SOME FIXED POINT THEOREMS FOR MAPPINGS ATISFYING CONTRACTIVE CONDITIONS OF INTEGRAL TYPE IN MODIFIED INTUITIONISTIC UZZY METRIC SPACES

In this paper, we prove some common fixed point theorems for occasionally weakly compatible mappings satisfying contractive conditions of integral type in modified intuitionistic fuzzy metric spaces. AMS Subject Classification: 47H10, 54H25


Introduction
Atanassove [2] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets.
In 2004, Park [9] defined the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norms and continuous t-conorms.
Recently, in 2006, Alaca et al. [1] using the idea of intuitionistic fuzzy sets, defined the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norm and continuous t-conorms as a generalization of fuzzy metric space due to Kramosil and Michálek [8].In 2006, Türkoǧlu et al. [13] proved Jungck's common fixed point theorem ( [6]) in the setting of intuitionistic fuzzy metric spaces for commuting mappings.In 2006, Gregori et al. [5] showed that the topology induced by fuzzy metric coincides with topology induced by intuitionistic fuzzy metric.
In view of this observation, Saadati et al. [11], in 2008, reframed the idea of intuitionistic fuzzy metric spaces and proposed a new notion under the name of modified intuitionistic fuzzy metric spaces with the help of the notion of continuous t-representable.
In this paper, we prove some common fixed point theorems for occasionally weakly compatible mappings satisfying contractive conditions of integral type in modified intuitionistic fuzzy metric spaces.
Remark 2.4.( [10]) In a modified intuitionistic fuzzy metric space (X, ζ M,N , ℑ), for any t > 0, ζ M,N (x, y, t) is non-decreasing with respect to t in (L * , ≤ L * ) for all x, y ∈ X. Definition 2.5.( [12]) Let f and g be self-mappings of a modified intuitionistic fuzzy metric space (X, ζ M,N , ℑ).Then the pair (f, g) is said to be for all x ∈ X and t > 0.
Definition 2.6.( [12]) Let f and g be self-mappings of a modified intuitionistic fuzzy metric space (X, ζ M,N , ℑ).Then the pair (f, g) is said to be weakly commuting if for all x ∈ X and t > 0.
Definition 2.7.( [11], [12]) Let f and g be self-mappings of a modified intuitionistic fuzzy metric space (X, ζ M,N , ℑ).Then the pair (f, g) is said to be compatible if lim Definition 2.8.( [11], [12]) Let f and g be self-mappings of a modified intuitionistic fuzzy metric space (X, ζ M,N , ℑ).Then the pair (f, g) is is said to be weakly compatible if they commute at the coincidence points, that is, if f u = gu for some u ∈ X, then f gu = gf u.
It is easy to see that compatible mappings are weakly compatible but converse is not true.Definition 2.9.([3], [7]) Let f and g be self-mappings of a modified intuitionistic fuzzy metric space (X, ζ M,N , ℑ).Then the pair (f, g) is said to be occasionally weakly compatible if there exists a point x ∈ X which is coincidence point of f and g at which f and g commute.

Main Results
Now, we prove common fixed point theorems for three mappings.
Theorem 3.1.Let (X, ζ M,N , ℑ) be a modified intuitionistic fuzzy metric space.Suppose that f, g and h are three self-mappings on X satisfying the following conditions: for all x, y ∈ X and α, β, γ are non-negative reals numbers with α+2β +2γ < 1, where and φ : R + → R is a Lebesgue-integrable mapping which is summable, nonnegative and ǫ 0 φ(t)dt > 0 for each ǫ > 0. Assume that the pair (f, h) or (g, h) is occasionally weakly compatible.Then f, g and h have a unique common fixed point.
Proof.Suppose that the pair (f, h) is occasionally weakly compatible.Then there exists an element u ∈ X such that f u = hu and f hu = hf u.Now, we prove that f u = gu.Indeed, by inequality (C1), we get where which is a contradiction and hence gu = f u = hu.Again, suppose that f f u = f u.Then by (C1), we have where which is a contradiction and hence f f u = f u = hf u.Now, suppose that gf u = f u.Then by (C1), we have where which is a contradiction and hence gf u = f u.Put f u = gu = hu = z.Therefore, z is a common fixed point of f, g and h.
Similarly, if the pair (g, h) is occasionally weakly compatible, then f, g and h have a common fixed point.
Finally, let z and w (z = w) be two common fixed points of f, g and h.
Then from (C1), we have where which is a contradiction and hence z = w.Thus the common fixed point is unique.This completes the proof.
If we put φ(t) = 1 in Theorem 3.1, we get the following corollary: Corollary 3.2.Let (X, ζ M,N , ℑ) be a modified intuitionistic fuzzy metric space.Suppose that f, g and h are three self-mappings on X satisfying the following conditions: for all x, y ∈ X and α, β, γ are non-negative reals numbers with α+2β +2γ < 1.
Assume that the pair (f, h) or (g, h) is occasionally weakly compatible.Then f, g and h have a unique common fixed point.
Next, we prove common fixed point theorems for four mappings.
Theorem 3.3.Let (X, ζ M,N , ℑ) be a modified intuitionistic fuzzy metric space.Suppose that f, g, h and k are four self-mappings on X satisfying the following conditions: for all x, y ∈ X and α, β, γ are non-negative reals numbers with α+2β +2γ < 1, where and φ : R + → R is a Lebesgue-integrable mapping which is summable, nonnegative and ǫ 0 φ(t)dt > 0 for each ǫ > 0. Assume that the pairs (f, h) and (g, k) are occasionally weakly compatible.Then f, g, h and k have a unique common fixed point.
Proof.Since pairs of mappings (f, h) and (g, k) are occasionally weakly compatible.Then there exists two points u, v ∈ X such that f u = hu and f hu = hf u, gv = kv and gkv = kgv.Now, we prove that f u = gv.Indeed, by (C2), we get which is a contradiction and hence gv = f u = hu = kv.Again, suppose that f f u = f hu = hf u = f u.Then by (C2), we have for all t > 0 and x, y ∈ X. Define x ∈ [0, 1), Clearly the pairs (f, h) and (g, k) are occasionally weakly compatible.Also if we define φ(x) = 3x 2 and by taking α = 1 4 , β = 1 5 and γ = 1 6 , then all the hypotheses of Theorem 3.3 are satisfied and x = 1 is a unique common fixed point of f, g, h and k.