We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.

By using the fixed point method, we obtain a version of a stability result of Baker in probabilistic metric and quasimetric spaces under triangular norms of Hadžić type. As an application, we prove a theorem regarding the stability of the additive Cauchy functional equation in random normed spaces.

Let $F$ be a Riesz algebra with an extended norm $||·|{|}_u$ such that $(F,||·|{|}_u)$ is complete. Also, let $||·|{|}_v$ be another extended norm in $F$ weaker than $||·|{|}_u$ such that whenever (a) $x_n\to x$ and $x_n·y\to z$ in $||·|{|}_v$, then $z=x·y$; (b) $y_n\to y$ and $x·y_n\to z$ in $||·|{|}_v$, then $z=x·y$. Let $\epsilon$ and $\delta >$ be two nonnegative real numbers. Assume that a map $f:F\to F$ satisfies $||f(x+y)-f(x)-f(y)|{|}_u\le \epsilon$ and $||f(x·y)-x·f(y)-f(x)·y|{|}_v\le \delta$ for all $x,y\in F$. In this paper, we prove that there exists a unique derivation $d:F\to F$ such that $||f(x)-d(x)|{|}_u\le \epsilon$, ($x\in F$). Moreover, $x·(f(y)-d(y))=0$ for all $x,y\in F$.

We present two theorems describing the structure of the set of all regular points and the set of all irregular points for a Brouwer homeomorphism which is embeddable in a flow. The theorems are counterparts of structure theorems proved by Homma and Terasaka. To obtain our results, we use properties of the codivergence relation.

The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equation $f(ax+by)+af(x-by)=(a+1)b^{2}f(y)+a(a+1)f(x)$, in $(\beta ,p)$-Banach space, where $a,b$ are fixed rational numbers such that $a\ne -\mathrm{1,0}$ and $b\ne 0$.

We prove the Hyers-Ulam stability of the following Jensen functional inequality $\parallel f((x-y)/n+z)+f((y-z)/n+x)+f((z-x)/n+y)\parallel \le \parallel f((x+y+z)\parallel$ in $p$-Banach spaces for any fixed nonzero integer $n$.

Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.

In this paper we prove a fixed-point theorem for a class of operators with suitable properties, in very general conditions. Also, we show that some recent fixed-points results in Brzdęk et al., (2011) and Brzdęk and Ciepliński (2011) can be obtained directly from our theorem. Moreover, an affirmative answer to the open problem of Brzdęk and Ciepliński (2011) is given. Several corollaries, obtained directly from our main result, show that this is a useful tool for proving properties of generalized Hyers-Ulam stability for some functional equations in a single variable.

We study the stability of cubic ∗-derivations on Banach ∗-algebras. We also prove the superstability of cubic ∗-derivations on a Banach ∗-algebra A, which is left approximately unital.

We take account of the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation. In addition, we deal with the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.

We study the Hyers-Ulam-Rassias stability of ${(m,n)}_{(\sigma ,\tau )}$-derivations on normed algebras.

We prove the Ulam-Hyers stability of Cauchy fractional differential equations in the unit disk for the linear and non-linear cases. The fractional operators are taken in sense of Srivastava-Owa operators.

We investigate the generalized Ulam-Hyers stability of the Cauchy functional equation and pose two open problems in fuzzy Banach space.

We find out the general solution of a generalized Cauchy-Jensen functional equation and prove its stability. In fact, we investigate the existence of a Cauchy-Jensen mapping related to the generalized Cauchy-Jensen functional equation and prove its uniqueness. In the last section of this paper, we treat a fixed point approach to the stability of the Cauchy-Jensen functional equation.

The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.

The oldest quartic functional equation was introduced by J. M. Rassias in (1999), and then was employed by other authors. The functional equation $f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f\left(x\right)-6f\left(y\right)$ is called a quartic functional equation, all of its solution is said to be a quartic function. In the current paper, the Hyers-Ulam stability and the superstability for quartic functional equations are established by using the fixed-point alternative theorem.