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... In the rest of this paper, unless otherwise explicitly stated, we assume that G is an Abelian group, E a vector space, n a positive integer and k a fixed integer with k = ±1. We follow the notation used in [1,13,16]. ...

... A family S(E) of subsets of E is called linearly invariant (see [1]) if it is closed under the addition and scalar multiplication defined as usual sense and translation invariant, i.e., x + A ∈ S(E), for all x ∈ E and A ∈ S(E). It is easy to verify that S(E) contains all singleton subsets of E. In particular, CB(E) the family of all closed balls is a linearly invariant family in a normed vector space E. ...

... Definition 1 [1,13]. We say that B (G n , S(E)) admits a multi-symmetric left invariant mean (MSLIM, in short) if the family S(E) is linearly invariant and there exists a linear operator M : ...

A mapping F:Gn→E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F:G^n\rightarrow E$$\end{document}, where G is an Abelian group, E a vector space, and n a positive integer, is called generalized multi-quadratic if it is generalized quadratic in each variable. In this paper, we prove the stability of generalized multi-quadratic mappings in Lipschitz spaces. The results of the present paper improve and extend some existing results.

... Later, it was also used by Badora [8] in 1993 and Czerwik [42, pp. 333-338] in 2002. ...

... Concerning the operation ∧ , we can also easily prove the following Theorem 10. 8 We have ...

... Theorem 11. 8 We have ...

In the first part of this paper, we provide several historical facts on the famous Hyers-Ulam stability theorems, Hahn-Banach extension theorems, and their set-valued generalizations with numerous references. These generalizations will clearly show that the essence of the above-mentioned theorems is nothing but the statement of the existence of a certain homogeneous, additive, or linear selection function of a particular relation. In the second part of this paper, motivated by the above generalizations, we briefly review the most basic additivity and homogeneity properties of relations and investigate, in greater detail, some elementary operations on relations. More concretely, for any relation F on one group X to another Y, we define two relations -F and F ˇ on X to Y such that F ˇ(x)=F(-x) and (-F)(x)=-F(x) for all x∈X. Moreover, we also define F ^=-F ˇ and F =F∩F ˇ. Furthermore, if in particular Y is a vector space over ℚ, then for any k∈ℤ, with k≠0, we also define a relation F k on X to Y such that F k (x)=k -1 F(kx) for all x∈X. Moreover, we also define F ☆ =⋂ n=1 ∞ F n and F * =F ☆ . The above operations and the intersection convolutions of relations, which can only be sketched here, will certainly allow for instructive treatments of some hoped-for common relational generalizations of the Hyers-Ulam and Hahn-Banach theorems.

... Any solution of (1) in the space of real numbers is of the form g(x) = ax 2 for all x ∈ , where a ∈ . The stability problem of the functional equation (1) has been verified in other spaces (see [17,2,4]). ...

... Let G × G × G be the Cartesian product of an abelian group G with itself and denote by G 3 . We denote By B (G 3 , S (V)) the subset of all functions f : [1]), and there exists a linear operator Γ : [ ] : ...

Stability of functional equations is a classical problem proposed by Ulam. In this paper, we prove the stability of the 3-quadratic functional equations in Lipschitz spaces.

... The first one, the so-called direct method, or iterative method was already invented by Hyers in [16] (see also [6], [14], [24], [28]). Results using the technique of invariant means (over amenable semigroups) were first proved by L. Székelyhidi [29] (see also [2], [3], [4]). The third method is to use variants of the Hahn-Banach separation theorem or sandwich theorems, or more generally selections theorems, see [23], [22]. ...

... Taking the limit η → ε in the last inequality, we arrive at (2), which was to be proved. ...

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations.

... Since then, the method of invariant means has been used by a number of authors for solving stability problems (see the recent book [6,Chapter 4] for an exposition; the papers [10,11] contain further information). Our purpose with this note is to give a simple approach to vector-valued invariants means which extends (and, we hope, clarifies) previous results by Gajda [3], Badora [1] and Ger [4]. To some extent, our results complement Zhang's illuminating monograph [14] (see also [15]). ...

... More precisely, we show that a Banach space is complemented in its ultraproducts (equivalently, in its bidual) if and only if for every amenable semi-E-mail address: fcabello@unex.es. 1 Supported in part by DGICYT project PB97-0377. group S the space B(S, Y ) of Y -valued bounded functions on S admits an invariant average; this is equivalent to admit what we call "an admissible assignment." ...

We show that a Banach space X is complemented in its ultraproducts if and only if for every amenable semigroup S the space of bounded X-valued functions defined on S admits (a) an invariant average; or (b) what we shall call “an admissible assignment”. Condition (b) still provides an equivalence for quasi-Banach spaces, while condition (a) necessarily implies that the space is locally convex.

... Analogously we can define so-called right invariant mean. For more information about spaces which admit LIM see, e.g., [1,6,7]. Now we will introduce d-Lipschitz functions (see [12]). ...

... In this section we are going to introduce some basic definitions and notations needed for further considerations. Definition 1 [1]. Let E be a vector space and S(E) a family of subset of E. We say that this family is linearly invariant if ...

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y) for x,y∈G. Under some assumptions about f and Qf we prove that if Qf is Lipschitz, then there exists a quadratic function K:G→E such that f−K is Lipschitz. Moreover, some results concerning the stability of the quadratic functional equation in the Lipschitz norms are presented.

... Besides the shift operator, there are classes of bounded linear operators C on ' 1 and Banach limits which are invariant with respect to C in the sense that BH ¼ B for all B 2 B and H 2 C: A useful choice for is the subset of all operators T 2 Lð' 1 Þ satisfying: ...

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence ð1; 1;. . .; 1;. . .Þ is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces. The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of 'simultaneous continuous linear' operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces. Some open problems that naturally arise in this circle of ideas shall also be included at their appropriate places.

... Later, it was also used by Badora [8] in 1993 and Czerwik [42, pp. 333-338] in 2002. ...

... Analogously we can define so-called right invariant mean. For more information about spaces which admit LIM see, e.g., [2,12,13]. ...

Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the formula
$$\Lambda {\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right): = 2{\rm{f}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{y}} \right) + {\rm{f}}\left( {{\rm{ - y}}} \right) - {\rm{f}}\left( {{\rm{x + y}}} \right) - {\rm{f}}\left( {{\rm{x - y}}} \right)$$
for all x, y ∈ G. In this article, we prove that if ˄f is Lipschitz, then there exists a Drygas function D : G → E such that f − D is Lipschitz with the same constant. Moreover, some results concerning the approximation of the Drygas functional equation in the Lipschitz norms are presented.

... (ii) Injective ( [2]). The proof in each of the above cases depends upon the notion and the existence of vector-valued Banach limits on certain special spaces of X-valued bounded functions on a semigroup S and a certain 'selection procedure'. ...

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces. The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of simultaneous continuous linear operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces. Some open problems that naturally arise in the study have also been included.

... Besides the shift operator, there are classes of bounded linear operators C on ' 1 and Banach limits which are invariant with respect to C in the sense that BH ¼ B for all B 2 B and H 2 C: A useful choice for is the subset of all operators T 2 Lð' 1 Þ satisfying: ...

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1, 1,. .. , 1,. . .) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces. The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of 'simultaneous continuous linear' operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Ba-nach spaces.

... Analogously we can define so-called right invariant mean. For more information about spaces which admit LIM see, e.g., [3,11,12]. ...

... There are results when, as in Sect. 3 on quantifier weakening, a property such as additivity or subadditivity holds off some exceptional set (say, almost everywhere), and the conclusion is also similarly restricted. This goes back to work of Hyers and Ulam [1,20]. See also de Bruijn [25], Ger [32,33]. ...

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

... 61-62], see also [4,Theorem 3.3]). Quite clearly, the possibility of averaging vectors in an infinitedimensional space is a desirable titbit and so has been considered, for instance, in the theory of functional equations ( [2,3,8,9]). ...

Banach spaces that are complemented in the second dual are characterised precisely as those spaces $X$ which enjoy the property that for every amenable semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined on the Banach space of all bounded $X$-valued functions on $S$. This was first observed by Bustos Domecq (J. Math. Anal. Appl., 2002), however the original proof was slightly flawed as remarked by Lipecki. The primary aim of this note is to present a corrected version of the proof. We also demonstrate that universally separably injective spaces always admit invariant means with respect to countable amenable semigroups, thus such semigroups are not rich enough to capture complementation in the second dual as spaces falling into this class need not be complemented in the second dual.

... The proof of Theorem 9 makes use of some results due to Z. Gajda ([44]) and R. Badora ([4]) extending the notion of amenability to vector–valued mappings. The case (i) has also been treated by B.E. Johnson in[61]. ...

The paper is a survey about Hyers—Ulam stability of functional equations and systems in several variables.
Its content is divided in the following chapters:
1.
—Introduction. Historical background.
2.
—The additive Cauchy equation; Jensen’s equation.
3.
—The quadratic equation and the polynomial equation.
4.
—The multiplicative Cauchy equation. Superstability.
5.
—Approximately multiplicative linear maps in Banach algebras.
6.
—Other equations and systems.
7.
—Final remarks and open problems.
The bibliography contains 120 items.

... The technique presented in the foregoing proof has been developed further and applied to several stability problems (see e.g. [7,8]). We will mention here two more contributions, in which the authors deal with conditional stability problems of approximate additivity almost everywhere or on some large sets. ...

The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them.

... The existence of an invariant mean in the space of weakly almost periodic functions was investigated by de-Leew and Glicksberg [14], [15]. Vector-valued invariant means have been used by a number of authors for the study of some vectorvalued function spaces, functional equations , a linear topological classification of spaces of continuous functions and for solving stability problems [1], [2], [6], [23]- [25]. ...

A definition of an invariant averaging for a linear representation of a group in a locally convex space is given. Main results: A group $H$ is finite if and only if every linear representation of $H$ in a locally convex space has an invariant averaging. A group $H$ is amenable if and only if every almost periodic representation of $H$ in a quasi-complete locally convex space has an invariant averaging. A locally compact group $H$ is compact if and only if every strongly continuous linear representation of $H$ in a quasi-complete locally convex space has an invariant averaging.

... This element m is then called the mean of f and is denoted by M. f /. Another approach is due to Badora [1] who extends M to normed spaces with the Hahn-Banach extension property. Ger [6] considered boundedly complete Banach lattices with a strong unit element and showed that a mean M on .S;Ê/ admits a continuous linear extension on the space of bounded lattice-valued functions. ...

The main result of this paper offers a necessary and sufficient condition for the existence of an additive selection of a weakly compact convex set-valued map defined on an amenable semigroup. As an application, we obtain characterisations of the solutions of several functional inequalities, including that of quasi-additive functions.

... The proof of this theorem uses some results of Z. Gajda in [29] and R. Badora in [5] on vector-valued invariant means. The case i) has been treated also by B.E. Johnson in [55]. ...

The present paper is devoted to the memory of D.H. Hyers, who contributed to the theory of stability of functional equations and inequalities with fundamental ideas and results. Here we try to present a summary of this field, attaching a list of references, in order to help the interested reader to gather information. The author apologizes for leaving unmentioned further important results-due to lack of time and space.

... The proof of Theorem 9 makes use of some results due to Z. Gajda ([44]) and R. Badora ([4]) extending the notion of amenability to vector–valued mappings. The case (i) has also been treated by B.E. Johnson in [61]. ...

The paper is a survey about Hyers—Ulam stability of functional equations and systems in several variables.
Its content is divided in the following chapters:1.—
Introduction. Historical background.
2.—
The additive Cauchy equation; Jensen's equation.
3.—
The quadratic equation and the polynomial equation.
4.—
The multiplicative Cauchy equation. Superstability.
5.—
Approximately multiplicative linear maps in Banach algebras.
6.—
Other equations and systems.
7.—
Final remarks and open problems.
The bibliography contains 120 items.

It is well-known that the Lipschitz stability originated from the paper that appeared in J. Tabor (1997) [32]. In this work, we establish the general solution of the new class of generalized multi-quadratic functional equationf(x1,…,xi−1,xi+kyi,xi+1,...,xn)+f(x1,…,xi−1,xi−kyi,xi+1,…,xn)=f(x1,…,xi−1,xi+ℓyi,xi+1,…,xn)+f(x1,…,xi−1,xi−ℓyi,xi+1,…,xn)+2(k2−ℓ2)f(x1,...,xi−1,yi,xi+1,...,xn),xi,yi∈G,i∈{1,...,n} where k,ℓ are two fixed integers with k≠±ℓ and G is an Abelian group. Under some natural conditions, we prove the stability of the above equation in Lipschitz spaces. Moreover, some results concerning the stability of the generalized multi-quadratic type functional equation in the Lipschitz norms are presented. Our main results improve and generalize results obtained in [5], [10], [11], [12], [25], [26], [27].

After some preparations, we prove several useful theorems on Hyers sequences and their pointwise and uniform limits in quite natural ways which make a straightforward generalization of Hyers's stability theorem rather plausible.

By working out an appropriate technique of relations and relators and extending the ideas of the direct methods of Z. Gajda and R. Ger, we prove some generalizations of the stability theorems of D. H. Hyers, T. Aoki, Th. M. Rassias and P. Gavru¸tˇaGavru¸tˇGavru¸tˇa in terms of the existence and unicity of 2-homogeneous and additive approximate selections of generalized subadditive relations of semigroups to vector relator spaces. Thus, we obtain generalizations not only of the selection theorems of Z. Gajda and R. Ger, but also those of the present author.

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic diﬀerence of the function f by the formula
Qk f (x, y) := f (x + ky) + f (x − ky) − f (x + y) − f (x − y) − 2(k² − 1)f (y)
for all x, y ∈ G and for any integer k with k ≠ 1, −1. In this paper, we achieve the general solution of equation Qk f (x, y) = 0, after it, we show that if Qk f is Lipschitz, then there exists a quadratic function K : G → E such that f − K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].

Let ℕ be the set of all positive integers, G an Abelian group with a metric d and E a normed space. For any f : G → E we define the k-quadratic difference of the function f by the formula
for x, y ∈ G and k ∈ ℕ. Under some assumptions about f and Qk f we prove that if Qk f is Lipschitz, then there exists a quadratic function K : G → E such that f − K is Lipschitz with the same constant. Moreover, some results concerning the stability of the k-quadratic functional equation in the Lipschitz norms are presented.

In this paper we approximate the quartic functional equations in Lipschitz spaces.

We consider the problem of the separation of a pair of n-subadditive and n-superadditive functions defined on a product of amenable semigroups with values in a complete vector lattice by a n-additive mapping.

Let G be an amenable metric semigroup with nonempty center, let E be a reflexive Banach space, and let ƒ: G → E be a given function. By Cƒ: G × G → E we understand the Cauchy difference of the function /, i.e.:
$$ {\cal C}f(x,y):=f(x+y)- f(x)- f(y)\ {\rm for}\ x,y\in G. $$
We prove that if the function C(f) is Lipschitz then there exists an additive function A: G → E such that f − A is Lipschitz with the same constant. Analogous result for Jensen equation is also proved.
As a corollary we obtain the stability of the Cauchy and Jensen equations in the Lipschitz norms.

We study the stability of the Drygas functional equation: g(xy) + g(xy(-1)) = 2g(x) + g(y) + g(y(-1)) mixing the direct method of the proof with the method of the invariant means. Dropping the assumption about the domain to be an Abelian group, we assume that the function we are dealing with is central (i.e., g(xy) = g(yx)), is approximatively central (i.e., vertical bar g(xy) - g(yx)vertical bar <= delta), satisfies the Kannappan condition (i.e., g(xyz) = g(xzy)), or that the group is amenable.

The main result of this paper offers a necessary and sufficient condition for the existence of a multimonomial selection of a set-valued map with weakly compact convex values defined on an amenable semigroup.

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