Theorem 2.1 Let be a partially ordered set, and let be a complete FNS such that the t-norm ∗ is of H-type. Let be a mapping such that F is non-decreasing and
(2.1)
for which and , where . Suppose either
-
(a)
F is continuous, or
-
(b)
if is a non-decreasing sequence and , then for all .
If there exists
such that
then F has a fixed point in X.
Proof Let such that , and let , , then we have that
Now, put
Then, by using (2.1), we have
Thus, it follows that , and so
(2.2)
On the other hand, we have
By Definition 1.4, we get that
(2.3)
It follows from (2.2) and (2.3) that
(2.4)
By the hypothesis, the t-norm ∗ is of H-type; for all , there exists such that
for all and for all p. Note that
for all and , we have that there exists such that
for all . Thus, is a Cauchy sequence. Since X is complete, there exists such that
If the assumption (a) holds, then by the continuity of F, we get that
If the assumption (b) holds, then we have that for all . It follows from (2.1) that
Thus, , that is, . Therefore, x is a fixed point of F. The proof is completed. □
Theorem 2.2 Let be a partially ordered set, and let be a complete FNS such that the t-norm ∗ is of H-type and for any . Let be a mapping such that F has the mixed monotone property and
(2.5)
for which and , where . Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if is a non-decreasing sequence and , then for all ,
-
(ii)
if is a non-increasing sequence and , then for all .
If there exist
such that
then F has a coupled fixed point , that is,
Proof First, we define a partial order ≼ on as follows: if and only if and . Second, we define a fuzzy set on as follows: for any and any . Since is a complete FNS, we can easily prove that is a complete FNS. Lastly, we define a mapping by
Since F has the mixed monotone property, if , we have that
that is, . Therefore, is a non-decreasing mapping. Since and
we have that . If , by (2.5) we have that
Thus, all the assumptions of Theorem 2.1 hold for and . By Theorem 2.1 we get that has a fixed point , that is, . This implies that , , that is, is a coupled fixed point of F. The proof is completed. □
By using Theorem 2.2, we can prove the following coupled fixed point theorem in intuitionistic fuzzy normed spaces.
Theorem 2.3 Let be a partially ordered set, and let be a complete IFNS such that the t-norm ∗ is of H-type and for any . Let be a mapping such that F has the mixed monotone property and
(2.6)
for which and , where . Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if is a non-decreasing sequence and , then for all ,
-
(ii)
if is a non-increasing sequence and , then for all .
If there exist
such that
then F has a coupled fixed point , that is,
Proof Assume that is a sequence in . Let , . If , then by Definition 1.5(i) we can deduce that . Thus, a sequence in is Cauchy if and only if is Cauchy in . By Lemma 1.1, we know that the topology of is the same as the topology of . This implies that is a complete IFNS if and only if is a complete FNS. Therefore, by using Theorem 2.2 to and F, we get that F has a coupled fixed point . The proof is completed. □
Theorem 2.4 Let be a partially ordered set, and let be a complete IFNS such that the t-norm ∗ is of H-type and for any . Let , be two mappings such that F has the mixed g- monotone property and
(2.7)
for which , and , where , and g is continuous. Suppose either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if is a non-decreasing sequence and , then for all ,
-
(ii)
if is a non-increasing sequence and , then for all .
If there exist
such that
then there exist
such that
that is, F and g have a coupled coincidence point in X.
Proof The conclusion of Theorem 2.4 can be proved by using Lemma 1.2 and Theorem 2.3. Since the proof is similar to the proof of Theorem 3.2 in [17], we delete the details of the proof. The proof is completed. □
Remark 2.1 It follows from the proof of the above theorems that the following implications hold: Theorem 2.1 ⟹ Theorem 2.2 ⟹Theorem 2.3 ⟹ Theorem 2.4. Conversely, it is clear that the following implications hold: Theorem 2.4 ⟹ Theorem 2.3 ⟹ Theorem 2.2. Thus, we have the following conclusion.
Theorem 2.5 Theorem 2.2-Theorem 2.4 are equivalent.
Remark 2.2 In [13] and [17], the condition for all is used. But this condition cannot hold in intuitionistic fuzzy normed spaces. In fact, if this condition holds, by using (iii) and (iv) in the definition of IFNS, we can get , which yields a contradiction. Furthermore, the proofs of the results in [13] and [17] have the same errors as noted in [18]. Therefore, our results improve and correct results in [13] and [17].
In the following, we give an example to show that our contractive conditions are a real improvement over the contractive conditions used in [13] and [17].
Example 2.1 Let , , for every , and let , , for all . Then is a complete intuitionistic fuzzy normed linear space, and the t-norm ∗ and t-conorm ⋆ are of H-type. If X is endowed with the usual order , then is a partially ordered set. Let , and define , for any . Then is a mixed g-monotone mapping, , and g is continuous. Let and , then
For any , with , we have
Thus, all the conditions of Theorem 2.4 are satisfied. By Theorem 2.4, there is such that and . But υ does not satisfy the contractive conditions in [13] and [17]. In fact, for , ,
This shows that υ does not satisfy the contractive conditions in [13] and [17].