#### Abstract

The intent of this paper is to prove a coupled fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings, satisfying -contractive conditions in a fuzzy metric space. We also furnish some illustrative examples to support our results.

#### 1. Introduction

The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy sets by Zadeh [1], where the concept of uncertainty was introduced in the theory of sets, in a nonprobabilistic manner. Fuzzy set theory has applications in applied sciences such as mathematical programming, model theory, engineering sciences, image processing, and control theory. In 1975, Kramosil and Michalek [2] introduced the concept of fuzzy metric space as a generalization of the statistical (probabilistic) metric space. Afterwards, Grabiec [3] defined the completeness of the fuzzy metric space and extended the Banach contraction principle to fuzzy metric spaces. Since then, many authors contributed to the development of this theory, also in relation to fixed point theory (e.g., [4–9]).

Mishra et al. [10] extended the notion of compatible mappings (introduced by Jungck [11] in metric spaces) to fuzzy metric spaces and proved common fixed point theorems in presence of continuity of at least one of the mappings, completeness of the underlying space, and containment of the ranges amongst involved mappings. Further, Singh and Jain [12] weakened the notion of compatibility by using the notion of weakly compatible, mappings in fuzzy metric spaces and showed that every pair of compatible mappings is weakly compatible but converse is not true. Inspired by Bouhadjera and Godet-Thobie [13, 14], Gopal and Imdad [15] extended the notions of subcompatibility and subsequential continuity to fuzzy metric spaces and proved fixed point theorems using these notions together due to Imdad et al. [16]. In recent past, several authors proved various fixed point theorems employing more general contractive conditions (e.g., [17–26]).

On the other hand, Bhaskar and Lakshmikantham [27] and Lakshmikantham and Ćirić [28] gave some coupled fixed point theorems in partially ordered metric spaces (see also [29–31]). In 2010, Sedghi et al. [32] proved common coupled fixed point theorems in fuzzy metric spaces for commuting mappings. Motivated by the results of [33], Hu [34] proved a coupled fixed point theorem for compatible mappings satisfying -contractive conditions in fuzzy metric spaces with continuous t-norm of H-type and generalized the result of Sedghi et al. [32]. In an interesting note, Zhu and Xiao [35] showed that the results contained in Sedghi et al. [32] are not true in their present form.

Inspired by the work of Zhu and Xiao [35], we prove coupled common fixed point theorems for two pairs of mappings satisfying a general contractive condition in fuzzy metric spaces, by using the notions of compatibility and subsequential continuity (alternately subcompatibility and reciprocal continuity). Our results improve many known common coupled fixed point theorems available in the existing literature. We support our results with two illustrative examples.

#### 2. Preliminaries

In this section, we collect some basic notions and results. In the sequel will denote the set of all positive real numbers while will denote the set of natural numbers.

*Definition 1 (see [1]). *Let be any set. A fuzzy set in is a function with domain and values in .

*Definition 2 (see [36]). *A binary operation is a continuous -norm if satisfies the following conditions: (a) is commutative and associative; (b) is continuous; (c) for all ; (d) whenever and for all .

*Definition 3 (see [37]). *One says that a -norm is of H-type if the family of its iterates is equicontinuous at ; that is, for any , there exists such that implies , for all .

The -norm for all is an example of -norm of H-type, but there are some other -norms of H-type (see [37]).

*Definition 4 (see [2]). *A -tuple is said to be a fuzzy metric space if is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set in satisfying the following conditions, for each and : (a); (b) for all if and only if ; (c); (d); (e) is continuous.

*Example 5 (see [7]). *Let be a metric space. Define the -norm for all , and, for all and ,
Then is a fuzzy metric space, and the fuzzy metric induced by the metric is often referred, as the standard fuzzy metric.

*Example 6 (see [32]). *Let be a metric space and be an increasing and continuous function from into such that . Four typical examples of these functions are , , , and . Let for all , and, for each and , define
It is easy to see that is a fuzzy metric space.

*Definition 7 (see [34]). *Define such that satisfies the following conditions: -1) is nondecreasing; (-2) is upper semicontinuous from the right; (-3) for all , where , .

Clearly if , then for all .

*Definition 8 (see [27]). *An element is called (a)a coupled fixed point of the mapping if
(b)a coupled coincidence point of the mappings and if
(c)a common coupled fixed point of the mappings and if

*Definition 9 (see [27]). *An element is called a common fixed point of the mappings and if

*Definition 10 (see [34]). *The mappings and are called compatible if
for all , whenever and are sequences in such that
for some .

Now we introduce the following notions.

*Definition 11. *The mappings and are said to be reciprocally continuous if, for sequences , in , one has
whenever
for some .

If two self-mappings are continuous, then they are obviously reciprocally continuous, but the converse is not true. Moreover, in the setting of common fixed point theorems for compatible pairs of self mappings satisfying contractive conditions, continuity of one of the mappings implies their reciprocal continuity but not conversely (see [38]).

*Definition 12. *The mappings and are said to be subsequentially continuous if and only if there exist sequences , in such that
for some , and

One can easily check that if two self mappings and are both continuous, hence also reciprocally continuous mappings but and are not sub-sequentially continuous (see [38, Example 1]).

*Definition 13. *The mappings and are said to be subcompatible if and only if there exist sequences , in such that
for some , and
for all .

#### 3. Results

In this section, we state and prove our fixed point results.

Theorem 14. *Let be a fuzzy metric space, where is a continuous t-norm of H-type such that as , for all . Let and be four mappings such that*(a)*the pairs and are compatible and subsequentially continuous; *(b)*there exists such that
for all and . ** Then there exists a unique point in such that . *

*Proof. *Since the mappings and are subsequentially continuous and compatible, there exist sequences , in such that
for all , and
that is and . Similarly, with respect to the pair , there exist sequences , in such that
for all , and
that is and . Hence is a coupled coincidence point of the pair , whereas is a coupled coincidence point of the pair .

Now we assert that , that is, and . Since is a -norm of H-type, for any , there exists an such that
for all .

Since is continuous and for all , there exists such that and .

On the other hand, since , by condition (-3), we have . Then for any , there exists such that . On using inequality (15) with , , , and , we have
Letting , we get
Again using inequality (15) with , , , and , we have
Letting , we get
From (22) and (24), we obtain
In general, for all , we have
Then, we have
So for any , we have
for all , and so and . Therefore we have
Next, we show that and . Since is a -norm of H-type, for any , there exists an such that
for all .

Since is continuous and for all , there exists such that and .

Since , by condition (-3), we have . Then for any , there exists such that . On using inequality (15) with , , we have
and so
Similarly, we can obtain
From (32) and (33), we have
In general, for all , we get
Then, we have
So for any , we obtain
for all , and hence and . Therefore
Now we show that and . Since is a -norm of H-type, for any , there exists an such that
for all .

Since is continuous and for all , there exists such that and .

On the other hand, since , by condition (-3) we have . Then for any , there exists such that . On using inequality (15) with , , , , we have
Letting , we obtain
Similarly, we can get
Consequently, from (41) and (42), we have
In general, for all , we get
Then, we have
Therefore for any , we obtain
for all and so and . Thus
Finally, we assert that . Since is a -norm of H-type, for any , there exists an such that
for all .

Since is continuous and for all , there exists such that .

Also, since , by condition (-3), we have . Then for any , there exists such that . On using inequality (15) with , , we have
and so
Thus we have
which implies that . Therefore, we proved that there exists in such that
The uniqueness of such a point follows immediately from inequality (15) and so we omit the details.

*Remark 15. *The conclusion of Theorem 14 remains true if we substitute condition (a) with the following condition: (a′)the pairs and are subcompatible and reciprocally continuous.

From Theorem 14, taking and , we deduce the following natural result.

Corollary 16. *Let be a fuzzy metric space, where is a continuous t-norm of H-type such that as , for all . Let and be compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings such that
**
for all , and . Then there exists a unique point in such that . *

Next, we illustrate our results providing the following examples.

*Example 17. *Let , for all and for all . Then is a fuzzy metric space, where
for all and . Let , and let the mappings , be defined as
In view of Definition 10, to prove compatibility, we have only to consider sequences and converging to zero from the right. In such case we have
Next, we get

Consequently
for all .

On the other hand, to prove subsequential continuity, in view of Definition 12, we have only to consider sequences and converging to one from the right. In such case we have
Also, note that, for the same sequences, we get
but
Thus, the mappings and are compatible as well as subsequentially continuous but not reciprocally continuous. Next, by a routine calculation, one can verify that condition (53) holds true. For instance, for all and , we have

Therefore, all the conditions of Corollary 16 are satisfied and (0,0) is the unique common fixed point of the pair . It is noted that this example cannot be covered by those fixed point theorems which involve compatibility and reciprocal continuity both.

*Example 18. *In the setting of Example 17 (besides retaining the rest), let , and let the mappings , be defined as
In view of Definitions 11 and 13, to prove reciprocal continuity and subcompatibility, we have only to consider sequences and converging to one from the right. For such sequences, we get
Also, we deduce that
Therefore, we have
for all . Finally, to show that the mappings and are not compatible, it suffices to consider the particular sequences and in . In fact, in such case, we have
Next, we deduce that
Consequently, we obtain
for all . Thus, the mappings and are reciprocally continuous as well as subcompatible but not compatible. Next, by a routine calculation, one can verify that condition (53) holds true. For instance, for all and , we have

Therefore, all the conditions of Corollary 16 are satisfied, and (1, 1) is the unique common fixed point of the pair . It is also noted that this example cannot be covered by those fixed point theorems which involve compatibility and reciprocal continuity both.

*Remark 19. *The conclusions of Theorem 14 and Corollary 16 remain true if we assume , where .

#### 4. Conclusion

Theorem 14 is proved for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings in fuzzy metric spaces, wherein conditions on completeness (or closedness) of the underlying space (or subspaces) together with conditions on continuity in respect to anyone of the involved mappings are relaxed. Theorem 14 improves the results of Jain et al. [39, Theorem 3.2, Corollary 3.2, Theorem 3.3, Theorem 3.4, Theorem 4.1] and Hu [34, Theorem 1]. A natural result is also obtained for a pair of mappings (see Corollary 16). Finally, Examples 17 and 18 are furnished to demonstrate the usefulness of Corollary 16. In view of Remark 19, Theorem 14 and Corollary 16 improve the results of Sedghi et al. [32, Theorem 2.5, Corollary 2.6] and Jain et al. [39, Corollary 3.1].

#### Acknowledgments

The authors would like to express their sincere thanks to the editor and reviewer(s) for their valuable suggestions.