Abstract
New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side.
Highlights
Integral inequalities have been used as fabulous instruments to explore the qualitative properties of differential equations [1]
This section is devoted to provide our main results of this paper
We are in a position to state the main theorem, which is a new version of the Gronwall–Bellman inequality within the generalized proportional fractional Riemann–Liouville settings
Summary
Integral inequalities have been used as fabulous instruments to explore the qualitative properties of differential equations [1]. Fractional differential equations (FDEs) is a rich area of research that has widespread applications in science and engineering. New versions for a Gronwall–Bellman inequality in the frame of the newly defined generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features [26]. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo generalized proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side. Our results extend the classical inequalities and generalize the existing ones for non-integer-order equations
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