Abstract:
We investigate the superstability of generalized derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy functional equation.

Abstract:
We prove the generalized Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras. Moreover, we prove the superstability of homomorphisms on unital non-Archimedean Banach algebras and we investigate the superstability of derivations in non-Archimedean Banach algebras with bounded approximate identity. 1. Introduction and Preliminaries In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property. During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings [2]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition [3–9]. Let be a field. A non-Archimedean absolute value on is a function such that for any we have (i) and equality holds if and only if ,(ii) , (iii) . Condition (iii) is called the strict triangle inequality. By (ii), we have . Thus, by induction, it follows from (iii) that for each integer . We always assume in addition that is non trivial, that is, that there is an such that . Let be a linear space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions: (NA1) if and only if ; (NA2) for all and ; (NA3)the strong triangle inequality (ultrametric), namely, Then is called a non-Archimedean space. It follows from (NA3) that therefore a sequence is Cauchy in if and only if converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. A non-Archimedean Banach algebra is a complete non-Archimedean algebra which satisfies for all . For more detailed definitions of non-Archimedean Banach algebras, we can refer to [10]. The first stability problem concerning group homomorphisms was raised by Ulam [11] in 1960 and affirmatively solved by Hyers [12]. Perhaps Aoki was the first author who has generalized the theorem of Hyers (see [13]). T. M. Rassias [14] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. Theorem 1.1 (T. M. Rassias). Let be a mapping from a normed vector space into a Banach space subject to the inequality for all , where and are constants with and . Then the limit exists for all and is the unique additive mapping which satisfies for all . Also, if for each the mapping is continuous in

Abstract:
We investigate the superstability of the functional equation , where and are the mappings on Banach algebra . We have also proved the superstability of generalized derivations associated to the linear functional equation , where .

Abstract:
We investigate the superstability of the functional equation f(xy)=xf(y)+g(x)y, where f and g are the mappings on Banach algebra A. We have also proved the superstability of generalized derivations associated to the linear functional equation f(γx+βy)=γf(x)+βf(y), where γ,β∈ .

Abstract:
We investigate the stability and superstability of ternary quadratic higher derivations in non-Archimedean ternary algebras by using a version of fixed point theorem via quadratic functional equation.

Abstract:
We investigate the generalized Hyers-Ulam stability of the functional inequalities and in non-Archimedean normed spaces in the spirit of the Th. M. Rassias stability approach. 1. Introduction Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. Let be a group and let be a metric group with the metric . Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all , and for some . Then there exists a unique additive mapping such that for all . Moreover, if is continuous in for each fixed , then is linear. In 1978, Rassias [3] proved the following theorem. Theorem 1.1. Let be a mapping from a normed vector space into a Banach space subject to the inequality for all , where and p are constants with and . Then there exists a unique additive mapping such that for all . If then inequality (1.3) holds for all , and (1.4) for . Also, if the function from into is continuous in real for each fixed , then is linear. In 1991, Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations. The reader is referred to [5–13] for a number of results in this domain of research. In 1994, a generalization of the Rassias theorem was obtained by G？vru？a as follows [14]. Suppose is an abelian group, is a Banach space, and that the so-called admissible control function satisfies for all . If is a mapping with for all , then there exists a unique mapping such that and for all . During the last decades, several stability problems of functional equations have been investigated by a number of mathematicians, see [15–17] and references therein for more detailed information. By a non-Archimedean field we mean a field equipped with a function (valuation) from into such that if and only if , , and for all . Clearly and for all . Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm

Abstract:
We define the notion of an approximate generalized higher derivation and investigate the superstability of strong generalized higher derivations. 1. Introduction and Preliminaries The problem of stability of functional equations was originally raised by Ulam [1, 2] in 1940 concerning the stability of group homomorphisms. Hyers [3] gave an affirmative answer to the question of Ulam. Superstability, the result of Hyers, was generalized by Aoki [4], Bourgin [5], and Rassias [6]. During the last decades, several stability problems for various functional equations have been investigated by several authors. We refer the reader to the monographs [7–10]. Let be a complex normed space, and let . We denote by the linear space consisting of -tuples , where . The linear operations on are defined coordinatewise. The zero element of either or is denoted by 0. We denote by the set and by the group of permutations on symbols. Definition 1.1. A multi-norm on is a sequence such that is a norm on for each , for each , and the following axioms are satisfied for each with :( ) ;( ) ;( ) ;( ) .In this case, we say that is a multi-normed space. We recall that the notion of multi-normed space was introduced by Dales and Polyakov in [11]. Motivations for the study of multi-normed spaces and many examples are given in [11]. Suppose that is a multi-normed space, and . The following properties are almost immediate consequences of the axioms:(i) ;(ii) . It follows from that if is a Banach space, then is a Banach space for each . In this case, is a multi-Banach space. By (ii), we get the following lemma. Lemma 1.2. Suppose that and . For each , let be a sequence in such that . Then for each , one has Definition 1.3. Let be a multi-normed space. A sequence in is a multinull sequence if, for each , there exists such that Let . We say that if is a multi-null sequence. Definition 1.4. Let be a normed algebra such that is said to be a multi-normed space. Then is a multi-normed algebra if for and . Furthermore, if is a multi-Banach space, then is a multi-Banach algebra. Let be an algebra and . A family of linear mappings on is said to be a higher derivation of rank if the functional equation holds for all , . If , where is the identity map on , then is a derivation and is called a strong higher derivation. A standard example of a higher derivation of rank is , where is a derivation. The reader may find more information about higher derivations in [12–18]. A family of linear mappings on is called a generalized strong higher derivation if , and there exists a higher derivation such that for

Abstract:
In this paper, we consider the stability of generalized Cauchy functional equations such as Especially interesting is that such equations have the Hyers-Ulam stability or superstability whether g is identically one or not. 2000 Mathematics Subject Classification: 39B52, 39B82.

Abstract:
Using fixed point methods, we prove the superstability and generalized Hyers-Ulam stability of ring homomorphisms on non-Archimedean Banach algebras. Moreover, we investigate the superstability of ring homomorphisms in non-Archimedean Banach algebras associated with the Jensen functional equation.

Abstract:
Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in ？-algebras and Lie ？-algebras and of derivations on non-Archimedean ？-algebras and Non-Archimedean Lie ？-algebras for an -variable additive functional equation.