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

Cabada, Alberto; Iannizzotto, Antonio; Tersian, Stepan

J. Math. Anal. Appl. 356, No. 2, 418-428 (2009).

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.

Reviewer: Fozi Dannan (Damascus)

MSC:

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

Keywords:

difference equations; variational methods; boundary value problem; discrete \(p\)-Laplacian operator

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