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On an iterative method for nonlinear variational inequalities. (English) Zbl 0631.65070
The author considers a nonlinear variational inequality in the form \(D\Phi(u,v-u)\geq (\phi,v-u)\) for all \(v\in K\) where \(u\in K\) and K is a closed convex subset of a real Hilbert space H. Here \(\Phi\) is a functional on H with a Gâteaux differential defined by \(D\Phi\) and \(\phi\in H\). The basic method is to replace this equation by \(D\Phi (u_ n,v-u_{n+1})+A(u_ n,u_{n+1}-u_ n,v-u_{n+1})\geq (\phi,v- u_{n+1})\) where A is a symmetric, bounded, H-elliptic bilinear form and this reduces the problem to a sequence of linear problems. The convergence of the process is discussed and the necessary assumptions given. In particular weaker sufficient conditions are given for convergence when \(D\Phi(u,h)=B(u,n,h)\) with \(A=\lambda B\). In this case the equality problem reduces to \(B(u_ n,u_ n+\lambda (u_{n+1}-u_ n),v)=(\phi,v)\) for all \(v\in H\) which is the secant modules method. An application is made to a Signorini’s contact problem from the theory of plasticity.
Reviewer: B.Burrows

65K10 Numerical optimization and variational techniques
74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
49J40 Variational inequalities