Research

17 pages, 1423 KiB

Open AccessArticle

Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays

by Zhao Jiang, Azhar Halik and Ahmadjan Muhammadhaji

Axioms 2023, 12(5), 501; https://doi.org/10.3390/axioms12050501 - 19 May 2023

Cited by 1 | Viewed by 976

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison [...] Read more.

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat’s lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

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19 pages, 770 KiB

Open AccessArticle

Dynamic Behavior of a Predator–Prey Model with Double Delays and Beddington–DeAngelis Functional Response

by Minjuan Cui, Yuanfu Shao, Renxiu Xue and Jinxing Zhao

Axioms 2023, 12(1), 73; https://doi.org/10.3390/axioms12010073 - 11 Jan 2023

Viewed by 1371

Abstract

In the predator–prey system, predators can affect the prey population by direct killing and predation fear. In the present study, we consider a delayed predator–prey model with fear and Beddington–DeAngelis functional response. The model incorporates not only the fear of predator on prey [...] Read more.

In the predator–prey system, predators can affect the prey population by direct killing and predation fear. In the present study, we consider a delayed predator–prey model with fear and Beddington–DeAngelis functional response. The model incorporates not only the fear of predator on prey with an intraspecific competition relationship, but also fear delay and pregnancy delay. Apart from the local stability analysis of the equilibrium points of the model, we find that time delay can change the stability of the system and cause Hopf bifurcation. Taking time delay as the bifurcation parameter, the critical values of delays in several cases are derived. In addition, we extend it to the random environment and study the stochastic ultimate boundedness of the stochastic process. Finally, our theoretical results are validated by numerical simulation. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

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14 pages, 2560 KiB

Open AccessArticle

Stability Analysis of Fractional-Order Predator-Prey System with Consuming Food Resource

by Muhammad Shoaib Arif, Kamaleldin Abodayeh and Asad Ejaz

Axioms 2023, 12(1), 64; https://doi.org/10.3390/axioms12010064 - 07 Jan 2023

Cited by 4 | Viewed by 1639

Abstract

The cardinal element of ecology is the predator-prey relationship. The population of interacting organisms is based on many factors such as food, water, space, and protection. A key component among these factors is food. The presence of food for the organisms shapes the [...] Read more.

The cardinal element of ecology is the predator-prey relationship. The population of interacting organisms is based on many factors such as food, water, space, and protection. A key component among these factors is food. The presence of food for the organisms shapes the structure of the habitat. The present study considers a predator and two types of prey. It is assumed that one prey species utilizes the same food resource as the predator, whereas the other prey species depends on a different food resource. The existence and uniqueness of the model are studied using the Lipschitz condition. The fixed points for the fractional-order model are sorted out, and the existence of the equilibrium points is discussed. The stability analysis of the model for the biologically important fixed points is provided. These include the coexistence fixed point and the prey-free (using the same food resources as the predator does) fixed point. A fractional-order scheme is implemented to support theoretical results for the stability of equilibrium points. The time series solution of the model is presented in the form of plots. Moreover, the impact of some mathematically and biologically important parameters is presented. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

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15 pages, 338 KiB

Open AccessArticle

The Existence, Uniqueness, and Multiplicity of Solutions for Two Fractional Nonlocal Equations

by Yue Wang, Wei Wei and Ying Zhou

Axioms 2023, 12(1), 45; https://doi.org/10.3390/axioms12010045 - 01 Jan 2023

Cited by 2 | Viewed by 1269

Abstract

This paper establishes the existence of unique and multiple solutions to two nonlocal equations with fractional operators. The main results are obtained using the variational method and algebraic analysis. The conclusion is that there exists a constant

λ^{*} > 0

such that [...] Read more.

This paper establishes the existence of unique and multiple solutions to two nonlocal equations with fractional operators. The main results are obtained using the variational method and algebraic analysis. The conclusion is that there exists a constant

λ^{*} > 0

such that the equations have only three, two, and one solution, respectively, for

λ \in (0, λ^{*})

,

λ = λ^{*}

, and

λ > λ^{*}

. The main conclusions fill the gap in the knowledge of this kind of fractional-order problem. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

18 pages, 363 KiB

Open AccessArticle

Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type

by Ravi P. Agarwal and Snezhana Hristova

Axioms 2022, 11(12), 742; https://doi.org/10.3390/axioms11120742 - 18 Dec 2022

Cited by 4 | Viewed by 1105

Abstract

A boundary-value problem for a couple of scalar nonlinear differential equations with a delay and several generalized proportional Caputo fractional derivatives is studied. Ulam-type stability of the given problem is investigated. Sufficient conditions for the existence of the boundary-value problem with an arbitrary [...] Read more.

A boundary-value problem for a couple of scalar nonlinear differential equations with a delay and several generalized proportional Caputo fractional derivatives is studied. Ulam-type stability of the given problem is investigated. Sufficient conditions for the existence of the boundary-value problem with an arbitrary parameter are obtained. In the study of Ulam-type stability, this parameter was chosen to depend on the solution of the corresponding fractional differential inequality. We provide sufficient conditions for Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability for the given problem on a finite interval. As a partial case, sufficient conditions for Ulam-type stability for a couple of multi-term delay, Caputo fractional differential equations are obtained. An example is illustrating the results. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

11 pages, 280 KiB

Open AccessArticle

Existence Solutions for Implicit Fractional Relaxation Differential Equations with Impulsive Delay Boundary Conditions

by Varaporn Wattanakejorn, Panjaiyan Karthikeyann, Sadhasivam Poornima, Kulandhaivel Karthikeyan and Thanin Sitthiwirattham

Axioms 2022, 11(11), 611; https://doi.org/10.3390/axioms11110611 - 02 Nov 2022

Cited by 6 | Viewed by 960

Abstract

The aim of this paper is to study the existence and uniqueness of solutions for nonlinear fractional relaxation impulsive implicit delay differential equations with boundary conditions. Some findings are established by applying the Banach contraction mapping principle and the Schauder fixed-point theorem. An [...] Read more.

The aim of this paper is to study the existence and uniqueness of solutions for nonlinear fractional relaxation impulsive implicit delay differential equations with boundary conditions. Some findings are established by applying the Banach contraction mapping principle and the Schauder fixed-point theorem. An example is provided that illustrates the theoretical results. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

21 pages, 361 KiB

Open AccessArticle

Some New Integral Inequalities Involving Fractional Operator with Applications to Probability Density Functions and Special Means

by Bibhakar Kodamasingh, Soubhagya Kumar Sahoo, Wajid Ali Shaikh, Kamsing Nonlaopon, Sotiris K. Ntouyas and Muhammad Tariq

Axioms 2022, 11(11), 602; https://doi.org/10.3390/axioms11110602 - 29 Oct 2022

Cited by 2 | Viewed by 1218

Abstract

Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its [...] Read more.

Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator refined exponential type convexity. Fractional versions of the Hermite–Hadamard type inequality employing generalized R–L fractional operators are established. Additionally, some novel refinements of Hermite–Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

11 pages, 348 KiB

Open AccessArticle

Training Neural Networks by Time-Fractional Gradient Descent

by Jingyi Xie and Sirui Li

Axioms 2022, 11(10), 507; https://doi.org/10.3390/axioms11100507 - 26 Sep 2022

Cited by 2 | Viewed by 1265

Abstract

Motivated by the weighted averaging method for training neural networks, we study the time-fractional gradient descent (TFGD) method based on the time-fractional gradient flow and explore the influence of memory dependence on neural network training. The TFGD algorithm in this paper is studied [...] Read more.

Motivated by the weighted averaging method for training neural networks, we study the time-fractional gradient descent (TFGD) method based on the time-fractional gradient flow and explore the influence of memory dependence on neural network training. The TFGD algorithm in this paper is studied via theoretical derivations and neural network training experiments. Compared with the common gradient descent (GD) algorithm, the optimization effect of the time-fractional gradient descent algorithm is significant when the value of fractional

α

is close to 1, under the condition of appropriate learning rate

η

. The comparison is extended to experiments on the MNIST dataset with various learning rates. It is verified that the TFGD has potential advantages when the fractional

α

nears 0.95∼0.99. This suggests that the memory dependence can improve training performance of neural networks. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

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13 pages, 1309 KiB

Open AccessArticle

Almost Periodic Solution for Forced Perturbed Non-Instantaneous Impulsive Model

by Rui Ma and Mengmeng Li

Axioms 2022, 11(10), 496; https://doi.org/10.3390/axioms11100496 - 23 Sep 2022

Viewed by 1033

Abstract

In this paper we investigate a forced perturbed non-instantaneous impulsive model. Firstly, we prove the existence and uniqueness of an almost periodic solution for the model considered by the Banach contraction principle. Secondly, we prove that all solutions converge exponentially to the almost [...] Read more.

In this paper we investigate a forced perturbed non-instantaneous impulsive model. Firstly, we prove the existence and uniqueness of an almost periodic solution for the model considered by the Banach contraction principle. Secondly, we prove that all solutions converge exponentially to the almost periodic solution. In other words, the solution of the model considered is exponentially stable. Finally, we provide some simulations to show the effectiveness of the theoretical results. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

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15 pages, 324 KiB

Open AccessArticle

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

by Xinxin Feng and Sirui Li

Axioms 2022, 11(10), 495; https://doi.org/10.3390/axioms11100495 - 23 Sep 2022

Viewed by 1109

Abstract

The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties [...] Read more.

The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties is an important issue. In this work, we derived a reduced Q-tensor hydrodynamic theory that described smectic phases with density variations from the Doi–Onsager molecular theory using the Bingham closure approximation. The coefficients in the tensor model were derived from those in the molecular model. The energy dissipation law was inherited from the tensor model. Some special cases for the model were also discussed. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

17 pages, 359 KiB

Open AccessArticle

On (k,ψ)-Hilfer Fractional Differential Equations and Inclusions with Mixed (k,ψ)-Derivative and Integral Boundary Conditions

by Sotiris K. Ntouyas, Bashir Ahmad, Cholticha Nuchpong and Jessada Tariboon

Axioms 2022, 11(8), 403; https://doi.org/10.3390/axioms11080403 - 15 Aug 2022

Cited by 5 | Viewed by 1324

Abstract

In this paper we study single-valued and multi-valued (k,ψ)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k,ψ)-Hilfer-type derivative and integral operators. The results for single-valued case are established [...] Read more.

In this paper we study single-valued and multi-valued (k,ψ)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k,ψ)-Hilfer-type derivative and integral operators. The results for single-valued case are established by using Banach and Krasnosel’skiĭ fixed point theorems as well as Leray–Schauder nonlinear alternative. In the multi-valued case, we establish an existence result for the convex valued right-hand side of the inclusion via Leray–Schauder nonlinear alternative for multi-valued maps, while the second one when the right-hand side has non-convex values is obtained by applying Covitz–Nadler fixed point theorem for multi-valued contractions. Numerical examples illustrating the obtained theoretical results are also presented. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

14 pages, 318 KiB

Open AccessArticle

The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System

by Wenlin Zhang, Michal Fečkan and Jinrong Wang

Axioms 2022, 11(7), 347; https://doi.org/10.3390/axioms11070347 - 20 Jul 2022

Viewed by 1077

Abstract

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate [...] Read more.

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

19 pages, 309 KiB

Open AccessArticle

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

by Qing Yang, Chuanzhi Bai and Dandan Yang

Axioms 2022, 11(6), 283; https://doi.org/10.3390/axioms11060283 - 10 Jun 2022

Cited by 2 | Viewed by 1265

Abstract

In this paper, we investigate the controllability of a class of impulsive

ψ

-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are [...] Read more.

In this paper, we investigate the controllability of a class of impulsive

ψ

-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are generalizations and continuations of the recent results about controllability of a class of impulsive

ψ

-Caputo fractional evolution equations. Finally, an example is given to illustrate the effectiveness of the main results. Full article

(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)

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