Abstract
Abstract In this paper, we develop the theory of fractional hybrid differential equations with linear perturbations of second type involving Riemann-Liouville differential operators of order 0 < q < 1 . An existence theorem for fractional hybrid differential equations is proved under the φ-Lipschitz condition. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle which will be useful for further study of qualitative behavior of solutions is proved. MSC:34A40, 34A12, 34A99.
Highlights
1 Introduction Fractional differential equations have been of great interest recently
It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [ – ]
The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases
Summary
Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [ – ]. We initiate the basic theory of fractional hybrid differential equations of mixed perturbations of second type involving three nonlinearities and prove the basic result such as the strict and nonstrict fractional differential inequalities, an existence theorem and maximal and minimal solutions etc. We consider fractional hybrid differential equations (in short FHDE) involving RiemannLiouville differential operators of order < q < , It is known that differential inequalities are useful for proving the existence of extremal solutions of ODEs and hybrid differential equations defined on J.
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