Fractional-order Differential Inclusions Research Articles

Fractional-order differential inclusion with pseudo-Lipschitz multi-valued map

In this paper, we investigate the existence of a solution for a fractional-order differential inclusion when the multi-valued map is supposed to be non-convex and satisfies a pseudo-Lipschitz property. For our aim, we use Filippov's theorem. An application to an optimal control problem is also presented for a fractional differential equation.

Analysis of Fractional Differential Inclusion Models for COVID-19 via Fixed Point Results in Metric Space

We examine in this paper some new problems on coincidence point and fixed point theorems for multivalued mappings in metric space. By applying the characterizations of a modified M T ~ -function, under the name D -function, a few novel fixed point results different from the existing fixed point theorems are launched. It is well-known that differential equation of either integer or fractional order is not sufficient to capture ambiguity, since the derivative of a solution to any differential equation inherits all the regularity properties of the mapping involved and of the solution itself. This does not hold in the case of differential inclusions. In particular, fractional-order differential inclusion models are more suitable for describing epidemics. Thus, as a generalization of a newly launched existence result for fractional-order model for COVID-19, using Banach and Shauder fixed point theorems, we investigate solvability criteria of a novel Caputo-type fractional-order differential inclusion model for COVID-19 by applying a standard fixed point theorem of multivalued contraction. Stability analysis of the proposed model in the framework of Ulam-Hyers is also discussed. Nontrivial comparative illustrations are constructed to show that our ideas herein complement, unify and, extend a significant number of existing results in the corresponding literature.

Fractional-order discontinuous systems with indefinite LKFs: An application to fractional-order neural networks with time delays

In this article, we discuss bipartite fixed-time synchronization for fractional-order signed neural networks with discontinuous activation patterns. The Filippov multi-map is used to convert the fixed-time stability of the fractional-order general solution into the zero solution of the fractional-order differential inclusions. On the Caputo fractional-order derivative, Lyapunov-Krasovskii functional is proved to possess the indefinite fractional derivatives for fixed-time stability of fragmentary discontinuous systems. Furthermore, the fixed-time stability of the fractional-order discontinuous system is achieved as well as an estimate of the new settling time.. The discontinuous controller is designed for the delayed fractional-order discontinuous signed neural networks with antagonistic interactions and new conditions for permanent fixed-time synchronization of these networks with antagonistic interactions are also provided, as well as the settling time for permanent fixed-time synchronization. Two numerical simulation results are presented to demonstrate the effectiveness of the main results

Boundary Value Problems for ψ-Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions

In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach fixed point theorem while an existence result is established via Krasnosel’skiĭ’s fixed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples.

Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operators

Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operators

On bounded solutions of semilinear fractional order differential inclusions in Hilbert spaces

On bounded solutions of semilinear fractional order differential inclusions in Hilbert spaces

Existence of solutions for a coupled system of Caputo type fractional-order differential inclusions with non-separated boundary conditions on multivalued maps

Existence of solutions for a coupled system of Caputo type fractional-order differential inclusions with non-separated boundary conditions on multivalued maps

Fractional order differential inclusions on an unbounded domain with infinite delay

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

New Results on Synchronization of Fractional-Order Memristor‐Based Neural Networks via State Feedback Control

This paper addresses the synchronization issue for the drive-response fractional-order memristor‐based neural networks (FOMNNs) via state feedback control. To achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method, Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization of the drive-response FOMNNs. Lastly, for illustrating the effectiveness of the obtained theoretical results, an example is given.

Finite-Time Projective Synchronization of Fractional-Order Memristive Neural Networks with Mixed Time-Varying Delays

This paper is concerned with the finite-time projective synchronization problem of fractional-order memristive neural networks (FMNNs) with mixed time-varying delays. Firstly, under the frame of fractional-order differential inclusion and the set-valued map, several criteria are derived to ensure finite-time projective synchronization of FMNNs. Meanwhile, three properties are established to deal with different forms of the finite-time fractional differential inequation, which greatly extend some results on estimation of settling time of FMNNs. In addition to the traditional Lyapunov function with 1-norm form in Theorem 1, a more general and flexible Lyapunov function based on p-norm is constructed in Theorem 2 to analyze the finite-time projective synchronization problem, and the estimation of settling time has been verified less conservative than previous results. Finally, numerical examples are provided to demonstrate the effectiveness of the derived theoretical results.

ON FRACTIONAL DIFFERENTIAL INCLUSION PROBLEMS INVOLVING FRACTIONAL ORDER DERIVATIVE WITH RESPECT TO ANOTHER FUNCTION

In this research work, we investigate the existence of solutions for a class of nonlinear boundary value problems for fractional-order differential inclusion with respect to another function. Endpoint theorem for [Formula: see text]-weak contractive maps is the main tool in determining our results. An example is presented in aim to illustrate the results.

Global Mittag–Leffler Stability and Stabilization Analysis of Fractional-Order Quaternion-Valued Memristive Neural Networks

This paper studies the global Mittag–Leffler stability and stabilization analysis of fractional-order quaternion-valued memristive neural networks (FOQVMNNs). The state feedback stabilizing control law is designed in order to stabilize the considered problem. Based on the non-commutativity of quaternion multiplication, the original fractional-order quaternion-valued systems is divided into four fractional-order real-valued systems. By using the method of Lyapunov fractional-order derivative, fractional-order differential inclusions, set-valued maps, several global Mittag–Leffler stability and stabilization conditions of considered FOQVMNNs are established. Two numerical examples are provided to illustrate the usefulness of our analytical results.

On the existence of solutions of a set-valued functional integral equation of Volterra\u2013Stieltjes type and some applications

This paper is concerned with the existence of continuous solutions of a set-valued functional integral equation of Volterra–Stieltjes type. The continuous dependence of the solution on the set of selections of the set-valued function will be proven. As an application, we study the existence of solutions to an initial-value problem of arbitrary fractional-order differential inclusion.

Finite-Time Synchronization of Memristor-Based Fractional Order Cohen-Grossberg Neural Networks

This paper is concerned with the finite-time synchronization (FTS) of memristor-based fractional order Cohen-Grossberg neural networks (MFCGNNs) with time-varying delays. Under the frame of fractional order differential inclusion and set-valued map, some new sufficient conditions to guarantee the FTS of MFCGNNs are established by means of constructing two different Lyapunov functions based on L1-norm in Theorem 1 and Lp-norm in Theorem 2. Via applying the asymptotic expansion property of Mittag-Leffler function, we propose a new estimation method of the settling time for synchronization which is less conservative than previous researches. Meanwhile, we deeply discuss the influence factor of settling time for synchronization. Finally, two numerical examples are given to demonstrate the effectiveness of obtained results.

Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations

Abstract This manuscript concerns quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional order BAM neural networks with time-varying delays and discontinuous neuron activations (FBAMNNDDAs). Firstly, under the framework of Filippov solution and fractional-order differential inclusions analysis for the initial value problem of FBAMNNDDAs is presented. Secondly, two kinds of novel pinning controllers according to pinning control technique are designed. By means of fractional order Lyapunov method and designed pinning control strategy, the sufficient criteria is given first to ensure the quasi-synchronization for the dynamic behavior of FBAMNNDDAs. Furthermore, the error bound of pinning synchronization is explicitly evaluated. Thirdly, via Kakutani s fixed point theorem of set-valued map analysis, Razumikhin condition, and a nonlinear pinning controller, the existence and β-exponential stabilization of FBAMNNDDAs equilibrium point is obtained in the voice of linear matrix inequality (LMI) technique. Fourthly, based on as well as Mittag-Leffler function and growth condition, the global existence of a solution in the Filippov sense of such system is guaranteed with detailed proof. At last, a numerical example with computer simulations are performed to illustrate the effectiveness of proposed theoretical consequences.

Stability analysis of fractional order memristive discontinuous neural networks with partial state control

This paper investigates exponential stability of fractional order memristive discontinuous neural networks (FMDNNs). Under the framework of the fractional order Filippov solution and differential inclusion theory, the global existence of the solution for the FMDNNs is studied by a given growth conditions. Based on fractional stability theoryand the properties of the Mittag Leffler function, some new criteria for the stability of FMDNNs are obtained by using effective partial state impulsive control, which only needs to control a small fraction of the states. At every impulsive moment, these states of the trajectory far away from the desired trajectory will be firstly controlled. The relations between stable region and fractional order α, control parameters and control rate are discussed. Finally, a numerical simulation is given to verify the effectiveness of the theoretical analysis.

Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay.

This paper addresses the global stabilization of fractional-order memristor-based neural networks (FMNNs) with time delay. The voltage threshold type memristor model is considered, and the FMNNs are represented by fractional-order differential equations with discontinuous right-hand sides. Then, the problem is addressed based on fractional-order differential inclusions and set-valued maps, together with the aid of Lyapunov functions and the comparison principle. Two types of control laws (delayed state feedback control and coupling state feedback control) are designed. Accordingly, two types of stabilization criteria [algebraic form and linear matrix inequality (LMI) form] are established. There are two groups of adjustable parameters included in the delayed state feedback control, which can be selected flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization. Since the existing LMI-based stability analysis techniques for fractional-order systems are not applicable to delayed fractional-order nonlinear systems, a fractional-order differential inequality is established to overcome this difficulty. Based on the coupling state feedback control, some LMI stabilization criteria are developed for the first time with the help of the newly established fractional-order differential inequality. The obtained LMI results provide new insights into the research of delayed fractional-order nonlinear systems. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

Global Mittag-Leffler synchronization of delayed fractional-order memristive neural networks

This paper deals with the global Mittag-Leffler synchronization of fractional-order memristive neural networks (FMNNs) with time delay. Since the FMNNs are essentially a class of switched systems with irregular switching laws, it is more difficult to achieve synchronization than with the traditional neural networks. First, under the framework of fractional-order differential inclusions and set-valued maps, the FMNNs are transformed into a continuous system with uncertainties. Then a linear state feedback combined with switching control law is designed in order to achieve the Mittag-Leffler synchronization. In addition, several synchronization criteria are obtained by constructing appropriate Lyapunov functionals, together with the help of some inequality techniques. Finally, an example is given to demonstrate the effectiveness of the obtained results.

Existence Results for Generalized Bagley-Torvik Type Fractional Differential Inclusions with Nonlocal Initial Conditions

In this article, we prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. It allows applying the noncompactness measure of Hausdorff, fractional calculus theory, and the nonlinear alternative for Kakutani maps fixed point theorem to obtain the existence results under the assumptions that the nonlocal item is compact continuous and Lipschitz continuous and multifunction is compact and Lipschitz, respectively. Our results extend the existence theorems for the classical Bagley-Torvik inclusion and some related models.

Finite-time synchronization of fractional-order memristive recurrent neural networks with discontinuous activation functions

Abstract This paper is concerned with the finite-time synchronization for a class of drive-response fractional-order memristive recurrent neural networks with discontinuous activation functions. By using the theories of fractional-order differential inclusions and set-valued map, the finite-time synchronization problem for a class of drive-response fractional-order memristive recurrent neural networks with discontinuous activation functions is formulated under the framework of Filippov solution. Then, two novel state feedback controllers are designed according to state feedback control technique. In particular, based on the fractional Lyapunov stability theory, the finite-time stability theory and Young inequality, some novel algebraic synchronization criteria are obtained to ensure the finite-time synchronization of a class of drive-response fractional-order memristive recurrent neural networks with discontinuous activation functions. Moreover, we give the estimation of the upper bound of the settling time for synchronization. Finally, a simulation example is given to show the effectiveness of our theoretical results.