Fixed point results in non-Archimedean generalized intuitionistic fuzzy metric spaces

: In this paper, we introduce the concept of non-Archimedean generalized intuitionistic fuzzy metric space and obtain some results for two semi compatible mappings in this newly defined space. Our results improve and generalize the results of Mustafa et al. [8] and Abbas and Rhoades [1] in non-Archimedean G- fuzzy metric space.


Introduction
In mathematics, the concept of fuzzy sets was introduced by Zadeh [13]. It is a new way to represent vagueness in our daily life. In 1975, Kramosil and Michalek [6] introduced the concept 49 of fuzzy metric spaces which opened a new way for further development of analysis in such spaces. George and Veeramani [4] modified the concept of fuzzy metric space. After that, several fixed point theorems proved in fuzzy metric spaces. In 2006, Mustafa and Sims [7] presented a definition of G-metric space. After that, several fixed point results proved in G-metric spaces. On the other hand, Atanassov [2] introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Park [9] has introduced and studied the notion of intuitionistic fuzzy metric spaces. Further, Saadati et al. [11] proposed the idea of a continuous trepresentable under the name modified intuitionistic fuzzy metric space which is a milestone in developing fixed point theory.
Motivated by the concepts of G-metric space, non-Archimedean metric space and Fuzzy metric space, we introduce the concept of non-Archimedean generalized intuitionistic fuzzy metric space and obtain two common fixed point theorems for two semi compatible mappings. In this paper, we introduce the notion of generalized intuitionistic fuzzy metric space and describe some of their properties of generalized fuzzy metric space. Our results improve and generalize the results of Mustafa et. al. [8] and Abbas and Rhoades [1] in non-Archimedean G-fuzzy metric space. We also establish properties P and Q for these mappings.

Preliminaries
Definition 2.1. A 5-tuple (X, G, H, *, ) is said to be a generalized intuitionistic f uzzy metric space, if X is an arbitrary non-empty set, * is a continuous t-norm, is a continuous t-conorm, G and H are fuzzy sets on X 3 × (0, ∞) satisfying the following conditions: for every x, y, z, a ∈ X and t, s > 0.
(i) In this case, the pair (G, H) is called a generalized intuitionistic fuzzy metric on X. Definition 2.2. A 5-tuple (X, G, H, *, ) is said to be a non-Archimedean generalized intuitionistic f uzzy metric space (shortly GIFM-Space), if X is an arbitrary set, * is a continuous t-norm, is a continuous t-conorm and G and H are fuzzy sets on X 3 × (0, ∞ ) satisfying the following conditions: for all x, y, z, a X and s, t > 0. The functions G(x, y, z, t) and H(x, y, z, t) denote the degree of nearness and degree of nonnearness between x, y and z with respect to t, respectively.

Remark 2.3.
In generalized intuitionistic fuzzy metric space X. G(x, y, z, .) is non-decreasing and H(x, y, z, .) is non-increasing for all x, y, z X .
In the above definition, if the triangular inequality (GIFM 5) and (GIFM 10) are replaced by the following:
So, that the sequence {pn} is a decreasing sequence of positive real numbers in [0, 1] and tends to a limit p ≥ 0. We claim that p = 0. If p >1, on taking n → ∞ in (2.6.1), we get p ≤ (p) < p, which is a contradiction. Hence p = 0. Now, for any positive integer p, we have Taking the limit as n → ∞, we get lim It follows that (f , g) is semi-compatible and f y = gy, then f gy = gf y. Note that every pair of semi-compatible maps are weakly compatible, but the converse need not be true.

Main results
Now, we generalize the results of Abbas and Rhoades [1] to non-Archimedean generalized intuitionistic fuzzy metric space for semi-compatible maps as follows: where ϕ Ф, * , t > 0. If f (X) ⊂ g(X) and g(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are semi-compatible, then f and g have a unique common fixed point.
Proof . Let x0 be a point in X. Since f(X) ⊂ g(X), so we choose a point x1 in X such that f (x0) = g(x1). Continuing this process, having chosen xn in X, we can find xn+1 in X such that yn = f xn = gxn+1, n = 0, 1, 2, ….
Then, Lemma (2.6), {yn} is a Cauchy sequence. This implies that {gxn} is a Cauchy sequence. Since g(X) is complete, so there exists u g(X) such that Since u g(X), so there exists p X such that gp = u. Let f p ≠ u. From (3.1.1) , f p, f p, t) ≥ ϕ(G(gxn, gp, gp, t)) as n → ∞, we get G (u, f p, f p, t) ≥ ϕ(G(gp, gp, gp, t) This implies that G(u, fp, fp, t) = 1, H(fxn, fp, fp, t) ≤ (H(gxn, gp, gp, t)), as n → ∞, we get This implies that H(u, f p, f p, t) = 0, which is a contradiction, since f p ≠ u. Thus, f p = gp = u. Hence, p is a coincidence point of f and g. Now, we will show that p is unique. Assume that there exists another point q in X such that fq = gq. If fp ≠ fq, then
Moreover, if f and g are semi-compatible, then from proposition (2.11), f and g have a unique common fixed point.
If we take g = 1 in Theorem 3.1, we obtain the following result:   (1t). If the mappings f , g : X → X satisfy either , f x, f x, t), G(gy, gy, f y, t), G(gz, f z, f z, t) for all x, y, z X where ϕ Ф and Ψ, t > 0. If f (X) ⊂ g(X) and g(x) is a complete subspace of X than f and g have a unique point of coincidence in X. Moreover, if f and g are semicompatible, then f and g have a unique common fixed point.
Proof . Suppose that f and g satisfy (3.3.1). Let x0 be an arbitrary point in X. Since f (X) ⊂ g(X), so we choose a point x1 in X such that f (x0) = g(x1). Continuing this process, having chosen xn in X, we can find xn+1 in X such that f (xn) = g(xn+1   Proof . From Theorem 3.1, F(f ) ∩F(g) ≠ ϕ. Therefore, F(f n )∩F(g n ) ≠ ϕ for each positive integer n. Let n be a fixed positive integer greater than 1 and suppose that U F(f n ) ∩ F(g n ). We claim that u F(f ) ∩ F(g).
Let u F(f n )∩ F(g n ). Then, for any positive integers i, j, k, r, l, s satisfying 0 ≤ i, j, r, k, l, s ≤ n, we have G(f i g j u, f r g l u, f s g k u, t) ≥ ϕ(G(g(f i-1 g j u), g(f i-1 g l u), g(f s-1 g k u), t)) ≥ ϕ(G(f i-1 g j+1 u, f r-1 g l+1 u, f s-1 g k+1 u, t)).
H(f i g j u, f r g l u, f s g k u, t) ≤ (H(g(f i-1 g j u), g(f i-1 g l u), g(f s-1 g k u), t)) ≤ (H(f i-1 g j+1 u, f r-1 g l+1 u, f s-1 g k+1 u, t)).
Hence f and g have property Q. , F(f ) ∩F(g) ≠ Ø. Therefore, F(f n )∩F(g n ) ≠ Ø for each positive integer n. Let n be a fixed positive integer greater than 1 and suppose that U F(f n ) ∩F(g n ). We claim that u F(f )∩F(g). Let u F(f n ) ∩ F(g n ).
Then, for positive integers i, j, r, l, s, k satisfying 0 ≤ i, r, j, l, s, k ≤ n, we have