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Every strongly minihedral normal cone is regular.

The definition of a normal cone is given below [6].

The proximal normal cone is closed as a set-valued mapping.

It is easy to see that every strongly minihedral normal cone is regular.

Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from T K (x) by polarity.

For all x ∈ U r), one has P K r ( x ) ≠ ∅ ; For all r′ ∈ (0, r, P K r is Lipschitz continuous with constant r r - r ′ on U ( r ′ ) = { u ∈ H : 0 < d K r ( u ) < r ′ } ; The proximal normal cone is closed as a set-valued mapping.

Common fixed point theorem under contractive condition of Ćirić's type (see [20]) on cone metric space in settings of a normal cone was proved in [21].

In [11], the concept of a set-valued contraction of Mizoguchi-Takahashi type was introduced and a fixed point theorem in setting of a normal cone was proved.

Recently, optimality conditions and duality for a strict minimizer of nonsmooth multiobjective optimization problems with normal cone were derived in [14].

For example, the projection formula onto a cone facilitates designing algorithms for solving conic programming problems [9 13]; the distance formula from an element to a cone is an important factor in the approximation theory; the tangent cone and normal cone are crucial in analyzing the structure of the solution set for optimization problems [14 16].

Let be a cone metric space, a normal cone, and is a Zamfirescu operator with and whenever is a sequence with as, then Picard's iteration is -stable.