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Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 792307, 7 pages

http://dx.doi.org/10.1155/2014/792307

## A Model for Influence of Nuclear-Electricity Industry on Area Economy

School of Mathematics and Physics, Nanhua University, Hengyang 421001, China

Received 11 November 2013; Accepted 3 January 2014; Published 18 February 2014

Academic Editor: Chuangxia Huang

Copyright © 2014 Zigen Ouyang and Hui Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A model for Influence of Nuclear-Electricity Industry on Area Economy is established in this paper; we obtain an existence condition of solutions. Further, the approximation conditions are also constructed; the economic aggregate will tends to the equilibriums point as time tend to infinity.

#### 1. Introduction

As a clean energy resource, nuclear electricity is developed more and more quickly in recent years; many nuclear-electricity stations are under construction or planned to construct in China, which will be more beneficial to the development of the economy of the country. On one hand, the development of economy will not be restricted by the absence of electricity. On the other hand, the output of coal electricity will be reduced; then our resource of coal can be saved largely.

With the development of nuclear-electricity speedup in China, many regional governments hope that there are some nuclear-electricity stations which could be established in their provinces or counties. This is because if a nuclear-electricity station is established in their provinces or counties, their regional economy will develop faster and faster.

By the time, many authors investigated the nuclear-electricity industry [1–11], and more and more authors are interested in area economy [12–20]. However, few authors have investigated how much influence of nuclear-electricity industry on area economy is. Recently, Lixin [21] investigated the economy of source energy of Jiangsu Province. Using the method of dynamics systems, he brought light to the relation of area economy of Jiangsu Province with energy resource.

In this paper, we investigate the influence of nuclear-electricity industry on area economy. First, we establish a mathematics model, show the existence of the solutions of the model, investigate the property of solutions, and obtain how much the influence is. For convenience, we consider the influence of nuclear-electricity industry on a county economy. For example, we consider the influence of nuclear-electricity station of Taojiang on the economy of Taojiang county.

The paper is arranged as follows. In Section 2, we will establish a model about nuclear-electricity economy. We predigest the model in Section 3. In Section 4, We will investigate the existence of the solutions of the above model. We will discuss the approximation of solutions in Section 5. Finally, we will give a conclusion.

#### 2. A Model for the Influence of Nuclear-Electricity Industry on Area Economy

As we know, the construction of nuclear-electricity station is a project which is mastered by the country; at the same time, it is also a big project. The construction of nuclear-electricity station is composed of the following four parts mainly: technology, equipment, component, and material.

It is easy to know that technology is usually fetched in, the type of Taojiang nuclear-electricity station is AP-1000, and this technology is fetched from America. Equipment and component are domestic; they will be assembled into a group. Therefore, technology, equipment, and component will not affect the economy of Taojiang county. But material is only associate domain, such as cement and rock. Because there is a big industry of cement in Taojiang, so the export of cement and rock can advance the development of economy of Taojiang county, but it is only for a moment; once the nuclear-electricity station has been built, their influence on the economy of Taojiang county will disappear for a long time.

Now, we consider the deriving domains which includes real estate, service, and tour. These domains will influence the area economy of Taojiang county for a long time.

Through this paper, we suppose that when we begin to build the nuclear-electricity station.

First, we consider the real estate domain. The real estate only influences the economy of Taojiang until Taojiang nuclear-electricity station has been constructed. Once Taojiang nuclear-electricity station building starts, the influence of real estate on Taojiang will become little. Thus, we do not consider the influence of real estate on Taojiang in this paper and only need to consider service and tour domain.

Second, we consider the service domain. We suppose that the gross economy of the service domain is at time . The service domain will be developed when we start building the nuclear-electricity station; that is, , as . But, during the construction of building nuclear-electricity station, the gross of economy of the service domain is small. years later, the nuclear-electricity station begins running and the gross of economy of the service domain will stabilize. Thus, the gross of economy of the service domain can be expressed as the following: where is a positive function and is the upper limit of the gross of economy of the service domain ; that is, if , then will be decreasing.

Finally, we consider the tour domain. Since the type of pile of Taojiang nuclear-electricity station is AP-1000, which is first used in inner-continent nuclear-electricity station in China. Therefore, many tourists will visit Taojiang nuclear-electricity station. We suppose that the gross of economy of the tour domain is at time ; then where is a positive function, , which is the gross of economy of the tour domain for , and is the upper limit of the gross of economy of the tour domain ; that is, if , then will decreasing.

In fact, , influence each other. As the nuclear-electricity station begins running, the development of and will grow. If increases or decreases, then will also increase or decrease. But, if , then decreases. From (1), it follows that where , .

Similarly, where , .

According to (3)-(4), the the gross of area economy can be described as where , , , , , and , .

#### 3. The Predigestion of Model (5)

It is obvious that the equilibrium points of (5) are and , and is a positive equilibrium, so we only consider the positive equilibrium . Let Then

Since , , then , . It is obvious that if and only if . It is obvious that if and only if .

Substituting (7) into (5), we obtain It is easy to show that is an equilibrium point of (8). Let Then (8) can be rewritten as Let . We consider the following initial condition: Therefore, system (12) is equivalent to system (5); by the time, it is equivalent to (8) also. Therefore, we only need to investigate system (12)-(13).

It is obvious that vector function satisfies the following:(i)if , then ,(ii)if , then .

#### 4. The Continuous Solutions of Model (12)

Integrating both sides of (12) from to , it follows that where . Then is a continuous solution of (12)-(13) if and only if it is a continuous solution (14).

Theorem 1. *Assume that are continuous and positive function. Suppose that there exists a positive constant such that
**
where
**
Then (12) has a continuous solution in .*

*Proof. *Since is a continuous solution of (12)-(13) if and only if it is a continuous solution (14), we only need to show that (14) has a continuous solution.

Let
Define an operator as follows:
It is easy to show that is a nonempty closed convex set. Now, we prove that maps into itself; that is, for any , .

From (18), it is easy to show that
which implies
Combining (11) and (12) with (21), we have
From (14), (16), (22), and it follows that
which means that maps into itself.

Defining a sequence in ,

It is easy to show that is a sequence of measurable functions, and . Thus there exists a subsequence of and a function such that

By the continuity of , then the function is continuous for . It follows from Lebesgue's dominated convergence theorem that satisfies

which means that is a continuous solution of (14) for . Using the extension of solution repeatedly, satisfies

which is a continuous solution of (14) for . The proof is complete.

*Remark 2. *From conditions (15) and (16), we may choose the number small enough such that large enough. For example, we take , ; then . In fact, if , then , which implies that it is possible that . Therefore, we only need to control the number and not .

#### 5. Approximation of Model (12)

*Definition 3. *A solution of system (12) is oscillatory if and only if there is a nontrivial component of being oscillatory; one calls system (12) oscillatory if every solution of (12) is oscillatory.

*Remark 4. *If a vector solution is nonoscillatory, then every nontrivial component of (12) is eventually positive or eventually negative.

*Definition 5. *A solution of system (12) is oscillatory strongly if and only if every nontrivial component is oscillatory; one calls system (12) oscillatory strongly if every solution of (12) is oscillatory strongly.

For convenience, we give some hypothesis as follows: (H): are positive continuous functions, do not vanish to zero, and satisfy

Usually, we consider the influences of nuclear-electricity industry on area economy after the nuclear-electricity station has been established, and the nuclear-electricity station can be running many years. So, we may ignore the time delays and . Thus, we take in this section, and (12) can be rewritten as follows:

We have the following main theorem.

Theorem 6. *Assume that . Further, suppose that ( H) holds. Then, every solution of (12) satisfies .*

*Proof. *We divide the proof into the following two steps.*Step 1*. Prove that all solutions of (29)-(30) are nonoscillatory.

Assume that is an oscillatory solution of (29)-(30); for convenience, we suppose that is oscillatory.

Suppose that is also oscillatory. By defining oscillatory, there exist two sequences and such that , . Then, there exist two points or such that
For convenience, we suppose that (the proof of the case is similar). Assume that ; for convenience, we assume that (the case of is similar); then we have from (29)-(30) that , , which follows that

If , then ; it implies from (30) that , which contradicts the fact that and (32).

Thus, the point is a repeated zero point of and . We have from (29)-(30) that , , which means that the point is a maximum point or minimum point of and . We consider the following four cases:(a) is a maximum point of and a maximum point of ;(b) is a maximum point of and a minimum point of ;(c) is a minimum point of and a maximum point of ;(d) is a minimum point of and a minimum point of .

We only prove case (a) and case (b); since the proof of (b) is similar to (a), thus, we omit it.

First, we prove case (a). In the right neighborhood of , , and , , which contradicts (29)-(30); thus, case (a) is impossible.

Now, we prove case (b). In the right neighborhood of , , and , , which follows from (H) and (29) that
for near the right neighborhood of . Substituting the above into the right-hand side of (30), one gets
for near the right neighborhood of . This means from (30) that for near the right neighborhood of , which contradicts a minimum point of .

Similarly, it is easy to show that cases (c) and (d) are impossible.

By the available, the function is eventually positive or eventually negative. Suppose that is eventually negative; by the definition of oscillatory again, there exists a sequence such that is a local minimum point of ; that is, , , . From (29), one gets
which derive a contradiction. Similarly, we can prove the case of that is eventually positive. Thus, the function is nonoscillatory.

Similarly, we can prove that is nonoscillatory. Therefore, are all nonoscillatory.*Step 2* (approximation of model (29)-(30)). Since is a nonoscillatory solution of (12), then every component is nonoscillatory, which means that are eventually positive or eventually negative. Then there exists large enough, such that, when , satisfy the following four possible cases:(A),(B),(C),(D).

For the convenience, we only prove case (A) and case (B), the proof of the rest is similar.*Case (A).* Consider .

From and (29)-(30), it follows that , , which implies that , are decreasing; that is, for any , , . Using (29)-(30) again, one gets
that is,
Integrating both sides of the above from to , it follows that
which means
*Case (B).* Consider .

From (29) and (30), it is impossible that and are all positive, or and are all negative. So we consider the following three subcases: (B1) is oscillatory, or is oscillatory; (B2) is eventually negative and is eventually positive; (B3) is eventually positive and is eventually negative.*Subcase (B1).* is oscillatory, or is oscillatory.

Suppose that is oscillatory (the case that is oscillatory is similar), then for large enough such that . By (H), we have
Combining (30) to (40) and noticing that , one gets

From (40) and , then , which follows from (29) that

Since is oscillatory, then, there exists a sequence such that they are the minimum points of , and .

If eventually, then and exists, which implies from (30) that exists. Let
Then, we have from (29) and (30) that
which implies that and . Therefore,

If is also oscillatory, then there exists a sequence which are maximum points of , and . From (29) and (30), , and for any , . For convenience, we suppose that
Thus, for any large enough,
Also
Substituting (48) into (47), one gets
Repeating the above computing, we have
Noticing that , we have
which follows that , or . If , then ; it is impossible. Thus, . Similarly, . Therefore, (39) holds.*Subcase (B2).* is eventually negative, and is eventually positive.

Noticing that and , then , which follows that , exist. Let
According to (29), (30) and using a similar method of (44), we obtain
*Subcase (B3).* is eventually positive and is eventually negative.

From being eventually positive and , then exists; let . Since is eventually negative, we have from (41) that
for large enough, which implies that
Thus, exists; let ; then we have from (29) and (30) that
thus, which contradicts the fact that .

By the similar method of Case (A) and Case (B), we can prove Case (C) and Case (D). Concluding the above, , . The proof is complete.

#### 6. Conclusion

From Theorem 6, when , the solution of (12) satisfies , which follows that the solution of (5) satisfies , . This means that the gross of economy of the service domain and the gross of economy of the tour domain will tend to their equilibrium point and , respectively, when time is large enough.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was supported by the NSF of Hunan Province (no. 13JJ3074), Major project of Department of Education of Hunan Province (no. 13A088), the Scientific Research Foundation of Hengyang City (no. J1), and the Construct Program in USC.

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