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Generalized Auxiliary Problem Principle and Solvability of a Class of Nonlinear Variational Inequalities Involving Cocoercive and Co-Lipschitzian Mappings  
 
  Authors: Ram U. Verma,  
  Keywords: Generalized auxiliary variational inequality problem, Cocoercive mappings, Approximation-solvability, Approximate solutions, Partially relaxed monotone mappings.  
  Date Received: 31/12/00  
  Date Accepted: 15/03/01  
  Subject Codes:

49J40

  
  Editors: Drumi Bainov,  
 
  Abstract:

The approximation-solvability of the following class of nonlinear variational inequality (NVI) problems, based on a new generalized auxiliary problem principle, is discussed.

Find an element $ x^{ast }in $ $ K$ such that

$displaystyle leftlangle (S-T)left( x^{ast }right) ,x-x^{ast }rightrangle +f(x)-fleft( x^{ast }right) geq 0 $for all $displaystyle xin K, $
where $ S,T:Krightarrow H$ are mappings from a nonempty closed convex subset $ K$ of a real Hilbert space $ H$ into $ H$, and $ f:Krightarrow mathbb{R}$ is a continuous convex functional on $ K.$ The generalized auxiliary problem principle is described as follows: for given iterate $ x^{k}in K$ and, for constants $ rho>0$ and $ sigma>0$), find $ x^{k+1}$ such that

 $displaystyle leftlangle rho (S-T)left( y^{k}right) +h^{prime }left( x^{k+1}right) -h^{prime }left( y^{k}right) ,x-x^{k+1}rightrangle$   
 $displaystyle qquadqquad +rho (f(x)-f(x^{k+1}))geq 0$ for all  $displaystyle xin K,$
   

where





  $displaystyle leftlangle sigma (S-T)left( x^{k}right) +h^{prime }left( y^{k}right) -h^{prime }left( x^{k}right) ,x-y^{k}rightrangle$    
 $displaystyle qquadqquad +sigma (f(x)-f(y^{k}))geq 0 $ for all  $displaystyle xin K,$   

where $ h$ is a functional on $ K$ and $ h^{prime }$ the derivative of $ h$. ;


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