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UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

In this paper we define nearly continuity and nearly openness of linear operators between locally convex cones. Also we give some conditions on locally convex cones and linear operators which ensure that every nearly continuous (nearly open) mapping is continuous (open). We show by an example that a nearly continuous operator is not necessarily continuous.

We define the concept of completion for locally convex cones. We show that how a locally convex cone with (SP) can be embedded as an upper dense subcone in an upper complete locally convex cone with (SP). We prove that every upper complete locally convex cone with (SP) is also symmetric complete.

In this paper, we define dual (abstract) neighborhood systems for locally convex cones. Also we consider three types of different polars and study some relations of them with bounded sets in locally convex cones.

We define and study the projective and inductive limit notions for locally convex cones. We use convex quasiuniform structure method for this purpose. Also we study the barreledness in the locally convex cones and introduce the notion upper-barreled cones and prove that the inductive limit of upper-barreled cones is upper-barreled.

C. Park in his paper (C. Park, Isomorphisms between C ∗ -ternary algebras ,J. Math. Anal. Appl. 327 (2007) 101-115.), has proved the Hyers-Ulam-Rassias stability of homomor- phisms in C ∗ -ternary algebras and of derivations on C ∗ -ternary algebras for the following Cauchy-Jensen additive mappings: f x + y

In this paper, we investigate homomorphisms between C ∗ -ternary algebras, and derivations on C ∗ -ternary algebras associated with the following Cauchy–Jensen type additive functional equation: f ( x + y 2 + z ) + f ( x + z 2 + y ) + f ( y + z 2 + x ) = 2 ( f ( x ) + f ( y ) + f ( z ) ) .