Existence of nontrivial solutions for discrete nonlinear two point boundary value problems. (English) Zbl 1113.39023

This paper deals with the two point boundary value problem \[ \begin{cases} -\Delta^{2}x\left( k-1\right) =f\left( k,x\left( k\right) \right) ,\;k\in\mathbb{Z}\left( 1,T\right) \\ x\left( 0\right) =0=\Delta x\left( T\right) , \end{cases} \] where \(\mathbb{Z}\left( 1,T\right) =\left\{ 1,2,\dots,T\right\} ,\) \(T>0\) is an integer, \(\Delta x\left( k\right) =x\left( k+1\right) -x\left( k\right) \) is the forward difference operator, and \(f\) is a continuous function.
Using the critical point theory and the strongly monotone operator principle, the authors solve the existence of nontrivial solutions for the above problem. By virtue of Green’s function and the separation of a linear operator, they construct a variational framework of this problem. Some conditions for function \(f\) are found in order to guarantee that the boundary value problem has a unique solution or at least one nontrivial solution.


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