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In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness of solutions. We further employ the nonlinear analysis to reveal that this problem is Ulam–Hyers (UH) stable. Finally, some examples and numerical simulations are presented to illustrate the reliability and validity of our main results.

The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for differential equations and have wide applications. Therefore, this article considers a class of nonlinear Hadamard fractional coupling (p1,p2)-Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam–Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results.

Abstract The classical p $\mathcalligra{p}$ -Laplace equation is one of the special and significant second-order ODEs. The fractional-order p $\mathcalligra{p}$ -Laplace ODE is an important generalization. In this paper, we mainly treat with a nonlinear coupling ( p 1 , p 2 ) $(\mathcalligra{p}_{1},\mathcalligra{p}_{2})$ -Laplacian systems involving the nonsingular Atangana–Baleanu (AB) fractional derivative. In accordance with the value range of parameters p 1 $\mathcalligra{p}_{1}$ and p 2 $\mathcalligra{p}_{2}$ , we obtain sufficient criteria for the existence and uniqueness of solution in four cases. By using some inequality techniques we further establish the generalized UH-stability for this system. Finally, we test the validity and practicality of the main results by an example.

Prostate cancer is a serious disease that endangers men's health. The genetic mechanism and treatment of prostate cancer have attracted the attention of scientists. In this paper, we focus on the nonlinear mixed reaction diffusion dynamics model of neuroendocrine transdifferentiation of prostate cancer cells with time delays, and reveal the evolutionary mechanism of cancer cells mathematically. By applying operator semigroup theory and the comparison principle of parabolic equation, we study the global existence, uniqueness and boundedness of the positive solution for the model. Additionally, the global invariant set and compact attractor of the positive solution are obtained by Kuratowski's measure of noncompactness. Finally, we use the Pdepe toolbox of MATLAB to carry out numerical calculations and simulations on an example to check the correctness and effectiveness of our main results. Our results show that the delay has no effect on the existence, uniqueness, boundedness and invariant set of the solution, but will affect the attractor.

In this paper, we mainly take into account a nonlinear fractional coupled Laplacian equations with nonsingular exponential kernel. After discussing the Laplacian parameters in four cases, some new and easily verifiable sufficient criteria of solvability are obtained. We further prove that this system is generalized Ulam-Hyers (GUH) stable. Finally, an example is applied to explain the availability of our major results.

In this paper, we build a novel nonlinear diffusion online game addiction model with unsustainable control. The existence and boundedness of a solution are investigated by a $ C_0 $-semigroup and differential inclusion. Simultaneously, we study the global asymptotic stability of steady states of the model. Finally, a concrete example is theoretically analyzed and numerically simulated.

The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value conditions. To unify the two types of equations, we investigate a general nonlinear impulsive coupled implicit system. By cleverly constructing relevant operators involving impulsive terms, we establish the coincidence degree theory and obtain the solvability. We explore the stability of solutions using nonlinear analysis and inequality techniques. As the most direct application, we naturally obtained the solvability and stability of the Langevin and Sturm–Liouville equations mentioned above. Finally, an example is provided to demonstrate the validity and availability of our major findings.

Hadamard fractional calculus is one of the most important fractional calculus theories. Compared with a single Hadamard fractional order equation, Hadamard fractional differential equations have a more complex structure and a wide range of applications. It is difficult and challenging to study the dynamic behavior of Hadamard fractional differential equations. This manuscript mainly deals with the boundary value problem (BVP) of a nonlinear coupled Hadamard fractional system involving fractional derivative impulses. By applying nonlinear alternative of Leray-Schauder, we find some new conditions for the existence of solutions to this nonlinear coupled Hadamard fractional system. Our findings reveal that the impulsive function and its impulsive point have a great influence on the existence of the solution. As an application, we discuss an interesting example to verify the correctness and validity of our results.

This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.

In this article, we firstly establish a nonlinear population dynamical model to describe the changes and interaction of the density of patient population of China's primary medical institutions (PHCIs) and hospitals in China's medical system. Next we get some sufficient conditions of existence of positive singularity by utilising homotopy invariance theorem of topological degree. Meanwhile, we study the qualitative properties of positive singularity based on Perron's first theorem. Furthermore, we briefly analyze the significance and function of the mathematical results obtained in this paper in practical application. As verifications, some numerical examples are ultimately exploited the correctness of our main results. Combined with the numerical simulation results and practical application, we give some corresponding suggestions. Our research can provide a certain theoretical basis for government departments to formulate relevant policies.

The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space.

The Ayala-Gilpin (AG) kinetics system is one of the famous mathematical models of ecosystem. This model has been widely concerned and studied since it was proposed. This paper stresses on a nonlinear distributed delayed periodic AG-ecosystem with competition on time scales. In the sense of time scale, our model unifies and generalizes the discrete and continuous cases. Firstly, with the aid of the auxiliary function having only two zeros in the real number field, we apply inequality technique and coincidence degree theory to obtain some sufficient criteria which ensure that this model has periodic solutions on time scales. Meanwhile, the global asymptotic stability of the periodic solution is founded by employing stability theory in the sense of Lyapunov. Eventually, we provide an illustrative example and conduct numerical simulation by means of MATLAB tools.

Abstract In this paper, we mainly study a kind of fractional-order multiple point boundary value problem involving noninstantaneous impulse and abstract bounded operator. The existence and uniqueness is obtained by the Banach contraction principle. And by applying direct analysis methods, we establish some conditions of the Ulam–Hyers stability for this problem. Finally, an interesting application example is given to illustrate the validity of the results.

The number of patients in a hospital is a direct indicator of patients' choice of hospital, which is a complex process affected by many factors. Based on the national medical system and patients' preference for high-grade hospitals in China, this study establishes a three-dimensional differential equation model for calculating the time variation of the number of visits to three grades of hospitals. We performed a qualitative analysis of the system. We carried out a subsequent numerical simulation to analyze the impact on the system when the rate of leapfrog treatment and the maximum capacity of doctors and treatments changed. The results show that the sustainability of China's three levels of hospitals mainly depends on the level of hospital development. The strength of comprehensive health improvement at specific levels is the key to increasing the service efficiency of medical resources.

Fractional Langevin system has great advantages in describing the random motion of Brownian particles in complex viscous fluid. This manuscript deals with a delayed nonlinear fractional Langevin system with nonsingular exponential kernel. Based on the fixed point theory, some sufficient criteria for the existence and uniqueness of solution are established. We also prove that this system is UH- and UHR-stable attributed to the nonlinear analysis and inequality techniques. As applications, we provide some examples and simulations to illustrate the availability of main findings.

Based on the second derivative, this paper directly establishes the coincidence degree theory of a nonlinear periodic Sturm–Liouville (SL) system. As applications, we study the existence of periodic solutions to the S–L system with some special nonlinear functions by applying Mawhin’s continuation theorem. Some examples and simulations are furnished to inspect the correctness and availability of the chief findings.

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or Riemann–Liouville (RL) derivatives, the system with Mittag–Leffler (ML)-type fractional derivatives can eliminate singularity such that the solution of the system has better analytical properties. Therefore, we concentrate on a nonlinear Langevin system of ML-type fractional derivatives affected by time-varying delays and differential feedback control in the manuscript. We first utilize two fixed-point theorems proposed by Krasnoselskii and Schauder to investigate the existence of a solution. Next, we employ the contraction mapping principle and nonlinear analysis to establish the stability of types such as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) as well as generalized UH and UHR. Lastly, the theoretical analysis and numerical simulation of some interesting examples are carried out by using our main results and the DDESD toolbox of MATLAB.

We establish a non-linear diffusion partial differential equation (PDE) model to depict the dynamic mechanism of Internet gaming disorder (IGD). By constructing appropriate super- and sub-solutions and applying Schauder’s fixed point theorem and continuation method, we study the existence and asymptotic stability of traveling wave solutions to probe into the oscillating behavior of IGD. An example is numerically simulated to examine the correctness of our outcomes.

Abstract We deal with a class of coupled systems of neutral fractional integro-differential equations with infinite delay in a Banach space in this paper. Based on the Banach contraction principle and Krasnoselskii’s fixed point theorem, some new sufficient criteria are established to guarantee the existence of mild solutions. As an application, an example is given to illustrate the effectiveness of our main results.

Abstract This paper considers the boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses. Some sufficient conditions of the existence and uniqueness are obtained by means of the Banach contraction principle and Leray–Schauder alternative. We also give some interesting examples to illustrate the effectiveness of our main results.

Abstract In this paper, we study the nonlocal boundary value problem for a nonlinear fractional differential coupled system with fractional order impulses. Applying Nonlinear Alternative of Leray–Schauder, we obtain some new existence results for this system. As application, an interesting example is given to illustrate the effectiveness of our main result.

The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag–Leffler (ML) kernel. We first investigate the existence and uniqueness of the solution by employing some fixed-point theorems. Then, we apply direct analysis to obtain the Ulam–Hyers (UH) type stability. Finally, the theoretical analysis and numerical simulation of some interesting examples show that there is a great difference between the fractional Langevin equation and integer Langevin equation in describing the random motion of free particles.

The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense.

Abstract In this paper, we consider a class of impulsive Lotka–Volterra cooperative population models with time delay and harvesting control on time scales. Using the fixed point theorem of strict-set-contraction, we analyze the existence conditions of positive periodic solutions for this model. As applications, we analyze the existence conditions of positive periodic solutions for some common Lotka–Volterra systems on time scales.

Abstract This paper mainly considers a class of two-layer Gilpin–Ayala predator–prey models with time delays. By means of Mawhin’s continuation theorem of coincidence degree theory, some new sufficient criteria for the existence of positive almost periodic solutions have been established. We also obtain the global exponential stability of the positive almost periodic solution for this system by constructing appropriate Lyapunov functionals and smart transformations. As an application, an example is given to illustrate the validity of our main results.

Abstract This paper is considered with a class of nonlinear fractional differential coupled system with fractional differential boundary value conditions and impulses. By means of the Banach contraction principle and the Schauder fixed point theorem, some sufficient criteria are established to guarantee the existence of solutions. As applications, some interesting examples are given to illustrate the effectiveness of our main results.

In this paper, we study the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays. The existence of positive periodic solution is proved by employing the fixed point theorem on cones. By constructing appropriate Lyapunov functional, we also obtain the global exponential stability of the positive periodic solution of this system. As an application, an interesting example is provided to illustrate the validity of our main results.

The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results.

Abstract This paper is concerned with the Lotka-Volterra-like functional differential equations with impulses and infinite distributed time delays on time scales. By applying a fixed point theorem of strict-set contractions, some sufficient conditions have been established to guarantee the existence of periodic solutions. As applications, the existence of positive periodic solutions is given for some common Lotka-Volterra systems on time scales.

Abstract In this paper, we deal with a class of Gilpin-Ayala ecological models with discrete and distributed time delays. By employing a fixed point theorem of strict-set-contraction and inequality techniques, some sufficient conditions for the existence of periodic solutions are established. As an application, one example is given to illustrate the validity of our main results.

Abstract In this paper, we consider a two-dimensional predator-prey model with a time delay and square root response function. We analyze the stability of equilibria with the delay τ increasing and the critical value of τ when Hopf bifurcation occurs. Because the model has the term of square root, the zero point is a singularity. In order to clearly study the stability of the zero point, we rescale the variable x ( t ) $x(t)$ , say x ( t ) = X 2 ( t ) $x(t)=X^{2}(t)$ . The conclusion is that the zero point is not stable and the instability is not affected by the delay τ. We apply the normal form method and center manifold theorem to obtain the direction and stability of the Hopf bifurcation. Finally, we make several numerical simulations which is consistent with the conclusion of theoretical analysis.

In this article, we study a Lotka-Volterra cooperative system of equations with time-varying delays and distributed delays. By using Mawhin's continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive almost periodic solutions. Also we present an example to illustrate our results.

Using Mawhin's continuation theorem of coincidence degree theory, we establish the existence of $2^{n+m}$ positive periodic solutions for a non-autonomous Lotka-Volterra network-like predator-prey system with harvesting terms. Here n and m denote the number of prey and predator species respectively. An example is given to illustrate our results.

By using Mawhin's continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive almost periodic solutions for a non-autonomous delay competitive system with weak Allee effect.

A class of fuzzy neural networks (FNNs) with time-varying delays and impulses is investigated. With removing some restrictions on the amplification functions, a new differential inequality is established, which improves previouse criteria. Applying this differential inequality, a series of new and useful criteria are obtained to ensure the existence of global robust attracting and invariant sets for FNNs with time-varying delays and impulses. Our main results allow much broader application for fuzzy and impulsive neural networks with or without delays. An example is given to illustrate the effectiveness of our results.

This paper considers the existence of positive almost-periodic solutions for almost-periodic Lotka-Volterra cooperative system with time delay. By using Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost-periodic solutions are obtained. An example and its simulation figure are given to illustrate the effectiveness of our results.

By using the Mawhin continuation theorem of coincidence degree theory and some results on inequalities, we establish the existence of 2 n positive periodic solutions for n species non‐autonomous Lotka‐Volterra unidirectional food chains with harvesting terms. Two examples are given to illustrate the effectiveness of our results. First published online: 09 Jun 2011

The existence of equilibrium solutions to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales is proved by the topological degree theory and M-matrix method. Under some sufficient conditions, we obtain the uniqueness and global exponential stability of equilibrium solution to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales by constructing suitable Lyapunov functional and inequality skills. One example is given to illustrate the effectiveness of our results.

By using Mawhin's continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of at least 2n positive periodic solutions for n-species nonautonomous Lotka-Volterra type food chains with harvesting terms. An example is given to illustrate the effectiveness of our results.