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Nonlinear difference equations investigated via critical point methods. (English) Zbl 1166.39006

The authors obtain some critical point theorems in the setting of finite dimensional Banach spaces. Based on those theorems they establish multiple solutions for the following problem:
\[ \begin{cases} -\Delta (\phi_p(\Delta u(k-1))=\lambda f(k,u(k)),\quad k\in [1,T],\\ u(0)=u(T+1)=0,\end{cases} \]
where \(T\) is a fixed positive integer, \([1,T]\) is the discrete interval \(\{1,\dots T\}\), \(\lambda\) is a positive real parameter, \(\Delta u(k):=u(k+1)-u(k)\) is the forward difference operator, \(\phi_p(s)=|s|^{p-2}s,\;1<p<\infty\) and \(f:[1,T]\times\mathbb R\to\mathbb R\) is a continuous function.

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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[1] Agarwal, R.P., Difference equations and inequalities: theory, methods and applications, (2000), Marcel Dekker New York, Basel · Zbl 0952.39001
[2] Agarwal, R.P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular discrete \(p\)-Laplacian problems via variational methods, Adv. difference equ., 2005, 2, 93-99, (2005) · Zbl 1098.39001
[3] 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
[4] Averna, D.; Bonanno, G., A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. methods nonlinear anal., 22, 93-103, (2003) · Zbl 1048.58005
[5] Bai, D.; Xu, Y., Nontrivial solutions of boundary value problems of second-order difference equations, J. math. anal. appl., 326, 297-302, (2007) · Zbl 1113.39018
[6] Bonanno, G., Some remarks on a three critical points theorem, Nonlinear anal., 54, 651-665, (2003) · Zbl 1031.49006
[7] Bonanno, G., A critical points theorem and nonlinear differential problems, J. global optim., 28, 249-258, (2004) · Zbl 1087.58007
[8] Bonanno, G.; Candito, P., Non-differentiable functions with applications to elliptic equations with discontinuous nonlinearities, J. differential equations, 244, 3031-3059, (2008) · Zbl 1149.49007
[9] P. Candito, N. Giovannelli, Multiple solutions for a discrete boundary value problem, Comput. Math. Appl. (in press) · Zbl 1155.39301
[10] Chu, J.; Jiang, D., Eigenvalues and discrete boundary value problems for the one-dimensional \(p\)-Laplacian, J. math. anal. appl., 305, 452-465, (2005) · Zbl 1074.39022
[11] Fang, G., On the existence and the classification of critical points for non-smooth functionals, Canad. J. math., 47, 4, 684-717, (1981) · Zbl 0835.58007
[12] Faraci, F.; Iannizzotto, A., Multiplicity theorems for discrete boundary problems, Aequationes math., 74, 111-118, (2007) · Zbl 1128.39010
[13] 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
[14] Jiang, L.; Zhou, Z., Existence of nontrivial solutions for discrete nonlinear two point boundary value problems, Appl. math. comput., 180, 318-329, (2006) · Zbl 1113.39023
[15] Jiang, L.; Zhou, Z., Three solutions to Dirichlet boundary value problems for \(p\)-Laplacian difference equations, Adv. difference equ., 2008, 1-10, (2008)
[16] Kelly, W.G.; Peterson, A.C., Difference equations, an introduction with applications, (1991), Academic Press San Diego, New York
[17] Liang, H.; Weng, P., 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
[18] Marano, S.A.; Motreanu, D., On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear anal., 48, 37-52, (2002) · Zbl 1014.49004
[19] Ricceri, B., On a three critical points theorem, Arch. math. (basel), 75, 220-226, (2000) · Zbl 0979.35040
[20] Zhang, G.; Zhang, W.; Liu, S., Existence of \(2^n\) nontrivial solutions for a discrete two-point boundary value problems, Nonlinear anal., 59, 1181-1187, (2004) · Zbl 1062.39020
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