MATHEMATICAL MODEL FOR THE ENTERPRISE COMPETITIVE ABILITY AND PERFORMANCE THROUGH A PARTICULAR EMDEN-FOWLER EQUATION u′?n?q?1u(n)q=0?

Yue-Loong CHANG

Department of Phycology,National Chengchi University,Taipei,China

E-mail:cyl.88054@gmail.com

Meng-Rong LI?

Department of Mathematical Sciences,National Chengchi University,Taipei,China

E-mail:liwei@math.nccu.edu.tw

C.Jack YUE Yong-Shiuan LEE

Department of Statisitics,National Chengchi University,Taipei,China

E-mail:csyue@nccu.edu.tw;99354501@nccu.edu.tw

Tsung-Jui CHIANG-LIN

Graduate Institute of Finance,National Taiwan University of Science and Technology,Taipei,China

E-mail:peter006189@gmail.com

Abstract In this article,we work with the ordinary equation u′′?n?q?1u(n)q=0 and learn some interesting phenomena concerning the blow-up and the blow-up rate of solution to the equation.

Key words estimate;life-span;blow-up;blow-up rate;competitive ability;performance

1 Introduction

In the field of industrial and organizational psychology,understanding and modelling the relationship between the performance and the competitive ability of the company and how to achieve the maximum performance and the maximum competitive ability are critical issues.We try to apply Emden-Fowler equation to propose the reasonable and appropriate model to describe them.In the existing study,Chang and Li[11]tried a relatively simpler model of Emden-Fowler equation and obtained some results depicting the interesting phenomena of the surveying companies on some conditions.In this study,we try to propose a more generalized model through a particular Emden-Fowler equation to characterize the relationship between the performance and the competitive ability.560 enterprises are surveyed to study their competitive abilities and the performances.We prove in the following that no global solutions exist and the life-span is finite under three different conditions of the energy respectively.From the economic viewpoint,the positive energy represents the economic growth or the expansion;the zero energy represents the economic stagnation;the negative energy represents the recession.Therefore,the results shown below means that the benchmark enterprise would attain its maximum competitive ability and performance on certain condition no matter what the economic environment is.

The relationship between the competitive ability(force,F(P(n)))and the performance(P(n))can be represented in the following form.

There exist positive constants k(n)>0 such that

where n is the number of the departments or the main unit commanders in the surveying benchmark enterprises,and F is proportional to the second derivative of P with respect to n.

Now,we consider stationary,0-dimensional semilinear wave equation

It is clear that the functions n?q?1u(n)qis locally Lipschitz,hence by the standard theory,the local existence of classical solutions is applicable to equation(1.1).

We discuss problem(1.1)in three parts:nonexistence of global solution with negative energy,positive energy,and zero energy.

Notation and Fundamental Lemmas

First,we make a substitution

Set s=lnn,v(n)=C(s),then nv′(n)=Cs(s),n2v′′(n)=Css(s)? Cs(s).

For a given function C in this work,we use the following abbreviations

Equation(1.1)can be transformed into the form

From some elementary calculations we obtain the following lemmas.

Lemma 1.1 Suppose that C is the solution of(1.2),then we have

Lemma 1.2 Suppose that C is the solution of(1.2),then we have

Thus(1.7)and(1.8)are obtained.

By(1.6)and(1.7),we get

Therefore(1.9)is followed.

Now to prove(1.10).According to(1.5)and(1.6),we obtain

To(1.11),by(1.6)and the definitions of

The last formulation is equivalent to the assertion(1.11).Thus Lemma 1.2 is completely proved. ?

Definition 1.3 A function g:R→R has a blow-up rate q means that g exists only infinite time,that is,there is a finite number T?such that the following is valid

and there exists a non-zero β∈R with

in this case β is called the blow-up constant of g.

The following lemma is easy to prove so we omit the arguments.

Lemma 1.4 If g(t)and h(t,r)are continuous with respect to their variables and theh(t,r)dr exists,then

2 Nonexistence of Global Solution in Time Under E<0

In this section,we want to show the global solution of problem(1.1)does not exist under negative energy E<0.We have the following result.

Theorem 2.1 If T is the life-span of u and u is the positive solution of problem(1.1)with E<0,then T is finite.This means that the global solution of(1.1)does not exist for

Remark 2.2 This means that such benchmark enterprises would attain their maximal competitive abilities and the performances under the condition

Proof By(1.10)of Lemma 1.2,we obtain

(i)as(0)>0.

for

According to(1.6)of Lemma 1,for s≥s0,

because that for s≥s1,

and

therefore,there exists

so that

This means that the solution for problem(1.1)does not exist for all t≥1 and the life-span T?of u is finite with T?≤lnS?1.

(ii)as(0)≤0.By(1.10)of Lemma 1.2,we obtain

for all large s≥s2,where s2is given by

and

for all s≥s3,s3can be obtained by

According to(1.6)of Lemma 1.1,for s≥s3,

thus it is obtained that there exists s4>s3such that for s≥s4,

therefore there exists

such that

This means that the solution for problem(1.1)does not exist for all t≥1 and the life-span T?of u is finite with T?≤ln?

3 Nonexistence of Global Solution in Time Under E>0

In this section,we want to show that the global solution of problem(1.1)does not exist under positive energy E>0.We have the following result.

Theorem 3.1 If T is the life-span of u and u is the positive solution of problem(1.1)with E>0 and u0(u1?u0)≥0,then T is finite.This means that the global solution of(1.1)does not exist for

If T is the life-span of u and u is the positive solution of problem(1.1)with E>0,u0(u1?u0)<0 and E+u0(u1?u0)>0,then T is finite.This means that the global solution of(1.1)does not exist for

Proof Using Lemma 1.2,(1.9),E>0,and u0(u1?u0)≥0,we obtain

for some large s4,s≥s5,because that as(0)+2E>0.

By the similar argumentations in Theorem 2.1,the assertions in Theorem 3.1 can be obtained.

4 Nonexistence of Global Solution in Time Under E=0

In this section,we want to show the global solution of problem(1.1)does not exist under zero energy E=0.We have the following result.

Theorem 4.1 If T is the life-span of u and u is the positive solution of problem(1.1)with E=0 and u0(u1?u0)>0,then T is finite.This means that the global solution of(1.1)does not exist for

Proof Using Lemma 1.2,(1.10),E=0 and u0(u1?u0)>0,we obtain

By(1.7)of Lemma 1.2,

thus we obtain the fact that there exists s7>s6such that for s≥s7,

therefore there exists

such that

This means that the solution for problem(1.1)does not exist for all t≥1 and the life-span T?of u is finite with T?≤ln?

AcknowledgementsWe want to thank Prof.Long-Yi Tsai and Prof.Tai-Ping Liu for their continuously encouragement and their opinions to this work;thanks to NSC and Grand Hall for their financial support and to the referees for their interest and helpful comments to this work.