Can anyone put forward a condition that under that, the following quasi variational inequality have solution ?
(Y-X)^T.F(X)>0 , Y \in K(X) \in R^N
K(X) is a set valued function that has the following structure
K(X)=m(X)+ U(X)
where m(X) is a function and U(X) is a conic valued set valued function.
It looks like the model in my script Optimization and variational inequalities;
cf. pg. 48, corollary 3.5.4. There, one (among others) crucial supposition consists in lower semicontinuity of your cone-mapping U, and Kakutani's FP-Theorem is used.
the mentioned corollary in your text restrict K(x)=M is not set valued function of X, in equation 3.5.6, however in my question, K(x) is a set valued function as defined in my question. K(X)=m(X)+ U(X)
m(X) is a function and U(X) is a conic valued set valued function
in eq 3.5.7 however K(X)=\Gamma(X) is what I mean which is not the result of corollary 3.5.4!
Ok. the number of the corollary you need is 3.5.3, four lines above corollary 3.5.4 (a specialisation).
Is this a problem for you ?
In cor. 3.5.3 "quasi (VI) (3.5.7)" is explicitly written, in the other "(VI)"
You are right by setting Gamma = m + U, g = F in my text
(y-x)F(x)>=0 forall y in K(x) := m(x)+U(x) = Gamma(x) \subset R^n.
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