一類宏觀經濟學中的二階離散方程的正解的有界性

摘 要 利用LeraySchauder 延拓定理，使用對角化方法考慮了來源于宏觀經濟學中的一類二階差分方程的有界性.

關鍵詞 正解；對角化方法；邊值問題

中圖分類號 O175.7 文獻標識碼 A

Boundedness of Positive Solutions for Nonlinear Second Order Discrete Equations from Macroeconomics

MA Huili1，XU Jia2，MA Huifang3

(1.College of Economics and Management， Northwest Normal University，Lanzhou，Ganshu 730010，China;

2.College of Physical Education， Northwest Normal University，Lanzhou，Ganshu 730070，China;

3.Coolege of Mathematics and In for mation Science，Northwest Normal University，Lazhou，Gansu 730070，China)

Abstract By using the LeraySchauder continuation theorem and diagonalization method，this paperstudied the existence and boundedness of the positive solutions for the equation from macroeconomics.

Key words positive solution; diagonalization arguments; boundary value problems

1 Introduction

Set Z+={1，2，…}，N={0，1，…}，N(a)={a，a+1，…}，［a，b］={a，a+1，…，b}， a，b∈N，a＜b＜，［a，b］R={x∈R|a≤x≤b，a，b∈R)}.

Let X［a，b］={ψ|ψ:［a，b］→R} with the norm ‖ψ‖X［a，b］=max{|ψ(t)|t∈［a，b］}.

Let Y［a，b］={ψ|ψ:［a，b］→R} with the norm ‖ψ‖L［a，b］=∑bt=a|ψ(t)|.

Consider the following equation

Δ2u(k－1)+g(k，u(k)，Δu(k))=0.(1)

Equation of (1) and similar， arose in some of the earliest mathematical modelsof the macroeconomic “trade cycle”. For example， equation (1) generalizes the classic HasenSamuelson’s acceleratormultiplier model［1］， namely，

Yn+1=cYn+αc(Yn－Yn－1)+A0，

where the constant A0=C0+I0+G0 represents the sum of the minimum consumption， the “autonomous” investment and the fixed government spending in period n， and Yn is the outputGNP or national incomein period n. The net investment amount in the same period is given as In=αc(Yn－1－Yn－2).The constant c∈(0，1)represents Keynes’ “marginal propensity to consume” or the MPC， while the coefficient α ＞ 0 is the “accelerator”.

The linear model above improved the earlier Keynesian models in substantial new research. However， this model was soon found to be unsatisfactory and certain nonlinear models were subsequently proposed. For instance， rather than the linear Keynesian consumption C(Y)=cY+C0，Samuelson considered a nonlinear consumption function［2］. Some years later， Hicks proposed a model in which consumption was linear， but investment and output were both piecewise linear［3］. For other similar nonlinear models， the reader can refer the work［4-5］ and the references therein.

Few of the existing results in the literature seem directly applicable to equation (1) in setting basic questions such as boundedness and convergence. The objective of this paper is to investigate some of the mathematical properties ofequation(1). Specifically， we study the existence and boundedness of the positive solutions for the equation (1). To be convenient， wediscuss the existence of positive solution of the following corresponding boundary value problem

Δ2u(k－1)+g(k，u(k)，Δu(k))=0，k∈Z+，

u(0)=0，u is bounded on N.(2)

The main result of this paper is the following theorem.

Theorem 1Let g: Z+×R+×R+→R+. Assume we have the following

(H1) For any constant H＞0， there exists a nonnegative function ψH(k)，k∈(0， and a constant o≤γ＜1 with

g(k，u，v)≥ψH(k)vγ on Z+×0，H2R;

(H2) There exists functions p，q，r:R+→R+ such that

Q=∑k=1q(k)＜，Q1=∑k=1kq(k)＜

P1=∑k=1kp(k)＜

R=∑ R1=∑

and

|g(k，u，v)|≤p(k)|u|+q(k)|v|+r(k)， (k，u，v)∈R+3.

Then equation (2) has at least one solution provided P1+Q＜1.

2 Proof of the main results

Lemma 1Let e∈Y［1，n］ and x be a function such that

Δ2x(k－1)+e(k)=0，k∈［1，n］

and

x(0)=0，Δx(n)=0.

Then

‖Δx‖x［0，n］≤‖e‖L［1，n］.

Proof Since －Δ2x(k－1)=e(k)，k∈［1，n］ can be extended to

 －Δx(k)+Δx(k－1)=e(k)，k∈［1，n］， 

summing from k to n for both sides results in

Δ x(k－1)=∑nt=k［－Δ2x(t－1)］=∑nt=ke(t).

that is，

‖Δx‖x［0，n］=‖Δx‖x［0，n－1］≤‖e‖L［1，n］.

Lemma 2Let (H1) and (H2) hold. Let n be a positive integer and consider the boundary value problem

Δ2u(k－1)+g(k，u(k)，Δu(k))=0，k∈［1，n］，u(0)=0，Δu(n)=0.(3)

Then equation (3) has at least one positive solution yn∈X［0，n+1］ and there is a constant M＞0 independent ofn such that

∑kt=0［∑ns=t+1ψM(s)(b(s))γ］11－γ≤yn(k)≤M，

k∈［0，n］，

經 濟 數 學第 28卷第1期馬慧莉等：一類宏觀經濟學中的二階離散方程的正解的有界性

where b is some function satisfies 0≤b(t)＜1，t∈［1，n］ which will be defined later.

Proof Define a linear operator

Ln:D(Ln)X［0，n+1］→Y［1，n］

by setting

D(Ln)={x∈X［0，n+1］:x(0)=Δx(n)=0}，and for y∈D(Ln)，Lny(k)=－Δ2y(k－1).

We also define a nonlinear mapping N:X［0，n+1］→Y［1，n］by setting

(Nu)(k)=g(k，u(k)，Δu(k)).

We have from the fact N is bounded， mapping from X［0，n+1］ to Y［1，n］.

Next， it is easy to see that L:D(Ln)X［0，n+1］→Y［1，n］ is one to one mapping. Moreover， it follows easily by using ArzélaAscoli theorem that

(Ln)－1N:X［0，n+1］→X［0，n+1］ is a compact mapping.

We note y is a solution of equation (3) if and only if y is a fixed point of the equation

y=(Ln)－1Ny.

We apply the LeraySchauder continuation theorem［3］［6］ to obtain the existence of a solution for y=(Ln)－1Ny. To do this， it suffices to verify that the set of all possible solutions of the family of equations

Δ2u(k－1)+λg(k，u(k)，Δu(k))=0，k∈［1，n］，

u(0)=0，Δu(n)=0(4)

has a prior， bounded in X by a constant independent of λ∈(0，1).

Let y∈X［0，n+1］ be any solution of equation (4)， then

Δy(k)≥0， k∈［0，n］，

y(k)≥0， k∈［0，n+1］.

Moreover， we have from y(k+1)=∑kt=0Δy(t) that |y(k+1)|≤k‖Δy‖X［0，n］.

Applying lemma 1 and using equation (4)， we can get that

‖Δy‖X［0，n］≤‖g(k，y(k)，Δy(k))‖L［1，n］≤

‖p(k)y(k)‖L［1，n］+‖q(k)Δy(k)‖L［1，n］+

‖r(k)‖L［1，n］≤(‖(k－1)p(k)‖L［1，n］+

‖q‖L［1，n］)‖Δy‖X［0，n］+‖r‖L［1，n］≤

(P1+Q)‖Δy‖X［0，n］+R. 

So we have consequently

‖Δy‖X［0，n］≤R1－P1－Q:=M1.

From equation (4)， we have that

y(n)=λ∑nk=1∑nt=kg(t，y(t)，Δy(t))

=λ∑nt=1tg(t，y(t)，Δy(t)).

Moreover，

y(n+1)≤∑n+1t=1tg(t，y(t)，Δy(t))=

‖tg(t，y(t)，Δy(t))‖L［1，n+1］ ≤

‖tp(t)‖L［1，n+1］‖y‖X［0，n+1］ +

‖tq(t)‖L［1，n+1］‖Δy‖X［0，n］+

‖tr(t)‖L［1，n+1］≤P1‖y‖X［0，n+1］+

Q1‖Δy‖X［0，n］+R1，

‖y‖X［0，n+1］≤Q1M1+R11－P1:=M2，

Thus equation (4) has a solution yn with ‖yn‖X［0，n+1］≤M2. In fact，

0≤yn(k)≤M2， k∈［0，n+1］;

0≤Δyn(k)≤M1， k∈［0，n］.(5)

Finally， it’s obvious that M1 and M2 are independent of n∈Z+. Now (H1) guarantees the existence of a function ψM(k)， which is positive on (0，

SymboleB@ )，and a constant γ∈［0，1］with g(k，yn(k)，Δyn(k))≥ψM(k)［Δyn(k)］γ for (k，yn(k)，Δyn(k))∈［1，n］×［0，M］2R，where M=max{M1，M2}.

Of course， from equation(4) and the fact that Δyn(k)≥0 on ［0，n］， we have

－Δ2yn(k－1)≥ψM(k)［Δyn(k)］γ.

Sum from k+1 to n to obtain

－∑nt=k+1Δ2yn(t－1)≥∑nt=k+1ψM(t)［Δyn(t)］γ，

that is，

Δyn(k)≥∑nt=kψM(t)［Δyn(t)］γ. (6)

While Δ2yn(t－1)≤0 implies that there exists 0≤b(t)＜1 such that

Δyn(t)≥b(t)Δyn(k)， t∈{k+1，…，n}，

which combined with equation (6) lead to

Δyn(k)≥∑nt=k+1ψM(t)(b(t))γ［Δyn(k)］γ，

Δyn(k)≥［∑nt=k+1ψM(t)(b(t))γ］11－γ.

Thus，

yn(k)≥∑kt=0［∑ns=t+1ψM(s)(b(s))γ］11－γ.

Proof of the main theorem. From equation (4) and (5)， we know that

0≤－Δ2yn(k－1)≤φ(k)，k∈［1，n］，

where φ(k)=［p(k)+q(k)］M+r(k).

In addition， we have

Δyn(k－1)≤∑nt=kφ(t)≤∑

To show equation (2) has a solution， we will apply the diagonalization argument. Let

un(k)=yn(k)，k∈［0，n+1］，

yn(n)，k∈［n+1，∞］.

Note that

0≤un(k)≤M，k∈［0，n+1］;

0≤Δun(k)≤M，k∈［0，n］.

From the definition of un， we get

|Δun(k1)－Δun(k2)|≤∑k2t=k1|φ(t)|， k1，k2∈N.

References

［1］ P A Samuelson. Interaction between the multiplier analysis and the principle of acceleration［J］. Review of Econ. Stat.， 1939， 21(2): 75-78.

［2］ P ASamuelson. A synthesis of the principle of acceleration and the multiplier［J］. JPolitical Ecno， 1939，47(6):786-797.

［3］ J R Hicks. A contribution to the theory of the trade cycle［M］. Oxford:Oxford University Press，1950.

［4］ P N V Tu. Dynamical systems: an introduction with applications in economics and biology， 2nd edn.［M］. New York:Springer，1994.

［5］ Hassan Sedaghat. A class of nonlinear second order difference equations from macroeconomics［J］. Nonlinear Anal.， 1997， 29(5): 593-603.

［6］ D Guo.Nonlinear functional analysis［M］.Jinan:Shandong Science and Technology Press，2002.(In Chinese)

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