A note on a fourth order discrete boundary value problem (Q2901937)

The authors consider the following Dirichlet boundary value problem for a fourth order discrete equation NEWLINE\[NEWLINE \Delta^2(p(k)\Delta^2x(k-2)) + \Delta(q(k)\Delta x(k-1)) + f(k,x(k))=g(k) NEWLINE\]NEWLINE NEWLINE\[NEWLINE x(0)=x(1)=x(T+1)=x(T+2)=0 .NEWLINE\]NEWLINE They define the action functional \(J:E\mapsto \mathbb R\) on the space of the sequences \(x(k)\), \(0\leq k\leq T+2\) as follows NEWLINE\[NEWLINEJ(y) = \sum_2^{T+2}\left(-{{p(k)}\over{2}}(\Delta^2y(k-2))^2\right) + \sum_2^{T+1}{{q(k)}\over{2}}(\Delta y(k-1))^2 + \sum_2^T(-F(k,y(k))+g(k)y(k))NEWLINE\]NEWLINE where NEWLINE\[NEWLINEF(k,y(k))=\int_0^{y(k)}f(k,t)dt.NEWLINE\]NEWLINE Using the variational approach, existence, uniqueness and continuous dependence on the parameters are obtained for the solutions.