Hyers–Ulam stability of hypergeometric differential equations

Aequationes mathematicae - In the present paper by applying the series method we prove the Hyers–Ulam stability of the homogeneous hypergeometric differential equation in a subclass of...     

Ulam–Hyers–Mittag-Leffler stability of fractional-order delay differential equations

In this paper, we first prove two existence and uniqueness results for fractional-order delay differential equation with respect to Chebyshev and Bielecki norms. Secondly, we prove the above equation is Ulam–Hyers–Mittag-Leffler stable on a compact interval. Finally, two examples are also provided to illustrate our results.     

Hyers–Ulam stability of Euler’s equation

We prove that Euler’s equation $x_1\frac{?u}{?x_1}+x_2\frac{?u}{?x_2}+?+x_n\frac{?u}{?x_n}=\alpha u$, characterising homogeneous functions, is stable in Hyers–Ulam sense if and only if $\alpha \in \mathbb{R}?\left\{0\right\}$.     

Hyers–Ulam stability of Sahoo–Riedel’s point

In this paper, we construct a counter example to show that “Theorem” of Hyers–Ulam Stability of Flett’s Point in [M. Das, T. Riedel, P.K. Sahoo, Hyers-Ulam stability of Flett’s points, Applied Mathematics Letters. 16 (3) (2003), 269–271] is incorrect. At the same time, we give the correct theorem and generalize it.     

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.     

The properties of functional inclusions and Hyers–Ulam stability

We prove that a set-valued function satisfying some functional inclusions admits, in appropriate conditions, a unique selection satisfying the corresponding functional equation. As a consequence we obtain the result on the Hyers–Ulam stability of that functional equation.     

Fixed points of a mapping and Hyers–Ulam stability

We show that many general results on Hyers–Ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem. The theorem is formulated for self-maps of some subsets of the space of functions from a nonempty set into the set of reals. We also give some applications of that theorem, e.g., in investigations of solutions of some difference equations and functional inequalities.     

On a Jensen-cubic functional equation and its Hyers–Ulam stability

In this paper, we obtain the general solution and stability of the Jensen-cubic functional equation f((x₁+x₂)/2, 2y₁+y₂)+f((x₁+x₂)/2, 2y₁-y₂) = f(x₁, y₁+y₂)+f(x₁, y₁-y₂)+6f(x₁, y₁)+f(x₂, y₁+y₂)+f(x₂, y₁-y₂)+6f(x₂, y₁).     

Hyers–Ulam–Rassias Stability of Derivations in Proper JCQ*–triples

In this paper, we investigate stability of derivations in proper JCQ*–triples associated to the following Pexiderized functional equation $$f(x + y + z) = f_{0}(x) + f_{1}(y) + f_{2}(z)$$ .     

Hyers–Ulam Stability of General Jensen-Type Mappings in Banach Algebras

Using the fixed point method, we prove the Hyers–Ulam stability of homomorphisms in complex Banach algebras and complex Banach Lie algebras and also of derivations on complex Banach algebras and complex Banach Lie algebras for the general Jensen-type functional equation f(α x + β y) + f(α x ? β y) = 2α f(x) for any \({\alpha, \beta \in \mathbb{R}}\) with \({\alpha, \beta \neq 0}\) . Furthermore, we prove the hyperstability of homomorphisms in complex Banach algebras for the above functional equation with α = β = 1.     

On Hyers–Ulam stability of two functional equations in non-Archimedean spaces

In this paper we prove, using the fixed point method, the generalized Hyers–Ulam stability of two functional equations in complete non-Archimedean normed spaces. One of these equations characterizes multi-Cauchy–Jensen mappings, and the other gives a characterization of multi-additive-quadratic mappings.     

Hyers–Ulam stability of linear partial differential equations of first order

In this work, we will prove the Hyers–Ulam stability of linear partial differential equations of first order.     

Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces

In this paper, we investigate the Hyers–Ulam stability of the differential operators \(T_\lambda \) and D on the weighted Hardy spaces \(H_\beta ^2\) with the reproducing property. We obtain a necessary and sufficient condition in order that D is stable on \(H_\beta ^2\), and construct an example concerning the stability of \(T_\lambda \) on \(H_\beta ^2\). Moreover, we also investigate the Hyers–Ulam stability of the partial differential operators \(D_i\) on the several variables reproducing kernel space \(H_f^2(\mathbb {B}_d)\).     

A Uniform Method to Ulam–Hyers Stability for Some Linear Fractional Equations

In this paper, we first utilize fractional calculus, the properties of classical and generalized Mittag-Leffler functions to prove the Ulam–Hyers stability of linear fractional differential equations using Laplace transform method. Meanwhile, Ulam–Hyers–Rassias stability result is obtained as a direct corollary. Finally, we apply the same techniques to discuss the Ulam’s type stability of fractional evolution equations, impulsive fractional evolutions equations and Sobolev-type fractional evolution equations.