Cauchy Rassias Stability Research Articles

GENERALIZED (θ, ø)-DERIVATIONS ON BANACH ALGEBRAS

We introduce the concept of generalized $(\theta, \phi)$-derivations on Banach algebras, and prove the Cauchy-Rassias stability of generalized $(\theta, \phi)$-derivations on Banach algebras.

A FIXED POINT APPROACH TO THE CAUCHY-RASSIAS STABILITY OF GENERAL JENSEN TYPE QUADRATIC-QUADRATIC MAPPINGS

In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: f( sx + ty 2 + rz) + f(

Cauchy-Rassias Stability of Sesquilinear $n$-Quadratic Mappings in Banach Modules

Cauchy-Rassias Stability of Sesquilinear $n$-Quadratic Mappings in Banach Modules

Cauchy-Rassias stability of homomorphisms associated to a Pexiderized Cauchy-Jensen type functional equation

We use a fixed point method to prove the Cauchy–Rassias stability of homomorphisms associated to the Pexiderized Cauchy–Jensen type functional equation

Poisson Banach Modules over a Poisson C*-Algebr

Abstract. It is shown that every almost linear mapping h: A → B of a unital PoissonC ∗ -algebra A to a unital Poisson C ∗ -algebra B is a Poisson C -algebra homomorphismwhen h(2 n uy) = h(2 n u)h(y) or h(3 n uy) = h(3 n u)h(y) for all y∈ A, all unitary elementsu∈ A and n= 0,1,2,···, and that every almost linear almost multiplicative mappingh: A → B is a Poisson C ∗ -algebra homomorphism when h(2x) = 2h(x) or h(3x) = 3h(x)for all x∈ A. Here the numbers 2,3 depend on the functional equations given in thealmost linear mappings or in the almost linear almost multiplicative mappings. We provethe Cauchy–Rassias stability of Poisson C ∗ -algebra homomorphisms in unital Poisson C -algebras, and of homomorphisms in Poisson Banach modules over a unital Poisson C ∗ -algebra. 1. IntroductionA Poisson C ∗ -algebra A is a C ∗ -algebra with a C-bilinear map {·,·} : A×A →A, called a Poisson bracket, such that (A,{·,·}) is a complex Lie algebra and{ab,c} = a{b,c}+{a,c}bfor all a,b,c ∈ A. Poisson algebras have played an important role in many mathe-matical areas and have been studied to find sympletic leaves of the correspondingPoisson varieties. It is also important to find or construct a Poisson bracket in thetheory of Poisson algebra.A Poisson Banach module X over a Poisson C

On the Cauchy–Rassias stability of a generalized additive functional equation

Let X and Y be Banach spaces and f : X → Y an odd mapping. We solve the following generalized additive functional equation r f ( ∑ j = 1 d x j r ) + ∑ ι ( j ) = 0 , 1 ∑ j = 1 d ι ( j ) = l r f ( ∑ j = 1 d ( − 1 ) ι ( j ) x j r ) = ( C l d − 1 − C l − 1 d − 1 + 1 ) ∑ j = 1 d f ( x j ) for all x 1 , … , x d ∈ X . Moreover we deal with the above functional equation in Banach modules over a C ∗ -algebra and obtain generalizations of the Cauchy–Rassias stability. The concept of Cauchy–Rassias stability for the linear mapping was originated from Th.M. Rassias's stability theorem that appeared in his paper: [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300].

Cauchy–Rassias Stability of Cauchy–Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f : X → Y satisfies $$ f{\left( {\frac{{x + y}} {2} + z} \right)} + f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( x \right)} + 2f{\left( z \right)}, $$ (0.1) $$ f{\left( {\frac{{x + y}} {2} + z} \right)} - f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( y \right)}, $$ (0.2) or $$ 2f{\left( {\frac{{x + y}} {2} + z} \right)} = f{\left( x \right)} + f{\left( y \right)} + 2f{\left( z \right)} $$ (0.3) for all x, y, z ∈ X, then the mapping f : X → Y is Cauchy additive. Furthermore, we prove the Cauchy–Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.

ON A GENERALIZED TRIF'S MAPPING IN BANACH MODULES OVER A C*-ALGEBRA

Let X and Y be vector spaces. It is shown that a mapping <TEX>$f\;:\;X{\rightarrow}Y$</TEX> satisfies the functional equation <TEX>$$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$</TEX> <TEX>$(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<...<i_k{\leq}mn}\;f(\frac {x_{i_1}+...+x_{i_k}} {k})$$</TEX> if and only if the mapping <TEX>$f : X{\rightarrow}Y$</TEX> is additive, and we prove the Cauchy-Rassias stability of the functional equation <TEX>$(\ddagger)$</TEX> in Banach modules over a unital <TEX>$C^*-algebra$</TEX>. Let A and B be unital <TEX>$C^*-algebra$</TEX> or Lie <TEX>$JC^*-algebra$</TEX>. As an application, we show that every almost homomorphism h : <TEX>$A{\rightarrow}B$</TEX> of A into B is a homomorphism when <TEX>$h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$</TEX> for all unitaries <TEX>${\mu}{\in}A,\;all\;y{\in}A$</TEX>, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping <TEX>$h:\;A{\rightarrow}B$</TEX> is a homomorphism when h(2x)=2h(x) for all <TEX>$x{\in}A$</TEX>. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in <TEX>$C^*-algebras$</TEX> or in Lie <TEX>$JC^*-algebras$</TEX>, and of Lie <TEX>$JC^*-algebra$</TEX> derivations in Lie <TEX>$JC^*-algebras$</TEX>.

On the Cauchy-Rassias inequality and linear n-inner product preserving mappings

We prove the Cauchy-Rassias stability of linear n-inner product preserving mappings in $n$-inner product Banach spaces. We apply the Cauchy-Rassias inequality that plays an influencial role in the subject of functional equations. The inequality was introduced for the first time by Th.M.Rassias in his paper entitled: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.Soc. 72(1978), 297-300.

Cauchy–Rassias stability of a generalized Trif's mapping in Banach modules and its applications

Let X and Y be vector spaces. It is shown that a mapping f : X → Y satisfies the functional equation (0.1) mn mn - 2 C k - 2 f x 1 + ⋯ + x mn mn + mn - 2 C k - 1 ∑ i = 1 mn f x i + ⋯ + x i + m - 1 m = k ∑ 1 ⩽ i 1 < ⋯ < i k ⩽ mn f x i 1 + ⋯ + x i k k if and only if the mapping f : X → Y is additive, and we prove the Cauchy–Rassias stability of the functional equation (0.1) in Banach modules over a unital C * -algebra. As an application, we show that every almost homomorphism h : A → B of a unital C * -algebra A onto a unital C * -algebra B is a C * -algebra isomorphism when h ( 2 d uy ) = h ( 2 d u ) h ( y ) for all unitaries u ∈ A , all y ∈ A , and d = 0 , 1 , 2 , … .

Generalized quadratic mappings of r-type in several variables

Let X , Y be vector spaces. It is shown that if an even mapping f : X → Y satisfies f ( 0 ) = 0 , and (∗) ( 2 C l − 1 d − 2 − C l d − 2 − C l − 2 d − 2 ) r 2 f ( ∑ j = 1 d x j r ) + ∑ ι ( j ) = 0 , 1 ∑ j = 1 d ι ( j ) = l r 2 f ( ∑ j = 1 d ( − 1 ) ι ( j ) x j r ) = ( C l d − 1 + C l − 1 d − 1 + 2 d − 2 C l − 1 − C l d − 2 − C l − 2 d − 2 ) ∑ j = 1 d f ( x j ) for all x 1 , … , x d ∈ X , then the even mapping f : X → Y is quadratic. Furthermore, we prove the Cauchy–Rassias stability of the functional equation (∗) in Banach spaces.

On the Cauchy–Rassias stability of the Trif functional equation in [formula omitted]-algebras

Let A , B be two unital C ∗ -algebras, and let q:= k( n−1)/( n− k) for given integers k, n with 2⩽ k⩽ n−1. Consider an almost unital approximately linear mapping h : A→ B . We prove that h is a homomorphism when h( q − j xu)= h( x) h( q − j u) for all x∈ A , all unitaries u∈ A , and all sufficiently large integers j. Moreover, when A has real rank zero, we give conditions in order for h to be a ∗-homomorphism. Furthermore, we investigate the Cauchy–Rassias stability of the Trif functional equation associated with ∗-homomorphisms between unital C ∗ -algebras and ∗-derivations of a unital C ∗ -algebra.

Generalized quadratic mappings in several variables*1

Let X,Y be vector spaces. It is shown that if an even mapping f:X→Y satisfies f(0)=0, and (0.1)(2d−2Cl−1−d−2Cl−d−2Cl−2)f∑j=1dxj+∑ι(j)=0,1∑j=1dι(j)=lf∑j=1d(−1)ι(j)xj=(d−1Cl+d−1Cl−1+2d−2Cl−1−d−2Cl−d−2Cl−2)∑j=1df(xj)for all x1,…,xd∈X, then the even mapping f:X→Y is quadratic. Furthermore, we prove the Cauchy–Rassias stability of the functional equation (0.1) in Banach spaces.

Generalized quadratic mappings in several variables

Let X, Y be vector spaces. It is shown that if an even mapping f : X→Y satisfies f(0)=0, and (0.1) (2 d−2C l−1− d−2C l− d−2C l−2)f ∑ j=1 d x j + ∑ ι(j)=0,1 ∑ j=1 d ι(j)=l f ∑ j=1 d (−1) ι(j)x j = ( d−1C l+ d−1C l−1+2 d−2C l−1− d−2C l− d−2C l−2) ∑ j=1 d f(x j) for all x 1,…, x d ∈ X, then the even mapping f : X→Y is quadratic. Furthermore, we prove the Cauchy–Rassias stability of the functional equation (0.1) in Banach spaces.