Advances in Difference Equations

http://www.advancesindifferenceequations.com/

List of Papers (Total 2,909)

On fractional Hahn calculus

In this paper, the new concepts of Hahn difference operators are introduced. The properties of fractional Hahn calculus in the sense of a forward Hahn difference operator are introduced and developed.

Local stable manifold of Langevin differential equations with two fractional derivatives

In this paper, we investigate the existence of local center stable manifolds of Langevin differential equations with two Caputo fractional derivatives in the two-dimensional case. We adopt the idea of the existence of a local center stable manifold by considering a fixed point of a suitable Lyapunov-Perron operator. A local center stable manifold theorem is given after deriving ...

Compact almost automorphic solutions for some nonlinear integral equations with time-dependent and state-dependent delay

We study the existence of compact almost automorphic solutions for a class of integral equations with time-dependent and state-dependent delay. An application to a blowflies model and a transmission lines model is carried out to support the theoretical finding.

Synchronization of fractional chaotic systems based on a simple Lyapunov function

In this paper the synchronization of fractional-order chaotic systems and a new property of fractional derivatives are studied. Then we propose a new fractional-order extension of Lyapunov direct method to control the fractional-order chaotic systems. A new synchronization method and a linear feedback controller are given to achieve the synchronization of fractional-order chaotic ...

Alternate-continuous-control systems with double-impulse

We propose a mathematical model that can control the stability of an unstable system. Periodicity is an important feature of the system. We add a continuous control to the first half of each period of the system and then add an impulse control J1 at the 1 / 2 $1/2$ period time. Again, we do not control the rest half of each period of the system. Finally, we add an impulse control ...

A stochastic prey-predator model with time-dependent delays

A stochastic predator-prey system with time-dependent delays is considered. Firstly, we show the existence of a global positive solution and stochastically ultimate boundedness. Secondly, the critical value between weak persistence and extinction of the prey is obtained and we also give the asymptotic pathwise estimation. Finally, we simulate the model to illustrate our results.

Qualitative behaviors of the high-order Lorenz-Stenflo chaotic system arising in mathematical physics describing the atmospheric acoustic-gravity waves

The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic ...

Mathematical modeling of infectious disease transmission in macroalgae

Understanding the infectious diseases outbreak of algae can provide significant knowledge for disease control intervention and/or prevention. We consider here a disease caused by highly pathogenic organisms that can result in the death of algae. Even though a great deal of understanding about diseases of algae has been reached, studies concerning effects of the outbreak at the ...

Discrete Schrödinger equations in the nonperiodic and superlinear cases: homoclinic solutions

Using variational methods, we study the existence and multiplicity of homoclinic solutions for a class of discrete Schrödinger equations in infinite m-dimensional lattices with nonlinearities being superlinear at infinity. Our results generalize some existing results in the literature by using some weaker conditions.

A general class of noninstantaneous impulsive fractional differential inclusions in Banach spaces

In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild ...

An exponential B-spline collocation method for the fractional sub-diffusion equation

In this article, we propose an exponential B-spline approach to obtain approximate solutions for the fractional sub-diffusion equation of Caputo type. The presented method is established via a uniform nodal collocation strategy by using an exponential B-spline based interpolation in conjunction with an effective finite difference scheme in time. The unique solvability is rigorously ...

Local uniqueness of positive solutions for a coupled system of fractional differential equations with integral boundary conditions

In this paper, we study a coupled system of fractional boundary value problems subject to integral boundary conditions. By applying a recent fixed point theorem in ordered Banach spaces, we investigate the local existence and uniqueness of positive solutions for the coupled system. We show that the unique positive solution can be found in a product set, and that it can be ...

Existence of positive mild solutions for a class of fractional evolution equations on the half line

The equivalent integral equation of a new form for a class of fractional evolution equations is obtained by the method of Laplace transform, which is different from those given in the existing literature. By the monotone iterative method without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract ...

Generalized finite difference/spectral Galerkin approximations for the time-fractional telegraph equation

We discuss the numerical solution of the time-fractional telegraph equation. The main purpose of this work is to construct and analyze stable and high-order scheme for solving the time-fractional telegraph equation efficiently. The proposed method is based on a generalized finite difference scheme in time and Legendre spectral Galerkin method in space. Stability and convergence of ...

The roles of maturation delay and vaccination on the spread of Dengue virus and optimal control

A mathematical model of Dengue virus transmission between mosquitoes and humans, incorporating a control strategy of imperfect vaccination and vector maturation delay, is proposed in this paper. By using some analytical skills, we obtain the threshold conditions for the global attractiveness of two disease-free equilibria and prove the existence of a positive equilibrium for this ...

An effective collocation technique to solve the singular Fredholm integral equations with Cauchy kernel

In this paper, an effective numerical method to solve the Cauchy type singular Fredholm integral equations (CSFIEs) of the first kind is proposed. The collocation technique based on Bernstein polynomials is used for approximation the solution of various cases of CSFIEs. By transforming the problem into systems of linear algebraic equations, we see that this approach is ...

Oscillation criteria for difference equations with non-monotone arguments

This paper is concerned with the oscillatory behavior of first-order retarded [advanced] difference equation of the form Δ x ( n ) + p ( n ) x ( τ ( n ) ) = 0 , n ∈ N 0 [ ∇ x ( n ) − q ( n ) x ( σ ( n ) ) = 0 , n ∈ N ] , where ( p ( n ) ) n ≥ 0 [ ( q ( n ) ) n ≥ 1 ] is a sequence of nonnegative real numbers and τ ( n ) [ σ ( n ) ] is a non-monotone sequence of integers such that τ ...

Systems of generalized Sturm-Liouville and Langevin fractional differential equations

In this paper, we study anti-periodic boundary value problems for systems of generalized Sturm-Liouville and Langevin fractional differential equations. Existence and uniqueness results are proved via fixed point theorems. Examples illustrating the obtained results are also presented. MSC: 34A08, 34A12, 34B15.

Stability and traveling waves in a Beddington-DeAngelis type stage-structured predator-prey reaction-diffusion systems with nonlocal delays and harvesting

The goal of this paper is to study the stability and traveling waves of stage-structured predator-prey reaction-diffusion systems of Beddington-DeAngelis functional response with both nonlocal delays and harvesting. By analyzing the corresponding characteristic equations, the local stability of various equilibria is discussed. We reduce the existence of traveling waves to the ...

Neutral stochastic functional differential equations with Lévy jumps under the local Lipschitz condition

In this paper, a general neutral stochastic functional differential equations with infinite delay and Lévy jumps (NSFDEwLJs) is studied. We investigate the existence and uniqueness of solutions to NSFDEwLJs at the phase space C g under the local Carathéodory type conditions. Meanwhile, we also give the exponential estimates and almost surely asymptotic estimates of solutions to ...

Sustained oscillation induced by time delay in a commodity market model

In this paper, the existence of local and global Hopf bifurcation for a delay commodity market model is studied in detail. As time delay increases, the commodity price will fluctuate periodically. Furthermore, such fluctuations will occur even if the time delay is sufficiently large. MSC: 91B55, 34K18.

Exponential stability criteria for fuzzy bidirectional associative memory Cohen-Grossberg neural networks with mixed delays and impulses

This paper is concerned with fuzzy bidirectional associative memory (BAM) Cohen-Grossberg neural networks with mixed delays and impulses. By constructing an appropriate Lyapunov function and a new differential inequality, we obtain some sufficient conditions which ensure the existence and global exponential stability of a periodic solution of the model. The results in this paper ...

On the nonexistence of global solutions for a class of fractional integro-differential problems

We study the nonexistence of (nontrivial) global solutions for a class of fractional integro-differential problems in an appropriate underlying space. Integral conditions on the kernel, and for some degrees of the involved parameters, ensuring the nonexistence of global solutions are determined. Unlike the existing results, the source term considered is, in general, a convolution ...

Long time behavior of a tumor-immune system competition model perturbed by environmental noise

This paper investigates the long time behavior of tumor cells evolution in a tumor-immune system competition model perturbed by environmental noise. Sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are derived by constructing Lyapunov functions. The study results show that environmental noise can accelerate the ...

Solvability and Mann iterative approximations for a higher order nonlinear neutral delay differential equation

The purpose of this paper is to study solvability of the higher order nonlinear neutral delay differential equation d n d t n [ x ( t ) + c ( t ) x ( t − τ ) ] + ( − 1 ) n + 1 f ( t , x ( σ 1 ( t ) ) , x ( σ 2 ( t ) ) , … , x ( σ k ( t ) ) ) = g ( t ) , t ≥ t 0 , where n and k are positive integers, τ > 0 , t 0 ∈ R , f ∈ C ( [ t 0 , + ∞ ) × R k , R ) , c , g , σ i ∈ C ( [ t 0 , + ∞ ...