Abstract
In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.
Highlights
Over the years, human beings have needed to be acquainted with various natural phenomena more and more
Numerous fractional operators have been introduced during years and their applicability is becoming increasingly apparent to researchers every day that passes
In much of the literature we can see various complicated fractional modelings in which one of the well-known fractional Caputo or the Riemann–Liouville operators has been utilized. Some generalizations of these operators such as the Hadamard, Caputo–Hadamard and Hilfer fractional operators were utilized by other researchers in the period and different modelings are investigated using these new operators
Summary
Human beings have needed to be acquainted with various natural phenomena more and more. Jarad et al [44] proceeded to answer this key problem if we can generalize the usual fractional Riemann–Liouville integral provided that we obtain a unification to remaining useful operators such as Caputo, Riemann–Liouville, Hadamard, and Caputo– Hadamard derivatives [45] To achieve this purpose, they tried to derive two corresponding integration and differentiation operators of arbitrary order based on the existing conformable operators. By taking into account the aforementioned new operators introduced by Jarad et al [44] and inspired by some existing ideas in the above articles, in the current manuscript, for the first time, we formulate an inclusion version of the pantograph boundary problem in the fractional Caputo conformable settings subject to three-point Riemann–Liouville conformable integral conditions as follows:. We derive desired existence results for three different structures considered on the set-valued maps and this cover a vast range of multifunctions satisfying our given conditions. the last part of the present research is devoted to proposing two numerical simulative examples to demonstrate the consistency of the analytical findings
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