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.


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


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