2010
1
2
2
0
Isomorphisms in unital $C^*$algebras
2
2
It is shown that every almost linear bijection $h : Arightarrow B$ of a unital $C^*$algebra $A$ onto a unital$C^*$algebra $B$ is a $C^*$algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries $u in A$, all $y in A$, and all $nin mathbb Z$, andthat almost linear continuous bijection $h : A rightarrow B$ of aunital $C^*$algebra $A$ of real rank zero onto a unital$C^*$algebra $B$ is a $C^*$algebra isomorphism when $h(3^n u y) =h(3^n u) h(y)$ for all $u in { v in A mid v = v^*, v=1, v text{ is invertible} }$, all$y in A$, and all $nin mathbb Z$.Assume that $X$ and $Y$ are left normed modules over a unital$C^*$algebra $A$. It is shown that every surjective isometry $T : Xrightarrow Y$, satisfying $T(0) =0$ and $T(ux) = u T(x)$ for all $xin X$ and all unitaries $u in A$, is an $A$linear isomorphism.This is applied to investigate $C^*$algebra isomorphisms in unital$C^*$algebras.
1

1
10


C.
Park
Department of Mathematics, Hanyang University,
Seoul 133791, Republic of Korea
Department of Mathematics, Hanyang University,
Iran


Th. M.
Rassias
Department of Mathematics,
National Technical
University of Athens,
Zografou Campus, 15780 Athens, Greece
Department of Mathematics,
National Technical
Univ
Iran
generalized HyersUlam stability
$C^*$algebra isomorphism
real rank zero
isometry
A new method for the generalized HyersUlamRassias stability
2
2
We propose a new method, called the textit{the weighted space method}, for the study of the generalized HyersUlamRassias stability. We use this method for a nonlinear functional equation, for Volterra and Fredholm integral operators.
1

11
18


P.
Gavruta
Department of Mathematics,
University "Politehnica" of Timisoara, 300006, Timisoara, Romania.
Department of Mathematics,
University "Politehnica
Iran


L.
Gavruta
Department of Mathematics,
University "Politehnica" of Timisoara, 300006, Timisoara, Romania.
Department of Mathematics,
University "Politehnica
Iran
Hyers–UlamRassias stability
functional equation
Volterra integral operator
Fredholm integral operator
Weighted space method
HyersUlam stability of Volterra integral equation
2
2
We will apply the successive approximation method forproving the HyersUlam stability of a linear integral equation ofthe second kind.
1

19
25


M.
Gachpazan
Department of Applied Mathematics, Faculty of Mathematical Sciences,
Ferdowsi University of Mashhad, Mashhad, Iran.
Department of Applied Mathematics, Faculty
Iran


O.
Baghani
Department of Applied Mathematics, Faculty of Mathematical Sciences,
Ferdowsi University of Mashhad, Mashhad, Iran.
Department of Applied Mathematics, Faculty
Iran
HyersUlam stability
Banach's fixed point theorem
Volterra integral equation
Successive approximation method
stability of the quadratic functional
equation
2
2
In the present paper a solution of the generalizedquadratic functional equation$$f(kx+ y)+f(kx+sigma(y))=2k^{2}f(x)+2f(y),phantom{+} x,yin{E}$$ isgiven where $sigma$ is an involution of the normed space $E$ and$k$ is a fixed positive integer. Furthermore we investigate theHyersUlamRassias stability of the functional equation. TheHyersUlam stability on unbounded domains is also studied.Applications of the results for the asymptotic behavior of thegeneralized quadratic functional equation are provided.
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26
35


E.
Elqorachi
Department of
Mathematics, Faculty of Sciences, University Ibn Zohr, Agadir,
Morocco
Department of
Mathematics, Faculty of Sciences,
Iran


Y.
Manar
Department of
Mathematics, Faculty of Sciences, University Ibn Zohr, Agadir,
Morocco
Department of
Mathematics, Faculty of Sciences,
Iran


Th. M.
Rassias
Department of Mathematics, National
Technical University of Athens, Zografou Campus, 15780, Athens
Greece
Department of Mathematics, National
Technical
Iran
HyersUlamRassias stability
quadratic functional equation
Approximately higher Hilbert $C^*$module derivations
2
2
We show that higher derivations on a Hilbert$C^{*}$module associated with the Cauchy functional equation satisfying generalized HyersUlam stability.
1

36
43


M. B.
Ghaemi
Department of Mathematics, Iran
University of Science and Technology, Tehran, Iran
Department of Mathematics, Iran
University
Iran


B.
Alizadeh
PhD and Graduate Center, Payame Noor University,
Shahnaz Alley Haj Mahmood Norian Street,
$$AND$$
Tabriz College of
Technology, P. O. Box 51745135, Tabriz, Iran.
PhD and Graduate Center, Payame Noor University,
S
Iran
HyersUlam stability
Hilbert $C^{*}$modules
Derivation
Higher derivation
fixed point theorem
Fuzzy approximately additive mappings
2
2
Moslehian and Mirmostafaee, investigated the fuzzystability problems for the Cauchy additive functional equation, the Jensen additivefunctional equation and the cubic functional equation in fuzzyBanach spaces. In this paper, we investigate thegeneralized Hyers–UlamRassias stability of the generalizedadditive functional equation with $n$variables, in fuzzy Banachspaces. Also, we will show that there exists a close relationshipbetween the fuzzy continuity behavior of a fuzzy almost additivefunction, control function and the unique additive function whichapproximate the almost additive function.
1

44
53


H.
Khodaei
Department of Mathematics,
Semnan University P. O. Box 35195363, Semnan, Iran.
Department of Mathematics,
Semnan University
Iran


M.
Kamyar
Department of Mathematics,
Semnan University P. O. Box 35195363, Semnan, Iran.
Department of Mathematics,
Semnan University
Iran
Fuzzy stability
Additive functional equation
Fuzzy normed space
Generalized additive functional
inequalities in Banach algebras
2
2
Using the HyersUlamRassias stability method, weinvestigate isomorphisms in Banach algebras and derivations onBanach algebras associated with the following generalized additivefunctional inequalitybegin{eqnarray}af(x)+bf(y)+cf(z) le f(alpha x+ beta y+gamma z) .end{eqnarray}Moreover, we prove the HyersUlamRassias stability of homomorphismsin Banach algebras and of derivations on Banach algebras associatedwith the generalized additive functional inequality (0.1).
1

54
62


C.
Park
Department of Mathematics, Hanyang University,
Seoul 133791, Republic of Korea.
Department of Mathematics, Hanyang University,
Seo
Iran


A.
Najati
Faculty of Sciences, Department of Mathematics,
University of Mohaghegh Ardabili,
Ardabil,
Islamic Republic of Iran.
Faculty of Sciences, Department of Mathematics,
Iran
HyersUlamRassias stability
generalized additive functional inequality
algebra homomorphism in Banach algebra
derivation on Banach algebra
Lie $^*$double derivations on Lie $C^*$algebras
2
2
A unital $C^*$  algebra $mathcal A,$ endowed withthe Lie product $[x,y]=xy yx$ on $mathcal A,$ is called a Lie$C^*$  algebra. Let $mathcal A$ be a Lie $C^*$  algebra and$g,h:mathcal A to mathcal A$ be $Bbb C$  linear mappings. A$Bbb C$  linear mapping $f:mathcal A to mathcal A$ is calleda Lie $(g,h)$  double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all $a,bin mathcal A.$ In this paper, our main purpose is to prove thegeneralized Hyers – Ulam – Rassias stability of Lie $*$ double derivations on Lie $C^*$  algebras associated with thefollowing additive mapping:begin{align*}sum^{n}_{k=2}(sum^{k}_{i_{1}=2} sum^{k+1}_{i_{2}=i_{1}+1}...sum^{n}_{i_{nk+1}=i_{nk}+1}) f( sum^{n}_{i=1, ineqi_{1},..,i_{nk+1} } x_{i}&sum^{nk+1}_{ r=1}x_{i_{r}})+f(sum^{n}_{ i=1} x_{i})&=2^{n1} f(x_{1}) end{align*} for a fixed positive integer $n$ with $n geq 2.$
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63
71


N.
Ghobadipour
Department of Mathematics,
Urmia University, Urmia, Iran.
Department of Mathematics,
Urmia University,
Iran
Generalized Hyers  Ulam  Rassias stability
$*$  double derivation
Lie $C^*$  algebra
Stability of the quadratic functional equation in nonArchimedean Lfuzzy normed spaces
2
2
In this paper, we prove the generalized HyersUlam stability of the quadratic functionalequation$$f(x+y)+f(xy)=2f(x)+2f(y)$$in nonArchimedean $mathcal{L}$fuzzy normed spaces.
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72
83


S.
Shakeri
Department of Mathematics,
Islamic Azad UniversityAiatollah Amoli Branch, Amol, P.O. Box 678, Iran.}
Department of Mathematics,
Islamic Azad
Iran


R.
Saadati
Department of Mathematics,
Islamic Azad UniversityAiatollah Amoli Branch, Amol, P.O. Box 678, Iran.}
Department of Mathematics,
Islamic Azad
Iran


C.
Park
Department of Mathematics, Research Institute for Natural Sciences,
Hanyang University,
Seoul 133791, Korea.
Department of Mathematics, Research Institute
Iran
$mathcal{L}$fuzzy metric and normed spaces
intuitionistic fuzzy metric and normed spaces
generalized HyersUlam stability
quadratic functional equation
nonArchimedean $mathcal{L}$fuzzy normed space
Stability of generalized QCAfunctional equation in PBanach spaces
2
2
In this paper, we investigate the generalizedHyersUlamRassias stability for the quartic, cubic and additivefunctional equation$$f(x+ky)+f(xky)=k^2f(x+y)+k^2f(xy)+(k^21)[k^2f(y)+k^2f(y)2f(x)]$$ ($k in mathbb{Z}{0,pm1}$) in $p$Banach spaces.
1

84
99


S.
Zolfaghari
Department of Mathematics,
Urmia University, Urmia, Iran.
Department of Mathematics,
Urmia University,
Iran
Stability
QCAfunctional equation
$p$Banach space
Intuitionistic fuzzy stability of a
quadratic and quartic functional equation
2
2
In this paper, we prove the generalized HyersUlamstability of a quadratic and quartic functional equation inintuitionistic fuzzy Banach spaces.
1

100
124


S.
Abbaszadeh
Department of Mathematics, Semnan
University, P. O. Box 35195363,
Semnan, Iran.
Department of Mathematics, Semnan
University,
Iran
Intuitionistic fuzzy normed space
Mixed functional equation
Intuitionistic fuzzy stability