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Open AccessFeature PaperArticle

Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses

1
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
2
Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA
3
Department of Applied Mathematics and Modeling, University of Plovdiv “Paisii Hilendarski”, 4000 Plovdiv, Bulgaria
4
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 607; https://doi.org/10.3390/math8040607
Received: 26 March 2020 / Revised: 12 April 2020 / Accepted: 14 April 2020 / Published: 16 April 2020
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of a changeable lower limit at each impulsive time. Integral representations of the solutions in all considered cases are obtained. Existence results on finite time intervals are proved using the Banach principle. View Full-Text
Keywords: Riemann-Liouville fractional derivative; delay; impulses; initial value problem; existence Riemann-Liouville fractional derivative; delay; impulses; initial value problem; existence
MDPI and ACS Style

Agarwal, R.; Hristova, S.; O’Regan, D. Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses. Mathematics 2020, 8, 607.

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