Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses

Abstract

In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and their mild solutions are given via integral representations. In the linear case the equivalence of the solution and mild solutions is established. Conditions for existence and uniqueness of initial value problems are presented. Several examples are provided to illustrate the influence of impulsive functions and the interpretation of impulses in the RL fractional case.

Scalar linear impulsive Riemann-Liouville fractional differential equations with constant delay-explicit solutions and finite time stability

“…3 Interpretations of the impulses in the RL fractional equations (which is not true for the ordinary case = q 1) leads to two basic interpretations of the solution (see, e.g., [10,18]): -Fixed lower limit of the RL fractional derivativein this case the lower limit of the fractional derivative is kept equal to the initial time t 0 on the whole interval of consideration. At points of impulses the amount of jump is taken into account.…”

“…For some explanations about the presence of impulses in the fractional differential equation without any delays and Caputo fractional derivative we refer to [18,19]. In the case of the RL fractional derivative, impulses and no delays, a discussion about the interpretation of the solutions is given in [10].…”

“…The question about Riemann-Liouville (RL) fractional differential equations is still at the initial stage of investigations (see, e.g., [9][10][11]). Li and Wang introduced the concept of a delayed Mittag-Leffler-type matrix function, and then they presented the finite-time stability results by virtue of a delayed Mittag-Leffler-type matrix in [12][13][14].…”

Explicit solutions of initial value problems for systems of linear Riemann–Liouville fractional differential equations with constant delay

“…However, little is known regarding Riemann-Liouville (RL) fractional differential equations with delays. For some related contributions about RL factional differential equations, one can refer to previous works [1,3]. Note that linear systems of RL fractional differential equations without any delay are studied in [18] and explicit formulas for the solutions are obtained.…”

A study on fractional differential equations using the fractional Fourier transform

“…[44]) Given function U ∈ (R), the definition of the Riemann-Liouville derivative with fractional order ν > 0, m = [ν] + 1 is…”