## 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.

### MSC:

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

  Agarwal, R. P., Difference Equations and Inequalities (2000), Marcel Dekker Inc. · Zbl 1006.00501  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  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  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  Cai, X.; Yu, J., Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl., 320, 649-661 (2006) · Zbl 1113.39019  Faraci, F.; Iannizzotto, A., Multiplicity theorems for discrete boundary value problems, Aequationes Math., 74, 111-118 (2007) · Zbl 1128.39010  Guo, Z.; Ma, M., Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67, 1737-1745 (2007) · Zbl 1120.39007  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  Lakshmikantham, V.; Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications (1988), Academic Press · Zbl 0683.39001  Ricceri, B., Well-posedness of constrained minimization problems via saddle-points, J. Global Optim., 40, 389-397 (2008) · Zbl 1194.49034  Ricceri, B., A multiplicity theorem in $$R^n$$, J. Convex Anal., 16, 3-4 (2009)  Struwe, M., Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2000), Springer-Verlag · Zbl 0939.49001
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