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Stability property of a system of inequalities

Some general criteria are given for the stability of a system of convex inequalities and then extended to an arbitrary system by the procedure of convex approximation. The stability is understood in the following sense: given a system x∊D G(x)∊M, where D is a subset of a locally convex space X, M a closed convex cone in a locally convex space Y, G a mapping from D to Y, a solution [xbar] is said to be stable if for every neighbourhood W of xbar; there exists a neighbourhood V of 0 ∊ Y such that for any conditinuous mapping A: D → Y satisfying A(D ∩ W) ⊂V+M the perturbed system x∊D G(x) A(x)∊M has at lease one solution in W; the system is said to be stable if every its solution is stable.