A Note on Gronwall’s Inequality on Time Scales

Lin, Xueru

doi:10.1155/2014/623726

Cited by 6 publications

(4 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

A Gronwall inequality via the generalized proportional fractional derivative with applications

et al. 2019

View full text Add to dashboard Cite

show abstract

“…Theorem 6.6 Suppose p ∈ R + ν . Let γ ∈ (0, 1], a ∈ T and y 0 ∈ R. The unique solution of the initial value problem Proof Using Theorem 6.3 and Lemma 6.5, we get Proof We use Lemma 6.5 and the properties in Theorem 5.9 to find γ ν p(t) e γ p a, ρ(t) For the introduction to the Gronwall's inequality, we refer to [34,38]. The following dynamic inequality is useful in proving the Gronwall's inequality in this new setting.…”

Section: Conformal Dynamic Equations and Inequalitiesmentioning

confidence: 99%

“…If we take T = hN 0 and a = 0 in Gronwall's inequality, we obtain the following example. Finally, we present a new version of Gronwall's inequality in this new setting (see [38]). The proof of the following theorem is similar to that in [38].…”

Section: Conformal Dynamic Equations and Inequalitiesmentioning

confidence: 99%

A new conformable nabla derivative and its application on arbitrary time scales

Rahmat

Noorani

2021

Adv Differ Equ

View full text Add to dashboard Cite

In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function $\widehat{\mathcal{G}}_{\eta }(t, a)$ G ˆ η ( t , a ) for $t,a\in \mathbb{T}$ t , a ∈ T . The time scale power function takes the form $(t-a)^{\eta }$ ( t − a ) η for $\mathbb{T}=\mathbb{R}$ T = R which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form $(t-a)^{(\eta )}$ ( t − a ) ( η ) which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form $\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))$ ▽ a γ y ( t ) = y ( t , f ( t ) ) , and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

A Note on Gronwall’s Inequality on Time Scales

Abstract: This paper gives a new version of Gronwall’s inequality on time scales. The method used in the proof is much different from that in the literature. Finally, an application is presented to show the feasibility of the obtained Gronwall’s inequality.

Cited by 6 publications

References 19 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 conformable nabla derivative and its application on arbitrary time scales

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

Contact Info

Product

Resources

About