A discrete version of the boundary value problem \[ \begin{cases} u''(t) \in Au(t) \text{ a.e. } t\in(0,\infty) \\ u(0)=a,\;\sup_{t\geq0}\|u(t)\|<\infty \end{cases} \] is studied. Here \(A\) is a maximal monotone (possibly multivalued) operator in a Hilbert space \(H\) and \(a\) belongs to the domain \(D(A)\) of \(A.\)
The proposed version of the above problem is \[ \begin{cases} u_{n+1}-(1+\theta_n) u_n+\theta_n u_{n-1}\in c_n Au_{n},\;n\geq1\\ u_0=a,\;\|u_{n}\|\leq c,\;(\forall) n\geq1, \end{cases} \] where \(a\in H,\) \(c>0,\) \(c_n>0,\) \(\theta_n\geq1,\) \((\forall)n\geq1,\) \((\theta_n)\) nonincreasing sequence.
It is shown that, if \(A:D(A) \subseteq H\rightarrow H\) is a maximal monotone operator in \(H,\) with \(A^{-1}0\neq\Phi,\) then the above difference scheme has a unique solution \((u_n) \subset D(A).\)
The asymptotic behavior is also studied. In the subdifferential case, if \(\sum_{n=1}^{\infty} c_n/\theta_n=\infty,\) then the weak convergence of the solution to an element \(u\in A^{-1}0\) is proved. If \(A\) is univoque, maximally monotone and strongly monotone, then the convergence of the solution is strong. If \((I+A)^{-1}\) is a compact operator and \(\sum_{n=1}^{\infty}c_n^2/(1+\theta_n)^2=\infty,\) then the solution is also strongly convergent to an element \(u\in A^{-1}0.\)
The case \(\theta_n\equiv1\) was treated in some papers due to E. Mitidieri and G. Morosanu.