Common Fixed Point Theorems for Weakly Subsequentially Continuous Mappings in Modified Intuitionistic Fuzzy Metric Spaces

The notion of intuitionistic fuzzy sets was been introduced by Atanassov [1], it can be considered as a generalization to fuzzy sets concept due to Zadeh [30]. Later Çoker [6] introduced a topology on intuitionistic fuzzy sets. Park [21] introduced the notion of intuitionistic fuzzy metric spaces as a generalization to fuzzy metric spaces, which is a combination between intuitionistic fuzzy sets and the concept of a fuzzy metric space given by George and Veeramani [9], many authors established some results concerning fixed point in such spaces, see for example [2, 5, 12, 14, 19, 29]. Meanwhile, jungck[16] defined the concept of compatible mappings, Jungck and Roadhes [17] generalized the last concept to the weakly compatible mappings, which is weaken than the compatible ones. Mishra et al [20] generalized the concept of compatibility in the setting of fuzzy metric spaces, he obtained some common fixed point theorems for compatible mappings in such spaces. Recently Bouhadjera and Godet Tobie [4] introduced the concept of subsequential continuity and utilized it with the concept of subcompatible mappings to establish a common fixed point, later Imdad et al.[13] improved these results and replaced subcompatibility by compatibility and subsequential continuity by reciprocal continuity, more recently, Gopal and Imdad [11] combined subsequential continuous maps with compatible maps concept to obtain some results in fuzzy metric spaces. In present work, we will generalize certain definitions to intuitionistic fuzzy metric spaces in order to obtain some common fixed point theorems by combining the concept of weakly subsequentially continuous mappings due to second author [3] with compatible of type (E) mappings given by Singh et al.[27, 28].


Introduction
The notion of intuitionistic fuzzy sets was been introduced by Atanassov [1], it can be considered as a generalization to fuzzy sets concept due to Zadeh [30]. Later Ç oker [6] introduced a topology on intuitionistic fuzzy sets. Park [21] introduced the notion of intuitionistic fuzzy metric spaces as a generalization to fuzzy metric spaces, which is a combination between intuitionistic fuzzy sets and the concept of a fuzzy metric space given by George and Veeramani [9], many authors established some results concerning fixed point in such spaces, see for example [2,5,12,14,19,29]. Meanwhile, jungck [16] defined the concept of compatible mappings, Jungck and Roadhes [17] generalized the last concept to the weakly compatible mappings, which is weaken than the compatible ones. Mishra et al [20] generalized the concept of compatibility in the setting of fuzzy metric spaces, he obtained some common fixed point theorems for compatible mappings in such spaces. Recently Bouhadjera and Godet Tobie [4] introduced the concept of subsequential continuity and utilized it with the concept of subcompatible mappings to establish a common fixed point, later Imdad et al. [13] improved these results and replaced subcompatibility by compatibility and subsequential continuity by reciprocal continuity, more recently, Gopal and Imdad [11] combined subsequential continuous maps with compatible maps concept to obtain some results in fuzzy metric spaces. In present work, we will generalize certain definitions to intuitionistic fuzzy metric spaces in order to obtain some common fixed point theorems by combining the concept of weakly subsequentially continuous mappings due to second author [3] with compatible of type (E) mappings given by Singh et al. [27,28].

For every
c j = 1, then we have: Its units are given by 0 L * = (0, 1) and 1 L * = (1, 0). A triangular norm T = * on [0, 1] is considered as an commutative, associative and increasing mapping T : 1]. The same, a triangular co-norm S = is a commutative, associative and increasing mapping S : Definition 2.2. [7,8] Let T be a continuous t-norm on L * , then T is a continuous t-representable if and only if there exist a continuous t-norm * and a continuous t-conorm on [0, 1] satisfy: for all x = (x 1 , x 2 ), y = y 1 , y 2 ) ∈ L * . Now, let {T n } be a sequence such that {T 1 = T } and for n > 1 and x(i) ∈ L * ,  3. T (x, T (y, z)) = T ((T (x, y)), z).
4. x ≤ L * x , y ≤ L * y =⇒ T (x, y) ≤ mathcalT (x , y ), (monotonicity). Definition 2.5. [24] Let M, N : X 2 × (0, ∞) → [0, 1] be two fuzzy sets satisfying M (x, y, t) + N (x, y, t) ≤ 1, for all x, y ∈ X and t > 0. The triplet (X, M M,N , T ) is called to be an intuitionistic fuzzy metric space if X is an arbitrary and non-empty set, T is a continuous t-representable and M M,N is a mapping (an intuitionistic fuzzy set)from X 2 × (0, ∞) into L * satisfying the following conditions for every x, y ∈ X and t, s > 0: Example 2.1. [24] Let (X, d) be a metric space and T (a, b) = (a 1 , b 1 ), min(a 2 + b 2 , 1)) for all a = (a 1 , a 2 ) and b = (b 1 , b 2 ) ∈ L * . M and N are to fuzzy sets on X 2 × (0, ∞) defined by: ), Common Fixed Point Theorems for Weakly Subsequentially Continuous Mappings in Modified Intuitionistic Fuzzy Metric Spaces ) for all a = (a 1 , a 2 ) and b = (b 1 , b 2 ) ∈ L * . M, N : X 2 × (0, ∞) → L * are two fuzzy sets such that: Let (X, M, N, * , ) be an intuitionistic fuzzy metric space and let {x n } be a sequence in X, and for each n, m ≥ n 0 , where N s is the standard negator.
2. non compatible [29] if there exists at least one sequence {x n } in X such that or non-existent for at least one t > 0.
3. weakly compatible [24] if they commute at their coincidence points, i.e, if Au = Su for some u ∈ X, then ASu = SAu.
It is clear the discontinuity of A and S at 1.
Let {x n } be a sequence in X defined by: x n = 1 n , for each n ≥ 1. Hence (A, S) is A-subsequentially continuous.
Motivated by [27,28], define: Remark that if the pair (A, S) is compatible of type (E), so it is S-compatible and A-compatible of type (E), but the converse may be not true.  1 , min(a 2 + b 2 , 1)) for all a = (a 1 , a 2 ) and b = (b 1 , b 2 ) ∈ L * . We define A, S as follows: Let Ψ be the set of all continuous functions F : (L * ) 6 → L * satisfying the conditions: Example 2.6.