Several New Types of Fixed Point Theorems and Their Applications to Two-Point Ordinary Differential Equations ()
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.
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
, 
for
and
,
.
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 
if
whenever 
and
.
Definition 10. Let 
be a fuzzy metric space and 
a fuzzy set on
. 
is said to satisfies the 
property on 
if
for 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 whenever
for all 
where
satisfying 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
, and
for 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 
, then
is 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, and
for 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.