Abstract
In this paper, fractional differential inequalities and systems of fractional differential inequalities involving fractional derivatives in the sense of Caputo are investigated. Namely, necessary conditions for the existence of global solutions are obtained. Our approach is based on the test function method and some integral inequalities.
Highlights
Introduction and Main ResultsIn several studies, the usefulness of fractional derivatives in the mathematical modeling of various phenomena from physics and engineering has been demonstrated
The study of fractional differential equations has received a great deal of attention from many researchers
The study of sufficient conditions for the existence of solutions has been investigated by many authors using different approaches from functional analysis
Summary
The usefulness of fractional derivatives in the mathematical modeling of various phenomena from physics and engineering has been demonstrated (see, e.g., [1,2,3,4,5,6,7], and the references therein). The study of necessary conditions for the existence of global solutions in the context of fractional differential equations has been initiated by Kirane and his collaborators (see, e.g., [19,20,21,22,23,24], and the references therein). Motivated by Furati and Kirane [19], in this paper, we first consider the fractional differential inequality By a global solution to (13), we mean a pair of functions ðu, u2Þ ∈ ACð1⁄20,∞ÞÞ × ACð1⁄20,∞ÞÞ satisfying the fractional differential inequalities in (13) for almost everywhere t > 0, and the initial condition ðu1ð0Þ, u2ð0ÞÞ = ðuð10Þ, uð20ÞÞ.
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