## 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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