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

Zhang, Xingyong; Tang, Xianhua

Math. Slovaca 61, No. 5, 769-778 (2011).

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)].

Reviewer: Fukun Zhao (Kunming, Yunnan)

MSC:

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

Keywords:

critical point; boundary value problem; discrete system; Clarks theorem

Citations:

Zbl 0228.58006; Zbl 0902.34036

References: