C\u{a}dariu and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we adopt the idea of C\u{a}dariu and Radu to prove the Hyers-Ulam-Rassias stability of a functional equation of the square root spiral, $f\!\left( \!\sqrt{r^2 + 1}\, \right) = f(r) + \tan^{-1} (1/r)$.

Some recent inequalities for the norm and the numerical radius of linear operators in Hilbert spaces are surveyed.

In this paper, we obtain the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation $$ \sum_{k\in K} f(x+k\cdot y)=|K|g(x)+|K|h(y), \; x, y \in G ,$$ where $G$ is an abelian group, $K$ is a finite abelian subgroup of the group of automorphism of $G$.

The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias' Stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300.

It is well known that the quasi variational inequalities are equivalent to the fixed point problems. We this equivalent alternative formulation to discuss the existence of a solution of quasi variational inequalities under some mild conditions. Since the quasi variational inequalities include variational inequalities, implicit complementarity problems and optimization problems as special cases, results proved in this paper continue to hold these problems. This shows that results proved in this paper can be viewed as an important and significant improvement and refinement of the previous results.

It is shown that every locally idempotent (locally $m$-pseudoconvex), Hausdorff algebra $A$ with pseudoconvex von Neumann bornology,is a regular (respectively, bornological) inductive limit of metrizable,locally $m$-($k_B$-convex) subalgebras $A_B$ of $A$. In the case where $A$, in addition, is sequentially $\mathcal{B}_A$-complete (sequentially advertibly complete), then every subalgebra $A_B$ is a locally $m$-($k_B$-convex) Frechet algebra (respectively, an advertibly complete metrizable locally $m$-($k_B$-convex) algebra) for some $k_B\in (0,1]$. Moreover, for a commutative unital locally $m$-pseudoconvex Hausdorff algebra $A$ over $\mathbb{C}$ with pseudoconvex von Neumann bornology, which at the same time is sequentially $\mathcal{B}_A$-complete and advertibly complete, the statements (a)-(j) of Proposition 3.2 are equivalent.

In this note we discuss the convergence of greedy approximants for trigonometric Fourier expansion in $L_p(\mathbb{T})$, $1\leq p \leq2, p \neq 2$.

We study in a unitary way the Schur-convexity or concavity of the Stolarsky and Gini means $D_{a,b}(x,y)$ and $S_{a,b}(x,y)$, for fixed $x,y>0$, $x\ne y$.

In this paper we investiagte Steiner loops introduced by N.S. Mendelsohn [Aeq. Math. 6 (1991), 228-230] and provide six (seven) equivalent identities to characterize it. We also prove the power associativity of Bol loops by using closure (Hexagonal) conditions.

At the Fourty-forth International Symposium on Functional Equations in 2006, Louisville, Kentucky, USA, T.M.K. Davison presented a talk under the title "D'Alembert's functional equation on compact groups" in which he offered a solution method for this equation on any compact group. In this paper we offer another approach based on some very recent results on spectral synthesis, which method seems to be useful in the case of other equations.

The aim of this paper is to study the superstability problem of the mixed trigonometric functional equations and the Hyers-Ulam-Rassias stability for a Jensen type functional equation.

Let $X,Y$ be the normed spaces, $C\subset X$ a convex set, and $T:C\rightarrow Y$ a continuous mapping. Some weak conditions implying the Lipschitz continuity of $T$ are presented. Applications to the fixed point theory and theory of composition operators are presented.

In this paper we investigate the problem of the Hyers-Ulam stability of the generalized quadratic functional equation $$f(x+y)+f(x-y)=g(x)+g(y),$$ where $f,g$ are functions defined on a group with values in a linear topological space.

While the authors were visiting Athens in 2006 and 2005, respectively, they interviewed Professor Themistocles M. Rassias concerning his contributions to Mathematics. This article presents those interviews.