On an Approximate Solution of Variational Inequalities

✍ Scribed by Jozef Kačur

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 741 KB
• Volume: 123
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19851230119

✦ Synopsis

By means of time discretization, we approximate evolution variational inequalities by the corresponding elliptic variational inequalities. Using ROTHE'S method (method of lines), an approximate solution is constructed by means of direct variational methods. Existence, uniqueness and regularity of solutions as well a8 convergence of the approximate solutions are proved.

Introduction.

In this paper we shall be concerned with approximate solutions of the following types of variational inequalities

for all v E I' and for a.e. t E (0, T) (T < oo), where (u, v) is the scalar product in a HILBERT space H and a(u, v), b(u, v) are continuous bilincar forms on the corresponding HILBERT spaces V , V,, respectively, with the continuous imbedding V 4 V , 4 H . In the case (4), A is a monotone operator from a real reflexive BANACH space B into its dual space V*. The functional YJ(v) is convex on V with values in [-oo, 003. The obtained results hold true also in the more general form of fsee Remark 7. Instead of f ( t ) a LIrscHrrz continuous operator f ( t , u) : [0, T] X V -+ H can be considered.

The existence of solutions of (1)-( 4) can be found (in the case b = 0, f ( t ) ) , e.g., in the works of H. RREZIS [l], [2], J. L. LIOXS [12], G. DWAUT -J. L. LIONS [4] etc.