Volume 2,  Issue 3, 2001

Article 27

GENERALIZED AUXILIARY PROBLEM PRINCIPLE AND SOLVABILITY OF A CLASS OF NONLINEAR VARIATIONAL INEQUALITIES INVOLVING COCOERCIVE AND CO-LIPSCHITZIAN MAPPINGS

RAM U. VERMA

UNIVERSITY OF TOLEDO
DEPARTMENT OF MATHEMATICS
TOLEDO, OHIO 43606, USA
E-Mail: verma99@msn.com

Received _____; accepted 15 March, 2001.
Communicated by: D. 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 \left\langle (S-T)\left( x^{\ast }\right) ,x-x^{\ast }\right\rangle
+f(x)-f\left( x^{\ast }\right) \geq 0   $for all $\displaystyle x\in K,
$
where $ S,T:K\rightarrow H$ are mappings from a nonempty closed convex subset $ K$ of a real Hilbert space $ H$ into $ H$, and $ f:K\rightarrow \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
  $\displaystyle \left\langle \rho (S-T)\left( y^{k}\right) +h^{\prime }\left( x^{k+1}\right) -h^{\prime }\left( y^{k}\right) ,x-x^{k+1}\right\rangle$    
  $\displaystyle \qquad\qquad +\rho (f(x)-f(x^{k+1}))\geq 0$ for all $\displaystyle x\in K,
$    
where
  $\displaystyle \left\langle \sigma (S-T)\left( x^{k}\right) +h^{\prime }\left( y^{k}\right) -h^{\prime }\left( x^{k}\right) ,x-y^{k}\right\rangle$    
  $\displaystyle \qquad\qquad +\sigma (f(x)-f(y^{k}))\geq 0\,\,$ for all $\displaystyle x\in K,
$    
where $ h$ is a functional on $ K$ and $ h^{\prime }$ the derivative of $ h$.
Key words:
Generalized auxiliary variational inequality problem, Cocoercive mappings, Approximation-solvability, Approximate solutions, Partially relaxed monotone mappings.

2000 Mathematics Subject Classification:
49J40.


Download this article (PDF):

Suitable for a printer:    

Suitable for a monitor:        

To view these files we recommend you save them to your file system and then view by using the Adobe Acrobat Reader. 

That is, click on the icon using the 2nd mouse button and select "Save Target As..." (Microsoft Internet Explorer) or "Save Link As..." (Netscape Navigator).

See our PDF pages for more information.

Other papers in this issue

Generalized Auxiliary Problem Principle and Solvability of a Class of Nonlinear Variational Inequalities Involving Cocoercive and Co-Lipschitzian Mappings
Ram U. Verma

On Some Generalizations of Steffensen's Inequality and Related Results
P. Cerone

A Weighted Analytic Center for Linear Matrix Inequalities
I. Pressman and S. Jibrin

Good Lower and Upper Bounds on Binomial Coefficients
Pantelimon Stanica 

Improvement of an Ostrowski Type Inequality for Monotonic Mappings and its Application for Some Special Means
S.S. Dragomir and M.L. Fang

On the Utility of the Telyakovskii's Class S
Laszlo Leindler 

L'Hospital Type Rules for Oscillation, with Applications
Iosif Pinelis

Matrix and Operator Inequalities
Fozi M. Dannan

Consequences of a Theorem of Erdös-Prachar
Laurentiu Panaitopol 

On a Reverse of Jessen's Inequality for Isotonic Linear Functionals
S.S. Dragomir 

Lp-Improving Properties for Measures on R4 Supported on Homogeneous Surfaces in Some Non Elliptic Cases
E. Ferreyra, T. Godoy and M. Urciuolo 

Some Properties of the Series of Composed Numbers
Laurentiu Panaitopol


Other issues

 

 

© 2000 School of Communications and Informatics, Victoria University of Technology. All rights reserved.
JIPAM is published by the School of Communications and Informatics which is part of the Faculty of Engineering and Science, located in Melbourne, Australia. All correspondence should be directed to the editorial office.

Copyright/Disclaimer