Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

The generalized Hyers-Ulam-Rassias stability of a cubic functional equation

In this paper, we obtain the general solution and the generalized Hyers-Ulam stability for a cubic functional equation f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+12f(x).     

Stability Problem for Jensen-type Functional Equations of Cubic Mappings

In this paper, we establish the general solution and the generalized Hyers-Ulam-Rassias stability problem for a cubic Jensen-type functional equation,4f（（3x＋y）/4）＋4f（（x＋3y）/4）=6f（（x＋y）/2）＋f（x）＋f（y）,9f（（2x＋y/3）＋9f（（x＋2y）/3）=16f（（x＋y）/2＋f（x）＋f（y）in the spirit of D. H. Hyers, S. M. Ulam, Th. M. Rassias and P. Gaevruta.     

On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces

In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for Euler-Lagrange type cubic functional equations

f(ax+y)+f(x+ay)=(a+1)²(a−1)[f(x)+f(y)]+a(a+1)f(x+y)     

A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations

In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder ? is defined by f(x∗y)+f(x∗y−1)−2g(x)−2g(y)=?(x,y), f(x∗y)+g(x∗y−1)−2h(x)−2k(y)=?(x,y), where (G,∗) is a group, X is a real or complex Hausdorff topological vector space and f,g,h,k are functions from G into X.     

Stability problem of Ulam for generalized forms of Cauchy functional equation

Let G₁ be a vector space and G₂ a Banach space. In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for a generalized form

     

Extended Hyers–Ulam stability for Cauchy–Jensen mappings†

In 1940, Ulam proposed the famous Ulam stability problem. In 1941, Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. In 2003–2006, the last author of this paper investigated the Hyers–Ulam stability of additive and Jensen type mappings. In this paper, we improve results obtained in 2003 and 2005 for Jensen type mappings and establish new theorems about the Ulam stability of additive and alternative additive mappings. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.     

On the stability of approximately additive mappings

In this paper we prove a generalization of the stability of approximately additive mappings in the spirit of Hyers, Ulam and Rassias.

     

Solution of Hyers-Ulam stability problem for generalized Pappus' equation

The purpose of this paper is to solve the stability problem of Ulam for an approximate mapping of the following generalized Pappus' equation:

n²Q(x+my)+mnQ(x−ny)=(m+n)[nQ(x)+mQ(ny)]     

Ulam stability problem for generalized A-quadratic mappings

It is the purpose of this paper to investigate the stability problem of Ulam for an approximate mapping for the following generalized quadratic functional equation: