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Approximate generalized derivations close to derivations in Lie C*-algebras

Eshaghi M., Abbaszadeh S.

Journal of Applied Analysis

2015

Vol. 21, nr 1

37--43

We apply a fixed point theorem to prove that there exists a unique derivation close to an approximately generalized derivation in Lie C*-algebras. Also, we prove the hyperstability of generalized derivations. In other words, we find some conditions under which an approximately generalized derivation becomes a derivation.

Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings

Filippis V. de, Dhara B.

Demonstratio Mathematica

2014

Vol. 47, nr 1

22--36

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, f(x1,…,xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(f(r))f(r) – f(r)H(f(r)) € C, for all r = (r1,…,rn) € In. Then one of the following holds: 1. there exist a; b; p (…) ; 2. R satisfies s4, the standard identity of degree 4, and there exist a; (…) ; 3. R satisfies s4 and there exist a; (…) ; 4. R satisfies s4 and there exist a; (…) ; 5. there exists e2 = e (…) and one of the following holds: (a) (…) is an identity for I; (b) char(R) = 2 and s4(x1, x2, x3, x4)x5 is an identity for I; (c) (…) is an identity for I and there exist a, a’, b, b’ (…) , a derivation of R, such that G(x) = ax + xa’ + d(x), H(x) = bx + xb’ – d(x), for all x (…) R, with (a – b’ – α)I = (0) = (b – a’ – α)I.

Examples and counterexamples in Lipschitz analysis

Fusek P., Klatte D., Kummer B.

Control and Cybernetics

2002

Vol. 31, no 3

471-492

In the analysis of functions and multi-valued mappings of Lipschitzian type, there are many different notions of Lipschitz behavior, regularity and generalized derivatives. We collect relevant examples illustrating the interrelations between various concepts, the differences with the smooth case, and the importance of certain assumptions and special classes of Lipschitz mappings in applications.

On a certain criterion of the existence of generalized derivative from Orlicz Class

Borsuk M.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 2

173-178

In terms of difference quotients it is established the existence of generalized derivative of the function.