Existence of solutions to operator equations and inequalities

Let us formulate assertions related to a solvability of problems which are not variational ones in general (Lions, 1969). 

Theorem 1.14. Let V be a reflexive separable Banach space. Assume that an operator A E — E possesses the following properties  

Then the equation (1.85) has at least one solution u G V for every fixed u gV. This solution is unique if A is strictly monotonous. 

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

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