Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions

Abstract

We consider the discrete boundary value problem (P):

where the nonlinear term has an oscillatory behaviour near the origin or at infinity. By a direct variational method, we show that (P) has a sequence of non-negative, distinct solutions which converges to 0 (respectively ) in the sup-norm whenever f oscillates at the origin (respectively at infinity).

Keywords:

• difference equations
• oscillatory nonlinearities
• small solutions
• large solutions

AMS Subject Classification::

• 46E39
• 34K10
• 35B38

Acknowledgements

A. Kristály is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and PN II IDEI 527/2007 of CNCSIS. M. Mihăilescu and V. Rădulescu are supported by Grant CNCSIS, IDEI 79/2007.