#### Abstract

We first introduce the new real function class satisfying an implicit Lipschitz-type condition. Then, by using -type real functions, some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in fuzzy metric spaces (in the sense of Kaleva and Seikkala) are established. As applications, we obtain the corresponding common fixed point theorems in metric spaces. Also, some examples are given, which show that there exist mappings which satisfy the conditions in this paper but cannot satisfy the general contractive type conditions.

#### 1. Introduction

In 1984, Kaleva and Seikkala [1] introduced the concept of a fuzzy metric space by setting the distance between two points to be a nonnegative fuzzy real number and studied some of its properties. From then on, some important results for single-valued and multivalued mappings in fuzzy metric spaces, such as coincidence theorems, various fixed point theorems, and so forth, were stated in subsequent work (see [1–11], etc.). Recently, Zhang [12, 13] established some new common fixed point theorems for generalized contractive type mappings in metric spaces and for Lipschitz-type mappings in cone metric spaces. These theorems extended the original contractive type conditions. Moreover, various real function classes satisfying an implicit relation were introduced in [10, 14–23], and some common fixed point theorems for composite mappings satisfying an implicit relation were established in metric spaces and fuzzy metric spaces, respectively.

It is well known that the fuzzy metric space is an important generalization of the ordinary metric space (see [1]). Inspired by [13–23], we establish some common fixed point theorems for new contractive type mappings in fuzzy metric spaces in this paper. In Section 3, we first introduce the new real function class satisfying an implicit Lipschitz-type condition. Then, in Section 4, by using -type real functions, some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in fuzzy metric spaces are established. In Section 5, as their applications, we obtain the corresponding common fixed point theorems in metric spaces. Also, some examples are given, which show that there exist mappings which satisfy the conditions in this paper but cannot satisfy the general contractive type conditions.

#### 2. Preliminaries and Lemmas

Throughout this paper, let be the set of all positive integers, and . For the details of fuzzy real number, we refer the reader to Kaleva and Seikkala [1], Dubois and Prade [24], and Bag and Samanta [25].

*Definition 1 (cf. Dubois and Prade [24]). *A mapping is called a fuzzy real number or fuzzy interval, whose -level set is denoted by , if it satisfies two axioms.(1)There exists such that .(2) is a closed interval of for each , where . The set of all such fuzzy real numbers is denoted by . If and whenever , then is called a nonnegative fuzzy real number, and by we mean the set of all nonnegative fuzzy real numbers. If and are admissible, then, for the sake of clarity, is called a generalized fuzzy real number. The sets of all generalized fuzzy real numbers or all generalized nonnegative fuzzy real numbers are denoted by and , respectively. In that case, if , for instance, then means the interval .

The notation stands for the fuzzy number satisfying and if . Clearly, . can be embedded in : if , then satisfies .

Lemma 2 (Xiao et al. [8]). *Let , , and . Then*(1)*.*(2)* is a left continuous and nonincreasing function for .*(3)

*is a left continuous and nonincreasing function for*.*Definition 3 (cf. Kaleva and Seikkala [1]). *Suppose that is a nonempty set and that is a mapping from into . Let be two symmetric and nondecreasing functions such that and . For and , define the mapping
The quadruple is called a fuzzy metric space (briefly, FMS), and is called a fuzzy metric, if (FM-1) if and only if ; (FM-2) for all ; (FM-3) for all (FM-3L) , , whenever , and ; (FM-3R) , , whenever , and .

If is a mapping from into and satisfies (FM-1)–(FM-3), then is called a generalized fuzzy metric space (briefly, GFMS).

From Lemma 2 and Definition 3, we obtain the following consequences.

Lemma 4. *Let be a FMS, for , where are any two fixed elements. Then *(1)*. *(2)* is a left continuous and nonincreasing function for . *(3)

*is a left continuous and nonincreasing function for*.Lemma 5 (Xiao et al. [8]). *Let be a FMS, and suppose that* *(R-1) ;* *(R-2) .* *Then *(R-1) (R-2).

Lemma 6. *Let be a FMS. Then*(1)*(R-1) for each , for all (cf. [4, 5]).*(2)*(R-2) for each there exists such that for all (cf. [5, 6, 8]).*

Lemma 7 (Kaleva and Seikkala [1]). *Let be a FMS with (R-2). Then the family of sets forms a basis for a Hausdorff uniformity on . Moreover, the sets
**
form a basis for a Hausdorff topology on and this topology is metrizable.*

According to Lemma 7, convergence in a FMS can be defined by sequences. A sequence in is said to be convergent to (we write or ) if ; that is, for each ; is called a Cauchy sequence in if ; equivalently, for any given and , there exists such that , whenever ; is said to be complete, if each Cauchy sequence in is convergent to some point in .

Lemma 8 (Kaleva and Seikkala [1]). *Let be a FMS with . Then for each , is continuous at .*

#### 3. The Real Functions Satisfying an Implicit Lipschitz-Type Condition

*Definition 9. *A lower semicontinuous function is called a real function satisfying an implicit Lipschitz-type condition, if the following conditions are satisfied. (-1) is nonincreasing in . (-2) There exist with such that for all , we have (-2a) , whenever , (-2b) , whenever . (-3) For all with , we have . We denote by the collection of all real functions satisfying an implicit Lipschitz-type condition.

*Remark 10. *Let ; if or , then by condition (-2) of Definition 9, we have .

The following examples show that the collection is a largish class of real functions.

*Example 11. *Let . The function is defined by
then .

In fact, it is easy to see that is continuous. Also, (-1) and (-3) are easy to check. For any , if or , then we have , or respectively. Taking , we have ; that is, (-2) holds. Hence .

*Example 12. *Let with and . The function is defined by
then .

Obviously, is continuous, and (-1) and (-3) are easy to check. For any , if or , then we have or , respectively, which implies that or . Let and . By , we have and ; that is, . Thus (-2) holds. Hence .

*Example 13. *Let . The function is defined by
then .

Obviously, is continuous, and (-1) and (-3) are easy to check.

For any , if , then we have , which implies that
Similarly, if , then we also have . Let . Note that ; we have ; that is, ; thus . Therefore (-2) holds. Hence .

*Example 14. *Let : be five continuous functions satisfying the following conditions.(i) for all ,(ii)There exist , , , with , such that
We define the function as follows:
Then .

In fact, it is easy to see that is continuous, and (-1) is satisfied.

For any , if , then we have ++, which implies that +. From condition (ii), we obtain . Similarly, if , then we have +. Let , . Note that ; we have ; that is, (-2) holds.

Furthermore, for any , if , then we have , which implies that . Note that for all ; we have . This shows that (-3) holds. Hence .

*Example 15. *Let the function be defined by
then .

In fact, in Example 14, taking , , , and , we obtain five continuous functions from into satisfying the following conditions:

It is evident that , and so all conditions of Example 14 are satisfied. Therefore, .

*Example 16. *Let with and . There exists such that and . We define the function as follows:
then .

Obviously, in Example 14, taking , , , and , we obtain five continuous functions from into . Moreover, by , we have . Hence, ; that is, condition (i) of Example 14 holds.

Furthermore, , is obvious. Note that and ; it follows that By , we obtain which implies that . Hence, condition (ii) of Example 14 is satisfied. Thus, by Example 14, we have .

*Remark 17. *The numbers , and in Example 16 really exist. For example, if we take , , , , , and , then , , , , and ; that is, the conditions of Example 16 are satisfied.

*Example 18. *Define the function as follows:
where , and are nonnegative real numbers, with and either , or , . Then .

In fact, if we take , , , , and , then condition (i) of Example 14 is obviously satisfied. Note that , ; we have ; that is, condition (ii) of Example 14 is satisfied. Similarly, we can prove the case of , . Therefore, by Example 14, .

#### 4. Main Results

Theorem 19. *Let be a complete FMS with and let and be two self-mappings on . If there exists such that
**
for all , whenever , , , , , and , then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

*Proof. *Firstly, we use (15) to prove that the following inequality:
holds for all and .

In fact, for each and , if we set , , , , , , then, for any , it is obvious that , , , , , , and , , , , , . By (15), we have − and , which imply that
then by the arbitrariness of and the lower semicontinuity of , we have
for each and ; that is, the inequality (16) holds for all and .

For any , we construct an iterative sequence in as follows:
For , applying (16), we obtain for each
By the known condition and conclusion (1) of Lemma 6, we have , . Note that is nonincreasing in ; it is not difficult to see that
Since , there exists such that

Similarly, for , applying (16), we obtain for each
By and (1) of Lemma 6, we have +, . Note that is nonincreasing in ; we obtain
Since , there exists such that
Using inductive method, for , we can obtain

Next, we prove that the sequence is a Cauchy sequence. For , by and conclusion (1) of Lemma 6, we have for each
where . By the similar reasoning process, we have for each
Then there exists with for , such that for each
Since , it is evident that the sequence is a Cauchy sequence in . By the completeness of , we set . Applying (16), we have
for each . Let ; by the lower semicontinuity of and Lemma 8, we have
for each . By Remark 10, for each , which implies that is a fixed point of .

Similarly, for each , we have
Let ; by the lower semicontinuity of and Lemma 8, we have for each
By Remark 10, for each , which implies that is also a fixed point of . Thus is a common fixed point of .

Lastly, we prove the uniqueness of the common fixed point. If is another common fixed point of , then by (16), we have for each
Note that , and by (-3) of Definition 9, we obtain for each ; hence . The uniqueness is proved and we complete the proof of the theorem.

According to the proof of Theorem 19, we can easily obtain the following corollary.

Corollary 20. *Let be a complete FMS with and let and be two self-mappings on . If there exists such that (16) holds for all and , then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

Theorem 21. *Let be a complete FMS with . Let be five continuous functions which satisfy the following conditions:*(i)* for all ,*(ii)*, , , and .**Let and be two self-mappings on such that
**
for all and . Then and have a unique common fixed point in . Moreover, for any , the iterative process , converges to the fixed point.*

*Proof. *Taking , from the known conditions (i) and (ii) and Example 14, we obtain . Furthermore, from (35) we can easily derive the inequality (16). Then by Corollary 20, the theorem is proved.

Corollary 22. *Let be a complete FMS with and let and be two self-mappings on . If there exist and with , , , and , such that
**
for all and , then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

*Proof. *Taking , note that the known conditions and Example 16, we have . Furthermore, from (36) we can easily derive the inequality (16). Then by Corollary 20, the corollary is proved.

#### 5. Applications to the Ordinary Metric Spaces and Examples

In this section, we first establish some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in complete metric spaces. After that, we give two examples, by which we can claim that our conclusions are really generalizations of the early results.

Let be an ordinary metric space and Then is a FMS (cf. [1, 9]). It is easy to see that and are homeomorphic and for all .

Theorem 23. *Let be a complete metric space and let and be two self-mappings on . If there exists such that
**
for all . Then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

*Proof. *Note that the topology and completeness of and the induced FMS are coincident, as well as for all ; it is not difficult to see that the inequality (16) holds as a result of (38). Moreover, the other conditions of Corollary 20 are satisfied; thus by Corollary 20, the theorem is proved.

Applying the same method, we can obtain the following theorem and corollary by virtue of Theorem 21 and Corollary 22, respectively.

Theorem 24. *Let be a complete metric space. Let be five continuous functions which satisfy the following conditions:*(i)* for all ,*(ii)*, , , and .**Let and be two self-mappings on such that
**
for all . Then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

Corollary 25. *Let be a complete metric space and let and be two self-mappings on . If there exist , and with , , , , and , such that
**
for all , then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

By Theorem 23 and Example 18, we can also obtain the following corollary.

Corollary 26. *Let be a complete metric space and let and be two self-mappings on . If there exist with and either , or , , such that
**
for all , then and have a unique common fixed point in . Moreover, for any , the iterative process , , , converges to the fixed point.*

*Remark 27. *In the conditions of Theorem 1 of [13], if we suppose the functions , , , , and to be continuous, then the theorem is just Theorem 24 in this paper. Therefore, in some sense, Theorem 23 is the improvement and generalizations of Theorem 1 in [13]. Meanwhile, Corollaries 25 and 26 are just Corollaries 1 and 2 in [13], respectively.

If we assume , then, by Corollary 26, we can obtain the following corollary.

Corollary 28. *Let be a complete metric space and let be a self-mapping on . If there exist with and either , or , , such that
**
for all , then has a unique fixed point in . Moreover, for any , the iterative process , , converges to the fixed point.*

Corollary 29. *Let be a complete metric space and let be a self-mapping on . If there exist with , such that
**
for all , then has a unique fixed point in . Moreover, for any , the iterative process , converges to the fixed point.*

*Proof. *If , by , we can take , and let , , , , and ; then we have , , , and for ,
which imply that the conditions of Corollary 28 are satisfied. Similarly, we can prove the case of . Therefore, by Corollary 28, the corollary is proved.

*Remark 30. *Corollary 29 is just the fixed point theorem for Reich-type contraction mappings in [26, 27]. Meanwhile, we can easily see that the conditions of Corollary 28 generalize the Reich-type contractive conditions in Corollary 29.

*Remark 31. *If we take in Theorem 23, then Theorem 23 is the generalizations of the fixed point theorem for mappings satisfying Cirić-type [28] contractive conditions to the common fixed point theorem for a pair of self-mappings with .

Now we give two examples, by which we can claim that the theorems in this paper are really the generalizations of the general contractive type fixed point theorems.

*Example 32. *Suppose , is an ordinary metric; then is a complete metric space.

We define two mappings as follows: ; then