Nonlinear Gronwall–Bellman Type Inequalities and Their Applications

Wang, Weimin; Yong, Feng; Wang, Yuanyuan

doi:10.3390/math5020031

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…Such inequalities have been studied by many researches who in turn used diverse techniques for the sake of exploring and proposing these inequalities [1][2][3]. One of the most important inequalities is the distinguished Gronwall inequality [4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

A Gronwall inequality via the generalized proportional fractional derivative with applications

et al. 2019

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In this paper, we provide a new version for the Gronwall inequality in the frame of the generalized proportional fractional derivative. Prior to the main results, we introduce the generalized proportional fractional derivative and expose some of its features. As an application, we accommodate the newly defined derivative to prove the uniqueness and obtain a bound in terms of Mittag-Leffler function for the solutions of a nonlinear delay proportional fractional system. An example is presented to demonstrate the applicability of the theory.

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Section: Introductionmentioning

confidence: 99%

A Gronwall inequality via the generalized proportional fractional derivative with applications

et al. 2019

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“…The Gronwall-Bellman inequality allows one to provide an estimate for a function that is known to satisfy a certain integral inequality by the solution of the corresponding integral equation. In particular, it has been employed to provide a comparison that can be used to prove uniqueness of a solution to an initial value problem (see some recent relevant papers [6][7][8][9]).…”

Section: Introductionmentioning

confidence: 99%

A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

et al. 2019

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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.

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Analysis and numerical solution of the generalized proportional fractional Cauchy problem

Boucenna

Băleanu

Makhlouf

et al. 2021

Applied Numerical Mathematics

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Nonlinear Gronwall–Bellman Type Inequalities and Their Applications

Abstract: Abstract:In this paper, some nonlinear Gronwall-Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers-Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively.

Cited by 6 publications

References 14 publications

A Gronwall inequality via the generalized proportional fractional derivative with applications

A Gronwall inequality via the generalized proportional fractional derivative with applications

A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

Analysis and numerical solution of the generalized proportional fractional Cauchy problem

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