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Research Article | Open Access
-Tuplet Coincidence Point Theorems in Intuitionistic Fuzzy Normed Spaces
For an arbitrary n positive integer, we investigate the existence of n-tuplet coincidence points in intuitionistic fuzzy normed space. Results of the paper are more general than those of the coupled and the tripled fixed point works in intuitionistic fuzzy normed space.
One of the most important fields in mathematics is the fixed point theory. This theory is used to solve a variety of problems in many areas such as economics, chemistry, computer science, and engineering as well as many branches of mathematics. One of the main theorems in the fixed point theory is the Banach contraction theorem . This theorem states that contraction map in complete metric space has a unique fixed point. Many authors studied the generalization of the Banach contraction theorem. Generalization on the complete partial ordered metric space was given by Ran and Reurings  with a weaker condition. In their theorem, contraction condition is provided only for comparable elements with respect to partial order relation in complete metric space. Some fixed point theorems have been obtained by many authors based on . Bhaskar and Lakshmikantham  defined the concept of coupled fixed point and used a theorem associated with it for existence and uniqueness of solution of the periodic boundary value problem. Later on, Lakshmikantham and Ćirić  introduced the concept of coincidence point which is a generalization of fixed point. By inspiring these works, coupled fixed point theorems have been studied for different type contraction mappings (see [5–12]). The interest on coupled fixed point theorem has motivated the authors to generalize it as tripled fixed point theorem in [13, 14], afterwards as quadruple fixed point theorem in [15–17], and as -tuplet fixed point theorem in [18–21].
On the other hand, fuzzy theory was introduced by Zadeh  and it was generalized by Atanassov  as intuitionistic fuzzy theory. While fuzzy theory assigns degree of membership for each element, intuitionistic fuzzy theory assigns degree of membership and nonmembership for each element. Both of them were applied in many fields of sciences.
Introduction of the intuitionistic fuzzy metric space by Park  and of the intuitionistic fuzzy normed space by Saadati and Park  has enabled many subjects in functional analysis to be studied in intuitionstic fuzzy normed (metric) spaces. Fixed point theory is one of these subjects. Many fixed point theorems have been studied in intuitionistic fuzzy normed (metric) space. Numerous works have been produced since richness of fixed point theory and intuitionistic fuzzy functional analysis. Some of the articles concerning these fields can be found in the literature (such as [26–33]).
Coupled and tripled fixed point theorems in intuitionistic fuzzy normed space were proved via -property in [29, 30], respectively. The purpose of our paper is to study -tuplet coincidence point theorem without -property in intuitionistic fuzzy normed space, which is the generalization of coupled fixed point theorem  and tripled fixed point theorem  in intuitionistic fuzzy normed space.
Let us start by recalling some of the concepts used in this paper.
Definition 1 (see ). A binary operation is a continuous -norm if it satisfies the following conditions: (i) is associative and commutative; (ii) is continuous; (iii) for all ; (iv) whenever and for each .
Definition 2 (see ). A binary operation is a continuous -conorm if it satisfies the following conditions: (i) is associative and commutative; (ii) is continuous; (iii) for all ; (iv) whenever and for each .
Definition 3 (see ). Let be a continuous -norm, let be a continuous -conorm, and let be a linear space over the field ( or ). If and are fuzzy sets on satisfying the following conditions, the five-tuple is said to be an intuitionistic fuzzy normed space and is called an intuitionistic fuzzy norm. For every and , one has the following:(i),(ii),(iii),(iv) for each ,(v),(vi) is continuous,(vii) and ,(viii),(ix),(x) for each ,(xi),(xii) is continuous,(xiii) and ;
we further assume that satisfies the following axiom:(xiv) and for all .
Throughout this paper, expression “intuitionistic fuzzy normed space” will be denoted by IFNS for short.
Definition 4 (see ). Let be an IFNS. is called a cartesian product of intuitionistic fuzzy normed spaces if () is a cartesian product of intuitionistic fuzzy norms defined
where and .
Definition 5 (see ). Let be an IFNS. Then a sequence in is said to be Cauchy sequence if for each and there exists such that for each .
Definition 6 (see ). Let be an IFNS. is said to be complete if every Cauchy sequence in is convergent.
Definition 7 (see ). Let and be two IFNSs. is continuous at if in convergences to for any in converging to . If is continuous at each element of , then is said to be continuous on .
Definition 8 (see ). Let be partially ordered set and and . It is said that has the mixed -monotone property if is monotone -nondecreasing in its odd argument and it is monotone -nonincreasing in its even argument. That is, for any , Note that if is the identity mapping, this definition is reduced to Definition 1 in .
Definition 9 (see ). Let be a nonempty set and let be a given mapping. An element is called an -tuplet coincidence point of and if Note that if is the identity mapping, this definition is reduced to Definition 2 in .
Definition 10 (see ). Let and be two mappings. and are called commutative if for all .
2. Main Results
Theorem 11. Let be continuous map having mixed -monotone property on the complete having partial order relation denoted by . Also ; is continuous and commutes with . Suppose that and hold the following conditions, for all and : where , and . If there exist such that then there exist such that that is, and have an -tuplet coincidence point.
Proof. Proof of this theorem consists of four steps.
Step 1. In first step, let us define . Let be as in (8). Since , we construct the sequence , as in : for .
Step 2. We prove that the following inequalities hold: for all . This step is similar to a part of proof of Theorem 1 in . So, we omit it. However, since we use it in the proof, we express it again for the integrity of our proof. We can write (12) from (11) as follows:
Step 3. In this step, we show that are Cauchy sequences in .
To do this, we define
By considering (6) and (12), we have in the following inequalities: Using the property (iv) of -norm and property (xiv) in Definition 3 together with (14), we get
Again, by (6) and (12),
By the property (iv) of -norm together with (16), we get Thus, if we continue this process in this way, we have
Now we will do the same calculations for . By using (7) and (12), Using the property (iv) of -norm and property (xiv) in Definition 3 together with (19), we get
Again, by (7) and (12),
Hence, from (21), Thus, if we continue this process in this way, we have
Now, we can show are Cauchy sequences in by means of (18) and (23). For each and , tend to infinity when in (24). So, we get (25) with properties (vii) and (xiii) of intuitionistic fuzzy norm:
Hence, are Cauchy sequences in .
Step 4. In final step, we prove that and have an -tuplet coincidence point. Since is complete, then there exist such that
By using intuitionistic fuzzy continuity of , we write Since commutativity of and , it follows that
Using continuous of -norm and -conorm, we get
That is . Similarly . From the intuitionistic fuzzy continuous assumption of , we write in the followings
Considering (28) and (29), we have