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Solution of a finite-dimensional problem with \(M\)-mappings and diagonal multivalued operators. (English) Zbl 1098.49503

Summary: The finite-dimensional problem: find a triple \((u,\gamma,\delta) \in (\mathbb R^N)^3\) such that \[ Au + B\gamma+\delta = f, \gamma \in Cu, \delta \in Du.\tag{1} \] is studied. Here \(A, B: \mathbb R^N\to \mathbb R^N\) are the continuous \(M\)-mappings and \(C, D:\mathbb R^N\to 2^{\mathbb R^N}\) are the multivalued diagonal maximal monotone operators. The existence of a solution on an ordered interval, which is formed by the so-called subsolution and supersolution for problem (1) is proved. Under several additional assumptions on the operators \(A, B, C\) and \(D\) the monotone dependence of a solution upon the right-hand side is investigated. This result implies, in particular, the uniqueness of a solution and serves as a basis for the analysis of the convergence for a multisplitting iterative method. As an illustrative example, the finite difference scheme approximating a model variational inequality is studied by using the general results.

MSC:

49J40 Variational inequalities
65J05 General theory of numerical analysis in abstract spaces
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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