An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities
Tóm tắt
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 ∈ 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.
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