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Stability for conformable impulsive differential equations. (English) Zbl 1462.34040

The authors consider the stability of solutions to a conformable linear non-instantaneous differential equation. In the second section of the manuscript, they define the conformable fractional derivative with lower limit \(\alpha\), \(\mathfrak{D}_{\beta}^{\alpha}\), as well as the conformable integral. They also present lemmas in this section related to impulsive initial value problems which are used in the subsequent sections to invert the problem. The authors begin section 3 by defining what they mean by local asymptotic stability for the differential equation. They then establish conditions under which the equation is locally asymptotically stable. Next in the fourth section, the authors generalize the concept of Ulam-Hyers-Rassias stability and use a fixed point theorem to show that the differential equation satisfies this type of stability under suitable conditions. Finally in the fifth section, the authors provide two examples to illustrate their results.

MSC:

34A37 Ordinary differential equations with impulses
34A08 Fractional ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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