Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces
© Hu et al.; licensee Springer. 2013
Received: 1 March 2013
Accepted: 1 August 2013
Published: 19 August 2013
In this paper, we prove a common fixed point theorem for weakly compatible mappings under ϕ-contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716). We also indicate a minor mistake in Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716).
In 1965, Zadeh  introduced the concept of fuzzy sets. Then many authors gave the important contribution to development of the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of a fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space, which have very important applications in quantum particle physics, particularly, in connection with both string and E-infinity theory.
Bhaskar and Lakshmikantham , Lakshmikantham and Ćirić  discussed the mixed monotone mappings and gave some coupled fixed point theorems, which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al.  gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Jin-xuan Fang  gave some common fixed point theorems for compatible and weakly compatible ϕ-contractions mappings in Menger probabilistic metric spaces. Xin-Qi Hu  proved a common fixed point theorem for mappings under φ-contractive conditions in fuzzy metric spaces. Many authors [9–26] proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.
In this paper, we give a new coupled fixed point theorem under weaker conditions than in  and give an example, which shows that the result is a genuine generalization of the corresponding result in .
First, we give some definitions.
Definition 2.1 (see )
∗ is commutative and associative,
∗ is continuous,
for all ,
whenever and for all .
Definition 2.2 (see )
The t-norm is an example of t-norm of H-type, but there are some other t-norms Δ of H-type .
Obviously, Δ is a t-norm of H-type if and only if for any , there exists such that for all , when .
Definition 2.3 (see )
A 3-tuple is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on satisfying the following conditions for each and ,
(FM-2) if and only if ,
(FM-5) is continuous.
A subset is called open if for each , there exist and such that . Let τ denote the family of all open subsets of X. Then τ is called the topology on X, induced by the fuzzy metric M. This topology is Hausdorff and first countable.
Example 2.4 Let be a metric space. Define t-norm or and for all and , . Then is a fuzzy metric space.
Definition 2.5 (see )
- (1)a sequence in X is said to be convergent to x (denoted by ) if
for all .
- (2)A sequence in X is said to be a Cauchy sequence if for any , there exists , such that
for all and .
A fuzzy metric space is said to be complete if and only if every Cauchy sequence in X is convergent.
Remark 2.6 (see )
for all , is non-decreasing;
- (2)if , , , then
if for x, y in X, , , then we can find a , such that ;
for any , we can find a such that , and for any , we can find a such that ().
Define , where and each satisfies the following conditions:
(ϕ-1) ϕ is non-decreasing,
(ϕ-2) ϕ is upper semi-continuous from the right,
(ϕ-3) for all , where , .
It is easy to prove that if , then for all .
Lemma 2.7 (see )
for all , then .
Definition 2.8 (see )
Definition 2.9 (see )
Definition 2.10 (see )
Definition 2.11 (see )
Definition 2.12 (see )
for all are satisfied.
Definition 2.13 (see )
The mappings and are called weakly compatible mappings if , implies that , for all .
Remark 2.14 It is easy to prove that if F and g are compatible, then they are weakly compatible, but the converse need not be true. See the example in the next section.
3 Main results
for all .
Xin-Qi Hu  proved the following result.
Theorem 3.1 (see )
for all , .
Suppose that , g is continuous, F and g are compatible. Then there exist such that ; that is, F and g have a unique common fixed point in X.
Now we give our main result.
Theorem 3.2 Let be a FM-space, where ∗ is a continuous t-norm of H-type satisfying (2.2). Let and be two weakly compatible mappings, and there exists satisfying (3.1).
Suppose that and or is complete. Then F and g have a unique common fixed point in X.
The proof is divided into 4 steps.
Step 1: We shall prove that and are Cauchy sequences.
for all .
for all with and . So is a Cauchy sequence.
Similarly, we can prove that is also a Cauchy sequence.
Step 2: Now, we prove that g and F have a coupled coincidence point.
which implies that .
Similarly, we can show that .
Since F and g are weakly compatible, we get that and , which implies that and .
Step 3: We prove that and .
for all .
Since is continuous and for all , there exists such that and .
for all . Hence conclude that and .
Step 4: Now, we prove that .
for all .
Since is continuous and , there exists such that .
On the other hand, since , by condition (ϕ-3), we have . Then, for any , there exists such that .
which implies that .
Thus, we proved that F and g have a common fixed point in X.
The uniqueness of the fixed point can be easily proved in the same way as above. This completes the proof of Theorem 3.2. □
Taking (the identity mapping) in Theorem 3.2, we get the following consequence.
for all , . is complete.
Then there exist such that ; that is, F admits a unique fixed point in X.
We only need the completeness of or .
The continuity of g is relaxed.
The concept of compatible has been replaced by weakly compatible.
Remark 3.5 The Example 3 in  is wrong, since the t-norm is not the t-norm of H-type.
Next, we give an example to support Theorem 3.2.
Example 3.6 Let , , , for all , . Then is a fuzzy metric space.
so g and F are not compatible. From , , we can get , and we have , which implies that F and g are weakly compatible.
Now, we verify inequality (3.11). Let , . By the symmetry and without loss of generality, , have 6 possibilities.
Case 1: , . It is obvious that (3.11) holds.
Case 2: , . It is obvious that (3.11) holds.
which implies that (3.11) holds.
Case 4: , . It is obvious that (3.11) holds.
Case 6: , .
If , , (3.11) holds.
Then all the conditions in Theorem 3.2 are satisfied, and 0 is the unique common fixed point of g and F.
This work of Xin-qi Hu was supported by the National Natural Science Foundation of China (under grant No. 71171150). The research of B. Damjanović was supported by Grant No. 174025 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.
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