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

Laitinen, E.; Lapin, A.

Comput. Methods Appl. Math. 1, No. 3, 242-264 (2001).

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

Keywords:

maximal monotone operator; \(M\)-mapping; iterative solution; finite difference scheme; variational inequality