Theory Of Differential Inequalities Research Articles

Singular perturbation boundary and interior layers problems with multiple turning points

In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities.

A Simple Linear Controller Design to Ensure Global Exponential Stability for a Class of Uncertain Fifth-Order Nonlinear Systems

This paper intends to explore the controller design problem for a class of uncertain fifth-order nonlinear control systems. Combining the theory of differential and integral inequalities, a linear controller will be proposed to promote a class of nonlinear control systems with multiple uncertainties to achieve the goal of global exponential stability. In addition, the guaranteed exponential convergence rate of such uncertain nonlinear systems will also be precisely calculated. Finally, some numerical simulation results will also be presented to verify and illustrate the correctness of this main theorem and the design process of the controller

Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

It is well known that many natural phenomena and human activities do exhibit impulsive effects in the fields of epidemiology. At the same time, compared with a single control strategy, it is obvious that the multiple control strategies are more beneficial to restrain the spread of infectious diseases. In this paper, we consider pulse vaccination and pulse elimination strategies at the same time and establish an SIRS epidemic model with standard incidence. Firstly, according to the stroboscopic mapping method of the discrete dynamical system, the disease-free T periodic solution of the model under the condition of pulse vaccination and pulse elimination is obtained. Secondly, the basic reproductive number R0 is defined, and the local asymptotic stability of the disease-free T periodic solution is proved by Floquet theory for R0<1. Finally, based on the impulsive differential inequality theory, the global asymptotic stability of the disease-free T periodic solution is given for R0<1, and the disease dies out eventually. The results show that in order to stop the disease epidemic, it is necessary to choose the appropriate vaccination rate and elimination rate and the appropriate impulsive period.

Merdan-type Allee Effect on a Lotka-Volterra Commensal Symbiosis Model with Density-dependent Birth Rate

A Lotka-Volterra commensal symbiosis model with a density dependent birth rate and a Merdan-type Allee effect on the second species has been proposed and examined. The global attractivity of system’s equilibria is ensured by using the differential inequality theory. Our results show that the Allee effect has no effect on the existence or stability of the system’s equilibrium point. However, both species take longer to approach extinction or a stable equilibrium state as the Allee effect increases.

Geometric Properties of a Linear Complex Operator on a Subclass of Meromorphic Functions: An Analysis of Hurwitz-Lerch-Zeta Functions

Abstract Geometric function theory (GFT) is one of the richest research disciplines in complex analysis. This discipline also deals with the extended differential inequality theory, known as the differential subordination theory. Based on these theories, this study focuses on analyzing intriguing aspects of the geometric subclass of meromorphic functions in terms of a linear complex operator and a special class of Hurwitz-Lerch-Zeta functions. Hence, several of its geometric attributes are deduced. Furthermore, the paper highlights the different fascinating advantages and applications of various new geometric subclasses in relation to the subordination and inclusion theorems.

The Permanence of a Nonautonomous Single-species Model with Stage- Structure and Feedback Control

A nonautonomous single-species model with stage structure and feedback control is revisited in this paper. By applying the differential inequality theory, a set of delay-dependent conditions ensures the permanence of the system is obtained; Next, by further developing the analytical technique of Chen et al, we prove that the system is always permanent. Numeric simulation supports our findings. Also, the numeric simulation shows that the feedback control variable harms the final density of the species, and this may increase the chance of the extinction of the species. Our results supplement and complement some known results.

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.

Global Attractivity of Symbiotic Model of Commensalism in Four Populations with Michaelis–Menten Type Harvesting in the First Commensal Populations

This article revisits the stability property of a symbiotic model of commensalism with Michaelis–Menten type harvesting in the first commensal populations. By constructing some suitable Lyapunov functions, we provide a thorough analysis of the dynamic behaviors of the subsystem composed of the second and third species. After that, by applying the stability results of this subsystem and combining with the differential inequality theory, sufficient conditions which ensure the global attractivity of the equilibria are obtained. The results obtained here essentially improve and generalize some known results.

Leader-following quasi-bipartite synchronization of coupled heterogeneous harmonic oscillators via event-triggered control

This paper investigates the leader-following quasi-bipartite synchronization problem of coupled heterogeneous harmonic oscillators under an undirected communication topology. To save limited resources and alleviate communication burden, two asynchronous event-triggered protocols are proposed. Some sufficient criteria are derived to ensure quasi-bipartite synchronization by utilizing differential equation theory, inequality theory and Lyapunov stability method. Meanwhile, it is shown that there is no Zeno behavior for the proposed event-triggered protocols. Furthermore, an upper bound of quasi-bipartite synchronization errors is explicitly provided. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results.

Stability conditions for some classes of time-varying switched systems

In the paper, some classes of time-varying nonlinear switched systems are considered. The stability problem is investigated. The Lyapunov direct method and the theory of differential inequalities are used. The non-stationary switched comparison equation with hybrid degrees is constructed for the considered system. Switching in the comparison equation can be caused by changing the operating modes, as well as changing non-stationary parameters of the system. Sufficient conditions of the asymptotic and finite-time stability are obtained. The region of attraction for the given equilibrium position is estimated. Some examples are presented to illustrate the results. In particular, some classes of nonlinear mechanical system are studied.

Quadratic Singular Perturbation Problems With Nonmonotone Transition Layer Properties

In this article, we consider the quadratic singular perturbation problems with Nonmonotone Transition Layer Properties. Under certain conditions, solutions are shown to exhibit nonmonotone transition layer behavior at turning point t=0. The formal approximation of problems is constructed using composite expansions, and then approximation solutions of left and right sides at t=0 are joined by joint method which exhibits spike layer behavior and boundary layer behavior respectively. As a result, an approximate solution is formed which exhibits nonmonotone transition layer behavior. In addition, the existence and asymptotic behavior of solutions are proved by the theory of differential inequalities.

Stability analysis for a class of mechanical systems with piecewise constant coefficients

A nonlinear mechanical system described by a vector equation of the Lienard type is considered. It is assumed that the forces acting on the mechanical system are given by functions that are piecewise constant with respect to time. With the aid of a special variable substitution, the equation is reduced to a system that can be considered as an impulsive switched system with infinite numbers of operating modes. The stability problem for the trivial equilibrium position of the obtained system is studied. The stability analysis is carried out using a specially constructed discontinuous Lyapunov function. This function can be understood as a multiple Lyapunov function, consisting of an infinite number of partial Lyapunov functions, each of which is used for a certain time interval. Differentiating the chosen Lyapunov function with respect to solutions of the given system on the corresponding time intervals and applying the theory of differential inequalities, sufficient conditions of the asymptotic stability of the equilibrium position are established.

Synchronization Analysis of a Class of Neural Networks with Multiple Time Delays

In this paper, we study the synchronization of a new fractional-order neural network with multiple delays. Based on the control theory of linear systems with multiple delays, we get the controller to analyse the synchronization of the system. In addition, a suitable Lyapunov function is constructed by using the theory of delay differential inequality, and some criteria ensuring the synchronization of delay fractional neural networks with Caputo derivatives are obtained. Finally, the accuracy of the method is verified by a numerical example.

New results on dynamics of neutral type HCNNs with proportional delays

We deal with the problem of the convergence of HCNNs (High-order Cellular Neural Networks) involving neutral type time-proportional delays and D operators in this paper. By combining Lyapunov function approach and differential inequality theory, we establish some novel assertions to check the global exponential convergence for the proposed model, which lay the foundation to devise a stable HCNNs and extend some known relevant results. Meanwhile, the outcomes of numerical simulations are highly consistent with the theoretical results.

Global Exponential Convergence of Neutral Type Competitive Neural Networks with D Operator and Mixed Delay

The models of competitive neural network (CNN) was in recent past proposed to describe the dynamics of cortical cognitive maps with unsupervised synaptic modifications, where there are two types of memories: Long-term memories (LTM) and short-term memories (STM), LTM presents unsupervised and slow synaptic modifications and STM characterize the fast neural activity. This paper is concerned with a class of neutral type CNN’s with mixed delay and D operator. By employing the appropriate differential inequality theory, some sufficient conditions are given to ensure that all solutions of the model converge exponentially to zero vector. Finally, an illustrative example is also given at the end of this paper to show the effectiveness of the proposed results.

New criteria for finite-time stability of fractional order memristor-based neural networks with time delays

In this paper, the finite-time stability (FTS) problems for a class of fractional order memristor-based neural network (FOMNN) with time delays are investigated. Based on the method of steps, set-valued mapping, the theory of differential inclusion and fractional order Gronwall inequality, a new delay dependent criterion for the FTS of FOMNN with time delays and the fractional order 0<μ<1 is derived, which is less conservative than the existing criteria. Furthermore, a sufficient condition to ensure the FTS of FOMNN with the fractional-order 1<μ<2 is proposed. Finally, two examples are given to illustrate the validity of the proposed results.

Impact of Michaelis\u2013Menten type harvesting in a Lotka\u2013Volterra predator\u2013prey system incorporating fear effect

We propose and study a Lotka–Volterra predator–prey system incorporating both Michaelis–Menten-type prey harvesting and fear effect. By qualitative analysis of the eigenvalues of the Jacobian matrix we study the stability of equilibrium states. By applying the differential inequality theory we obtain sufficient conditions that ensure the global attractivity of the trivial equilibrium. By applying Dulac criterion we obtain sufficient conditions that ensure the global asymptotic stability of the positive equilibrium. Our study indicates that the catchability coefficient plays a crucial role on the dynamic behavior of the system; for example, the catchability coefficient is the Hopf bifurcation parameter. Furthermore, for our model in which harvesting is of Michaelis–Menten type, the catchability coefficient is within a certain range; increasing the capture rate does not change the final number of prey population, but reduces the predator population. Meanwhile, the fear effect of the prey species has no influence on the dynamic behavior of the system, but it can affect the time when the number of prey species reaches stability. Numeric simulations support our findings.

Analysis for a Delayed Three-Species Predator-Prey Model with Feedback Controls and Prey Diffusion

In this paper, we study a class of 3-species multidelay Lotka–Volterra ratio-dependent predator-prey model with feedback controls and prey diffusion. By using the theory of delay differential inequalities and developing some new analysis methods as well as constructing a suitable Lyapunov function, some sufficient conditions are obtained to guarantee the permanence of the system and the global attractivity of the positive solution for the predator-prey system. Furthermore, the corresponding periodic system is discussed, and some conditions are established about the existence, uniqueness, and stability of the positive periodic solution for the periodic system by using the fixed-point theory and theoretical analysis. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria. Finally, the corresponding stochastic time-delay predator-prey model with multiplicative noise sources is solved numerically, and some interesting new dynamics behavior is obtained.

New model of May cooperative system with strong and weak cooperative partners

In this paper, based on the model of Zhao, Qin, and Chen [Adv. Differ. Equ. 2018:172, 2018], we propose a new model of the May cooperative system with strong and weak cooperative partners. The model overcomes the drawback of the corresponding model of Zhao, Qin, and Chen. By using the differential inequality theory, a set of sufficient conditions that ensure the permanence of the system are obtained. By combining the differential inequality theory and the iterative method, a set of sufficient conditions that ensure the extinction of the weak partners and the attractivity of the strong partners and the other species is obtained. Numeric simulations show that too large transform rate will lead to more complicated fluctuation; however, the system is still permanent.

Note on the persistence and stability property of a stage-structured prey\u2013predator model with cannibalism and constant attacking rate

We revisit the persistence and stability property of a stage-structured prey–predator model with cannibalism and constant attacking rate. By using the differential inequality theory and Bendixson–Dulac criterion, we show that if the system without cannibalism is permanent, then the system with cannibalism is also permanent. By developing some new analysis technique, we obtain a new set of sufficient conditions which ensure the global asymptotic stability of the nonnegative equilibrium, which means that, under some suitable assumption, prey cannibalism has no influence on the stability property of the predator free equilibrium. Our results essentially improve the corresponding results of Limin Zhang and Chaofeng Zhang.