Abstract. The aim of this note is to give a short proof that 2-local derivations on Mn, the n × n matrix algebra over the complex numbers are derivations and to give a shorter proof that 2-local *-automorphisms on Mn are *-automorphisms.A mapping φ of an algebra A into itself is called a local automorphism (respectively, local derivation) if for every A ∈ A there exists an automorphism (respectively, local derivation) φ A of A, depending on A, such that φ(A) = φ A (A). These notions were introduced by Kadison [Kad] and Larson and Sourour [LaSo]. In fact, their definitions were stronger. They have assumed that these mappings are also linear. Larson and Sourour proved that every local derivation on B(X), the algebra of all bounded linear operators on a Banach space X, is a derivation, and provided that X is infinite dimensional, every surjective linear local automorphism of B(X) is an automorphism. In [BrSe], they proved that the surjectivity assumption in the last result can be dropped if X is a separable Hilbert space.It is easy to see that if we drop the assumption of linearity of the local maps, then the corresponding statements are no longer true. However, in [KoSl]

In this paper, we noticed that the existence of fixed points of F-contractions, in F -metric space, can be ensured without the third condition (F3) imposed on the Wardowski function F : ( 0 , ∞ ) → R . We obtain fixed points as well as common fixed-point results for Reich-type F-contractions for both single and set-valued mappings in F -metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented. The application shows the role of fixed-point theorems in dynamic programming, which is widely used in computer programming and optimization. Our results extend and generalize the previous results in the existing literature.

Abstract. It is shown that continuous 2-local derivations on AF C * -algebras are derivations and surjective 2-local *-automorphisms on prime C * -algebras or on C * -algebras such that the identity element is properly infinite are *-automorphisms.A mapping φ of an algebra A into itself is called a local derivation (respectively, local automorphism) if for every A ∈ A there exists a derivation (respectively, automorphism) φ A of A, depending on A, such that φ(A) = φ A (A). These notions were introduced by Kadison [Kad] and Larson and Sourour [LaSo]. In fact, their definitions were stronger. They have assumed that these mappings are also linear. Larson and Sourour proved that every local derivation on B(X), the algebra of all bounded linear operators on a Banach space X, is a derivation, and provided that X is infinite-dimensional, every surjective linear local automorphism of B(X) is an automorphism. In [BrSe], they proved that the surjectivity assumption in the last result can be dropped if X is a separable Hilbert space.It is easy to see that if we drop the assumption of linearity of the local maps, then the corresponding statements are no longer true. However, in [KoSl]

In this article, we prove the generalized Hyers-Ulam stability of the following Pexider functional inequalitiesin non-Archimedean Banach spaces.

No abstract