Institute of Mathematics, University of Debrecen,
H-4010 Debrecen, Pf. 12,
Hungary
szaz@math.klte.hu

Department of
Mathematics, Faculty of Sciences,
Ibn Tofail University,
Kenitra, Morocco
charifi2000@yahoo.fr bbouikhalene@yahoo.fr

Department of
Mathematics, Faculty of Sciences,
Ibn Zohr University,
Agadir, Morocco
elqorachi@hotmail.com Redouani-ahmed@yahoo.fr
 

Institute of Mathematics
 Silesian University of Technology
 Kaszubska 23, 44-100 Gliwice,
Poland
Stefan.Czerwik@polsl.pl
Krzysztof.Krol@polsl.pl

Department of Mathematics
Chungnam National University
Daejeon, 305-764,
Republic of Korea
hmkim@cnu.ac.kr

Faculty of Engineering, Osaka Electro-Communication University,
Neyagawa 572-8530,
Japan

Faculty of Engineering, Ibaraki University,
Hitachi 316-8511,
Japan

Department of Applied Mathematics and Physics, Graduate School of Science and Engineering,
Yamagata University,
Yonezawa 992-8510
Japan

oka@mx.ibaraki.ac.jp
miura@yz.yamagata-u.ac.jp
sin-ei@emperor.yz.yamagata-u.ac.jp

Department of Mathematics, Hanyang University,
Seoul 133-791,
Republic of Korea

Department of Mathematics, National Technical University of Athens,
Zografou Campus, 15780 Athens,
Greece

baak@hanyang.ac.kr
trassias@math.ntua.gr

Mathematics Section, College of Science and Technology
 Hong-Ik University, 339-701 Chochiwon
 Republic of Korea.
smjung@hongik.ac.kr

Abstract:

In 1940 (and 1968) S. M. Ulam proposed the well-known Ulam stability problem. In 1941 D.H. Hyers solved the Hyers-Ulam problem for linear mappings. In 1951 D. G. Bourgin has been the second author treating the Ulam problem for additive mappings. In 1978 according to P.M. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982-2004 we established the Hyers-Ulam stability for the Ulam problem for different mappings. In 1992-2000 J.M. Rassias investigated the Ulam stability for Euler-Lagrange mappings. In this article we solve the Ulam problem for Euler-Lagrange type quadratic functional equations. These stability results can be applied in mathematical statistics, stochastic analysis, algebra, geometry, as well as in psychology and sociology.

Paper's Title:

Stability of a Mixed Additive, Quadratic and Cubic Functional Equation In Quasi-Banach Spaces

Author(s):

A. Najati and F. Moradlou

Department of Mathematics, Faculty of Sciences,
University of Mohaghegh Ardabili, Ardabil,
Iran
a.nejati@yahoo.com

Faculty of Mathematical Sciences,
University of Tabriz, Tabriz,
Iran
moradlou@tabrizu.ac.ir
 

Abstract:

In this paper we establish the general solution of a mixed additive, quadratic and cubic functional equation and investigate the Hyers--Ulam--Rassias stability of this equation in quasi-Banach spaces. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297--300.

Paper's Title:

Asymptotic Behavior of Mixed Type Functional Equations

Author(s):

J. M. Rassias

Pedagogical Department, E.E., National and Capodistrian University of Athens, Section of Mathematics And Informatics, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342,Greece
jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/~jrassias/

Abstract:

In 1983 Skof [24] was the first author to solve the Ulam problem for additive mappings on a restricted domain. In 1998 Jung [14] investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve the bounds and thus the results obtained by Jung [14], in 1998 and by the author [21], in 2002. Besides we establish new theorems about the Ulam stability of mixed type functional equations on restricted domains. Finally, we apply our recent results to the asymptotic behavior of functional equations of different types.

Paper's Title:

Solution of the Hyers-Ulam Stability Problem for Quadratic Type Functional Equations in Several Variables

Author(s):

John Michael Rassias

Pedagogical Department, E.E., National and Capodistrian University of Athens,
Section of Mathematics and Informatics,
4, Agamemnonos Str., Aghia Paraskevi,
Athens 15342,
Greece jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/~jrassias/

Abstract:

In 1940 (and 1968) S. M. Ulam proposed the well-known Ulam stability problem. In 1941 D. H. Hyers solved the Hyers-Ulam problem for linear mappings. In 1951 D. G. Bourgin has been the second author treating the Ulam problem for additive mappings. In 1978 according to P. M. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982-2004 we established the Hyers-Ulam stability for the Ulam problem for different mappings. In this article we solve the Hyers-Ulam problem for quadratic type functional equations in several variables. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.

Paper's Title:

Stability of a Pexiderized Equation

Author(s):

Maryam Amyar

Department of Mathematics,
Islamic Azad University, Rahnamaei Ave,
Mashhad 91735, Iran,
and Banach Mathematical Research Group (BMRG)
amyari@mshdiau.ac.ir

URL: http://amyari.mshdiau.ac.ir

Abstract:

The aim of the paper is to prove the stability of the Pexiderized equation f(x)=g(y+x)-h(y-x), for any amenable abelian group.

Paper's Title:

On a Method of Proving the Hyers-Ulam Stability of Functional Equations on Restricted Domains

Author(s):

Janusz Brzdęk

Department of Mathematics
 Pedagogical University Podchorąźych 2,
30-084 Kraków,
Poland
jbrzdek@ap.krakow.pl

Abstract:

We show that generalizations of some (classical) results on the Hyers-Ulam stability of functional equations, in several variables, can be very easily derived from a simple result on stability of a functional equation in single variable

Paper's Title:

Hyperbolic Barycentric Coordinates

Author(s):

Abraham A. Ungar

Abstract:

A powerful and novel way to study Einstein's special theory of relativity and its underlying geometry, the hyperbolic geometry of Bolyai and Lobachevsky, by analogies with classical mechanics and its underlying Euclidean geometry is demonstrated. The demonstration sets the stage for the extension of the notion of barycentric coordinates in Euclidean geometry, first conceived by Möbius in 1827, into hyperbolic geometry. As an example for the application of hyperbolic barycentric coordinates, the hyperbolic midpoint of any hyperbolic segment, and the centroid and orthocenter of any hyperbolic triangle are determined.

Paper's Title:

The Superstability of the Pexider Type Trigonometric Functional Equation

Author(s):

Gwang Hui Kim and Young Whan Lee

Department of Mathematics, Kangnam
University Yongin, Gyeonggi, 446-702, Korea.
 ghkim@kangnam.ac.kr

Department of Computer and Information Security
Daejeon University, Daejeon 300-716, Korea.
ywlee@dju.ac.kr
 

Abstract:

The aim of this paper is to investigate the stability problem for
the Pexider type (hyperbolic) trigonometric functional equation
f(x+y)+f(x+σy)=λg(x)h(y) under the conditions :
|f(x+y)+f(x+σy)- λg(x)h(y)|≤φ(x),
φ(y), and min {φ(x), φ (y)}.

As a consequence, we have generalized the results of stability for
the cosine(d'Alembert), sine, and the Wilson functional equations by J.
Baker, P. Găvruta, R. Badora and R. Ger, Pl.~Kannappan, and G.
H. Kim

Paper's Title:

On Some Mapping Properties of Möbius Transformations

Author(s):

Nihal Yilmaz Özgür

Abstract:

We consider spheres corresponding to any norm function on the complex plane and their images under the Möbius transformations. We see that the sphere preserving property is not an invariant characteristic property of Möbius transformations except in the Euclidean case.

Paper's Title:

On the Hyers-Ulam Stability of Homomorphisms and Lie Derivations

Author(s):

Javad Izadi and Bahmann Yousefi

Department of Mathematics, Payame Noor University,
P.O. Box: 19395-3697, Tehran,
Iran.
E-mail: javadie2003@yahoo.com, b_yousefi@pnu.ac.ir

Abstract:

Let A be a Lie Banach^*-algebra. For each elements (a, b) and (c, d) in A²:= A * A, by definitions

 (a, b) (c, d)= (ac, bd),
 |(a, b)|= |a|+ |b|,
(a, b)^*= (a^*, b^*),

A² can be considered as a Banach^*-algebra. This Banach^*-algebra is called a Lie Banach^*-algebra whenever it is equipped with the following definitions of Lie product:

for all a, b, c, d in A. Also, if A is a Lie Banach^*-algebra, then D: A²→A² satisfying

 D ([ (a, b), (c, d)])= [ D (a, b), (c, d)]+ [(a, b), D (c, d)]

for all $a, b, c, d∈A, is a Lie derivation on A². Furthermore, if A is a Lie Banach^*-algebra, then D is called a Lie^* derivation on A² whenever D is a Lie derivation with D (a, b)^*= D (a^*, b^*) for all a, b∈A. In this paper, we investigate the Hyers-Ulam stability of Lie Banach^*-algebra homomorphisms and Lie^* derivations on the Banach^*-algebra A².

Paper's Title:

A Stability of the G-type Functional Equation

Author(s):

Gwang Hui Kim

Department of Mathematics, Kangnam University
Suwon 449-702, Korea.
ghkim@kangnam.ac.kr

Abstract:

We will investigate the stability in the sense of Găvruţă for the G-type functional equation f(φ(x))=Γ(x)f(x)+ψ(x) and the stability in the sense of Ger for the functional equation of the form f(φ(x))=Γ(x)f(x). As a consequence, we obtain a stability results for G-function equation.

Paper's Title:

Power and Euler-Lagrange Norms

Author(s):

Mohammad Sal Moslehian and John Michael Rassias

Department of Mathematics,
Ferdowsi University,
P. O. Box 1159, Mashhad 91775,
Iran;
Department of Pure Mathematics,
University of Leeds,
Leeds LS2 9JT,
United Kingdom.
moslehian@ferdowsi.um.ac.ir
URL: http://www.um.ac.ir/~moslehian/

Pedagogical Department, E.E., Section of Mathematics and Informatics
National and Capodistrian University of Athens,
4, Agamemnonos str., Aghia Paraskevi, Attikis 15342, Athens,
Greece.
jrassias@primedu.uoa.gr
URL: http://www.primedu.uoa.gr/~jrassias/

Abstract:

We introduce the notions of power and Euler-Lagrange norms by replacing the triangle inequality, in the definition of norm, by appropriate inequalities. We prove that every usual norm is a power norm and vice versa. We also show that every norm is an Euler-Lagrange norm and that the converse is true under certain condition.

Paper's Title:

Stability Problems for Generalized Additive Mappings and Euler-Lagrange Type Mappings

Author(s):

M. Todoroki, K. Kumahara, T. Miura and S.-E. Takahasi

The Open University of Japan,
Chiba, 261-8586,
Japan
tomamiyu3232@sky.sannet.ne.jp
kumahara@ouj.ac.jp

 Yamagata University,
Yonezawa 992-8510,
Japan
miura@yz.yamagata-u.ac.jp

Toho University, Yamagata University,
Chiba, 273-0866,
Japan
sin_ei1@yahoo.co.jp

Abstract:

We introduce a generalized additivity of a mapping between Banach spaces and establish the Ulam type stability problem for a generalized additive mapping. The obtained results are somewhat different from the Ulam type stability result of Euler-Lagrange type mappings obtained by H. -M. Kim, K. -W. Jun and J. M. Rassias.

Paper's Title:

Superquadracity, Bohr's Inequality and Deviation from a Mean Value

Author(s):

S. Abramovich, J. Barić, and J. Pečarić

Department of Mathematics, University of Haifa,
Haifa, 31905,
Israel
abramos@math.haifa.ac.il 

FESB, University of Split,
Rudera Bošcovića,
B.B., 21000, Split,
Croatia
jbaric@fesb.hr  

Faculty of Textile Technology, University of Zagreb,
Prilaz Baruna Filipovića,
30, 10000 Zagreb,
Croatia.
pecaric@hazu.hr  

Abstract:

Extensions of Bohr's inequality via superquadracity are obtained, where instead of the power p=2 which appears in Bohr's inequality we get similar results when we deal with p≥ 2 and with p≤ 2. Also, via superquadracity we extend a bound for deviation from a Mean Value.

Paper's Title:

A Note on the Ulam stability of Reciprocal Difference and Adjoint Functional Equations

Author(s):

K. Ravi, J. M. Rassias, M. E. Gordji, and B. V. Senthil Kumar

Department of Mathematics,
Sacred Heart College, Tirupattur - 635601,
India
shckavi@yahoo.co.in
 

Pedagogical Department E. E.,
Section of Mathematics and Informatics,
National and Capodistrian University of Athens,
4, Agamemnonos Str., Aghia Paraskevi,
Athens, Attikis 15342,
GREECE
jrassias@primedu.uoa.gr

Department of Mathematics, Semnan University,
P.O. Box 35195-363, Semnan,
Iran
madjid.eshaghi@gmail.com

Department of Mathematics,
C.Abdul Hakeem College of Engineering and
Technology, Melvisharam - 632 509,
India
bvssree@yahoo.co.in

Abstract:

This note is an erratum to previous work published as Volume 8, Issue 1, Paper 13, 2011 of The Australian Journal of Mathematical Analysis and Applications.

Paper's Title:

Ostrowski Type Inequalities for Lebesgue Integral: a Survey of Recent Results

Author(s):

Sever S. Dragomir^1,2

¹Mathematics, School of Engineering & Science
Victoria University, PO Box 14428
Melbourne City, MC 8001,
Australia
E-mail: sever.dragomir@vu.edu.au

 
²DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences,
School of Computer Science & Applied Mathematics,
University of the Witwatersrand,
Private Bag 3, Johannesburg 2050,
South Africa
URL: http://rgmia.org/dragomir 

Abstract:

The main aim of this survey is to present recent results concerning Ostrowski type inequalities for the Lebesgue integral of various classes of complex and real-valued functions. The survey is intended for use by both researchers in various fields of Classical and Modern Analysis and Mathematical Inequalities and their Applications, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.

Paper's Title:

The Functional Equation With an Exponential Polynomial Solution and its Stability

Author(s):

Young Whan Lee¹, Gwang Hui Kim^2*, and Jeong Il Kim³

¹Department of Computer Hacking and Information Security,
College of Engineering, Daejeon University,
Daejeon, 34520,
Korea(Republic of).
E-mail: ywlee@dju.ac.kr

²Department of Mathematics,
Kangnam University,
Yongin, Gyeonggi, 16979,
Korea(Republic of).
E-mail: ighkim@kangnam.ac.kr

³Department of Statistics,
Daejeon University,
Daejeon 34520,
Korea(Republic of).
E-mail: jlkim@dju.ac.kr

Abstract:

In this paper, we prove that the unique continuous solution of the functional equation

is

where p_n(x) is a polynomial with degree n and

We also obtain the superstability and stability of the functional equation with the following forms, respectively:

and

Paper's Title:

Norm Estimates for the Difference between Bochner’s Integral and the Convex Combination of Function’s Values

Author(s):

P. Cerone, Y.J. Cho, S.S. Dragomir, J.K. Kim, and S.S. Kim

School of Computer Science and Mathematics,
Victoria University of Technology,
Po Box 14428, Mcmc 8001, Victoria, Australia.
pietro.cerone@vu.edu.au
URL: http://rgmia.vu.edu.au/cerone/index.html

Department of Mathematics Education, College of Education,
Gyeongsang National University, Chinju 660-701, Korea
yjcho@nongae.gsnu.ac.kr

School of Computer Science and Mathematics,
Victoria University of Technology,
Po Box 14428, Mcmc 8001, Victoria, Australia.
sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html

Department of Mathematics, Kyungnam University,
Masan,, Kyungnam 631-701, Korea
jongkyuk@kyungnam.ac.kr

Department of Mathematics, Dongeui University,
Pusan 614-714, Korea
sskim@dongeui.ac.kr

Abstract:

Norm estimates are developed between the Bochner integral of a vector-valued function in Banach spaces having the Radon-Nikodym property and the convex combination of function values taken on a division of the interval [a, b].

Paper's Title:

Orthogonality and ε-Orthogonality in Banach Spaces

Author(s):

H. Mazaheri and S. M. Vaezpour

Faculty of Mathematics, Yazd University, Yazd, Iran
vaezpour@yazduni.ac.ir 
hmazaheri@yazduni.ac.ir 

Abstract:

A concept of orthogonality on normed linear space was introduced by Brickhoff, also the concept of ε-orthogonality was introduced by Vaezpour. In this note, we will consider the relation between these concepts and the dual of X. Also some results on best coapproximation will be obtained.