**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**186

# Search results for: continuation theorem

##### 186 Periodic Solutions for a Third-order p-Laplacian Functional Differential Equation

**Authors:**
Yanling Zhu,
Kai Wang

**Abstract:**

By means of Mawhin’s continuation theorem, we study a kind of third-order p-Laplacian functional differential equation with distributed delay in the form: ϕp(x (t)) = g t, 0 −τ x(t + s) dα(s) + e(t), some criteria to guarantee the existence of periodic solutions are obtained.

**Keywords:**
p–Laplacian,
distributed delay,
periodic solution,
Mawhin's continuation theorem.

##### 185 Multiple Positive Periodic Solutions to a Predator-prey system with Harvesting Terms and Holling II Type Functional Response

**Authors:**
Pan Wang,
Yongkun Li

**Abstract:**

In this paper, a periodic predator-prey system with harvesting terms and Holling II type functional response is considered. Sufficient criteria for the existence of at least sixteen periodic solutions are established by using the well known continuation theorem due to Mawhin. An example is given to illustrate the main result.

**Keywords:**
Periodic solution,
predator-prey system,
harvesting terms,
continuation theorem.

##### 184 On the Maximum Theorem: A Constructive Analysis

**Authors:**
Yasuhito Tanaka

**Abstract:**

**Keywords:**
Maximum theorem,
Constructive mathematics,
Sequentially
locally at most one maximum.

##### 183 Existence of Periodic Solution for p-Laplacian Neutral Rayleigh Equation with Sign-variable Coefficient of Non Linear Term

**Authors:**
Aomar Anane,
Omar Chakrone,
Loubna Moutaouekkil

**Abstract:**

As p-Laplacian equations have been widely applied in field of the fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Mawhin’s continuation theorem, we study the existence of periodic solutions for p-Laplacian neutral Rayleigh equation (ϕp(x(t)−c(t)x(t − r))) + f(x(t)) + g1(x(t − τ1(t, |x|∞))) + β(t)g2(x(t − τ2(t, |x|∞))) = e(t), It is meaningful that the functions c(t) and β(t) are allowed to change signs in this paper, which are different from the corresponding ones of known literature.

**Keywords:**
periodic solution,
neutral Rayleigh equation,
variable
sign,
Deviating argument,
p-Laplacian,
Mawhin’s continuation.

##### 182 Fermat’s Last Theorem a Simple Demonstration

**Authors:**
Jose William Porras Ferreira

**Abstract:**

This paper presents two solutions to the Fermat’s Last Theorem (FLT). The first one using some algebraic basis related to the Pythagorean theorem, expression of equations, an analysis of their behavior, when compared with power and power and using " the “Well Ordering Principle” of natural numbers it is demonstrated that in Fermat equation . The second one solution is using the connection between and power through the Pascal’s triangle or Newton’s binomial coefficients, where de Fermat equation do not fulfill the first coefficient, then it is impossible that:

z^{n}=x^{n}+y^{n} for n>2 and (x, y, z) E Z^{+} - {0}

**Keywords:**
Fermat’s Last Theorem,
Pythagorean Theorem,
Newton Binomial Coefficients,
Pascal’s Triangle,
Well Ordering Principle.

##### 181 Fourier Spectral Method for Analytic Continuation

**Authors:**
Zhenyu Zhao,
Lei You

**Abstract:**

The numerical analytic continuation of a function f(z) = f(x + iy) on a strip is discussed in this paper. The data are only given approximately on the real axis. The periodicity of given data is assumed. A truncated Fourier spectral method has been introduced to deal with the ill-posedness of the problem. The theoretic results show that the discrepancy principle can work well for this problem. Some numerical results are also given to show the efficiency of the method.

**Keywords:**
Analytic continuation,
ill-posed problem,
regularization method Fourier spectral method,
the discrepancy principle.

##### 180 Periodic Solutions for a Delayed Population Model on Time Scales

**Authors:**
Kejun Zhuang,
Zhaohui Wen

**Abstract:**

This paper deals with a delayed single population model on time scales. With the assistance of coincidence degree theory, sufficient conditions for existence of periodic solutions are obtained. Furthermore, the better estimations for bounds of periodic solutions are established.

**Keywords:**
Coincidence degree,
continuation theorem,
periodic solutions,
time scales

##### 179 Multiple Periodic Solutions for a Delayed Predator-prey System on Time Scales

**Authors:**
Xiaoquan Ding,
Jianmin Hao,
Changwen Liu

**Abstract:**

This paper is devoted to a delayed periodic predatorprey system with non-monotonic numerical response on time scales. With the help of a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results improve and generalize some known ones.

**Keywords:**
Predator-prey system,
periodic solution,
time scale,
delay,
coincidence degree.

##### 178 Static Voltage Stability Assessment Considering the Power System Contingencies using Continuation Power Flow Method

**Authors:**
Mostafa Alinezhad,
Mehrdad Ahmadi Kamarposhti

**Abstract:**

According to the increasing utilization in power system, the transmission lines and power plants often operate in stability boundary and system probably lose its stable condition by over loading or occurring disturbance. According to the reasons that are mentioned, the prediction and recognition of voltage instability in power system has particular importance and it makes the network security stronger.This paper, by considering of power system contingencies based on the effects of them on Mega Watt Margin (MWM) and maximum loading point is focused in order to analyse the static voltage stability using continuation power flow method. The study has been carried out on IEEE 14-Bus Test System using Matlab and Psat softwares and results are presented.

**Keywords:**
Contingency,
Continuation Power Flow,
Static Voltage Stability,
Voltage Collapse.

##### 177 Analysis for a Food Chain Model with Crowley–Martin Functional Response and Time Delay

**Authors:**
Kejun Zhuang,
Zhaohui Wen

**Abstract:**

This paper is concerned with a nonautonomous three species food chain model with Crowley–Martin type functional response and time delay. Using the Mawhin-s continuation theorem in theory of degree, sufficient conditions for existence of periodic solutions are obtained.

**Keywords:**
Periodic solutions,
coincidence degree,
food chain model,
Crowley–Martin functional response.

##### 176 Multiple Positive Periodic Solutions to a Periodic Predator-Prey-Chain Model with Harvesting Terms

**Authors:**
Zhouhong Li,
Jiming Yang

**Abstract:**

In this paper, a class of predator-prey-chain model with harvesting terms are studied. By using Mawhin-s continuation theorem of coincidence degree theory and some skills of inequalities, some sufficient conditions are established for the existence of eight positive periodic solutions. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.

**Keywords:**
Positive periodic solutions,
Predator-prey-chain model,
coincidence degree,
harvesting term.

##### 175 Four Positive Almost Periodic Solutions to an Impulsive Delayed Plankton Allelopathy System with Multiple Exploit (or Harvesting) Terms

**Authors:**
Fengshuo Zhang,
Zhouhong Li

**Abstract:**

**Keywords:**
Almost periodic solutions,
plankton allelopathy
system,
coincidence degree,
impulse.

##### 174 Periodic Solutions for a Higher Order Nonlinear Neutral Functional Differential Equation

**Authors:**
Yanling Zhu

**Abstract:**

In this paper, a higher order nonlinear neutral functional differential equation with distributed delay is studied by using the continuation theorem of coincidence degree theory. Some new results on the existence of periodic solutions are obtained.

**Keywords:**
Neutral functional differential equation,
higher order,
periodic solution,
coincidence degree theory.

##### 173 Periodic Solutions for a Two-prey One-predator System on Time Scales

**Authors:**
Changjin Xu

**Abstract:**

In this paper, using the Gaines and Mawhin,s continuation theorem of coincidence degree theory on time scales, the existence of periodic solutions for a two-prey one-predator system is studied. Some sufficient conditions for the existence of positive periodic solutions are obtained. The results provide unified existence theorems of periodic solution for the continuous differential equations and discrete difference equations.

**Keywords:**
Time scales,
competitive system,
periodic solution,
coincidence degree,
topological degree.

##### 172 Positive Almost Periodic Solutions for Neural Multi-Delay Logarithmic Population Model

**Authors:**
Zhouhong Li

**Abstract:**

In this paper, by applying Mawhin-s continuation theorem of coincidence degree theory, we study the existence of almost periodic solutions for neural multi-delay logarithmic population model and obtain one sufficient condition for the existence of positive almost periodic solution for the above equation. An example is employed to illustrate our result.

**Keywords:**
Almost periodic solution,
Multi-delay,
Logarithmic population model,
Coincidence degree.

##### 171 2n Positive Periodic Solutions to n Species Non-autonomous Lotka-Volterra Competition Systems with Harvesting Terms

**Authors:**
Yongkun Li,
Kaihong Zhao

**Abstract:**

By using Mawhin-s continuation theorem of coincidence degree theory, we establish the existence of 2n positive periodic solutions for n species non-autonomous Lotka-Volterra competition systems with harvesting terms. An example is given to illustrate the effectiveness of our results.

**Keywords:**
Positive periodic solutions,
Lotka-Volterra competition system,
coincidence degree,
harvesting term.

##### 170 Positive Periodic Solutions in a Discrete Competitive System with the Effect of Toxic Substances

**Authors:**
Changjin Xu,
Qianhong Zhang

**Abstract:**

In this paper, a delayed competitive system with the effect of toxic substances is investigated. With the aid of differential equations with piecewise constant arguments, a discrete analogue of continuous non-autonomous delayed competitive system with the effect of toxic substances is proposed. By using Gaines and Mawhin,s continuation theorem of coincidence degree theory, a easily verifiable sufficient condition for the existence of positive solutions of difference equations is obtained.

**Keywords:**
Competitive system,
periodic solution,
discrete time delay,
topological degree.

##### 169 Existence of Periodic Solutions in a Food Chain Model with Holling–type II Functional Response

**Authors:**
Zhaohui Wen

**Abstract:**

In this paper, a food chain model with Holling type II functional response on time scales is investigated. By using the Mawhin-s continuation theorem in coincidence degree theory, sufficient conditions for existence of periodic solutions are obtained.

**Keywords:**
Periodic solutions,
food chain model,
coincidence degree,
time scales.

##### 168 An Algorithm for Computing the Analytic Singular Value Decomposition

**Authors:**
Drahoslava Janovska,
Vladimir Janovsky,
Kunio Tanabe

**Abstract:**

A proof of convergence of a new continuation algorithm for computing the Analytic SVD for a large sparse parameter– dependent matrix is given. The algorithm itself was developed and numerically tested in [5].

**Keywords:**
Analytic Singular Value Decomposition,
large sparse
parameter–dependent matrices,
continuation algorithm of a predictorcorrector
type.

##### 167 A New Method to Solve a Non Linear Differential System

**Authors:**
Seifedine Kadry

**Abstract:**

In this article, our objective is the analysis of the resolution of non-linear differential systems by combining Newton and Continuation (N-C) method. The iterative numerical methods converge where the initial condition is chosen close to the exact solution. The question of choosing the initial condition is answered by N-C method.

**Keywords:**
Continuation Method,
Newton Method,
Finite Difference Method,
Numerical Analysis and Non-Linear partial Differential Equation.

##### 166 Multiple Positive Periodic Solutions of a Competitor-Competitor-Mutualist Lotka-Volterra System with Harvesting Terms

**Authors:**
Yongkun Li,
Erliang Xu

**Abstract:**

In this paper, by using Mawhin-s continuation theorem of coincidence degree theory, we establish the existence of multiple positive periodic solutions of a competitor-competitor-mutualist Lotka-Volterra system with harvesting terms. Finally, an example is given to illustrate our results.

**Keywords:**
Positive periodic solutions,
competitor-competitor mutualist Lotka-Volterra systems,
coincidence degree,
harvesting term.

##### 165 Existence of Multiple Positive Periodic Solutions to n Species Nonautonomous Lotka-Volterra Cooperative Systems with Harvesting Terms

**Authors:**
Kaihong Zhao

**Abstract:**

In this paper, the existence of 2n positive periodic solutions for n species non-autonomous Lotka-Volterra cooperative systems with harvesting terms is established by using Mawhin-s continuation theorem of coincidence degree theory and matrix inequality. An example is given to illustrate the effectiveness of our results.

**Keywords:**
Multiple positive periodic solutions,
Nonautonomous
Lotka-Volterra cooperative system,
Coincidence degree,
Harvesting
term.

##### 164 Power Series Form for Solving Linear Fredholm Integral Equations of Second Order via Banach Fixed Point Theorem

**Authors:**
Adil AL-Rammahi

**Abstract:**

In this paper, a new method for solution of second order linear Fredholm integral equation in power series form was studied. The result is obtained by using Banach fixed point theorem.

**Keywords:**
Fredholm integral equation,
power series,
Banach fixed point theorem,
Linear Systems.

##### 163 Positive Solutions for Semipositone Discrete Eigenvalue Problems via Three Critical Points Theorem

**Authors:**
Benshi Zhu

**Abstract:**

**Keywords:**
Discrete eigenvalue problems,
positive solutions,
semipositone,
three critical points theorem

##### 162 Constructive Proof of Tychonoff’s Fixed Point Theorem for Sequentially Locally Non-Constant Functions

**Authors:**
Yasuhito Tanaka

**Abstract:**

We present a constructive proof of Tychonoff’s fixed point theorem in a locally convex space for uniformly continuous and sequentially locally non-constant functions.

**Keywords:**
sequentially locally non-constant functions,
Tychonoff’s
fixed point theorem,
constructive mathematics.

##### 161 Periodic Solutions of Recurrent Neural Networks with Distributed Delays and Impulses on Time Scales

**Authors:**
Yaping Ren,
Yongkun Li

**Abstract:**

In this paper, by using the continuation theorem of coincidence degree theory, M-matrix theory and constructing some suitable Lyapunov functions, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of recurrent neural networks with distributed delays and impulses on time scales. Without assuming the boundedness of the activation functions gj, hj , these results are less restrictive than those given in the earlier references.

**Keywords:**
Recurrent neural networks,
global exponential stability,
periodic solutions,
distributed delays,
impulses,
time scales.

##### 160 Existence and Global Exponential Stability of Periodic Solutions of Cellular Neural Networks with Distributed Delays and Impulses on Time Scales

**Authors:**
Daiming Wang

**Abstract:**

In this paper, by using Mawhin-s continuation theorem of coincidence degree and a method based on delay differential inequality, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of cellular neural networks with distributed delays and impulses on time scales. The results of this paper generalized previously known results.

**Keywords:**
Periodic solutions,
global exponential stability,
coincidence degree,
M-matrix.

##### 159 Multiple Positive Periodic Solutions of a Delayed Predatory-Prey System with Holling Type II Functional Response

**Authors:**
Kaihong Zhao,
Jiuqing Liu

**Abstract:**

In this letter, we considers a delayed predatory-prey system with Holling type II functional response. Under some sufficient conditions, the existence of multiple positive periodic solutions is obtained by using Mawhin’s continuation theorem of coincidence degree theory. An example is given to illustrate the effectiveness of our results.

**Keywords:**
Multiple positive periodic solutions,
Predatory-prey
system,
Coincidence degree,
Holling type II functional response.

##### 158 Weyl Type Theorem and the Fuglede Property

**Authors:**
M. H. M. Rashid

**Abstract:**

**Keywords:**
Fuglede Property,
Weyl’s theorem,
generalized
derivation,
Aluthge Transformation.

##### 157 Extremal Properties of Generalized Class of Close-to-convex Functions

**Authors:**
Norlyda Mohamed,
Daud Mohamad,
Shaharuddin Cik Soh

**Abstract:**

**Keywords:**
Argument of f ′(z) ,
Carathéodory Function,
Closeto-
convex Function,
Distortion Theorem,
Extremal Properties