Advances in Difference Equations

http://www.advancesindifferenceequations.com/

List of Papers (Total 3,024)

New oscillation theorems for second order quasi-linear difference equations with sub-linear neutral term

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the second order neutral difference equation Δ ( a n ( Δ z n ) β ) + q n x n − ℓ γ = 0 , n ≥ n 0 , $$\Delta \bigl( a_{n} ( \Delta z_{n} )^{\beta} \bigr) + q_{n}x_{n - \ell}^{\gamma} = 0,\quad n \ge n_{0}, $$ where z n = x n + p n x n − k α $z_{n} = x_{n} + p_{n}x_{n - k...

Singularities of attractive and repulsive type for p -Laplacian generalized Liénard equation

This paper is devoted to an investigation of a kind of p-Laplacian generalized Liénard equations with singularities of attractive and repulsive type, where the nonlinear term g has a singularity at the origin. The novelty of the present article is that we show that singularities of attractive and repulsive type enable the achievement of a new existence criterion of a positive...

Positive solutions for a system of second-order discrete boundary value problems

We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions, under some assumptions on the nonlinearities of the system which contains concave functions. In the proofs of our main results we use some theorems from the fixed point index theory.

Fujita-type theorems for a class of coupled semilinear parabolic systems with gradient terms

This paper concerns the asymptotic behavior of the solution to a class of coupled semilinear parabolic systems with gradient terms. The Fujita-type blow-up theorems are established and the critical Fujita curve is determined not only by the behavior of the coefficients of the gradient term and the source terms at infinity, but also by the spacial dimension.

On fractional derivatives with generalized Mittag-Leffler kernels

Fractional derivatives with three parameter generalized Mittag-Leffler kernels and their properties are studied. The corresponding integral operators are obtained with the help of Laplace transforms. The action of the presented fractional integrals on the Caputo and Riemann type derivatives with three parameter Mittag-Leffler kernels is analyzed. Integration by parts formulas in...

Pricing vulnerable options with variable default boundary under jump-diffusion processes

For the pricing of vulnerable options, we improve the results of Klein and Inglis [Journal of Banking and Finance] and Tian et al. [The Journal of Futures and Markets], considering the circumstances in which the writers of options face financial crisis. Our pricing model faces the risks of default and the occasional impact experienced by the underlying assets and counterparty’s...

Error estimates of a semi-discrete LDG method for the system of damped acoustic wave equation

We consider a system of acoustic wave equation possessing lower-order perturbation terms in a bounded domain in R 2 $\mathbb{R}^{2}$ . In this paper, we show the system is well-posed and stable with energy decays introducing a local discontinuous Galerkin (LDG) method. Also, we study an a priori L 2 $L^{2}$ -norm error estimate for the semi-discretized LDG method for the system...

Global threshold dynamics of a stochastic epidemic model incorporating media coverage

In this paper, we investigate the global threshold dynamics of a stochastic SIS epidemic model incorporating media coverage. We give the basic reproduction number R 0 s $\mathcal{R}_{0}^{s}$ and establish a global threshold theorem by Feller’s test: if R 0 s ≤ 1 $\mathcal{R}_{0}^{s}\leq 1$ , the disease will die out a.s.; if R 0 s > 1 $\mathcal{R}_{0}^{s}>1$ , the disease will...

Hyers–Ulam stability and existence criteria for coupled fractional differential equations involving p -Laplacian operator

In this article, by using nonlinear Leray–Schauder-type alternative and Banach’s fixed point theorem, we investigate existence and uniqueness of solutions. We also prove Hyers–Ulam stability for the proposed coupled system of fractional differential equations (FDEs) with the nonlinear p-Laplacian operator and Riemann–Liouville integral boundary conditions (IBCs). An illustrative...

Existence of solutions for a multi-point boundary value problem with a p ( r ) $p(r)$ -Laplacian

In this paper, we consider the existence of solutions to the p ( r ) $p(r)$ -Laplacian equation with multi-point boundary conditions. Under some new criteria and by utilizing degree methods and also the Leray–Schauder fixed point theorem, the new existence results of the solutions have been established. Some results in the literature can be generalized and improved. And as an...

Threshold dynamics of a delayed predator–prey model with impulse via the basic reproduction number

In this paper, we study a delayed predator–prey model with impulse and, in particular, the existence of the predator-free periodic solution. We employ the approach and techniques coming from epidemiology and calculate the basic reproduction number for the predator. Using the basic reproduction number, we consider the global attraction of the predator-free periodic solution and...

Predicting the effectiveness of drug interventions with ‘HIV counseling & testing’ (HCT) on the spread of HIV/AIDS: a theoretical study

In this paper, a dynamic compartmental model is constructed for the transmission of HIV/AIDS receiving drug treatment and knowledge from effective awareness programs through media. Using stability theory of differential equations the model is analyzed qualitatively. The equilibrium points in the local and global stability proof are found to be stable under certain conditions...

Dynamics of difference equation x n + 1 = f ( x n − l , x n − k ) $x_{n+1}=f( x_{n-l},x_{n-k})$

In this paper, we present the asymptotic behavior of the solutions for a general class of difference equations. We introduce general theorems in order to study the stability and periodicity of the solutions. Moreover, we use a new technique to study the existence of periodic solutions of this general equation. By using our general results, we can study many special cases that...

Weak convergence of the complex fractional Brownian motion

In this paper, we obtain two approximations in law of the complex fractional Brownian motion by processes constructed from a Poisson process and a Lévy process, respectively.

Global behavior of positive solutions for some semipositone fourth-order problems

In this paper, we study the global behavior of positive solutions of fourth-order boundary value problems { u ′′′′ = λ f ( x , u ) , x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , $$ \textstyle\begin{cases} u

The permanence and periodic solution of a competitive system with infinite delay, feedback control, and Allee effect

This paper is devoted to study the permanence and periodic solution of a competitive system with infinite delay, feedback control, and the Allee effect. We derive sufficient conditions for the permanence and existence of a periodic solution in a competitive system with infinite delay, feedback control, and the Allee effect by using the differential inequality theory and...

The Euler numbers and recursive properties of Dirichlet L -functions

The aim of this paper is using an elementary method and the properties of the Bernoulli polynomials to establish a close relationship between the Euler numbers of the second kind E n ∗ $E_{n}^{*}$ and the Dirichlet L-function L ( s , χ ) $L(s,\chi )$ . At the same time, we also prove a new congruence for the Euler numbers E n $E_{n}$ . That is, for any prime p ≡ 1 mod 8 $p\equiv...

Nonlocal evolution inclusions under weak conditions

We study evolution inclusions given by multivalued perturbations of m-dissipative operators with nonlocal initial conditions. We prove the existence of solutions. The commonly used Lipschitz hypothesis for the perturbations is weakened to one-sided Lipschitz ones. We prove an existence result for the multipoint problems that cover periodic and antiperiodic cases. We give examples...

Rumor spreading of a SEIR model in complex social networks with hesitating mechanism

In the process of rumor spreading, controlling and killing rumor problem is of great importance on social networks. In this paper, we present a new SEIR (susceptible-exposed-infected-removed) rumor spreading model with hesitating mechanism. By using mean-field theory, the equilibrium of the model and the basic reproduction number R 0 $R_{0}$ are obtained. The global stability of...

Oscillation of second order difference equation with several super-linear neutral terms

In this article,we present some new sufficient conditions for the oscillation of all solutions of a second order difference equation with several super-linear neutral terms. The results obtained here extend or complement some of the known results reported in the literature. Examples illustrating the importance of the main results are included.

Competition and coexistence of a stochastic Holling II n -predator one-prey model

In this paper, we discuss a stochastic Holling II predator–prey model with n-predator competing for one prey. The existence of a positive solution is established by using the comparison theorem. We get the stochastic break-even concentration R ˜ i $\tilde{R}_{i}$ of each predator which determines the competition outcomes. When the noise intensity of the prey is small, the...

Existence results for multi-term fractional differential equations with nonlocal multi-point and multi-strip boundary conditions

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and multi-strip boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples. Some new results are also deduced by fixing the parameters involved in the...

Iterative learning control for MIMO parabolic partial difference systems with time delay

In this paper, the iterative learning control (ILC) technique is extended to multi-input multi-output (MIMO) systems governed by parabolic partial difference equations with time delay. Two types of ILC algorithm are presented for the system with state delay and input delay, respectively. The sufficient conditions for tracking error convergence are established under suitable...

Multiplicity results for a class of fractional differential equations with impulse

In this paper, we apply Morse theory, local linking arguments and the Clark theorem to a study of the multiplicity of nontrivial solutions for a class of impulsive fractional differential equations with Dirichlet boundary conditions.

Global Mittag-Leffler synchronization of delayed fractional-order memristive neural networks

This paper deals with the global Mittag-Leffler synchronization of fractional-order memristive neural networks (FMNNs) with time delay. Since the FMNNs are essentially a class of switched systems with irregular switching laws, it is more difficult to achieve synchronization than with the traditional neural networks. First, under the framework of fractional-order differential...