## Existence of nontrivial solutions for boundary value problems of second-order discrete systems.(English)Zbl 1274.39018

This paper considers the following second-order discrete systems: $\begin{cases} \Delta ^2 x(n-1)+\lambda \nabla F\left (n,x(n)\right)=0,\quad n\in\mathbb Z(1,M),\\ x(0)=x(M+1)=0. \end{cases}\tag{P}$ Suppose that $$F:\mathbb Z(0,M)\times\mathbb R^m\rightarrow\mathbb R$$ is sublinear, i.e., there are $$\alpha \in [0,2)$$, $$\beta \geq 0$$ and $$\gamma >0$$ such that $F(n,y)\leq \beta | y| ^\alpha +\gamma , \tag{1}$
and $$F(n,\cdot)$$ is even and satisfies some other assumptions. The authors prove that (P) has $$mM$$ distinct pairs of nontrivial solutions for $$\lambda >\lambda _0$$. The argument is variational and the main tool is a critical point theorem due to D. C. Clark [Indiana Univ. Math. J. 22, 65–74 (1972; Zbl 0228.58006)]. Also, some examples are included.
The sublinear condition (1) appeared in the paper by C.-L. Tang [Proc. Am. Math. Soc. 126, No. 11, 3263–3270 (1998; Zbl 0902.34036)].

### MSC:

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

### Citations:

Zbl 0228.58006; Zbl 0902.34036
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