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In this article, we study impulsive differential equations with conformable derivatives. Firstly, we derive suitable formulas for solving linear impulsive conformable Cauchy problems. Then, we show that the linear problem has asymptotic stability, and the nonlinear problem has generalized Ulam-Hyers-Rassias stability. Also we illustrate our results with examples.

This article discusses the application of a fractional-like derivative of Lyapunov-type functions in the stability analysis of solutions of perturbed motion equations with a fractional-like derivative of the state vector. The main theorems of the direct Lyapunov method for this class of motion equations are established.

In this short note we indicate some improvement of an article published in this journal by Crespo, Ivorra and Ramos [6]. The techniques used are connected with several smoothing effects associated with linear partial differential operators which give rise to some accretive operators in $L^1(\Omega )$, as well as with some $H^2(\Omega )$ estimates independent on time.

We present new criteria for asymptotic stability of two classes of nonlinear neutral delay differential equations. By using two auxiliary functions on a contraction condition, we extend the results in [12]. Also we give two examples that illustrate our results.

In this article, we study an SIR model with nonlinear incidence rate. By defining the Filippov solution for the model and constructing suitable Lyapunov functions, we show that the global dynamics are fully determined by the basic reproduction number $R_0$, under certain conditions on the incidence rate and treatment functions. When $R_0\leq 1$ the disease-free equilibrium is globally asymptotically stable, and when $R_0>1$ the unique endemic equilibrium is globally asymptotically stable.

This article discusses the application of a fractional-like derivative of Lyapunov-type functions in the stability analysis of solutions of perturbed motion equations with a fractional-like derivative of the state vector. The main theorems of the direct Lyapunov method for this class of motion equations are established. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic stability of solutions for the initial value problems of second order evolution equations in Banach spaces, which can model elastic systems with structural damping. The discussion is based on exponentially stable semigroups theory. Applications to the vibration equation of elastic beams with structural damping are also considered.

In this short note we indicate some improvement of an article published in this journal by Crespo, Ivorra and Ramos 6. The techniques used are connected with several smoothing effects associated with linear partial differential operators which give rise to some accretive operators in L¹(Ω), as well as with some H²(Ω) estimates independent on time. [ABSTRACT FROM AUTHOR]

In this work, we present an asymptotic analysis of a coupled system of two advection-diffusion-reaction equations with Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacteria), called biomass, and a diluted organic contaminant (e.g., nitrates), called substrate, in a continuous ow bioreactor. This system exhibits, under suitable conditions, two stable equilibrium states: one steady state in which the biomass becomes extinct and no reaction is produced, called washout, and another steady state, which corresponds to the partial elimination of the substrate. We use the linearization method to give sufficient conditions for the linear asymptotic stability of the two stable equilibrium configurations. Finally, we compare our asymptotic analysis with the usual asymptotic analysis associated to the continuous bioreactor when it is modeled with ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotical stability in p-th moment of mild solutions to a class of fractional impulsive partial neutral stochastic integro-differential equations with state-dependent delay in Hilbert spaces. We assume that the linear part of this equation generates an alpha-resolvent operator and transform it into an integral equation. Sufficient conditions for the existence and asymptotic stability of solutions are derived by means of the Krasnoselskii-Schaefer type fixed point theorem and properties of the alpha-resolvent operator. An illustrative example is also provided.

In this article, we present the nonlinear Mecking-Lucke-Grilhe model describing the temporal evolution for simple and multi-instabilities of plastic deformation of stressed monocristal. This model extends the linear problem considered in [9, 13, 14]. Using a nonlinear analysis, we present some results of existence and stability of the solution with respect to the characteristics of the material and the retarded times. Numerical examples validating the theoretical results are also investigated in this study.

In this paper, we study a non-autonomous Lotka-Volterra model with two predators and one prey. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution.

This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation $$ u_{tt}-Delta u +|u|^kj'(u_t)=|u|^{p-1}u quad hbox{in }Omega imes (0,T), $$ where $p>1$ and $j'$ denotes the derivative of a $C^1$ convex and real value function $j$. We prove that every weak solution is asymptotically stability, for every $m$ such that $0k+m$ and the initial data is positive, but appropriately bounded.

We prove asymptotical stability and instability results for a general second-order differential equations with complex-valued functions as coefficients. To prove asymptotic stability of linear second-order differential equations, we use the technique of asymptotic representations of solutions and error estimates. For nonlinear second-order differential equations, we extend the asymptotic stability theorem of Pucci and Serrin to the case of complex-valued coefficients.

In this article we study the homogenization and uniform decay of the nonlinear hyperbolic equation $$ partial_{tt} u_{varepsilon} -Delta u_{varepsilon} +F(x,t,partial_t u_{varepsilon}, abla u_{varepsilon})=0 quadhbox{in }Omega_{varepsilon}imes(0,+infty) $$ where $Omega_{varepsilon}$ is a domain containing holes with small capacity (i. e. the holes are smaller than a critical size). The homogenization's proofs are based on the abstract framework introduced by Cioranescu and Murat [8] for the study of homogenization of elliptic problems. Moreover, uniform decay rates are obtained by considering the perturbed energy method developed by Haraux and Zuazua [10].

In this paper, we consider the impulsive second order system [ ddot{x}+f(x)=0quad (teq t_{n});quad dot{x}(t_{n}+0)=b_{n}dot{x}(t_{n}) quad (t=t_{n}) ] where $t_n=t_0+n,p$ $(p>0, n=1,2dots )$. In a previous paper, the authors proved that if $f(x)$ is strictly nonlinear, then this system has infinitely many periodic solutions. The impulses account for the main differences in the attractivity properties of the zero solution. Here, we prove that these periodic solutions are attractive in some sense, and we give good estimates for the attractivity region.

Slow and fast systems gain their special structure from the presence of two time scales. Their analysis is achieved with the help of Singular Perturbation Theory. The fundamental tool is Tykhonov's theorem which describes the limiting behaviour, for compact interval of time, of solutions of the perturbed system which is a one-parameter deformations of the so-called unperturbed system. Our aim here is to extend this description to the solutions of all systems that belong to a small neighbourhood of the unperturbed system. We investigate also the behaviour of solutions on the infinite time interval. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is an axiomatic approach to Nonstandard Analysis.

We study initial value problems for scalar, nonlinear, delay differential equations with distributed, possibly infinite, delays. We consider the initial value problem $$cases{ x(t) = varphi(t), & $t leq 0$cr x'(t)+int_0^infty g(t, s, x(t), x(t-s)), d mu(s) = f(t), & $tgeq 0$,cr} $$ where $varphi$ and $f$ are bounded and $mu$ is a finite Borel measure. Motivated by the nonresonance condition for the linear case and previous work of the authors, we introduce conditions on $g$. Under these conditions, we prove an existence and uniqueness theorem. We show that under the same conditions, the solutions are globally asymptotically stable and, if $mu$ satisfies an exponential decay condition, globally exponentially asymptotically stable.

In this article, we present the nonlinear Mecking-Lücke-Grilhé model describing the temporal evolution for simple and multi-instabilities of plastic deformation of stressed monocristal. This model extends the linear problem considered in [9, 13, 14]. Using a nonlinear analysis, we present some results of existence and stability of the solution with respect to the characteristics of the material and the retarded times. Numerical examples validating the theoretical results are also investigated in this study. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-autonomous Lotka-Volterra model with two predators and one prey. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution. [ABSTRACT FROM AUTHOR]

This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation utt - Δu + |u|kj′0(ut) = |u|p-1u in Ω (0, T), where p > 1 and j′ denotes the derivative of a C1 convex and real value function j. We prove that every weak solution is asymptotically stability, for every m such that 0 < m < 1, p < k + m and the the initial energy is small; the solutions blows up in finite time, whenever p > k +m and the initial data is positive, but appropriately bounded. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of periodic solutions and the stability of the zero solution of the nonlinear neutral differential equation d/dt x(t) = -a(t)x(t) + d/dt Q(t, x(t - g(t))) + G(t, x(t), x(t - g(t))). In the process we use integrating factors and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution provided that Q(0, 0) = G(t, 0, 0) = 0. [ABSTRACT FROM AUTHOR]

We prove asymptotical stability and instability results for a general second-order differential equations with complex-valued functions as coefficients. To prove asymptotic stability of linear second-order differential equations, we use the technique of asymptotic representations of solutions and error estimates. For nonlinear second-order differential equations, we extend the asymptotic stability theorem of Pucci and Serrin to the case of complex-valued coefficients. [ABSTRACT FROM AUTHOR]

In this article we study the homogenization and uniform decay of the nonlinear hyperbolic equation... where Ω∈ is a domain containing holes with small capacity (i. e. the holes are smaller than a critical size). The homogenization's proofs are based on the abstract framework introduced by Cioranescu and Murat [8] for the study of homogenization of elliptic problems. Moreover, uniform decay rates are obtained by considering the perturbed energy method developed by Haraux and Zuazua [10]. " [ABSTRACT FROM AUTHOR]