91 research outputs found
Approximate Homomorphisms of Ternary Semigroups
A mapping between ternary semigroups will be
called a ternary homomorphism if . In this paper,
we prove the generalized Hyers--Ulam--Rassias stability of mappings of
commutative semigroups into Banach spaces. In addition, we establish the
superstability of ternary homomorphisms into Banach algebras endowed with
multiplicative norms.Comment: 10 page
On the Orthogonal Stability of the Pexiderized Quadratic Equation
The Hyers--Ulam stability of the conditional quadratic functional equation of
Pexider type f(x+y)+f(x-y)=2g(x)+2h(y), x\perp y is established where \perp is
a symmetric orthogonality in the sense of Ratz and f is odd.Comment: 10 pages, Latex; Changed conten
Asymptotic stability of the Cauchy and Jensen functional equations
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
Some functional equations related to the characterizations of information measures and their stability
The main purpose of this paper is to investigate the stability problem of
some functional equations that appear in the characterization problem of
information measures.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1307.0657,
arXiv:1307.0631, arXiv:1307.0664, arXiv:1307.065
Normal Cones and Thompson Metric
The aim of this paper is to study the basic properties of the Thompson metric
in the general case of a real linear space ordered by a cone . We
show that has monotonicity properties which make it compatible with the
linear structure. We also prove several convexity properties of and some
results concerning the topology of , including a brief study of the
-convergence of monotone sequences. It is shown most of the results are
true without any assumption of an Archimedean-type property for . One
considers various completeness properties and one studies the relations between
them. Since is defined in the context of a generic ordered linear space,
with no need of an underlying topological structure, one expects to express its
completeness in terms of properties of the ordering, with respect to the linear
structure. This is done in this paper and, to the best of our knowledge, this
has not been done yet. The Thompson metric and order-unit (semi)norms
are strongly related and share important properties, as both are
defined in terms of the ordered linear structure. Although and
are only topological (and not metrical) equivalent on , we
prove that the completeness is a common feature. One proves the completeness of
the Thompson metric on a sequentially complete normal cone in a locally convex
space. At the end of the paper, it is shown that, in the case of a Banach
space, the normality of the cone is also necessary for the completeness of the
Thompson metric.Comment: 36 page
Orthogonalities and functional equations
In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations
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