G-Contractive Sequential Composite Mapping Theorem in Banach or Probabilistic Banach Space and Application to Prey-Predator System and A & H Stock Prices

Theorems of iteration g-contractive sequential composite mapping and periodic mapping in Banach or probabilistic Bannach space are proved, which allow some contraction ratios of the sequence of mapping might be larger than or equal to 1, and are more general than the Banach contraction mapping theorem. Application to the proof of existence of solutions of cycling coupled nonlinear differential equations arising from prey-predator system and A & H stock prices are given.


Introduction
Fixed point theorems play an important role for the proof of existence of solution of equations of algebra, differential, and integral, etc.It has been applied to many areas such as mathematical economics, game theory, dynamic optimization, functional analysis, etc.A lot of research work on fixed point theorems of various mappings on different spaces have been done (see [1,2] and their references).Among these works, the Banach contraction mapping theorem is a basic theorem for many research works.However, the Banach contraction mapping theorem needs a serious restriction, that is, all contraction mapping ratios must less than a constant less than 1.Instead of this restriction, a loosed restriction which allows some contraction mapping ratios to be greater or equal to 1 but the geometric mean of contraction mapping ratios of the sequential mapping (simplifying as "g-contraction mapping") must less than a constant less than 1, is proposed by the author [2].In this paper, the iterative g-contraction mapping and periodic mapping theorems in Banach or probabilistic Banach space are proved and application to cycling coupled nonlinear differential equations arising from prey-predator system and A & H stock prices are given..The A-stock market in mainland China is a new de-veloping market and is going to connect the rule with international market.The H-stock market in Hong Kong is a district international stock market.Many companies have their shares in both A-and H-markets, e.g., the China Petrol (601857) and Petro China (HK0857), a significant share in A-stock market.Although there are some papers on computational stock price based on certain model [3,4], however, no paper on quantitative analysis of stock prices on different stock markets has been found.This paper establishes a cycling coupled differential equations of stock prices of A & H shares, and uses the theorem to prove the existence of solutions and further more find the solution as well as the preypredator problem.

Main Results
In the following, let us consider a sequential mapping , M is a probability Banach space, i.e., a complete nonempty metric space satisfied probabilistic requirements.
1 By induction, and define 1 1 11 where 1 1 1 and , The symbol F G represents the composition of mapping F and mapping G.
shown in ( 4) is called the contraction ratio of .
where , x y are continuous random variables,   d , x y is the distance between x and in y M , and 0 1 T  .Obviously, we have , Obviously, we have Definition 2.4 A sequential composite mapping is called the g-contraction mapping, if for each i T i N  , there exits a constant G, such that the geometric man contraction ratio satisfies.
Definition 2.5.A sequential composite mapping is called the iterative g-contraction mapping U, U if for each , there exists a constant G, such that the iterative geometric mean contraction ratio satisfies.
Obviously, condition (9) is weaker than condition (8).Theorem 2.6.Any sequential composite iterative gcontraction mapping of a complete nonempty metric space M into M has a unique fixed point in M.
Proof: Suppose that the sequential composite mapping satisfies (9).
By the triangle inequality, we have for Since, M complete, the sequence has a limit z in M. i.e., . This fixed point in unique, if there are two fixed points z and w, i.e., i.e., z = w. Obviously, this theorem allows part of contraction ratio if (9) holds.But the Banach contraction mapping theorem needs each contraction ratio r less than1, so this theorem is more general.If , only 1 Theorem 1 reduces to the Banach contraction mapping theorem.Definition 2.7.A sequential composite mapping and is denoted by Where the superscript n denotes the times of cycling, the subscript  indicates the mapping at the corresponding space j X , we have  , : , , Theorem 2.8.Any periodic iterative g-contraction mapping of complete nonempty metric space M has a unique set of k related fixed points in M. That is  Proof: For a periodic iterative g-contraction mapping, (7) becomes where The same treatment as Theorem 1 for each x is proportional to the product of x and the amount of food ; and is decreasing with the product of a x and .Equation ( 22) shows that the increasing rate of is proportional to the product of y y x and ; and is decreasing with (natural death).
Then, Equations ( 25) and ( 26) can be solved by iteration method, we have Then, (16) becomes P y P y P y P y P y P y G P y y G P y y where the constant is chosen then, (23) is satisfied, and (32), (33) are solutions of ( 21) and ( 22).It should be mention that the supposition of food amount to be a periodic function is reasonable and confirm with the fact than that of the supposition of to be a constant.

a a
The exact solution of (21), ( 22) has a referent meaning for understanding the relationship among food, prey and predator quantitatively.The number of predator mainly depends on the number of prey y x ; and x mainly depends on the food amount ., a a x and are periodic functions with the same period y  .

Application to the Prey-Predator System
Problem in Probabilistic Banach Space Equation ( 21) shows that the growing rate of prey number x is proportional to the product of the food amount and the number of prey a x , if other elements keep unchanged.The proportion constant is 1 .Equation ( 22 Banach space M , we found that there is no differ-ence between the distances defined by Sup norm, i.e., Mappings 1 , 2 and 1 , 2 of (25) and ( 26) in Banach space are continuous mappings, so these mappings are still continuous mappings in probabilistic Banach space ((Lemma 4.3 continuous mapping) of [2]) and the proof of existence of fixed points in Section 3.1.2is still suited for the case of probabilistic Banach space.
T T P P

Cycling Coupled Differential Equations of Stock Prices of A & H Shares
  y y t  be the stock prices of China Petrol (601857) and Petrol China (HK0857) respectively.Follows the set up of differential equation of stock price [3,4], we have: 1) Equations of amount of purchasing and selling of x t ay t  .2) Assumes that the changing rate of stock price is proportion to the difference of demand and supply, we have where constant g keeps the same of dimensions in both sides of (40).
3) Substituting ( 38) and ( 39) into (40), we have Similarly, we have In which, the right hand side of (36) is changed to where Equation (45) has exact solution, depended on the root of characteristic function .
where , are arbitrary constants, and 3) When , the characteristic function has complex roots and there is no real meaning for has complex value and thus is out of discussion.The determination of coefficients via market data may be referred to [3], or we can directly use the share prices both in temporary equilibrium states (the so-called "Doji" or the Chinese stock market saying "cross star") of A and H stock markets at the same time to find .For example, 2008-04-10, a "Doji" for China Petrol (601857) (opening price 17.11, closing price 17.35 RMB) and a near "Doji" for Petrol China (HK0857) (opening price 10.20, closing price 9.82 HKD).Then, = a a x y = 17.35×0.88/9.82= 1.554.(where 0.88 is the changing rate for RMB to HKD) The analysis of this example might be useful to decision making of operators and is referred [6] for details.

Considering the Following Cycling Non-Linear Coupled Differential Equations Arising from a Model of Prey-Predator System [5]
3.1.Application to the Prey-Predator SystemProblem in Banach Space 3.1.1.3 k 28)

.3. An Exact Solution of (21), (22)
 , then it is the equilibrium state, in which no profits can be made by speculating the difference of stock prices in A-stock or H-stock markets.