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Zbl 1113.39018
Bai, Dingyong; Xu, Yuantong
Nontrivial solutions of boundary value problems of second-order difference equations.
(English)
[J] J. Math. Anal. Appl. 326, No. 1, 297-302 (2007). ISSN 0022-247X

The authors are concerned with second order boundary value problem $$\cases \Delta^{2}u_{k-1}+\lambda h\left( k,u_{k}\right) =0,\quad k=1,2,\dots,T,\\ u_{0}=u_{T+1}=0, \endcases$$ 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.
[N. C. Apreutesei (Iaşi)]
MSC 2000:
*39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems of ODE

Keywords: Palais-Smale condition; bounded solutions; boundary value problem; nontrivial solution; critical point theory

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