Abstract
The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of first order with constant coefficients using the Aboodh transform method. We also obtain the Hyers–Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.
Highlights
In 1940, Ulam [1] proposed the following stability problem: When is the statement of the theorem still true or nearly true, despite slight variations on the theorem’s hypotheses? In the following year, Hyers [2] gave the first positive answer to Ulam’s question by proving the stability of the additive functional equation in Banach spaces
3 Hyers–Ulam stability of (1.1) we prove several types of Hyers–Ulam stability of the homogeneous firstorder linear differential equation (1.1) using the Aboodh transform
We prove the Mittag-Leffler–Hyers–Ulam stability of the non-homogeneous linear differential equation (1.2) using the Aboodh transform method
Summary
In 1940, Ulam [1] proposed the following stability problem: When is the statement of the theorem still true or nearly true, despite slight variations on the theorem’s hypotheses? In the following year, Hyers [2] gave the first positive answer to Ulam’s question by proving the stability of the additive functional equation in Banach spaces. Z(n)) = 0 has the Hyers–Ulam stability if there exists a constant K > 0 such that the following statement is true: for every ε > 0, if an n times continuously differentiable function z : I → K satisfies the inequality ψ f , z, z , z , . Z(n) ≤ ε for all t ∈ I, there exists a solution y : I → K of the differential equation that satisfies the inequality |z(t) – y(t)| ≤ Kε for all t ∈ I. In 1998, Alsina and Ger [10] continued the study of Obłoza’s Hyers–Ulam stability of differential equations.
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