On a fixed point theorem in 2-Banach spaces and some of its applications

The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.     

Bifurcation analysis of the flour beetle population growth equations

In this article, we study a discrete delayed flour beetle population equation. Firstly, we study the existence of period-doubling bifurcation and Neimark–Sacker bifurcations for the system by analysing its characteristic equations. Secondly, we investigate the direction of the two bifurcations and the stability of the bifurcation periodic solutions by using normal form theory. Finally, some numerical simulations are carried out to support the analytical results.     

On the stability of explicit finite difference methods for advection–diffusion equations

In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi-discretization of the ADE leads to a conditionally stable fully discretized method as long as the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples.     

On stability in terms of two measures for impulsive systems of functional differential equations

This paper considers the stability problem for a class of impulsive systems of functional differential equations. By using Lyapunov functions and the Razumikhin technique, several stability criteria in terms of two measures are established. Some examples are worked out to illustrate the theorems.     

Meshless formulation to two-dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence

In this study, the singular boundary method (SBM) is employed for the simulation of nonlinear generalized Benjamin–Bona–Mahony–Burgers problem with initial and Dirichlet-type boundary conditions. The θ-weighted finite difference method is used to discretize the time derivatives. Then the original equations are split into a system of partial differential equations. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. To solve this system, the method of particular solution (MPS) in combination with the SBM is used where the SBM is used for homogeneous solution and MPS is used for particular solution. Furthermore, the stability and convergence of the proposed method is conducted. Finally, several numerical examples with different domains are provided and compared with the exact analytical solutions to show the accuracy and efficiency in comparison with existing other methods.     

Note on the stability of reaction–diffusion systems with delays by Lyapunov functional

This paper aims to investigate the stability of reaction–diffusion equations with delays. We extend a stability theorem on FDEs introduced by Hale to reaction–diffusion equations with time delay.     

Stability of functional equations in single variable

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the Hyers-Ulam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.     

On Hyers-Ulam stability for a class of functional equations

Summary In this paper we prove some stability theorems for functional equations of the formg[F(x, y)]=H[g(x), g(y), x, y]. As special cases we obtain well known results for Cauchy and Jensen equations and for functional equations in a single variable. Work supported by M.U.R.S.T. Research funds (60%).     

Regularization and general methods in the theory of functional equations

Summary The aim of this paper is to give a survey of two fields of the theory of non-iterated functional equations with two variables. One is the application of new, general methods of functional analysis, harmonic analysis and other topics to get a unified treatment of several kinds of equations. The other includes general regularity results for non-iterated functional equations with two variables.     

RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.     

On contraction of nonlinear difference equations with time‐varying delays

General nonlinear difference equations with time‐varying delays are considered. Explicit criteria for contraction of such equations are presented. Then some simple sufficient conditions for global exponential stability of equilibria and for stability of invariant sets are derived. Furthermore, explicit criteria for existence, uniqueness and global exponential stability of periodic solutions are derived. Finally, the obtained results are applied to time‐varying discrete‐time neural networks with delay.     

Stability of a three-dimensional diffusive Lotka–Volterra system of type-K with delays

A three-dimensional diffusive Lotka–Volterra system of type-K with delays is investigated. We give a stability analysis in detail for all equilibria of the system and obtain some threshold conditions for linear instability and linear asymptotic stability of each equilibrium. We develop the analytical method for stability analysis of reaction–diffusion equations with multi-delays.     

Ulam-type stability for differential equations driven by measures

Based on a notion of Stieltjes derivative of a function with respect to another function, we provide Ulam–Hyers type stability results for nonlinear differential equations driven by measures on compact or on unbounded intervals, in the lack of Lipschitz continuity assumptions. In particular, one can deduce stability results for generalized differential equations, dynamic equations on time scales or impulsive differential equations (including the case of an infinite number of impulses that accumulate in the considered interval, thus allowing the study of Zeno hybrid systems).     

STABILITY OF DELAY DIFFERENCE EQUATIONS IN BANACH SPACES

1 IntroductionDiscrete reaction--diffusion type partial difference equations llave recentlybeen introduced by a number of authors as modeIs for the study of spatiotem-poral chaos (see e.g. [2,3j). Stability criteria have also been derived fOr suchequatioIls which invoIve two time-level processes (see e.g. [1l) as well as three-level processes (see e.g. [9]). In this paper, we will study no11linear three--levc1partia1 diffcrence equations in an abstract setting and derive stability criteriafo…     

A fixed point approach to stability of functional equations

We prove a simple fixed point theorem for some (not necessarily linear) operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable.     

A Branching Method for Studying Stability of a Solution to a Delay Differential Equation

We study stability of antisymmetric periodic solutions to delay differential equations. We introduce a one-parameter family of periodic solutions to a special system of ordinary differential equations with a variable period. Conditions for stability of an antisymmetric periodic solution to a delay differential equation are stated in terms of this period function.     

关于运用Liapunov函数的滞后型系统部分稳定性的注释

Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.     

有限时滞微分系统依照双测度的实际稳定性

本文对具有有限时滞的泛函微分方程建立了关于依照两种测度的实际稳定性的Razumikhin型判定定理，其中未采用通常的辅助函数，且可运用多个含有状态变量x的部分变元的Lyapunov函数，得出部分变远实际稳定性的判定定理，从而改进了已有的结果。