Gronwall Inequality Research Articles

Ulam type stability for Caputo–Hadamard fractional functional stochastic differential equations with delay

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples.

Global fractional Halanay inequalities approach to finite-time stability of nonlinear fractional order delay systems

In this paper, we investigate the robust finite time stability of fractional order systems with time varying delay and nonlinear perturbation. We improve the sufficient condition for general fractional delay systems by utilizing the special structure of a singular Gronwall inequality. For stable fractional delay systems, our approach bases on a global Halanay type inequality in differential and integral forms. A sharper delay dependent sufficient condition for robust finite time stability of such systems is formulated in terms of the Mittag-Leffler functions and the delayed size. The connection between the new sufficient condition and the previous results is compared and discussed thoroughly.

Iterative learning consensus control of nonlinear impulsive distributed parameter multi-agent systems

This paper studies the consensus control of nonlinear impulsive distributed parameter multi-agent systems (IDPMASs), whose dynamic behavior depends on time and space. First, to achieve the complete consensus for nonlinear parabolic IDPMAS, a distributed P-type iterative learning consensus control protocol is proposed that includes network topologies information and nearest neighbor knowledge. Using impulsive Gronwall inequality and mild solution formula based on operator semigroup, the rigorous convergence analysis of consensus errors is provided, and convergence conditions are established as well. Furthermore, the consensus control of nonlinear second-order hyperbolic IDPMAS is investigated that employ an open-closed-loop P-type consensus control protocol, which uses both currents iterative consensus error and last iterative consensus error. The theoretical result of this paper shows the consensus errors between any 2 agents can converge to 0 with the increase in iterations under given conditions. Finally, two numerical simulations are given to demonstrate the effectiveness of the proposed methods.

A Gronwall inequality and its applications to the Cauchy-type problem under ψ-Hilfer proportional fractional operators

In this paper, we propose a generalized Gronwall inequality in the context of the ψ-Hilfer proportional fractional derivative. Using Picard’s successive approximation and the definition of Mittag–Leffler functions, we construct the representation formula of the solution for the ψ-Hilfer proportional fractional differential equation with constant coefficient in the form of the Mittag–Leffler kernel. The uniqueness result is proved by using Banach’s fixed-point theorem with some properties of the Mittag–Leffler kernel. Additionally, Ulam–Hyers–Mittag–Leffler stability results are analyzed. Finally, numerical examples are provided to demonstrate the theory’s application.

A new approach on approximate controllability of Sobolev-type Hilfer fractional differential equations

The approximate controllability of Sobolev-type Hilfer fractional control differential systems is the main emphasis of this paper. We use fractional calculus, Gronwall's inequality, semigroup theory, and the Cauchy sequence to examine the main results for the proposed system. The application of well-known fixed point theorem methodologies is avoided in this paper. Finally, a fractional heat equation is discussed as an example.

Existence and finite-time stability results of fuzzy Hilfer-Katugampola fractional delay differential equations1

In this article, we explore the question of existence and finite time stability for fuzzy Hilfer-Katugampola fractional delay differential equations. By using the generalized Gronwall inequality and Schauder’s fixed point theorem, we establish existence of the solution, and the finite time stability for the presented problems. Finally, the effectiveness of the theoretical result is shown through verification and simulations for an example.

On the uniqueness of mild solutions to the time-fractional Navier–Stokes equations in $$L^{N} \left( \mathbb {R} ^{N}\right) ^{N}$$

In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier–Stokes equations in class \(C\left( [0,\infty );L^{N}\left( \mathbb {R}^{N}\right) ^{N}\right) \) by means of the estimates \(L^{p}-L^{q}\) of Giga-Shor inequality and the Gronwall inequality.

Exponential stability analysis for nonlinear time-varying perturbed systems on time scales

<abstract><p>This paper is concerned with the stability of nonlinear time-varying perturbed system on time scales under the assumption that the corresponding linear time-varying nominal system is uniformly exponentially stable. Some less conservative sufficient conditions for uniform exponential stability and uniform practical exponential stability are proposed by imposing different assumptions on the perturbation term. Compared with the traditional exponential stability results of perturbed systems, the time derivatives of related Lyapunov functions in this paper are not required to be negative definite for all time. The main tools employed are two Gronwall's inequalities on time scales. Some examples are also given to illustrate the effectiveness of the theoretical results.</p></abstract>

Boundary value problems of higher order fractional integro-differential equations involving Gronwall's inequality in Banach spaces

Boundary value problems of higher order fractional integro-differential equations involving Gronwall's inequality in Banach spaces

Dynamics of the positive almost periodic solution to a class of recruitment delayed model on time scales

<abstract><p>By employing the operator theory, the Lyapunov function on time scales and the famous Gronwall's inequality, this paper addresses some dynamic properties of almost periodic solutions for a class of two species co-existence delayed model on time scales with almost periodic coefficients and Ricker, as well as the Beverton-Holt type function. First, we establish the existence and uniqueness of the almost periodic solution with a positive infimum by transforming the initial model into an equivalent integral equation. Second, we investigate the global exponential stability and uniformly asymptotic stability of the positive almost periodic solution. Finally, we give two examples to illustrate the main presented results.</p></abstract>

New interpretation of topological degree method of Hilfer fractional neutral functional integro-differential equation with nonlocal condition

<abstract><p>This manuscript deals with the concept of Hilfer fractional neutral functional integro-differential equation with a nonlocal condition. The solution representation of a given system is obtained from the strongly continuous operator, linear operator and bounded operator, as well as the Wright type of function. The sufficient and necessary conditions for the existence of a solution are attained using the topological degree method. The uniqueness of the solution is attained by Gronwall's inequality. Finally, we employed some specific numerical computations to examine the effectiveness of the results.</p></abstract>

A Pontryagin maximum principle for terminal state-constrained optimal control problems of Volterra integral equations with singular kernels

<abstract><p>We consider the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. A singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of $ \alpha \in (0, 1) $. Our state equation covers various state dynamics such as any types of classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. We prove the maximum principle for the corresponding state-constrained optimal control problem. In the proof of the maximum principle, due to the presence of the (terminal) state constraint and the control space being only a separable metric space, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance function and the generalized Gronwall's inequality, to obtain the desired necessary conditions for optimality. The maximum principle of this paper is new in the optimal control problem context and its proof requires a different technique, compared with that for classical Volterra integral equations studied in the existing literature.</p></abstract>

Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D

<abstract><p>We study the Benjamin-Bona-Mahony model with finite distributed delay in 3D, which depicts the dispersive impact of long waves. Based on the well-posedness of model, the family of pullback attractors for the evolutionary processes generated by a global weak solution has been obtained, which is unique and minimal, via verifying asymptotic compactness in functional space with delay $ C_V $ and topological space $ V\times C_V $, where the energy equation method and a retarded Gronwall inequality are utilized.</p></abstract>

FINITE-TIME STABILITY OF NON-INSTANTANEOUS IMPULSIVE SET DIFFERENTIAL EQUATIONS

In this paper, we investigate the finite-time stability of non-instantaneous impulsive set differential equations. By using the generalized Gronwall inequality and a revised Lyapunov method, the finite-time stability criteria for such equations are obtained. Finally, an example is given to illustrate the validity of the results.

A map-type Gronwall inequality on functional differential equations with state-dependence

A map-type Gronwall inequality on functional differential equations with state-dependence

The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

<abstract><p>In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.</p></abstract>

Attractors of Modified Coupled Ginzburg–Landau Model

Bose–Einstein condensation is a gaseous, superfluid state of matter exhibited by bosons as they cool to near absolute zero, which was discovered as early as 1924 but was not experimentally realized until 1995. In 2006, Machida and Koyama developed the corresponding Ginzburg–Landau model for superfluid and Bose–Einstein condensation‐spanning phenomena. We mainly consider the global attractor for the initial boundary value problem of the modified coupled Ginzburg–Landau equations, which come from the BCS‐BEC crossover model. Combining Gronwall inequality, properties of the binomial function, with some suitable a priori estimates, we establish the existence of global attractors. The attractor results obtained in this paper can provide a strong theoretical basis for the experimental realization of the BCS‐BEC spanning phenomenon, and the adopted research method can also serve as a reference for analysing other types of partial differential equation attractors.

Existence of solutions for a class of fractional dynamical systems with two damping terms in Banach space

<abstract><p>This paper studies the existence of solutions for fractional dynamical systems with two damping terms in Banach space. First, we generalize the well-known Gronwall inequality. Next, according to fixed-point theorems and inequalities, the existence results for the considered system are obtained. At last, an example is used to support the main results.</p></abstract>

Global well-posedness of the viscous Camassa–Holm equation with gradient noise

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa--Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-It\^{o} correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional It\^{o} formula and a DiPerna--Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

Controllability results of neutral Caputo fractional functional differential equations

<abstract><p>In this paper, using the properties of the phase space on infinite delay, generalized Gronwall inequality and fixed point theorems, the existence and controllability results of neutral fractional functional differential equations with multi-term Caputo fractional derivatives were obtained under Lipschitz and non-Lipschitz conditions.</p></abstract>