ABSTRACT

The present paper is mainly concerned with several new types of fixed point theorems in different spaces such as cone metric spaces and fuzzy metric spaces. By using these obtained fixed point theorems, we then prove the existence and uniqueness of the solutions to two classes of two-point ordinary differential equation problems.

1. Introduction

The theory of the fixed point has important applications in fields such as differential equations, equilibrium problems, variational inequality, optimization problems, maxmin problems etc. (cf. Klaus Deimling [1], Congjun Zhang [2] for example), which has attracted many scholars’ attention and became a hot topic in mathematics and applied mathematics field for a long time. In recent decades, many new types of fixed point theorems have been proposed (see [3-6] and the reference therein) and the generalizations of the existing ones have been dramatically developed in many ways. In [7], LongGuang Huang and Xian Zhang have introduced the notion of cone metric spaces and proved some fixed point theorems of contractive mappings on cone metric spaces. For fixed point theorems in fuzzy metric spaces, see [8-12]. In [13-16], some scholars have proved the fixed point theorem in partial order metric space, and applied them to prove the existence and uniqueness of the solution to the two-point ordinary differential equation problems. Inspired by the recent progress in this fields, we will study in the present paper the existence and uniqueness of the fixed point for some special mappings in cone metric spaces and fuzzy metric spaces as well as their applications to the following two-point ordinary differential equations.

Problem (1):

where, is a continuous function satisfying some conditions which will be given explicitly later.

Problem (2):

where, is a continuous function satisfying some conditions which will be given explicitly later.

The paper is organized as follows. For the reader’s convenience, we recall in Section 2 some definitions and lemmas in cone metric spaces and fuzzy metric spaces that will be used in the sequel. Section 3 is devoted to the investigation on the existence and uniqueness of the fixed point for some special mappings in cone metric spaces and fuzzy metric spaces. In last section, two-point ordinary differential equation problems are studied by using the results obtained in Section 3 and the existence and uniqueness of the solutions to such equations is established.

2. Preliminaries and Abstract Results

We recall in this section some definitions and lemmas in cone metric spaces and fuzzy metric spaces that will be used in the sequel.

Definition 1 [6]. Let be a metric space and a mapping from to . For any , define , for . The sequence is called the orbit of f and the n iterate of f.

Definition 2. A function is called an ω-function if it is a monotone increasing function and satisfies that and for any , there exists M > 0, such that , for every .

For example:, defined on , is an ω-function.

Definition 3 [7]. Let be a nonempty set. Let be a real Banach space, a cone of satisfying , where denotes the interior of . Define a partial order on based on as follows: for any , if and only if , while means and , and means . And the following convention is assumed: if and only if , if and only if and .

If a mapping satisfies:

1), for all if and only if ;

2), for all ;

3), for all then is called a cone metric on and is called a cone metric space with respect to the Banach space and the cone in .

Definition 4 [2]. 1) A cone in a Banach space is called normal, if there exists a number M > 0 such that for all, implies, where is the zero element of the Banach space. The smallest satisfying that inequality is denoted by, and it is called the normal constant of; 2) A cone in a Banach space is called regular if every increasing sequence which is bounded from above is convergent. That is, if is sequence such that for some, then there is such that.

Remark 1. 1) For any normal cone in a Banach space E, M^* exists and (see [2]); 2) Equivalently, a cone is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.

Definition 5 [7]. Let be a cone metric space with respect to a Banach space and a cone in. Let be a sequence in (see [7]).

1) is called a convergent sequence with limit, if for any, there exists such that for every, holds. In this case, we denote the limit of by, or.

2) is called a Cauchy sequence on, if for any with, there exists such that for each holds.

3) We call a complete cone metric space with respect to the Banach space and the cone in, if every Cauchy sequence is convergent in.

Remark 2. If K is a normal cone, then converges to x if and only if, as. is a Cauchy sequence on if and only if as (see[7]).

Definition 6 [7]. Let be a cone metric space with respect to a Banach space and a cone in. If for any sequence in, there exists a subsequence of, such that is convergent in. Then the cone metric space is said to be sequentially compact.

Definition 7 [9,10]. A binary operation is called a continuous t-norm, if the following conditions are satisfied: 1) * is associative and commutative; 2) is continuous; 3) for all; 4) whenever and, for each. If it only satisfies conditions 1), 2) and 4), then it is called a t-norm.

Four typical examples of continuous t-norms are, forand,.

Definition 8 [9,10]. Let be an arbitrary nonempty set. Let be a continuous t-norm and M a fuzzy set on. If the following conditions satisfy:

1);

2) if and only if;

3);

4);

5) is continuous, for any and, then the 3-tuple is called a fuzzy metric space.

Remark 3. For any, is a non-decreasing function (see [9,10]).

Definition 9 [9,10]. Let be a fuzzy metric space and M a fuzzy set on. is said to satisfies the n-property on ifwhenever and.

Definition 10. Let be a fuzzy metric space and a fuzzy set on. is said to satisfies the property on iffor all and.

Definition 11 [11]. A function is said to satisfy condition, if f is a strictly increasing function satisfying f(0) = 0 and for any, where.

Remark 4. If a function satisfies the condition, then the following inequalities hold (see [11]):

1), for all;

2), for each and for all.

Definition 12. Let be a fuzzy metric space, the fuzzy set is said to have property wheneverfor all wheresatisfying the condition.

Definition 13 [9,10]. Let be a fuzzy metric space and a fuzzy set on.

1) A sequence in is said to fuzzy-convergent to a point, if for all.

2) A sequence in is called a fuzzy-Cauchy sequence, if for each and, there exists, such that for each.

3) A fuzzy metric space is called fuzzy-complete, if every fuzzy-Cauchy sequence is fuzzy-convergent.

Definition 14 [9,10]. Let be a fuzzy metric space. The fuzzy set is said to be fuzzy-continuous on, whenever any in which fuzzy-converges to implies

.

Remark 5. M is a continuous function on (see [9,10]).

Definition 15 [12]. Let be a fuzzy metric space and M the fuzzy set on. Denote by the set of all compact subsets of and define a function by

for any and any, where

and

.

Lemma 1 [6]. Let be a complete metric space, , for the iterate of , the following statements hold:

1) If has a unique fixed point, then has a unique fixed point.

2) If there exists, such that the orbit of converges to, then the orbit of converges to.

3) If the orbit of is a bounded sequence, then the orbit of is a bounded sequence.

Lemma 2. Let be a complete metric space and an expansive and surjective mapping on, then has a unique fixed point.

Proof. We claim first that is injective. To show this claim, assume, by the way of contradiction, that there exist such that. Since, then holds. Since is an expansive mapping, it implies . It contradicts to, that is, , which implies is a bijection. Hence T^–1 exists and is a contraction mapping. By the contractive mapping priciple, there exists a unique, such that, that is . The proof is complete.

Lemma 3 [5]. Let be a complete metric space and f a self mapping on X. If the following condition satisfies, for any, there exists, such that implies, then f has a unique fixed point on, andfor any.

Lemma 4 [7]. Let be a sequentially compact cone metric space with respect to a Banach space and a regular cone in. Suppose a mapping satisfies the contractive condition: , for all, then has a unique fixed point in.

Lemma 5 [4]. Let be a compact metric space and a self mapping on. Assume that implies for any, then has a unique fixed point.

Lemma 6 [9,10]. Let be a fuzzy metric space, for all and M satisfy property. Let be a sequence in X such that for all, for every, then is a Cauchy sequence in X.

3. The Existence Theorem of Fixed Points

In this section, we apply the concepts and lemmas provided in Section 2 to prove some existence theorems of fixed points for some mappings. These results will be used in the following section.

Theorem 1. Let be a complete metric space and a surjective mapping. If there exist and such that

holds for any, then there exists a unique fixed point of f.

Proof. For each, since is a surjective, then there exists, such that, in the same way, there exist,such that, i.e. there exists, such that. We deduce by induction that is also surjective, which combining (I) shows that is an expansive mapping. By Lemma 2, there exists a unique fixed point of, then we know by Lemma 1 that there exists a unique fixed point of f. The proof is complete.

Remark 6. It is obvious that we can get Lemma 2 from Theorem 1. An example satisfying Theorem 1 is given below.

Example 1. Define by

it is clear that f is a surjective self-mapping on R and. satisfies condition (I), i.e., then f has a fixed point, 0 is the fixed point in this example.

Theorem 2. Let be a sequentially compact cone metric space with respect to a Banach space and a normal cone in with normal constant. Assume that is a self mapping on and satisfies for any, implies, then has a unique fixed point.

Proof. We claim first that where is defined by

Using reduction to absurdity, we suppose. Since is sequentially compact, we deduce from the definition of that there exists a sequence such that

and

for some. Observe that the normal constant, there exists such that for any the inequality

holds, which combining the given conditions shows that for any,

By calculations we then have

which contradicts to the definition of.

We prove next that T has a fixed point. We proceed once more by using reduction to absurdity and suppose that T has no fixed point. Then for each,

which implies that for each,

By the triangle inequality in cone metric spaces, we have

then,

We claim that at least one of the following two inequalities should be hold:

otherwise, we reach a contradiction by the following calculations:

If the first inequality of the above two holds, then

if the the other one holds, then

which show that in each case, and the proof of the existence of the fixed point is complete.

We finally prove the uniqueness of the fixed point. Suppose and. Since, then, we reach a contradiction which completes the proof.

Remark 7. In [7], Long-Guang Huang and Xian Zhang have established a fixed point theorem in a sequentially compact cone metric space with respect to a Banach space and a regular cone in (see Lemma 4), where the mapping satisfies the contractive condition. In [4], Tomonari Suzuki has established a fixed point theorem in a compact metric space where the mapping T satisfying a condition similarly to condition (II) of theorem 2 (see Lemma 5). Observe that any regular cone is always normal, Theorem 2 is established under a different and weaker condition when comparing with Lemma 4 and generalize the results of Lemma 5 from compact metric spaces to sequentially compact cone metric spaces.

Theorem 3. Let be a complete fuzzy metric space, where is defined by for any and M a fuzzy set on satisfying property. For a surjective function, if for any, the following inequality holds

then has a fixed point on. If inequality (II) is strict, then has a unique fixed point on.

Proof. By choosing, we deduce from (II) that for any,

Proceed by introduction on n, we have for any

For any, we have

Observe that satisfies property, then

which shows that is a fuzzy-Cauchy sequence. Since is complete, there exists, such that

Then by (II) and the nondecreasing property of M, we have

for any. Since

We therefore deduce

which shows has a fixed point on.

If there exist such that , then by condition (II),

It is a contradiction, hence. We have now proved the uniqueness which complete the proof.

Corollary 1. Let be a complete fuzzy metric space and a bijective mapping, where * is defined by for any and M a fuzzy set on satisfying property. If for any ,

then f has a fixed point on. If the above inequality is strict, then f has a unique fixed point on.

Proof. Since f is bijective, exists and satisfies for any ,

By Theorem 3, we know has fixed point, and the fixed point of is the same as that of f, then f has fixed point on . If the inequality is strict, then the proof is the same as that in Theorem 3.

Corollary 2. Let be a complete fuzzy metric space, where is defined by for any and M a fuzzy set on satisfying property, and a surjective mapping satisfying

for any, Then f has a fixed point on . If the inequality is strict, then f has a unique fixed point on .

Proof. Let, , then by Theorem 3 we can easily propose the results of Corollary 2. We omit the details.

Example 2. Assume, , and define M by

clearly M satisfies property. For any f satisfies the conditions of Corollary 2, i.e.

we have for any , hence is a contraction mapping which has a fixed point on .

In the following, we show an example to demonstrate the conditions in Corollary 2 are only sufficient condition, not necessary conditions.

Example 3. Assume, , and define M by

Obviously is a fuzzy set which doesn’t have property, hence it can’t be judged by Corollary 3. But if is a contraction mapping, a fixed point still exist on .

Theorem 4. Let be a complete fuzzy metric space, where is defined by for any and a fuzzy set on satisfying property. is a compact setvalued mapping, satisfies for any ,

then has a fixed point on .

Proof. By the choice axioms (see [6]), there exists a single-valued function, such that for any . Then for each , there exist . By the definition of , we have

Theorem 3 shows that has a fixed point , i.e ., which is also a fixed point of on .

Corollary 3. Let be a complete fuzzy metric space and a fuzzy set on satisfying property. is a compact setvalued mapping satisfying for every ,

Then has a fixed point on .

4. Applications to Differential Equations

This section is concerned with the proof of the existence and uniqueness of the solutions to the two-point ordinary differential equations by using the fixed point theorems obtained in Section 3. The following are the main results.

Theorem 5. Assume that is a continuous function. If there exists such that the following inequalities

hold for any with , where is an ω-function, then Problem(1) has a unique solution.

Proof. Problem (1) is equivalent to the integral equation

where

Define

by

Note that if is a fixed point of , then is a solution to Problem (1). Define a order relation in by if and only if for every , for every . Denote by

for any

the distance in . For each , by the left side of (IV), . Since , for each ,

which shows that is monotone increasing. For any , if , then

Since is a increasing function, then

for , and

By the definition of , for each , there exists such that

, let

hence . It demonstrates . By Lemma 3, F has a unique fixed point, and for each, is the fixed point of , i.e. the solution of Problem (1).

Assume is a lower solution of Problem (1), we can prove as Theorem 3.1 in [13] to obtain the uniqueness of the solution.

Remark 8. Contrasted with some related results in [13-15], the conditions in Theorem 5 is relatively clearer.

Theorem 6. Assume that is a continuous function. If there exists such that for any with , the following inequalities

hold, where is an ω-function, then the solution of Problem (2) exists.

Proof. Problem (2) is equivalent to the following integral equation

Define

by

for any. Note that is a fixed point of , thenis a solution of Problem (2). For, we define if and only if for any. Denote

for.

Then by (V), for any ,

which implies

and

By the definition of function , let , there exists , such that , there exists, such that , then , and

By Lemma 3, has a unique fixed point, andfor any , u is a fixed point of , which is also a solution of Problem (2). The proof is complete.

Define satisfying for any

,

then we have the following theorem:

Theorem 7. Let be a complete fuzzy metric space, . If the following conditions hold:

1) For any,

2) For any ,

then the solution of Problem (1) is unique.

Proof. By example 2, while, a mapping satisfying the above conditions is a contraction mapping, i.e. is a contraction mapping. Then we can proceed the proof with the same arguments as that in Theorem 5.

Remark 9. If we replace condition (1) by the inequality in Example 2 or Example 3 as well as the corresponding expression of M, then Theorem 7 can also make sure the uniqueness of t he solution of Problem (1).

Define satisfying for any

,

then we have the following theorem:

Theorem 8. Let be a complete fuzzy metric space, . If the following two conditions hold:

1) for any and ,

2) for any ,

then the solution of Problem (2) exists.

Proof. By Example 2, while, a mapping satisfying the conditions above is a contraction mapping, hence h is a contraction mapping. Then we can proceed the proof with the same arguments as that in Theorem 6 and complete the proof.

5. Conclusion

The paper is devoted to several new types of fixed point theorems in different spaces such as cone metric spaces and fuzzy metric spaces together with their applications. We have also proved the existence and uniqueness of the solutions to two classes of two-point ordinary differential equation problems by using these obtained fixed point theorems.

6. Acknowledgements

The authors of this paper would like to appreciate the referee’s helpful comments and valuable suggestions which have essentially improved this paper. This work is supported by the National Natural Science Foundation (11071109) of People’s Republic of China.

REFERENCES