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Abstract In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in C ∗ $C^{*}$ -algebras and Lie C ∗ $C^{*}$ -algebras, multi-stability of derivations in C ∗ $C^{*}$ -algebras, and Lie C ∗ $C^{*}$ -algebras for the following random operator equation via fixed point methods: μ f ( ð , x + y 2 ) + μ f ( ð , x − y 2 ) = f ( ð , μ x ) . $$ \mu f \biggl(\eth , \frac{x+y}{2} \biggr) + \mu f \biggl(\eth , \frac{x-y}{2} \biggr) = f(\eth , \mu x) . $$ In particular, for μ = 1 $\mu = 1$ , the above equation turns out to be Jensen’s random operator equation.

Abstract We stabilize pseudostochastic ( G 1 , G 2 ) $(\mathcal{G}_{1},\mathcal{G}_{2})$ -random operator inequality using a class of stochastic matrix control functions in matrix Menger Banach algebras. We get an approximation for stochastic ( G 1 , G 2 ) $(\mathcal{G}_{1},\mathcal{G}_{2})$ -random operator inequality by means of both direct and fixed point methods. As an application, we apply both stochastic Mittag-Leffler and H $\mathbb{H}$ -fox control functions to get a better approximation in a random operator inequality.

In this paper, we apply the well-known aggregation mappings on Mittag-Leffler-type functions to investigating new approximation error estimates of a W-Hilfer fractional differential equation, by a different concept of Ulam-type stability in both bounded and unbounded domains.

Abstract We introduce a class of stochastic matrix control functions and apply them to stabilize pseudo stochastic κ-random operator inequalities in matrix MB-algebras. We obtain an approximation for stochastic κ-random operator inequalities and calculate the maximum error of the estimate.

In this paper, we introduce the following $(a,b,c)$-mixed type functional equation of the form \\$g(ax_1+bx_2+cx_3 )-g(-ax_1+bx_2+cx_3 )+g(ax_1-bx_2+cx_3 )-g(ax_1+bx_2-cx_3 ) +2a^2 [g(x_1 )+g(-x_1)]+2b^2 [g(x_2 )+g(-x_2)]+2c^2 [g(x_3 )+g(-x_3)]+a[g(x_1 )-g(-x_1)]+b[g(x_2 )-g(-x_2)]+c[g(x_3 )-g(-x_3)]=4g(ax_1+cx_3 )+2g(-bx_2)+ 2g(bx_2)$\\ where $a,b,c$ are positive integers with $a>1$, and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.

In this paper, we investigate the generalized Hyers–Ulam stability for the generalized psi functional equation f ( x + p ) = f ( x ) + φ ( x ) by the direct method in the sense of P. Gǎvruta and the Hyers–Ulam–Rassias stability.

Some properties of isometric mappings as well as approximate isometries are studied.

Gauss' functional equation (used in the study of the arithmetic-geometric mean) is generalized by replacing the arithmetic mean and the geometric mean by two arbitrary means.

The Hyers-Ulam stability of mappings is in development and several authors have remarked interesting applications of this theory to various mathematical problems. In this paper some applications in nonlinear analysis are presented, especially in fixed point theory. These kinds of applications seem not to have ever been remarked before by other authors.

Gauss established a theory on the functional equation (Gauss' functional equation) f(a+b2,ab)=f(a,b) (a,b>0), where f:R+×R+→R is an unknown function of the above equation.

We will prove the stability of the functional equation 2f(∑i=1nxi)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑i=1nf(xi)+(n-1)∑i=1nf(-xi) in non-Archimedean normed spaces.

Using fixed point method and direct method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation 𝑟2𝑓((𝑥+𝑦+𝑧)/𝑟)+𝑟2𝑓((𝑥−𝑦+𝑧)/𝑟)+𝑟2𝑓((𝑥+𝑦−𝑧)/𝑟)+𝑟2𝑓((−𝑥+𝑦+𝑧)/𝑟)=4𝑓(𝑥)+4𝑓(𝑦)+4𝑓(𝑧), where 𝑟 is a positive real number, in non-Archimedean normed spaces.

We solve the inhomogeneous Chebyshev's differential equation and apply this result for approximating analytic functions by the Chebyshev functions.

We prove the Hyers-Ulam stability of the additive-cubic-quartic functional equation in multi-Banach spaces by using the fixed point alternative method. The first results on the stability in the multi-Banach spaces were presented in (Dales and Moslehian 2007).

In intuitionistic fuzzy normed spaces, we investigate some stability results for the functional equation ∑𝑛𝑖=1𝑓(𝑥𝑖∑−(1/𝑛)𝑛𝑗=1𝑥𝑗∑)=𝑛𝑖=1𝑓(𝑥𝑖∑)−𝑛𝑓((1/𝑛)𝑛𝑖=1𝑥𝑖) which is said to be a functional equation associated with inner products space.

We prove the generalized Hyers-Ulam stability of additive N-isometries on linear N-normed Banach spaces.