Closed convex subset

Let JsT be a closed convex subset of V. We consider the operator inequal-... 

Theorem 1.15. Let V be a reflexive separable Banach space, and K be a closed convex subset in V. Assume that an operator A V V is... 

Let K he a closed convex subset in a reflexive Banach space V let an operator A act from V into V and let f G V he given. Consider the variational inequality... 

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

The complement principle uses the idea of closures of a convex set X, conv(X). The closure of con(X), cl conv(X), is the smallest closed subset of conv(X). The closure of conv(X) is used to represent the set of all points in conv(X) including any points that might only be obtained in the limit of a process. cl conv(X) is used to include points in conv(X) that might not be physically achievable (i.e., equilibrium points). 

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