Common Fixed Point Theorems in Intuitionistic Generalized Fuzzy Cone Metric Spaces

In the present work, we study many fixed point results in intuitionistic generalized fuzzy cone metric space. Precisely, we prove new common fixed point theorems for three self mappings that do not require any commutativity or continuity but a generalized contractive condition. Our results are generalizations for many results in the literature. Some examples are given to support these results.


Introduction
In the year 1965, Zadeh [1] introduced the concept of fuzzy sets which permit the gradual assessment of the membership of the elements in a set. In contrast to classical sets, these sets are serving good in describing the vague and imprecise expressions in a formal way. As these sets have no means of incorporating the hesitation, Atanassov [2] brought out a possible solution with intuitionistic fuzzy sets in the year 1983. These sets serve as a powerful tool to deal with vagueness. In addition, in the year 1975, Kramosil and Michalek [3] first introduced a metric on fuzzy sets. Subsequently, kinds of fuzzy metrics [4][5][6] were introduced over fuzzy sets. In the year 1994, George and Veeramani [7] modified the definition of fuzzy metric space that was given by Kramosil and Michalek [3] and obtained a metrizable Hausdorff topology. As a consequence of these findings, several authors came up with generalized versions of these spaces in various settings. In 2007, Huang and Zhang [8] introduced cone and cone metric space, and, after that, Tarkan Oner et al. [9] defined fuzzy cone metric space as a generalization of fuzzy metric space [7]. Mohamed and Ranjith [10] came up with intuitionistic fuzzy cone metric space in the year 2017.
In the present paper, we study the Banach Contraction theorem in the setting of intuitionistic generalized fuzzy cone metric space [11] and to construct some common fixed point theorems for three self mappings which satisfy generalized contractive conditions in the intuitionistic generalized fuzzy cone metric spaces. The significant advantage of these theorems is that they work well where the Banach contraction theorem fails. Examples are provided to exhibit the novelty of the results given here.

Preliminaries
Let us begin the section with triangular norms which are kinds of binary operations introduced by Karl Menger and later revised by Schweizer and Sklar [25] with stronger axioms, as stated here.

Definition 3 ([9]
). Let B be a real Banach space and C be a subset of B. C is called a closed cone if and only if: The closed cones considered here are subsets of a real Banach space B and are with nonempty interiors.

Definition 4 ([11]
). An Intuitionistic Generalized Fuzzy Cone Metric Space (briefly, IGFCM Space) is a 5-tuple (Z, M, N, * , ) where Z is an arbitrary set, * is a continuous t-norm, is a continuous t-conorm, C is a closed cone and M, N are fuzzy sets in Z 3 × int(C) satisfying the following conditions: For all ζ, η, ω, u ∈ Z and c, c ∈ int(C), N(ζ, η, ω, c) = N(p{ζ, η, ω}, c), where p is a permutation function, The pair (M, N) is called Intuitionistic Generalized Fuzzy Cone Metric on Z. The functions M(ζ, η, ω, c) and N(ζ, η, ω, c) denote, respectively, the degree of nearness and the degree of non nearness between ζ, η and ω with respect to c.

Remark 1.
It is to be noted that: (i) The intuitionistic fuzzy setting provides both a membership degree and a nonmembership degree for an element, whereas the fuzzy settings provide only the membership degree alone and thus the space considered here will definitely provide a better environment than the latter to work with the applications. (ii) Reference [2] Every fuzzy setting can be generalized to intuitionistic fuzzy setting but not the converse.
(ii) {ζ n } is said to be a Cauchy sequence if, for all c ∈ int(C) and m ∈ N, for all c ∈ int(C).
The following theorem gives the extension of Banach Contraction Principle in the IGFCM Space.
Next, let us prove some common fixed point theorems for three self mappings satisfying generalized contractive conditions in a complete IGFCM Space.

Example 2.
Consider the metric space Z = [0, +∞) with metric d given by d(ζ, η) = |ζ − η| for all ζ, η ∈ Z. Let C = R + , * be a continuous t-norm, and be a continuous t-conorm. Define the M, N : for all ζ, η, ω ∈ Z and c ∈ int(C). Then, it is clear that (Z, M, N, * , ) is a complete IGFCM Space and that (M, N) is triangular. Consider the self mappings P, Q and R from Z to Z, given by Pζ = Here, P, Q and R together satisfy the condition (1) with Therefore, P, Q and R have a unique common fixed point and it is ζ = 14.
Remark 4. In the above example, P, Q and R are not k-FCC and hence Theorem 1 cannot assure the existence of fixed points of any of P, Q and R.  N) is triangular. If P, Q, R : Z → Z is such that for all ζ, η, ω ∈ Z and c ∈ int(C), where k ∈ (0, 1) and Then, P, Q and R have a unique common fixed point.
Therefore,ζ =ζ and we can conclude that P, Q and R have a unique common fixed point.

Example 3.
Consider the IGFCM Space given in Examsple 2 and the self mappings P, Q, R from Z to Z, given by for all ζ, η, ω ∈ Z. Thus, P, Q and R together satisfy the conditions (11) and (12) with k = 1 4 . Therefore, P, Q and R have a unique common fixed point, and it is ζ = 3. Then, P has a unique fixed point.

Conclusions
This work extended the Banach contraction theorem to intuitionistic generalized fuzzy cone metric spaces. This work also constructed and proved some common fixed point theorems for three self mappings under generalized fuzzy contractive conditions. It is clear from the examples that the common fixed point theorems given here assure the existence of fixed points of the mappings, but the Banach contraction theorem fails to prove the existence of the common fixed point of such examples. The results proved here can be further extended to various spaces in different settings by increasing the number of self mappings, by imposing conditions on/between them and by analyzing the generalized contractive conditions.