The aim of this paper is to consider the Hyers-Ulam stability of a class of parabolic equation { ∂u ∂t − a 2∆u+ b · ∇u+ cu = 0, (x, t) ∈ Rn × (0,+∞), u(x, 0) = φ(x), x ∈ Rn. We conclude that (i) it is Hyers-Ulam stable on any finite interval; (ii) if c 6= 0, it is Hyers-Ulam stable on the semi-infinite interval; (iii) if c = 0, it is not Hyers-Ulam stable on the semi-infinite interval by using ...

Introduction A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?”. Such a problem, called a stability problem of the functional equation, was formulated by Ulam [1] in 1940. In the next year, Hyers [2] gave a partial solution of Ulam’s probl...

Problem 1.1. Given a metric group (G,·,d), a positive number ε, and a mapping f : G→ G which satisfies the inequality d( f (xy), f (x) f (y)) ≤ ε for all x, y ∈ G, do there exist an automorphism a of G and a constant δ depending only on G such that d(a(x), f (x)) ≤ δ for all x ∈G? If the answer to this question is affirmative, we say that the equation a(xy) = a(x)a(y) is stable. A first answer ...

In this paper, we investigate the exact and approximate controllability, finite time stability, β–Hyers–Ulam–Rassias stability of a fractional order neutral impulsive differential system. The controllability criteria is incorporated with help fixed point approach. famous generalized Grönwall inequality used to study stability. Finally, main results are verified an example.

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, existence uniqueness of solution UHR stability system are rarely studied by using polynomial function matrix. In this paper, we firstly introduce a new concept impulsive delayed Mittag–Leffler type vector function, which helps us to construct representation an exact linear...

In this paper, we establish the existence and stability results for (?k,?k)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point integral boundary conditions. We achieve formulation of solution to differential equation constant coefficients in term Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory pr...

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of the generalized radical cubic functional equation[    fleft( sqrt[3]{ax^3 + by^3}right)=af(x) + bf(y),]    where $a,b in mathbb{R}_+$ are fixed positive real numbers, by using direct method in quasi-$beta$-Banach spaces. Moreover, we use subadditive functions to investigate stability of the generaliz...