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An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities

Authors:

Institution:

(1) International Publications, USA, Mathematical Sciences Division, 12046 Coed Drive, Suite A-29, Orlando, Florida, 32826. E-mail

Abstract:

Consider the convergence of the projection methods based on an extension of a special class of algorithms for the approximation--solvability of the following class of nonlinear quasivariational inequality (NQVI) problems: find an element $x* \in H$ such that $g\left( {x*} \right) \in K$ and

$\left\langle {T\left( {x*} \right),g\left( x \right) - g\left( {x{\kern 1pt} *} \right)} \right\rangle \geqslant 0{\text{foral}}{\kern 1pt} \operatorname{l} g\left( x \right) \in K,$

where $T,g:H \to H$ are mappings on H and K is a nonempty closed convex subset of a real Hilbert space H. The iterative procedure is characterized as a nonlinear quasivariational inequality: for any arbitrarily chosen initial point x ⁰ isin

K and, for constants $\rho >0$ and $\beta >0$ , we have $\left\langle {\rho T\left( {y^k } \right) + g\left( {x^{k + 1} } \right) - g\left( {y^k } \right),g\left( x \right) - g\left( {x^{k + 1} } \right)} \right\rangle \geqslant 0$

${\text{foral}}\operatorname{l} g\left( x \right) \in K{\text{andfor}}k \geqslant 0,$

where

$\left\langle {\beta T\left( {x^k } \right) + g\left( {y^k } \right) - g\left( {x^k } \right),g\left( x \right) - g\left( {y^k } \right)} \right\rangle \geqslant 0{\text{foral}}{\kern 1pt} \operatorname{l} g\left( x \right) \in K.$

This nonlinear quasivariational inequality type algorithm has an equivalent projection formula

$g\left( {x^{k + 1} } \right) = P_K \left {g\left( {y^k } \right) - \rho T\left( {y^k } \right)} \right],$

where

$g\left( {y^k } \right) = P_K \left {g\left( {x^k } \right) - \beta T\left( {x^k } \right)} \right],$

for the projection P _K of H onto K.

Keywords:

Cocoercive mapping projection method approximation-solvability

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