Generalized Gronwall Inequality Research Articles

Existence trajectory and optimal control of Clarke subdifferential stochastic integrodifferential inclusions suffered by non-instantaneous impulses and deviated arguments

In this article, the solvability, Trajectory(T-) and optimal controllability of stochastic integrodifferential inclusions with Clarke subdifferential along with deviated arguments and Poisson jumps are analyzed which are new and untreated topics in the literature. The theory of resolvent operators, stochastic analysis, semigroup theory and a multivalued fixed point theorem are used to prove the solvability of the proposed non-instantaneous impulsive stochastic integrodifferential inclusion in Hilbert space. In addition, the strongest notion of controllability called T-controllability of the system is determined using a generalized Gronwall inequality with some appropriate assumptions. Following that, we derive the existence of optimal control of the proposed problem using Balder’s theorem. A numerical example is given to validate the theoretical aspects. We study the numerical simulation of the proposed problem with challenges. An abstract application of a stochastic dam contamination model is studied to justify the developed theoretical result. This paper contains the study of T-control along with the optimal control. This work is the unique combination of theoretical and numerical simulation as well as the real life application.

Asymptotic behavior of fractional-order nonlinear systems with two different derivatives

This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag–Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results.

New finite-time stability result for a class of Itô-Doob stochastic fractional order systems

In this article, we study the Finite-Time Stability (FTS) of Linear Stochastic Fractional Differential Equations of Itô-Doob Type with Delay (LSFDEIDTwD) for a derivative order ρ∈(0,1). In fact, the (FTS) here consists in studying the stability of the (LSFDEIDTwD) in a finite-time domain [0,T]. To our knowledge, this work represents the first successful attempt to study the (FTS) for the (LSFDEIDTwD), where the fractional order ρ∈(0,1). According to the generalized Gronwall Inequality (GWI) and stochastic calculus theory, the (FTS) of (LSFDEIDTwD) is investigated. We illustrate the main results by two numerical examples.

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.

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.

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.

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>

Linkage-constraint criteria for robust exponential stability of nonlinear BAM system with derivative contraction coefficients and piecewise constant arguments

This paper presents linkage-constraint criteria to guarantee the robustness of global exponential stability (RoGES) for a class of generalized nonlinear bidirectional associative memory (BAM) systems in the presence of derivative contraction coefficients and piecewise constant arguments (NBAM-dp system). However, because of the appearance of the tricky derivative compression coefficients and coupling relation with piecewise constant arguments, the previous Gronwall inequality is not applicable here. Thus, by means of generalized Gronwall inequality and norm inequality techniques, an independent parameters and interdependent variables (IPIV) principle is designed to solve the implicit bivariate transcendental equations to provide the respective supremum of the two interference factors, and the major criteria for guaranteeing the RoGES for NBAM-dp systems are obtained accordingly. Meanwhile, according to the IPIV principle, the relationship between the above two interference factors reflected in the linkage-constraint criteria is ensured to be mutually constrained and dynamically interconnected. Finally, comparative simulations and auxiliary data are provided to demonstrate the effectiveness of the theoretical findings.

Concept and application of interval-valued fractional conformable calculus

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results.We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world.

Analysis of Fractional Order Differential Equation Using Laplace Transform

Exploring the functioning of analytical solutions of differential equations defined by the fractional order operators is one of the hottest topics in the field of research in stability problems. This article analyze the stability of a certain type of fractional order differential equation. Employing Banach contraction mapping principle, existence and uniqueness of solutions are obtained. A sufficient condition to assure the reliability of solving the fractional order differential equation by Laplace Transform method and Generalized Laplace Transform are presented. Exponential stability results for the solution is discussed using Gronwall inequality. Application of the generalized Gronwall inequality to fractional order differential equation under investigation yields new sufficient condition for stability of fractional order differential equation. Applicability of the theoretical results on stability are demonstrated with an example. In the analysis of fractional order differential equation, Laplace transform is proved to be a valid tool under certain conditions.

The existence and Ulam-Hyers stability results for generalized Hilfer fractional integro-differential equations with nonlocal integral boundary conditions

In this paper, we study the existence and uniqueness of mild solutions for nonlinear fractional integro-differential equations (FIDEs) subject to nonlocal integral boundary conditions (nonlocal IBC) in the frame of a ξ-Hilfer fractional derivative (FDs). Further, we discuss different kinds of stability of Ulam-Hyers (UH) for mild solutions to the given problem. Using the fixed point theorems (FPT's) together with generalized Gronwall inequality the desired outcomes are proven. Examples are given which illustrate the effectiveness of the theoretical results.

Caputo-Katugampola-type implicit fractional differential equation with anti-periodic boundary conditions

The given article describes the implicit fractional dierential equation with anti-periodic boundary conditionsin the frame of Caputo-Katugampola fractional derivative. We obtain an analogous integral equation of thegiven problem and prove the existence and uniqueness results of such a problem using the Banach andKrasnoselskii xed point theorems. Further, by applying generalized Gronwall inequality, the Ulam-Hyersstability results are discussed. To show the eectiveness of the acquired results, convenient examples arepresented.

Further results on finite-time stability of neutral nonlinear multi-term fractional order time-varying delay systems

In this paper, the finite-time stability for nonlinear neutral multi-term fractional order systems with time-varying input and state delays is investigated. By use of the generalized Gronwall inequality and extended form of the generalized Gronwall inequality, new sufficient conditions for finite-time stability of such systems are obtained. Finally, numerical examples are given to illustrate the effectiveness and applicability of the proposed theoretical results.

Besicovitch almost periodic solutions for fractional‐order quaternion‐valued neural networks with discrete and distributed delays

In this paper, we study a class of fractional‐order quaternion‐valued neural networks with discrete and distributed delays by direct method. We first investigate some properties of Besicovitch almost periodic functions, including the composition theorem with deviating arguments. Then, we obtain the existence and uniqueness of Besicovitch almost periodic solutions for such class of networks by considering an appropriate Banach space and using the contraction mapping principle. Furthermore, by making use of a generalized Gronwall inequality, we gain the finite‐time stability of the Besicovitch almost periodic solution. Even if the network under consideration degenerates into a real‐valued or complex‐valued networks, the results of this paper are still brand new. Finally, we use a numerical example to show the feasibility of our theoretical results.

Impulsive effects on fractional order time delayed gene regulatory networks: Asymptotic stability analysis

This paper address the fractional-order gene regulatory networks (FOGRNs) with both time delays and impulsive effects. Inspired by the integer-order gene regulatory network models, a general class of fractional-order gene regulatory network model is further investigated via fractional-order Lyapunov-Razumikhin stability theory. Firstly, impulsive effects, feedback regulation, and translation time delays are taken well into account and effective analysis techniques are used to reflect the system’s practically dynamic behavior. Secondly, the existence and uniqueness of the equilibrium point of the fractional-order gene regulatory networks are derived based on the Banach contraction mapping principle. Besides, we derived analytically the condition under which the solution of this network is bounded through generalized Gronwall inequality. Thirdly, some novel delay-free sufficient conditions are derived to ensure the global asymptotic stability of the considered model with the help of the fractional-order Lyapunov-Razumikhin method and properties of Mittag-Leffler functions. Finally, an example with numerical simulation is provided to illustrate the validity and effectiveness of the proposed results.

Existence, Uniqueness and Stability Results of Fractional Volterra-Fredholm Integro Differential Equations of ψ -Hilfer Type

In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinear $\psi$-Hilfer fractional Volterra-Fredholm integro differential equations with boundary conditions. In addition, the Ulam-Hyers stability for solutions of the given problem are also discussed. The desired results are proved by using generalized Gronwall inequality, aid of fixed point theorems due to Banach and Schauder in weighted spaces.

Finite-time stability of ABC type fractional delay difference equations

In this paper, finite-time stability of fractional delay difference equations with discrete Mittag-Leffler kernel are studied. Firstly, we establish a new generalized Gronwall inequality in sense of Atangana-Baleanu fractional difference sum operator. Then, based on this new generalized Gronwall inequality and the method of steps, finite-time stability criteria of fractional delay difference equations with discrete Mittag-Leffler kernel are induced respectively. Finally, examples are presented to illustrate the validity of main results.

Finite-time stability of linear stochastic fractional-order systems with time delay

This paper focuses on the finite-time stability of linear stochastic fractional-order systems with time delay for alpha in (frac{1}{2},1). Under the generalized Gronwall inequality and stochastic analysis techniques, the finite-time stability of the solution for linear stochastic fractional-order systems with time delay is investigated. We give two illustrative examples to show the interest of the main results.

Iterative learning consensus control with initial state learning for fractional order distributed parameter models multi‐agent systems

This paper considers the consensus control problem of multi‐agent systems (MAS) with distributed parameter models. Based on the framework of network topologies, a second‐order PI‐type iterative learning control (ILC) protocol with initial state learning is proposed by using the nearest neighbor knowledge. A discrete system for proposed ILC is established, and the consensus control problem is then converted to a stability problem for such a discrete system. Furthermore, by using generalized Gronwall inequality, a sufficient condition for the convergence of the consensus errors between any two agents is obtained. Finally, the validity of the proposed method is verified by two numerical examples.

Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary conditions, we extract an equivalent integral equation and then define new operators based on it. With the help of three distinct fixed-point theorems attributed to Krasnosel’skiĭ, Leray–Schauder, and Banach, we investigate desired uniqueness and existence results. Additionally, the dependence criterion of solutions for this CFBVP is checked via the generalized version of the Gronwall inequality. Next, three simulative examples are designed to examine our findings based on the procedures applied in the theorems.