Set-valued pseudomonotone maps and degenerate evolution inclusions. (English) Zbl 0959.34049

Kuttler, Kenneth L.; Shillor, Meir

Commun. Contemp. Math. 1, No. 1, 87-123 (1999).

The authors prove existence and uniqueness of a weak solution to the problem \[ (Bu)'+Au\ni f,\;u(0)=u_0, \] in a Banach space and then apply the theory to a frictional contact problem. \(B\) is assumed to be linear and is allowed to vanish. \(A\) is assumed to be a pseudomonotone set-valued operator. The proof of this theorem is based on the method of elliptic regularization from J. L. Lions [Quelques methods de resolution des problèmes aux limites nonlinéaires, Dunod, Paris (1969; Zbl 0189.40603)] adapted to the set-valued case. Notation, terminology and other preliminaries are given in section 2, while section 3 contains a number of technical lemmas needed for the proof of the main results, stated in theorems 4.3, 4.4 and 4.5. Sections 5-8 give a detailed explanation of the application, a problem of frictional contact between a deformable body and a moving rigid foundation. Theorems 4.4 and 4.5 are applied to the problem to obtain existence and uniqueness of a solution. The authors cite references for earlier work on this evolution inclusion and also information on pseudomonotone maps, plus give a number of references concerning related friction problems.

Reviewer: Daniel C.Biles (Bowling Green)

MSC:

34G25	Evolution inclusions
47H05	Monotone operators and generalizations
47H04	Set-valued operators
47N20	Applications of operator theory to differential and integral equations
49J40	Variational inequalities
74M15	Contact in solid mechanics
74M10	Friction in solid mechanics

Keywords:

pseudomonotone; degenerate evolution inclusions; frictional contact; elliptic regularization; weak solution

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