Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

Abstract

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems.

Fundamental Matrix, Integral Representation and Stability Analysis of the Solutions of Neutral Fractional Systems with Derivatives in the Riemann—Liouville Sense

“…Remark 2. We note that from Theorems 3-5 it follows that the requirement in Theorem 2 [11], that in the Lebesgue decomposition of U(t, θ) a singular part does not exist, is unnecessary. So, the results proved in Theorems 3-5 generalize the statement of Theorem 2 [11] even in the retarded case.…”

“…b . According to Lemma 1 in [11] and Lemma 4 in [22], since the initial function Φ 0 (t − a) is a continuous function for t ∈ [a − h, a), the second and the third addends on the right-hand side of (10) are continuous functions at t ∈ (a, b]. Taking into account that a is a critical jump point relative to the delays with numbers l ∈ ⟨l⟩, we conclude that the fourth addend is a continuous function at t ∈ (a, b] too.…”

“…It is worth noting that the standard definitions for stability in the Lyapunov or Ulam-Hyers senses introduced for the systems with integer-order derivatives (without or with delays) are applicable directly for systems with fractional derivatives only when the fractional derivative of a constant is equal to zero (as Caputo-type derivatives, etc.). As we have mentioned previously [11], in general when the fractional derivative of a constant is not equal to zero (as for RL-type derivatives), then the standard definitions are not applicable since the solutions of the systems with RL-type derivatives have singularity at a + 0 (the low terminal) of power order α − 1. That is why new types of definitions for the different kinds of stabilities applicable to systems with RL-type derivatives are needed.…”

“…It is well known that the existence of explicit solutions or an integral representation (variation of constants formula) of the solutions of linear fractional differential equations and/or systems (ordinary or delayed) is a main tool in executing their qualitative analysis. That is why establishing either explicit solutions (see [10]) or integral representation, for which existence of a fundamental matrix is needed (see [11]), are important tasks for stability analysis, especially in the case of equations with Riemann-Liouville derivatives. But it is surprising that there are not many articles devoted to this problem.…”

“…Modifying the approach in [11], for an arbitrary fixed number α ∈ (0, 1) we introduce the following real linear space:…”