International Journal Of Nonlinear Analysis And Applications
Volume:1 Issue: 2, Winter - Spring 2010

It is shown that every almost linear bijection h: A! B of a unital C-algebra A onto a unital C-algebra B is a C-algebra isomorphism when h(3nuy) = h(3nu)h(y) for all unitaries u 2 A, all y 2 A, and all n 2 Z, and that almost linear continuous bijection h: A! B of a unital C-algebra A of real rank zero onto a unital C-algebra B is a C-algebra isomorphism when h(3nuy) = h(3nu)h(y) for all u 2 {v 2 A | v = v, kvk = 1, v is invertible}, all y 2 A, and all n 2 Z. Assume that X and Y are left normed modules over a unital C-algebra A. It is shown that every surjective isometry T: X! Y, satisfying T(0) = 0 and T(ux) = uT(x) for all x 2 X and all unitaries u 2 A, is an A-linear isomorphism. This is applied to investigate C-algebra isomorphisms in unital C-algebras.

We propose a new method, called the the weighted space method, for the study of the generalized Hyers-Ulam-Rassias stability. We use this method for a nonlinear functional equation, for Volterra and Fredholm integral operators.

We will apply the successive approximation method for proving the Hyers–Ulam stability of a linear integral equation of the second kind

In the present paper a solution of the generalized quadratic functional equation f(kx + y) + f(kx + (y)) = 2k2f(x) + 2f(y), x, y 2 E is given where  is an involution of the normed space E and k is a fixed positive integer. Furthermore we investigate the Hyers-Ulam-Rassias stability of the functional equation. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

We show that higher derivations on a Hilbert C−module associated with the Cauchy functional equation satisfying generalized Hyers–Ulam stability.

Moslehian and Mirmostafaee, investigated the fuzzy stability problems for the Cauchy additive functional equation, the Jensen additive functional equation and the cubic functional equation in fuzzy Banach spaces. In this paper, we investigate the generalized Hyers-Ulam–Rassias stability of the generalized additive functional equation with n–variables, in fuzzy Banach spaces. Also, we will show that there exists a close relationship between the fuzzy continuity behavior of a fuzzy almost additive function, control function and the unique additive function which approximate the almost additive function

Using the Hyers-Ulam-Rassias stability method, we investigate isomorphisms in Banach algebras and derivations on Banach algebras associated with the following generalized additive functional inequality kaf(x) + bf(y) + cf(z)k  kf(x + y + z)k. (0.1) Moreover, we prove the Hyers-Ulam-Rassias stability of homomorphisms in Banach algebras and of derivations on Banach algebras associated with the generalized additive functional inequality (0.1).

A unital C – algebra A, endowed with the Lie product [x, y] = xy− yx on A, is called a Lie C – algebra. Let A be a Lie C – algebra and g, h: A! A be C – linear mappings. A C – linear mapping f: A! A is called a 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, b 2 A. In this paper, our main purpose is to prove the generalized Hyers - Ulam - Rassias stability of Lie  - double derivations on Lie C - algebras associated with the following additive mapping: Xn k=2(Xki1=2kX+1i2=i1+1...Xnin−k+1=in−k+1)f(Xni=1,i6=i1,..,in−k+1xi −n−Xk+1r=1xir) + f(Xni=1xi)= 2n−1f(x1) for a fixed positive integer n with n  2.

In this paper, we prove the generalized Hyers-Ulam stability of the quadratic functional equation f(x + y) + f(x − y) = 2f(x) + 2f(y) in non-Archimedean L-fuzzy normed spaces.

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability for the quartic, cubic and additive functional equation f(x+ky)+f(x−ky) = k2f(x+y)+k2f(x−y)+(k2−1)[k2f(y)+k2f(−y)−2f(x)] (k 2 Z − {0,±1}) in p−Banach spaces

In this paper, we prove the generalized Hyers–Ulam stability of a quadratic and quartic functional equation in intuitionistic fuzzy Banach spaces