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## Stabilization of company’s income modeled by a system of discrete stochastic equations

### Advances in Difference Equations (2014-11-17) 2014: 1-8 , November 17, 2014

The paper deals with a system of difference equations where the coefficients depend on Markov chains. The functional equations for a particular density and the moment equations for the system are derived and used in the investigation of mode stability of company’s income. An application of the results is illustrated by two models.

## Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales

### Advances in Difference Equations (2009-06-16) 2009: 1-12 , June 16, 2009

This paper studies Sturm-Picone comparison theorem of second-order linear equations on time scales. We first establish Picone identity on time scales and obtain our main result by using it. Also, our result unifies the existing ones of second-order differential and difference equations.

## The equivalence between singular point quantities and Liapunov constants on center manifold

### Advances in Difference Equations (2012-06-07) 2012: 1-12 , June 07, 2012

The algorithm of singular point quantities for an equilibrium of three-dimensional dynamics system is studied. The explicit algebraic equivalent relation between singular point quantities and Liapunov constants on center manifold is rigorously proved. As an example, the calculation of singular point quantities of the Lü system is applied to illustrate the advantage in investigating Hopf bifurcation of three-dimensional system.

*MR (2000) Subject Classification*: 34C23, 34C28, 37Gxx.

## Numerical scheme and dynamic analysis for variable-order fractional van der Pol model of nonlinear economic cycle

### Advances in Difference Equations (2016-07-25) 2016: 1-11 , July 25, 2016

Considering the fact that the memory in economic series changes with dynamic economic environment, this paper is devoted to the proposal of a variable-order fractional van der Pol model (VOFVDPM), where the order of the derivative is replaced by a time-dependent function. A numeric scheme for this model is designed by the Adams-Bashforth-Moulton method. The dynamic behaviors of the VOFVDPM with linear and periodic variable-order functions are investigated through numerical experiment. Some dynamic characteristics of the VOFVDPM that do not exist in a fractional order van der Pol model are discovered in the numerical simulation, such as existing limit point when the linear order functions have the same ranges and opposite slopes.

## Stability of Quartic Functional Equations in the Spaces of Generalized Functions

### Advances in Difference Equations (2009-02-10) 2009: 1-16 , February 10, 2009

We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.

## Existence results for a functional boundary value problem of fractional differential equations

### Advances in Difference Equations (2013-08-07) 2013: 1-25 , August 07, 2013

In this paper, a functional boundary value problem of fractional differential equations is studied. Based on Mawhin’s coincidence degree theory, some existence theorems are obtained in the case of nonresonance and the cases of and at resonance.

## Elementary Proof of Yu. V. Nesterenko Expansion of the Number Zeta(3) in Continued Fraction

### Advances in Difference Equations (2010-03-02) 2010: 1-11 , March 02, 2010

Yu. V. Nesterenko has proved that , , , , , , and for ; , , and , for His proof is based on some properties of hypergeometric functions. We give here an elementary direct proof of this result.

## Diffusive induced global dynamics and bifurcation in a predator-prey system

### Advances in Difference Equations (2017-10-11) 2017: 1-25 , October 11, 2017

In this paper, a diffusive Leslie-type predator-prey model is investigated. The existence of a global positive solution, persistence, stability of the equilibria and Hopf bifurcation are studied respectively. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain.

## Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equation

### Advances in Difference Equations (2010-03-01) 2010: 1-26 , March 01, 2010

The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.

## The modified two-dimensional Toda lattice with self-consistent sources

### Advances in Difference Equations (2017-09-11) 2017: 1-16 , September 11, 2017

In this paper, we derive the Grammian determinant solutions to the modified two-dimensional Toda lattice, and then we construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure. We show the integrability of the modified two-dimensional Toda lattice with self-consistent sources by presenting its Casoratian and Grammian structure of the N-soliton solution. It is also demonstrated that the commutativity between the source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice.