THE EXISTENCE AND UNIQUENESS AND STABILITY OF ALMOST PERIODIC SOLUTIONS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

This paper deals with the problems on the existence and uniqueness and stability of almost periodic solutions for functional differential equations with infinite delays.The author obtains some sufficient conditions which ganrantee the existence and uniqueness and stability of almost periodic solutions with module containment.The results extend all the results of the paper and solve the two open problems proposed in under much weaker conditions than that proposed in.

THE EXISTENCE, UNIQUENESS AND STABILITY OF ALMOST PERIODIC SOLUTION FOR A CLASS OF NONLINEAR SYSTEM

In this paper, we study the problems on the existence, uniqueness and stability of almost periodic solution for a class of nonlinear system. Using fixed point theorem and Lyapunov functional, the sufficient conditions are given which guarantee the existence, uniqueness and stability of almost periodic solution for the system.

On the Existence,Uniqueness and Stability of Periodic Solutions of Large Scale Systems with Unbounded Delay

In this paper,we consider the periodic solution problems for the systems with unbounded delay,and the existence,uniqueness and stability of the periodic solution are dealt with unitedly.First we establish the suitable delay-differential inequality,then study seperately the problems of periodic solution for the systems with bounded delay,with unbounded delay and the Volterra integral-dlfferentlal systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré’s periodic operator in the different phase spaces.A series of simple criteria for the existence,uniqueness and stability of these systems are obtained.

ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION

The purpose of this study is to acquire some conditions that reveal existence and stability for solutions to a class of difference equations with non-integer orderμ∈(1,2].The required conditions are obtained by applying the technique of contraction principle for uniqueness and Schauder’s fixed point theorem for existence.Also,we establish some conditions under which the solution of the considered class of difference equations is generalized Ulam-Hyers-Rassias stable.Example for the illustration of results is given.

EXISTENCE UNIQUENESS AND ASYMPTOTIC STABILITY OF PERIODIC SOLUTIONS OF A NONLINEAR EQUATION IN PHASE LOCKED TECHNOLOGY

In [1] and [2], the authors made a deep qualitative analysis of the equationwith the character of tangent detected phase and they mathematically provided atheoretical basis of why the phase looked loop has no look--losing point. However,according to many practical experts, it is rather difficult to put such a phaselooked loop into practice, though it has fine properties. W. C. Lindsey [3] made a

Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

Existence, uniqueness and stability of positive steady states to a prey-predator diffusion system

In the paper, we study the positive solutions of an elliptic system coming from a preypredator model with modified Leslie-Gower and Holling-Type II schemes. We study the existence, non-existence, bifurcation, uniqueness and stability of positive solutions. In particular, we obtain a continuum of positive solutions connecting a semitrivial solution to the unique positive solution of the limiting system.

Existence, uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems

In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R^3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular,our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.

Fractal Fractional Order Operators in Computational Techniques for Mathematical Models in Epidemiology

New fractional operators, the COVID-19 model has been studied in this paper. By using different numericaltechniques and the time fractional parameters, the mechanical characteristics of the fractional order model areidentified. The uniqueness and existence have been established. Themodel’sUlam-Hyers stability analysis has beenfound. In order to justify the theoretical results, numerical simulations are carried out for the presented methodin the range of fractional order to show the implications of fractional and fractal orders.We applied very effectivenumerical techniques to obtain the solutions of themodel and simulations. Also, we present conditions of existencefor a solution to the proposed epidemicmodel and to calculate the reproduction number in certain state conditionsof the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered foranalysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in theCommunity. For this reason, we employed the COVID-19 fractal fractional derivative model in the example ofWuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractionaloperators can facilitate the improvement of decision-making for measures to be taken in the management of anepidemic situation.

On existence,uniqueness and convergence of multi-valued stochastic diferential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic diferential equations driven by continuous semimartingales when the coefcients are stochastically Lipschitz continuous.We also show the convergence results when the random coefcients or the diferentials converge.

Positive solutions and stability of fuzzy Atangana–Baleanu variable fractional differential equation model for a novel coronavirus (COVID-19)

This work provides a new fuzzy variable fractional COVID-19 model and uses a variablefractional operator, namely, the fuzzy variable Atangana–Baleanu fractional derivativesin the Caputo sense. Next, we explore the proposed fuzzy variable fractional COVID-19 model using the fixed point theory approach and determine the solution’s existenceand uniqueness conditions. We choose an appropriate mapping and with the help ofthe upper/lower solutions method. We prove the existence of a positive solution for theproposed fuzzy variable fractional COVID-19 model and also obtain the result on theexistence of a unique positive solution. Moreover, we discuss the generalized Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability. Further, we investigate theresults on maximum and minimum solutions for the fuzzy variable fractional COVID-19model.

ON THE EXISTENCE OF PERIODIC SOLUTION OF HIGHER ORDER PERIODIC SYSTEM

In this paper, we deal with the problem on the existence of periodic solution of higher periodic system. Using the exponential dichotomy and the Schauder's fixed point theorem,we establish the sufficient conditions which guarantee the existence and uniqueness and stability of periodic solution.

A STABILIZED MIXED FINITE ELEMENT FORMULATION FOR THE NON-STATIONARY INCOMPRESSIBLE BOUSSINESQ EQUATIONS

In this study, we employ mixed finite element （MFE） method, two local Gauss integrals, and parameter-free to establish a stabilized MFE formulation for the non-stationary incompressible Boussinesq equations. We also provide the theoretical analysis of the existence, uniqueness, stability, and convergence of the stabilized MFE solutions for the stabilized MFE formulation.

Equilibria and Stability Analysis of Cohen-Grossberg BAM Neural Networks on Time Scale

This paper considers the Cohen-Grossberg BAM neural networks(CG-BAMNNs) on time scale, which can unify and generalize the continuous and discrete systems. First, the criteria for the existence and uniqueness of the equilibrium of CG-BAMNNs are derived on time scale. Then based on that, the authors give the criteria for the stability and estimation of equilibrium of the CG-BAMNNs on time scale. The method proposed in this paper unifies and generalizes the continuous and discrete CGBAMNNs systems, and is applicable to some other neural network systems on time scale with practical meaning. The effectiveness of the proposed criteria for delayed CG-BAMNNs is demonstrated by numerical simulation.

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

In this paper,we study the existence,uniqueness and asymptotic stabgility of the periodic solution for a class of the most,universal fourth-order nonlinear nonautonomous periodic systems.We give the relevant Liapunov function by using the method of analogical slowly changing coefficients.We also give a considerably accurate estimation of the slowly changing coefficients and obtain the sufficient conditions which guarantee the existence,uniqueness and asymptotic Stability of the periodci solutions.

Stability results for a nonlinear two-species competition model with size-structure

We formulate a system of integro-differential equations to model the dynamics of competition in a two-species community,in which the mortality,fertility and growth are sizedependent.Existence and uniqueness of nonnegative solutions to the system are analyzed.The existence of the stationary size distributions is discussed,and the linear stability is investigated by means of the semigroup theory of operators and the characteristic equation technique.Some sufficient conditions for asymptotical stability/instability of steady states are obtained.The resulting conclusion extends some existing results involving age-independent and age-dependent population models.

Analysis and Dynamics of Illicit Drug Use Described by Fractional Derivative with Mittag-Leffler Kernel

Illicit drug use is a significant problem that causes great material and moral losses and threatens the future of the society.For this reason,illicit drug use and related crimes are the most significant criminal cases examined by scientists.This paper aims at modeling the illegal drug use using the Atangana-Baleanu fractional derivative with Mittag-Leffler kernel.Also,in this work,the existence and uniqueness of solutions of the fractional-order Illicit drug use model are discussed via Picard-Lindelöf theorem which provides successive approximations using a convergent sequence.Then the stability analysis for both disease-free and endemic equilibrium states is conducted.A numerical scheme based on the known Adams-Bashforth method is designed in fractional form to approximate the novel Atangana-Baleanu fractional operator of order 0<a≤1.Finally,numerical simulation results based on different values of fractional order,which also serve as control parameter,are presented to justify the theoretical findings.

DIFFERENTIABILITY FOR THE HIGH DIMENSIONAL POLYNOMIAL-LIKE ITERATIVE EQUATION

This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.

DISCUSSION ON THE C^r－SOLUTIONS OF THE ITERATED EQUATION λ_1f（x）＋λ_2f^2（x）＝F（x）

In this paper, we consider the iterated equationλ1f(x) + λ2f2(x)=F(x)where f2(x)= f(f(x)), F (x) denotes known function and f(x) denotes the unknown function. There are given conditions for the existence, uniqueness and stability of C'-solutions ofthe iterated equation (*) and also there is a proved theorem for the continuous dependence of Cr-solutions of iterated equation (*) on the given function.

EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

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