},\) \(p>1\) is a real number, \(\Delta _{p}\) is the discrete \(p\)-Laplacian operator defined by \[ \Delta _{p}x(k-1)=\Delta [ \left| \Delta x(k-1)\right| ^{p-2}\Delta x(k-1)] \]
\(\Delta\) is the forward difference operator defined by \(\Delta x(k)=x(k+1)-x(k)\) and \(f:[1,T]\times \mathbb R\rightarrow \mathbb R\) is continuous, \(\lambda \in \mathbb R\) is a parameter. The authors prove the existence of a positive \(\lambda _{\ast }\) such that the problem (\(*\)) admits at least three solutions. Several examples are worked out.


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


[1] Agarwal, R. P., Difference Equations and Inequalities (2000), Marcel Dekker Inc. · Zbl 1006.00501
[2] Agarwal, R. P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal., 58, 69-73 (2004) · Zbl 1070.39005
[3] Agarwal, R. P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular discrete \(p\)-Laplacian problems via variational methods, Adv. Difference Equ., 2005, 93-99 (2005) · Zbl 1098.39001
[4] Cai, X.; Guo, Z.; Yu, J., Periodic solutions of a class of nonlinear difference equations via critical point method, Comput. Math. Appl., 52, 1639-1647 (2006) · Zbl 1134.39003
[5] Cai, X.; Yu, J., Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl., 320, 649-661 (2006) · Zbl 1113.39019
[6] Faraci, F.; Iannizzotto, A., Multiplicity theorems for discrete boundary value problems, Aequationes Math., 74, 111-118 (2007) · Zbl 1128.39010
[7] Guo, Z.; Ma, M., Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67, 1737-1745 (2007) · Zbl 1120.39007
[8] Jiang, L.; Zhou, Z., Three solutions to Dirichlet boundary value problems for \(p\)-Laplacian difference equations, Adv. Difference Equ., 2008 (2008), Article ID 345916, 10 pp
[9] Lakshmikantham, V.; Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications (1988), Academic Press · Zbl 0683.39001
[11] Ricceri, B., Well-posedness of constrained minimization problems via saddle-points, J. Global Optim., 40, 389-397 (2008) · Zbl 1194.49034
[12] Ricceri, B., A multiplicity theorem in \(R^n\), J. Convex Anal., 16, 3-4 (2009)
[13] Struwe, M., Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2000), Springer-Verlag · Zbl 0939.49001
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