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A recent paper Xu et al. (2018) studied the global stability and traveling wave solution of a vaccination model with nonlinear incidence. Furthermore, in this paper, we first study the local stability of this model, and then establish the uniform persistence result. Our results are a supplement to above paper. [ABSTRACT FROM AUTHOR]

Abstract: Recently, the micropositioner has become an important developing target for achieving the requirements of precision machinery. The piezo-actuating device plays a very important role in this application area. In this paper, a model-free adaptive sliding-mode controller is proposed for a 3D piezo-actuating system because of the system’s hysteresis nonlinearity and time-varying characteristics. This control strategy employs the functional approximation technique to establish the unknown function for releasing the model based requirements of the sliding-mode control. The update laws for the coefficients of the Fourier series function parameters are derived from a Lyapunov function to guarantee the control system stability. To verify the effectiveness of the proposed controller, drilling process control using the designed controller is investigated in this paper. [Copyright &y& Elsevier]

In this paper, we propose and investigate a reaction–diffusion virus infection model with humoral immunity and nonlinear incidence. In spatially heterogeneous case, the basic reproduction number of virus infection R 0 is calculated, when R 0 ≤ 1 the global asymptotical stability of the infection-free steady state is established, and when R 0 > 1 the uniform persistence of infected cells and viruses, as well as the existence of antibody-free infection steady state are also obtained. In spatially homogeneous case, the antibody response basic reproduction number R 1 is calculated, by using the Lyapunov functions method and the persistence theory of dynamical systems we obtain that when R 0 > 1 and R 1 ≤ 1 the antibody-free infection equilibrium is globally asymptotically stable, and when R 0 > 1 and R 1 > 1 the model is uniformly persistent and the infection equilibrium exists and is also globally asymptotically stable. Finally, the numerical examples are presented in order to verify the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we revisit a diffusive virus dynamics model with general incidence function and time delay. One novelty of our model is that we introduce cell-to-cell transmission via formation of virological synapses to reflect the fact that it may play a more important role in virus spreading in addition to virus-to-cell infection. We justify the well-posedness of the model and identify the basic reproduction number ℜ 0 for the model to be a sharp threshold value. The global stability of equilibria is determined by constructing suitable Lyapunov functionals in the sense that: the infection-free equilibrium is globally asymptotically stable if ℜ 0 ≤ 1 , and when ℜ 0 > 1 , the global asymptotic stability of infection equilibrium implies that the infection will persist. A significant impact of the cell-to-cell transmission is that they increase the basic reproduction number. If one neglects either the cell-to-cell transmission or virus-to-cell infection, the basic reproduction number of the model that is under-evaluated. Last, we perform numerical simulation to support our theoretic results. We set the domain of the viruses to be a two-dimensional square domain with the homogeneous Neumann boundary conditions to reflect the spatial spreading. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the wave equation of variable coefficients with Balakrishnan–Taylor damping, time-varying delay and source term. By establishing some Lyapunov functionals which are equivalent to the energy of the system and using the potential well method to deal with the source term, we prove a general stability result. [ABSTRACT FROM AUTHOR]

Abstract: The approximate solution of a system for variational inequality with different mapping in Hilbert spaces has been studied, based on the convergence of projection methods. However, little research has been done in Banach space. The primary reason is that projection mapping lacks preferable properties in Banach space. In this paper, we introduce generalized projection methods. By using these methods, the results presented in this paper have extended the main results of Chang et al.’s and Huang and Noor’s recent works [S.S. Chang, H.W. Joseph Lee, C.K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (2007) 329–334; Zhenyu Huang, Muhammad Aslam Noor, An explicit projection method for a system of nonlinear variational inequalities with different -cocoercive mappings, Appl. Math. Comput. 190 (2007) 356–361] from Hilbert spaces to Banach space. [Copyright &y& Elsevier]

Abstract: The discrete delayed Hopfield neural networks is an extension of the discrete Hopfield neural networks. In this paper, the convergence of discrete delayed Hopfield neural networks is mainly studied, and some results on the convergence are obtained by using Lyapunov function. Several new sufficient conditions for the delayed networks converging towards a limit cycle with period at most 2 are proved in parallel updating mode. Also, some conditions for the delayed networks converging towards a limit cycle with 2-period are investigated in parallel updating mode. All results established in this paper extend the previous results on the convergence of both the discrete Hopfield neural networks, and the discrete delayed Hopfield neural networks in parallel updating mode. [Copyright &y& Elsevier]

Abstract: This paper proposes a method for designing an state-feedback fuzzy controller for discrete-time Takagi–Sugeno (T–S) fuzzy systems with input saturation. To address the input saturation problem, this paper first formulates a set invariance condition for the T–S fuzzy systems. Then, based on the set invariance condition, this paper establishes stabilization conditions associated with a fuzzy weighting-dependent Lyapunov function, where the fuzzy controller is designed to be dependent on not only the current-time but also the one-step-past information on the time-varying fuzzy weighting functions. In the derivation, the stabilization conditions are first formulated in terms of parameterized linear matrix inequalities (PLMIs), and then reconverted into LMIs by an efficient relaxation technique. [Copyright &y& Elsevier]

In this paper, we consider a coupled Lamé system with a viscoelastic term and a strong damping. We prove well posedness by using Faedo–Galerkin method and establish an exponential decay result by introducing a suitable Lyapunov functional. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the finite iterative algorithm to solve a class of generalized coupled Sylvester-conjugate matrix equations. We prove that if the system is consistent, an exact generalized Hamiltonian solution can be obtained within finite iterative steps in the absence of round-off errors for any initial matrices; if the system is inconsistent, the least squares generalized Hamiltonian solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the minimum norm least squares generalized Hamiltonian solution of the system. Finally, numerical examples are presented to demonstrate the algorithm is efficient. [ABSTRACT FROM AUTHOR]

A diffusive and delayed viral dynamics model which incorporates cell-to-cell transmission, cell-mediated immune responses and general nonlinear incidence is investigated. By constructing Lyapunov functionals, it is shown that the global threshold dynamics of the model is fully determined by the basic reproduction numbers of the virus and the immune response, R 0 and R 1 . The result implies that both the basic reproduction number of the virus and the immune response might be underevaluated without considering either the virus-to-cell transmission or cell-to-cell transmission. The main results obtained in this paper extend some known results. [ABSTRACT FROM AUTHOR]

This paper considers the event-driven observer-based control for distributed parameter systems using mobile sensor and actuator. An observer is designed to estimate the states of the distributed parameter systems based on the measurement information provided by the mobile sensor. In order to reduce the frequency of the signal transmissions between the observer and the controller, an event-driven scheme is introduced. Once an event is generated, the event detector will send the newest observer state to the controller. Meanwhile, a guidance scheme is provided to drive the actuator to the position where the observer state reaches the maximum value to improve the control performance. For the event-driven control system, the global uniform ultimate boundedness can be guaranteed by the Lyapunov functional approach despite the reduced sampling frequency. A numerical example is finally presented to illustrate the effectiveness and the advantages of the proposed approaches. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions. Then, we consider the discretization of the model by using nonstandard finite difference scheme. It is then followed by the investigation of the global stability of the equilibria of the discrete model. Our study shows that if the basic reproduction number R 0 ≤ 1 , then the infection-free equilibrium is globally asymptotically stable; however if R 0 > 1 , then the infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a class of diffusive Lotka–Volterra equation with time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady state solution is investigated by applying Lyapunov–Schmidt reduction. The stability and nonexistence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the changes of a specific parameter are obtained by analyzing the distribution of the eigenvalues. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain. [ABSTRACT FROM AUTHOR]

In this paper, we study a diffusive and delayed virus dynamics model with Beddington–DeAngelis incidence and CTL immune response. By constructing Lyapunov functionals, we show that if the basic reproductive number is less than or equal to one, then the infection-free equilibrium is globally asymptotically stable; if the immune reproductive number is less than or equal to one and the basic reproductive number is greater than one, then the immune-free equilibrium is globally asymptotically stable; if the immune reproductive number is greater than one, then the interior equilibrium is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weak viscoelastic wave equation u t t − Δ u + δ Δ u t − σ ( t ) ∫ 0 t g ( t − s ) Δ u ( s ) d s = | u | p − 2 u with dynamic boundary conditions, and nonlinear delay term. First, we prove a local existence theorem by using the Faedo–Galerkin approximations combined with a contraction mapping theorem. Secondly, we show that, under suitable conditions on the initial data and the relaxation function, the solution exists globally in time, in using the concept of stable sets. Finally, by exploiting the perturbed Lyapunov functionals, we extend and improve the previous result from Gerbi and Said-Houari (2011). [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problem for a class of semilinear wave equations with damping and source terms. Under certain conditions we establish the global existence of solutions, and the asymptotic behavior of the solutions by introducing an appropriate Lyapunov functions. Moreover, we examine the blow-up in finite time when the initial data is sufficiently large. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic stability of the viscoelastic Kirchhoff equation with Balakrishnan–Taylor damping and a delay term. We establish general energy decay result by suitable Lyapunov functionals. [ABSTRACT FROM AUTHOR]

The vaccination period (i.e.,  the time period retaining acquired immunity following vaccination) and the latent period are two important factors affecting disease dynamics. Classical compartmental disease models unrealistically assume that these periods are exponentially distributed. In this paper, these two ages are assumed to have arbitrary distributions that are represented by age-specific rates leaving the vaccinated and the exposed classes, resulting in an SVEIR model with population dynamics. We investigate the global behavior of this model, and derive its basic reproduction number. The basic reproduction number completely determines the global dynamics of the model, i.e., the disease-free equilibrium is globally asymptotically stable and the disease always dies out when the basic reproduction number is less than unity; whereas when the basic reproduction number is larger than unity, there exists a unique endemic equilibrium which is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists. The contributions of the vaccine-wane rate to the basic reproduction number and the level of the endemic equilibrium are displayed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a three species food-chain model with diffusion. By means of analyzing eigenvalue spectrum and Lyapunov functional, we show that the positive equilibrium is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation. The instability of the Hopf bifurcation leads to the emergence of spatial patterns. Numerical calculations are performed to illustrate our theoretical results. [Copyright &y& Elsevier]

Abstract: This paper studies the stability problem for fractional order switching systems. The conditions for the stability of such systems are presented in terms of common Lyapunov theory, generalized to fractional order systems, and in frequency domain, an approach equivalent to the previous one. The effectiveness of the developed theory is shown through some illustrative examples. [Copyright &y& Elsevier]

Abstract: In this paper the variational Lyapunov method is developed for Caputo fractional differential equations. Further, the comparison theorems are proved with a relaxed hypothesis: the assumption of local Holder continuity is relaxed to continuity of the functions involved in the Riemann–Liouville fractional differential equations. In this process the Grünwald–Letnikov derivative is used to define Dini derivatives. Also, a relation between ordinary and fractional differential equations is given. [Copyright &y& Elsevier]

Abstract: In this paper we have investigated practical stability, boundedness criteria and Lagrange stability for fuzzy differential equations. These properties are unified with Lyapunov-like functions to establish comparison results. [Copyright &y& Elsevier]

Abstract: In this paper, we state and prove some new Lyapunov-type inequalities for a class of nonlinear systems, special cases of which contain the well-known Hamiltonian system, Emden–Fowler, half-linear and linear differential equations of second order. Our results improve and generalize these types of inequalities related to all existing ones. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of delay-dependent robust passivity analysis and robust passification of uncertain Markovian jump linear systems (MJLSs) with partially known transition rates and mode-dependent time-varying delays are investigated. In the deterministic model, the time-varying delay is in a given range and the uncertainties are assumed to be norm bounded. By constructing an appropriate Lyapunov–Krapunov functional (LKF) combining with Jensen’s inequality and the free-weighting matrix method, delay-dependent passification conditions are obtained in term of linear matrix inequalities(LMIs). For the robust passification problem, desired passification controllers are designed, which guarantee that the closed-loop MJLS is passive. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. [Copyright &y& Elsevier]

Abstract: This paper deals with the problems of robust stochastic stabilization and control for uncertain stochastic systems with time-varying delay and nonlinear perturbation. System uncertainties are assumed to be norm bounded and time delay is assumed to be bound and time varying with delay-derivative bounded by a constant, which may be greater than one. First, new delay-dependent criterion is proposed by exploiting delay-partitioned Lyapunov–krasovskii functional and by employing tighter integral equalities to estimate the upper bound of the stochastic differential of Lyapunov–krasovskii functional without ignoring some useful terms. Second, based on the criterion obtained, a delay-dependent criterion for the existence of a state feedback controller that ensures robust stochastic stability and a prescribed performance level of the closed-loop system for all admissible uncertainties is proposed. These developed results have advantages over some previous ones, in that they involve fewer matrix variables but have less conservatism and they also enlarge the application scope. New sufficient conditions are presented in terms of linear matrix inequality. Numerical examples are used to illustrate the effectiveness and feasibility of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, by using elementary analysis, we establish several new Lyapunov type inequalities for the following nonlinear dynamic system on an arbitrary time scale when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve all related existing ones including the continuous and discrete cases. [Copyright &y& Elsevier]

Abstract: In this paper, we establish several new Lyapunov-type inequalities for the first-order nonlinear Hamiltonian system which generalize or improve all related existing ones. [Copyright &y& Elsevier]

Abstract: In this paper, we develop the notions of -boundedness and practical -stability for difference equations via perturbing cone-valued Lyapunov functions, and obtain sufficient conditions of -boundedness and practical -stability. [Copyright &y& Elsevier]

Abstract: Employing the matrix measure approach and Lyapunov function, the author studies the global exponential stability of delayed neural networks with impulses in this paper. Some novel and sufficient conditions are given to guarantee the global exponential stability of the equilibrium point for such delayed neural networks with impulses. Finally, three examples are given to show the effectiveness of the results obtained here. [Copyright &y& Elsevier]

Abstract: In this paper, we establish several new Lyapunov-type inequalities for the nonlinear difference system when the end-points are not necessarily usual zeros, but rather, generalized zeros. Our results improve almost all related existing ones. [Copyright &y& Elsevier]

Abstract: In this paper, an HIV-1 infection model with distributed intracellular delays is investigated, where the intracellular delays account for the time the target cells are contacted by the virus particles and the time the contacted cells become actively infected meaning that the contacting virions enter cells and the time the virus has penetrated into a cell and the time the new virions are created within the cell and are released from the cell, respectively. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; and if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable. [Copyright &y& Elsevier]

Abstract: The paper presents the dynamical features of a vector–host epidemic model with direct transmission. First, we extended the model by taking into account the exposed individuals in both human and vector population with the impact of disease related deaths and total time dependent population size. Using Lyapunov function theory some sufficient conditions for global stability of both the disease-free equilibrium and the endemic equilibrium are obtained. For the basic reproductive number , a unique endemic equilibrium exists and is globally asymptotically stable. Furthermore, it is found that the model exhibits the phenomenon of backward bifurcation, where the stable disease-free equilibria coexists with a stable endemic equilibrium. Finally, numerical simulations are carried out to investigate the influence of the key parameters on the spread of the vector-borne disease, to support the analytical conclusion and illustrate possible behavioral scenarios of the model. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the nonlinear delay difference equations with positive and negative coefficients Sufficient conditions are obtained under which every solution of Eq. (∗) is bounded and tends to a constant as . [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a basic reproduction number for a multigroup epidemic model with nonlinear incidence. Then, we establish that global dynamics are completely determined by the basic reproduction number . It shows that, the basic reproduction number is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, a numerical example is also included to illustrate the effectiveness of the proposed result. [Copyright &y& Elsevier]

Abstract: In this paper, a strongly coupled system of partial differential equations in a bounded domain with the homogeneous Neumann boundary condition which models a predator–prey system with modified Holling–Tanner functional response is considered. First, the authors study the stability of the positive constant solution. Sufficient conditions are derived for the global stability of the positive equilibrium by constructing a suitable Lyapunov function. By using the Leray–Schauder theorem, the authors prove a number of existence and non-existence results about the non-constant steady states of the system. [ABSTRACT FROM AUTHOR]

Abstract: Like for smooth quadratic systems, it is important to determine the maximum order of a fine focus and the cyclicity of discontinuous quadratic systems. Previously, examples of discontinuous quadratic systems with five limit cycles bifurcated from a fine focus of order 5 have been constructed. In this paper we construct a class of discontinuous quadratic systems with a fine focus of order 9. In addition, by using a method similar to that developed by C. Christopher for smooth systems, which allows one to estimate the cyclicity just from the lower order terms of Lyapunov constants, we show that the cyclicity of discontinuous quadratic systems is at least 9, thus improving on previous results. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of global exponential stability for cellular neural networks (CNNs) with time-varying delays and fixed moments of impulsive effect is studied. A new sufficient condition has been presented ensuring the global exponential stability of the equilibrium points by using piecewise continuous Lyapunov functions and the Razumikhin technique combined with Young’s inequality. The results established here extend those given previously in the literature. Compared with the method of Lyapunov functionals as in most previous studies, our method is simpler and more effective for stability analysis. [Copyright &y& Elsevier]

Abstract: Very recently, a new theory known as set dynamic equations on time scales has been built. In this paper, notions of stability for the solutions of set dynamic equations on time scales, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotic stability, uniform and uniform asymptotic stability are developed. [Copyright &y& Elsevier]

Abstract: After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov’s method is a tedious task. If the fractional order is such that , the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension. [Copyright &y& Elsevier]

Abstract: In this paper, the permanence for the general nonautonomous -species discrete Kolmogorov systems with delays is studied. By using the method of multiple discrete Lyapunov-like functionals, a quite general sufficient condition is established. As application of this result, new sufficient conditions are obtained to ensure the permanence of all species for the discrete nonautonomous pure-delayed -species Lotka–Volterra competitive systems and food chain systems. Particularly, as special cases, a series of new sufficient conditions on the permanence for the discrete nonautonomous pure-delayed Lotka–Volterra two-species competitive systems, predator–prey systems and three-species food chain systems, and the discrete autonomous pure-delayed Lotka–Volterra competition systems and food chain systems are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and global exponential stability of periodic solutions for a class of numerical discretization neural networks are considered. Using coincidence degree theory and the Lyapunov method, sufficient conditions for the existence and global exponential stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results. [Copyright &y& Elsevier]

Abstract: In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function behaves as like a nontrivial self-similar profile. [Copyright &y& Elsevier]

Abstract: A model describing the dynamics of Cohen-Grossberg neural networks with time-delays and impulses is considered. By means of Lyapunov functionals and a differential inequality technique, criteria on global exponential stability of this model are derived. Many adjustable parameters are introduced in the criteria to provide flexibility for the design and analysis of the system. The results of this paper are new and they supplement previously known results. [Copyright &y& Elsevier]

Abstract: This paper studies discrete impulsive switching systems and uncertain linear discrete switching systems. It establishes some stability and robust stability criteria for such systems by using the method of Lyapunov functions and linear matrix inequalities (LMI). These criteria are verifiable by using the LMI toolbox of MATLAB. [Copyright &y& Elsevier]

Abstract: In this paper, the global asymptotic stability of Hopfield neural networks with delays is investigated. Distinct differences from other analytical approaches lie in transforming to an equivalent system by using a parameterized transformation which allows free variables in an operator. A novel, less conservative and restrictive criterion than those established in the earlier references is given in terms of several matrix inequalities by utilizing the Lyapunov theory and matrix inequality framework. The results are related to the size of delays. Numerical examples are given to show the effectiveness of our proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss impulsive high-order Hopfield type neural networks. Investigating their global asymptotic stability, by using Lyapunov function method, sufficient conditions that guarantee global asymptotic stability of networks are given. These criteria can be used to analyse the dynamics of biological neural systems or to design globally stable artificial neural networks. Two numerical examples are given to illustrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the global exponential stability of impulsive high-order Hopfield type neural networks with delays. By establishing the impulsive delay differential inequalities and using the Lyapunov method, two sufficient conditions that guarantee global exponential stability of these networks are given, and the exponential convergence rate is also obtained. A numerical example is given to demonstrate the validity of the results. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain some sufficient conditions for determining the asymptotic stability of discrete-time non-autonomous delayed Hopfield neural networks by utilizing the Lyapunov functional method. An example is given to show the validity of the results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a convex vector optimization problem of finding weak Pareto optimal solutions for an extended vector-valued map from a uniformly convex and uniformly smooth Banach space to a real Banach space, with respect to the partial order induced by a closed, convex and pointed cone with a nonempty interior. We propose an inexact vector-valued proximal-type point algorithm based on a Lyapunov functional when the iterates are computed approximately and prove the sequence generated by the algorithm weakly converges to a weak Pareto optimal solution of the vector optimization problem under some mild conditions. Our results improve and generalize some known results. [Copyright &y& Elsevier]