Gronwall-type Inequality Research Articles

Existence and stability of solution for a coupled system of Caputo–Hadamard fractional differential equations

In our present work, we study a coupled system of Caputo–Hadamard fractional differential equations supplemented with a novel set of initial value conditions involving the η=(tddt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\eta =(t\\frac{d}{dt})$\\end{document} derivatives. We provided sufficient criteria for the existence and stability of the solutions for a coupled system of fractional differential equations by applying the Hyers–Ulam stability theory, the fixed point theorems of Banach, Krasnoselskii, and the Leray–Schauder nonlinear alternative. When computing priori bounds in Leray–Schauder nonlinear alternative and stability of the solutions, a novel Gronwall type inequality related to Hadamard integral is employed. This study investigates the properties of a solution, such as existence, uniqueness, and stability, to a given problem without attempting to solve the exact solution, and its theoretical applications are illustrated by providing an example.

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

Gronwall-Type Inequalities and Qualitative Studies onHigher-Variable Orders of Atangana-Baleanu Fractional Operators via Increasing Functions

This paper introduces a novel extension of Caputo-Atangana-Baleanu and Riemann-Atangana-Baleanu fractional derivatives from constant to increasing variable order. We generalize the fractional order from a fixed value in (0, 1] to a time-dependent function in (k, k + 1], where k ≥ 0. The corresponding Atangana-Baleanu fractional integral is also extended. Key properties ofthese new definitions are explored, including a generalized Gronwall inequality. We then delve into the analysis of higher-variable initial fractional differential equations using the Caputo-Atangana-Baleanu operator with an increasing function, establishing existence and uniqueness results via Picard’s iterative method. The findings presented in this work are expected to stimulate further research on inequalities and fractional differential equations related to Atangana-Baleanu fractional calculus with respect to increasing functions. Concrete examples are provided to illustrate the practical applications of our results.

Further finite-time stability analysis of neural networks with proportional delay

Different from fixed delays between neurons, proportional delay is a time-varying unbounded delay. In this paper, the issue of finite-time stability of fractional-order neural networks with proportional delay is investigated. To address the proportional delay terms in the system, a new Gronwall-type inequality with delay is proposed. Based on this inequality, order-related criterion for finite-time stability of fractional-order neural networks with proportional delay is obtained. Finally, some numerical experiments demonstrate the effectiveness and superiority of the results.

Ψ-Bielecki-type norm inequalities for a generalized Sturm–Liouville–Langevin differential equation involving Ψ-Caputo fractional derivative

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems.

Practical stability analysis of stochastic perturbed linear time-varying systems via integral inequality

In this paper, we investigate the problem of stability of linear time-varying stochastic perturbed systems. We present sufficient conditions ensuring the global practical uniform exponential stability of a different class of stochastic perturbed systems based on generalized integral inequalities of Gronwall type. Finally, we provide numerical examples to prove the usefulness of the main results of this paper.

Gronwall type inequality on generalized fractional conformable integral operators

Abstract In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 {0<\alpha\leq{1}} (Riemann approach), yielding Hadamard fractional integrals when α = 0 {\alpha=0} . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for generalized fractional operators are established, as well as solutions to initial value problems in several new weighted spaces of functions.

Mean-square finite-time synchronization of stochastic competitive neural networks with infinite time-varying delays and reaction–diffusion terms

This paper focuses on the mean-square finite-time synchronization (MFTS) of stochastic competitive neural networks with infinite time-varying discrete delays and reaction–diffusion terms (IRSCNNs). Different from other articles, this paper presents a new approach, which does not use finite-time stability theorem but uses integral inequality, Gronwall-type inequality and comparison strategy to study MFTS. In addition, two control schemes are designed: a feedback control scheme and a new adaptive control strategy, which can achieve MFTS of IRSCNNs. Finally, the correctness of the results is verified by examples.

On Some Retarded Integral Inequalities of Gronwall Type: Application to Integral and Differential Equations

On Some Retarded Integral Inequalities of Gronwall Type: Application to Integral and Differential Equations

The delta [formula omitted]fractional Gronwall inequality on time scale

In this paper, we introduce delta q−fractional integral equations and inequalities. We obtain the equivalence of a Caputo delta q−fractional initial value problem of order 0<α≤1 and delta q−fractional integral equation. Following that, we provide an explicit solution to the linear delta q−fractional integral equation. This enables us to state and prove delta q−fractional Gronwall-type inequality on time scale. By using delta q−exponential type functions and delta q−Mittag Leffler functions, some particular cases are derived. To demonstrate the viability of the resulting inequality, an example is provided.

A higher-order extension of Atangana–Baleanu fractional operators with respect to another function and a Gronwall-type inequality

This paper aims to extend the Caputo–Atangana–Baleanu (ABC) and Riemann–Atangana–Baleanu (ABR) fractional derivatives with respect to another function, from fractional order mu in (0,1] to an arbitrary order mu in (n,n+1], n=0,1,2,dots . Also, their corresponding Atangana–Baleanu (AB) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall’s inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard’s iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an ABC operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function.

Existence and Hyers-Ulam stability for boundary value problems of multi-term Caputo fractional differential equations

The present paper is devoted to discussing a class of nonlinear Caputo-type fractional differential equations with two-point type boundary value conditions. We investigate the existence and uniqueness of the solutions by virtue of the classical Schauder alternative principle and the Banach contraction principle. Furthermore, by means of a novel Gronwall-type inequality, we prove the Hyers-Ulam stability of boundary value problems of multi-term Caputo fractional differential equations. Finally, some numerical examples are given to illustrate the results.

Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators

&lt;abstract&gt;&lt;p&gt;In this paper, the initial value problem of a nonlinear differential equation with higher order Caputo type modification of the Erdélyi-Kober fractional derivatives was studied. Based on the transmutation method, the well-posedness of initial value problem of the higher order linear model was proved and an explicit solution was presented. Then some new Gronwall type inequalities involving Erdélyi-Kober fractional integral were established. By applying these results and some fixed point theorems, the existence and uniqueness of the positive solution of the nonlinear differential equation were proved. The method is applicable to the fractional differential equation with any order $ \gamma\in (n-1, n] $.&lt;/p&gt;&lt;/abstract&gt;

Existence and global exponential stability of periodic solution for Cohen-Grossberg neural networks model with piecewise constant argument

In this paper, we introduce a Cohen-Grossberg neural networks model with piecewise alternately advanced and retarded argument. Some sufficient conditions are established for the existence and global exponential stability of periodic solutions. The approaches are based on employing Brouwer's fixed-point theorem and an integral inequality of Gronwall type with deviating argument. The criteria given are easily verifiable, possess many adjustable parameters, and depend on piecewise constant argument deviations, which provide flexibility for the design and analysis of Cohen-Grossberg neural networks model. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.

Stabilization by an estimated state controller of nonlinear time-varying systems

Abstract In linear systems, it is well known that any combination of stabilizing state feedback controller and a state observer results in the output feedback controller which asymptotically stabilizes the closed-loop system. This is known as the separation principle, which does not generally hold for nonlinear systems. For nonlinear systems, the output feedback problem is usually pursued by extending the results of observer synthesis, which itself is a challenging problem. In this paper, we investigate the stabilization via an observer configuration problem for some classes of nonlinear time-varying systems by using new integral inequalities of Gronwall type. Given that the sufficient conditions of the controller and observer problem are satisfied, we show that the proposed controller with estimated state feedback from the proposed practical observer will achieve the global practical exponential stabilization. Furthermore, the effectiveness of the proposed scheme is shown through simulation for two numerical examples.

Caputo delta weakly fractional difference equations

In this paper, solutions of fractional difference equations with Caputo-type delta-based fractional difference operator of order $$\mu \sim 1$$ are compared with solutions of corresponding limit difference equations with usual first-order forward difference. It is shown that the limit initial value problems differ substantially when $$\mu \rightarrow 1^-$$ and $$\mu \rightarrow 1^+$$ . To derive convergence results, Gronwall type inequalities are proved for suitable fractional sum inequalities of general noninteger order. An illustrative example is also given.

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α∈(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t→0+ albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results.

Uniqueness and Ulam–Hyers–Mittag–Leffler stability results for the delayed fractional multiterm differential equation involving the Φ-Caputo fractional derivative

The principal aim of this paper is to establish the uniqueness and Ulam–Hyers–Mittag–Leffler (UHML) stability of solutions for a new class of multiterm fractional time-delay differential equations in the context of the Φ-Caputo fractional derivative. To achieve this purpose, the generalized Laplace transform method alongside facet with properties of the Mittag–Leffler functions (M–LFs) are utilized to give a new representation formula of the solutions for the aforementioned problem. Besides that, the uniqueness of the solutions of the considered problem is also proved by applying the well-known Banach contraction principle coupled with the Φ-fractional Bielecki-type norm, while the Φ-fractional Gronwall type inequality and the Picard operator (PO) technique, combined with the abstract Gronwall lemma, are used to prove the UHML stability results for the proposed problem. Lastly, an example is offered to assure the validity of the obtained theoretical results.

Low regularity global well-posedness for 2D Boussinesq equations with variable viscosity

As a continuation of our previous work, we consider lower regularity global well-posedness for a model of the two-dimensional zero diffusivity Boussinesq equations with variable viscosity. More precisely, based on De Giorgi–Nash–Moser estimates and the refined logarithmic Gronwall-type inequality, we prove that it is globally well-posed, provided that the initial data belong to Hs with s &amp;gt; 1. Finally, we show that it is also valid for the two-dimensional zero diffusivity Boussinesq equations with variable viscosity in the non-divergence form.

On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

&lt;p style='text-indent:20px;'&gt;We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.&lt;/p&gt;