## Recent Development in Fixed-Point Theory, Optimization, and their Applications

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N. Hussain, S. Khaleghizadeh, P. Salimi, Afrah A. N. Abdou, "A New Approach to Fixed Point Results in Triangular Intuitionistic Fuzzy Metric Spaces", *Abstract and Applied Analysis*, vol. 2014, Article ID 690139, 16 pages, 2014. https://doi.org/10.1155/2014/690139

# A New Approach to Fixed Point Results in Triangular Intuitionistic Fuzzy Metric Spaces

**Academic Editor:**Ngai-Ching Wong

#### Abstract

The aim of this paper is to propose some fixed point theorems in complete parametric metric spaces. Using these theorems, we deduce as corollaries the recent results of Ionescu et al. Moreover, we suggest some new contractions and prove certain fixed point theorems in triangular intuitionistic fuzzy metric spaces. We also discuss some illustrative examples to highlight the realized improvements.

#### 1. Introduction and Preliminaries

The concept of fuzzy set was introduced by Zadeh [1] in 1965. In 1975, Kramosil and MichÃ¡lek [2] introduced the notion of fuzzy metric space, which can be regarded as a generalization of the statistical (probabilistic) metric space. This work has provided an important basis for the construction of fixed point theory in fuzzy metric spaces. Afterwards, Grabiec [3] defined the completeness of the fuzzy metric space (now known as a -complete fuzzy metric space) and extended the Banach contraction theorem to -complete fuzzy metric spaces. Successively, George and Veeramani [4] modified the definition of the Cauchy sequence introduced by Grabiec. Meanwhile, they slightly modified the notion of a fuzzy metric space introduced by Kramosil and MichÃ¡lek and then defined a Hausdorff and first countable topology on it. Since then, the notion of a complete fuzzy metric space presented by George and Veeramani has emerged as another characterization of completeness, and some fixed point theorems have also been proved on the basis of this metric space. From the above analysis, we can see that there are many studies related to fixed point theory based on the above two kinds of complete fuzzy metric spaces (see for more details [5â€“11] and the references therein). In 2004, Park introduced the notion of intuitionistic fuzzy metric space [12]. He showed that, for each intuitionistic fuzzy metric space , the topology generated by the intuitionistic fuzzy metric coincides with the topology generated by the fuzzy metric . For more details on intuitionistic fuzzy metric space and related results we refer the reader to [12â€“17].

*Definition 1. *A 5-tuple is said to be an intuitionistic fuzzy metric space if is an arbitrary set, is a continuous* t*-norm, is a continuous* t*-conorm, and are fuzzy sets on satisfying the following conditions, for all and : (i);(ii);(iii) for all if and only if ;(iv);(v);(vi) is left continuous;(vii);(viii);(ix) if and only if ;(x);(xi);(xii) is right continuous;(xiii).

Then is called an intuitionistic fuzzy metric on . The functions and denote the degree of nearness and the degree of nonnearness between and with respect to , respectively.

*Definition 2. *Let be an intuitionistic fuzzy metric space. Then, (i)a sequence is said to be Cauchy sequence whenever and for all ,(ii) is called complete whenever every Cauchy sequence is convergent with respect to the topology .

*Remark 3. *Note that, if is an intuitionistic fuzzy metric on and is a sequence in such that , then as, from (i) of Definition 1, we know that for all and all .

*Definition 4 (see [18, 19]). *Let be an intuitionistic fuzzy metric space. The fuzzy metric is called triangular whenever
for all and all .

#### 2. From Parametric Metric to Triangular Intuitionistic Fuzzy Metric

First we define the concept of parametric metric space.

*Definition 5. *Let be a nonempty set and let be a function. We say is a parametric metric on if (i) for all if and only if ;(ii) for all ;(iii) for all and all 0.

and one says the pair is a parametric metric space.

*Example 6. *Let denote the set of all functions . Define, by for all and all . Then is a parametric metric space.

*Example 7. *Let be an triangular intuitionistic fuzzy metric space. We consider the mapping defined by for all and all . Then by Definitions 1(iii) and 4, is a parametric metric on and hence is a parametric metric space.

Let be a parametric metric space. Let and ; then the set is called an open ball with center at and radius .

Now we have the following definitions.

*Definition 8. *Let be a parametric metric space, and let be a sequence of points of . A point is said to be the limit of the sequence , if for all , and one says that the sequence is convergent to and denotes it by as .

*Remark 9. *Note that if is a parametric metric space and , are two sequences in such that and as , then for all . That is, is continuous in its two variables.

*Definition 10. *Let be a parametric metric space. (S1)A sequence is called a Cauchy if and only if for all 0.(S2)A parametric metric space is said to be complete if and only if every Cauchy sequence in converges to .

*Definition 11. *Let be a parametric metric space and let be a mapping. One says is a continuous mapping at in , if, for any sequence in such that as , as .

Theorem 12. *Let be a complete parametric metric space and let be a continuous self-mapping such that
**
holds for all with and all where . Then has a unique fixed point.*

*Proof. *Let . Define the sequence by for all positive integers . If there exists such that , then is fixed point of and we have nothing to prove. Hence we assume for all . From (3) with and we get
Now if , then from the above inequality we get
which is a contradiction. Therefore,
for all and all . Hence,
Then for any by (7) we have
for all . By taking limit as in the above inequality we get . Therefore, is a Cauchy sequence in . Since is a complete parametric metric space, there exists such that as . Now since is continuous,
That is, has a fixed point. To prove the uniqueness of , suppose that is another fixed point of such that . From (3) we obtain
which is a contradiction. So, has a unique fixed point.

*Example 13. *Let be endowed with the parametric metric
for all and all . Define by

Clearly, is a complete parametric metric space and is a continuous mapping. Now we consider the following cases:(i)Let . Then, (ii)Let . Then, (iii)Let . Then, (iv)Let and . Then, (v)Let and . Then, (vi)Let and . Then, Therefore, for all with and all . Hence, all conditions of Theorem 12 hold and has a unique fixed point.

Corollary 14. *Let be a complete parametric metric space and let be a continuous self-mapping such that
**
holds for all and all where . Then has a unique fixed point.*

Corollary 15. *Let be a complete parametric metric space and let be a continuous self-mapping such that
**
holds for all with and all where , with . Then has a unique fixed point.*

*Proof. *Since
holds for all with and all , all conditions of Theorem 12 hold and has a unique fixed point.

Theorem 16. *Let be a complete parametric metric space and let be a self-mapping on . Assume that
**
holds for all and all where is a function. Then has a fixed point.*

*Proof. *Let . Define a sequence by for all positive integers . From (23) with and we get
which implies is a nonincreasing sequence and so
for all . Therefore from (24) we get
for all where . Now, it is easy to show that is a Cauchy sequence. The completeness of ensures that the sequence converges to some . From (23) we obtain
Taking limit as in the above inequality, we deduce that ; that is, .

*Example 17. *Let denote the set of all functions . Define by and by
for all and all where

Then is a complete parametric metric space. Define by . If or , then
and so,
which impliesNow we get
Also if and , then , , and . That is, and , and so Therefore, all conditions of Theorem 16 hold true and has a fixed point.

If we take in Corollaries 14 and 15, respectively, we deduce the following recent results as corollaries.

Corollary 18 (Theoremâ€‰â€‰2.1 in [19]). *Let be a complete triangular intuitionistic fuzzy metric space and let be a continuous mapping satisfying the contractive condition
**
for all and all where . Then has a fixed point.*

Corollary 19 (Theoremâ€‰â€‰2.3 in [19]). *Let be a complete triangular intuitionistic fuzzy metric space and let be a continuous mapping satisfying the contractive condition
**
for all and all where and . Then has a unique fixed point.*

By taking and in Theorem 16, we deduce the following result.

Corollary 20 (Theoremâ€‰â€‰2.2 in [19]). *Let be a complete triangular intuitionistic fuzzy metric space and let be a continuous mapping satisfying the contractive condition**for all and all . Then has a fixed point.*

#### 3. A New Fixed Point Theorem in Intuitionistic Fuzzy Metric Spaces

In this section we suggest new contraction and prove fixed point theorems in the framework of triangular intuitionistic fuzzy metric spaces which can not be obtained from the existing results in metric spaces.

Let denotes the class of those functions which satisfies the condition .

Theorem 21. *Let be a complete triangular intuitionistic fuzzy metric space and let be a self-mapping on such that
**
holds for all and all where ,
**
Then has a unique fixed point.*

*Proof. *Let . We define an iterative sequence in the following way:
From (38) with and we get
for all and all where
Now from (41) and (42) we have
which implies
where
Thus
Now if , then by (44) we obtain
which is a contradiction. Hence,
for all and all . Therefore is a nonincreasing sequence and so it converges to some . Suppose to the contrary that . Taking limit as in (48) we get . Due to property of we have . Equivalently, , which is a contradiction. Thus,
Now we want to show that for all . Suppose to the contrary

From (38) we have
On the other hand,
Similarly,
so from (52) and (53) we obtain
Taking limit supremum as in (54) and using (49) and (50) we deduce
which implies
Further,
Taking limit supremum as in (51) and applying (56) and (57) we get

On the other hand we have
By taking limit supremum as in the above inequality and using (49) and (58) we get
which implies . Thus,