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Let be an algebra, and let , be ring automorphisms of . An additive mapping is called a -derivation if for all . Moreover, an additive mapping is said to be a generalized -derivation if there exists a -derivation such that for all . In this paper, we investigate the superstability of generalized -derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation. 1. Introduction and Preliminaries In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [2, 3]. A non-Archimedean field is a field equipped with a function (valuation) from into such that if and only if ,？？ , and for all . An example of a non-Archimedean valuation is the mapping taking everything but 0 into 1 and . This valuation is called trivial (see [4]). Definition 1.1. Let be a vector space over a scalar field with a non-Archimedean non-trivial 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) for all (the strong triangle inequality). A sequence in a non-Archimedean space is Cauchy’s if and only if converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy’s sequence is convergent. A non-Archimedean-normed algebra is a non-Archimedean-normed space with a linear associative multiplication, satisfying for all . A non-Archimedean complete normed algebra is called a non-Archimedean Banach’s algebra (see [5]). Definition 1.2. Let be a nonempty set, and let satisfy the following properties: (D1) if and only if ,(D2) (symmetry),(D3) (strong triangle inequality), for all . Then is called a non-Archimedean generalized metric space. is called complete if every Cauchy’s sequence in is convergent. Definition 1.3. Let be a non-Archimedean algebra, and let , be ring automorphisms of . An additive mapping is called a derivation in case holds for all . An additive mapping is said to be a generalized derivation if there exists a derivation such that for all . We need the following fixed point theorem (see [6, 7]). Theorem 1.4 (Non-Archimedean Alternative Contraction Principle). Suppose is a non-Archimedean generalized complete metric space and is a strictly contractive mapping; that is, for some . If there exists a nonnegative integer such that for some , then the followings are true.(a)The sequence converges to a fixed point of .(b) is a unique fixed point of in (c)If , then A functional equation is superstable if every

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The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used.

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

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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 .

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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∈ .

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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

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Our aim is to study the Ulam's problem for Cauchy's functional equations. First, we present some new results about the superstability and stability of Cauchy exponential functional equation and its Pexiderized for class functions on commutative semigroup to unitary complex Banach algebra. In connection with the problem of Th. M. Rassias and our results, we generalize the theorem of Baker and theorem of L. Sze'kelyhidi. Then the superstability of Cauchy additive functional equation can be prove for complex valued functions on commutative semigroup under some suitable conditions. This result is applied to the study of a superstability result for the logarithmic functional equation, and to give a partial affirmative answer to problem 18, in the thirty-first ISFE. The hyperstability and asymptotic behaviors of Cauchy additive functional equation and its Pexiderized can be study for functions on commutative semigroup to a complex normed linear space under some suitable conditions. As some consequences of our results, we give some generalizations of Skof's theorem, S.-M. Joung's theorem, and another affirmative answer to problem 18, in the thirty-first ISFE. Also we study the stability of Cauchy linear equation in general form and in connection with the problem of G. L. Forti, in the 13th ICFEI (2009), we consider some systems of homogeneous linear equations and our aim is to establish some common Hyers-Ulam-Rassias stability for these systems of functional equations and presenting some applications of these results.

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We deal with the stability of the exponential Cauchy functional equation in the class of functions mapping a group ( , +) into a Riesz algebra . The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem. 1. Introduction In 1979 Baker et al. (cf. [1]) proved that the exponential functional equation in the class of functions mapping a vector space to the real numbers is superstable; that is, any function satisfying, with given , the inequality is either bounded or exponential (satisfies (1)). Then Baker generalized this famous result in [2]. We quote this theorem here since it will be used in the sequel. Theorem 1 (cf. [2, Theorem？？1]). Let be a semigroup and let be given. If a function satisfies the inequality for all , then either for all or for all . After that the stability of the exponential functional equation has been widely investigated (cf., e.g., [3–6]). This paper will primarily be concerned with the question if similar result holds true in the class of functions taking values in Riesz algebra with the common notion of the absolute value of an element stemming from the order structure of . The main aim of the present paper is to show that the superstability phenomenon does not hold in such an order setting. However, we prove that the exponential functional equation (1) is stable in the Ulam-Hyers sense; that is, for any given satisfying inequality (3) there exists an exponential function which approximates uniformly on in the sense that the set is bounded in . As a method of investigation we apply spectral representation theory for Riesz spaces; to be more precise, we use the Yosida Spectral Representation Theorem for Riesz spaces with a strong order unit. For some recent results concerning stability of functional equations in vector lattices we refer the interested reader to [7–12]. 2. Preliminaries Throughout the paper , , , and are used to denote the sets of all positive integers, integers, real numbers and nonnegative real numbers, respectively. For the readers convenience we quote basic definitions and properties concerning Riesz spaces (cf. [13]). Definition 2 (cf. [13, Definitions？？11.1 and？？22.1]). We say that a real linear space , endowed with a partial order , is a Riesz space if exists for all and We define the absolute value of by the formula . A Riesz space is called Archimedean if, for each , the inequality holds whenever the set is bounded above.

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We will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: , , , , which can be considered the mixed functional equations of the sine function and cosine function, of the hyperbolic sine function and hyperbolic cosine function, and of the exponential functions, respectively.

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We prove the Hyers-Ulam-Rassias stability of homomorphisms in real Banach algebras and of generalized derivations on real Banach algebras for the following Cauchy-Jensen functional equations: , , which were introduced and investigated by Baak (2006). The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias' stability theorem that appeared in his paper (1978).