Aoki,T. , (1950) "On stability of the linear transformation in Banach spaces," Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...

In 1940, Ulam [1] proposed the famous Ulam stability problem of linear mappings. In 1941, Hyers [2] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies Hyers inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = limn→∞ 2−n f (2nx) exists for all x ∈ E and that L : E→ E′ is the unique additive ...

In this paper, we prove the Hyers-Ulam stability in$beta$-homogeneous probabilistic modular spaces via fixed point method for the functional equation[f(x+ky)+f(x-ky)=f(x+y)+f(x-y)+frac{2(k+1)}{k}f(ky)-2(k+1)f(y)]for fixed integers $k$ with $kneq 0,pm1.$

and Applied Analysis 3 Clearly, every Menger PN-space is probabilistic metric space having a metrizable uniformity on X if supa<1T a, a 1. Definition 1.3. Let X,Λ, T be a Menger PN-space. i A sequence {xn} in X is said to be convergent to x in X if, for every > 0 and λ > 0, there exists positive integer N such that Λxn−x > 1 − λ whenever n ≥ N. ii A sequence {xn} in X is called Cauchy sequence ...

In this paper, we investigate the generalized Hyers-Ulam-Rassias and the Isac and Rassias-type stability of the conditional of orthogonally ring $*$-$n$-derivation and orthogonally ring $*$-$n$-homomorphism on $C^*$-algebras. As a consequence of this, we prove the hyperstability of orthogonally ring $*$-$n$-derivation and orthogonally ring $*$-$n$-homomorphism on $C^*$-algebras.

The question concerning the stability of group homomorphisms was posed by Ulam 1 . Hyers 2 solved the case of approximately additive mappings on Banach spaces. Aoki 3 provided a generalization of the Hyers’ theorem for additive mappings. In 4 , Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded see also 5 . The result of Rassias has bee...

in this paper, we obtain the general solution and the generalized  hyers--ulam--rassias stability in random normed spaces, in non-archimedean spacesand also in $p$-banach spaces and finally the stability viafixed point method for a functional equationbegin{align*}&amp;d_f(x_{1},.., x_{m}):= sum^{m}_{k=2}(sum^{k}_{i_{1}=2}sum^{k+1}_{i_{2}=i_{1}+1}... sum^{m}_{i_{m-k+1}=i_{m-k}+1}) f(sum^{m}_{i=1...

In this paper, we obtain sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of an abstract second–order nonlinear dynamic equation on bounded time scales. An illustrative example is given to show the applicability theoretical results.

This paper investigates the sufficient conditions for existence and uniqueness of a class Riemann-Liouville fractional differential equations variable order with boundary conditions. The problem is converted into constant orders by combining concepts generalized intervals piecewise functions. We derive required ensuring in to utilize Banach fixed point theorem. stability obtained solution Ulam-...

In this paper we are going to study the Hyers{Ulam{Rassias typesof stability for nonlinear, nonhomogeneous Volterra integral equations with delayon nite intervals.