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Nontrivial solutions of boundary value problems of second-order difference equations. (English) Zbl 1113.39018

The authors are concerned with second order boundary value problem
\[ \begin{cases} \Delta^{2}u_{k-1}+\lambda h\left( k,u_{k}\right) =0,\quad k=1,2,\dots,T,\\ u_{0}=u_{T+1}=0, \end{cases} \]
where \(\Delta u_{k-1}=u_{k}-u_{k-1},\) \(\Delta^{2}u_{k-1}=\Delta\left( \Delta u_{k-1}\right) ,\) \(\lambda>0\) is a parameter. Using the critical point theory, the existence of nontrivial solutions is proved, together with their boundedness.

MSC:

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