The Ulam-Hyers stability of functional equations is widely studied from various points of view by many authors. The present paper is concerned with local stability of the four Cauchy equations restricted on a bounded domain. These results can be easily adapted to the corresponding Pexiderized equations.

This is a survey paper concerning the notions of hyperstability and superstability, which are connected to the issue of Ulam’s type stability. We present the recent results on those subjects.

We establish the general solution of the functional inequality $∥f\left(x-y\right)+f\left(y-z\right)+f\left(x-z\right)-3f\left(x\right)-3f\left(y\right)-3f\left(z\right)∥\le ∥f\left(x+y+z\right)∥$ and then investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and in non-Archimedean Banach spaces.

We obtain the general solution of Euler-Lagrange-Rassias quartic functional equation of the following $f\left(ax+by\right)+f\left(bx+ay\right)+(\mathrm{1}/\mathrm{2})ab(a-b{)}^{\mathrm{2}}f(x-y)=(a^{\mathrm{2}}-b^{\mathrm{2}}{)}^{\mathrm{2}}\left[f\left(x\right)+f\left(y\right)\right]+(\mathrm{1}/\mathrm{2})ab(a+b{)}^{\mathrm{2}}f(x+y)\mathrm{}$. We also prove the Hyers-Ulam-Rassias stability in various quasinormed spaces when $b=\mathrm{1}\mathrm{}$.

We prove the generalized Hyers-Ulam stability of the heat equation, $\mathrm{\Delta }u=u_t$, in a class of twice continuously differentiable functions under certain conditions.

We obtain the general solution of the generalized mixed additive and quadratic functional equation $f\left(x+my\right)+f\left(x-my\right)=2f\left(x\right)-2m^{2}f\left(y\right)+m^{2}f\left(2y\right)$, $m$ is even; $f\left(x+y\right)+f\left(x-y\right)-2\left(m^2-1\right)f\left(y\right)+\left(m^2-1\right)f\left(2y\right)$, $m$ is odd, for a positive integer $m$. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces when $m$ is an even positive integer or $m=3$.

We establish the hyperstability of $n$-Jordan homomorphisms from a normed algebra to a Banach algebra, and also we show that an $n$-Jordan homomorphism between two commutative Banach algebras is an $n$-ring homomorphism.

We consider general solution and the generalized Hyers-Ulam stability of an Euler-Lagrange quadratic functional equation $f\left(rx+sy\right)+rsf\left(x-y\right)=\left(r+s\right)\left[rf\left(x\right)+sf\left(y\right)\right]$ in fuzzy Banach spaces, where $r$, $s$ are nonzero rational numbers with $r^2+rs+s^2-1\ne 0$, $r+s\ne 0$.

We prove the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in $n$-Banach spaces, using the socalled direct method. The corollaries from our main results correct some outcomes from Park (2011).

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equation $f(xy)=xf(y)+f(x)y$ in an intuitionistic fuzzy normed algebra with unit.

Let $ℝ$ be the set of real numbers, ${ℝ}^{+}=\{x\in ℝ\mathrm{}\mid \mathrm{}x>\mathrm{0}\}$, $ϵ\in {ℝ}_{+}$, and $f,g,h:{ℝ}^{+}\to ℂ$. As classical and $L^{\mathrm{\infty }}$ versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: $|f(x+y)-g(xy)-h(\mathrm{(1}/\mathrm{x)}+\mathrm{(1}/\mathrm{y)})|\le ϵ$, and ${∥f(x+y)-g(xy)-h\left(\mathrm{(1}/\mathrm{x)}+\mathrm{(1}/\mathrm{y)}\right)∥}_{L^{\mathrm{\infty }}({\mathrm{\Gamma }}_d)}\le ϵ$ in the sectors ${\mathrm{\Gamma }}_d=\{(x,y):x>\mathrm{0},\mathrm{}y>\mathrm{0},\mathrm{}\mathrm{(y}/\mathrm{x)}>d\}$. As consequences of the results, we obtain asymptotic behaviors of the previous inequalities. We also consider its distributional version ${∥u\circ S-v\circ \mathrm{\Pi }-w\circ R∥}_{{\mathrm{\Gamma }}_d}\le ϵ$, where $u,v,w\in 𝒟\text{'}({ℝ}^{+})$, $S(x,y)=x+y$, $\mathrm{\Pi }(x,y)=xy$, $R(x,y)=\mathrm{1}/x+\mathrm{1}/y$, $x,y\in {ℝ}^{+}$, and the inequality ${∥·∥}_{{\mathrm{\Gamma }}_d}\le ϵ$ means that $|\langle ·,\phi \rangle |\le ϵ\parallel \phi {\parallel }_{L^{\mathrm{1}}}$ for all test functions $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Gamma }}_d)$.

We will investigate some existence, uniqueness, and Ulam-Hyers stability results for fixed point problems via $\alpha$-$\psi$-contractive mapping of type-($b$) in the framework of $b$-metric spaces. The presented theorems extend, generalize, and unify several results in the literature, involving the results of Samet et al. (2012).

We study the stability and hyperstability of cubic Lie derivations on normed algebras. At the end, we write some additional observations about our results.

We investigate the stability of a functional equation $f\left(x+y+z\right)$ $+$ $f\left(x-y+z\right)$ $+$ $f\left(x+y-z\right)+f\left(-x+y+z\right)=3f\left(x\right)+f\left(-x\right)+3f\left(y\right)+f\left(-y\right)+3f\left(z\right)+f\left(-z\right)$ by applying the direct method in the sense of Hyers and Ulam.