On Dislocated Fuzzy Topologies and Fixed Points

The dislocated topology associated with a Dislocated Fuzzy Quasi Metric Space (DFqM − S pace) is discussed and a generalised common fixed point theorem for two mappings f : X → X and T : Xk → X in a DFqM − S pace is proved. Our result extends and generalises many well known results.


Introduction
Zadehs introduction (Zadeh, 1965) of the notion of fuzzy sets laid down the foundation of fuzzy mathematics.In the last two decades there were a tremendous growth in fuzzy mathematics.Many fixed point theorems for contractions in fuzzy metric spaces and quasi fuzzy metric spaces appeared (see Cho, Sedghi, & Shobe, 2009;Fang, 1992;George & Veeramani, 1997;Gregory & Romuguera, 2002;Gregory & Sapena, 2002;Mihet, 2004;Mishra, Sharma, & Singh, 1994;Radu, 2002;Romaguera, Sapena, & Tirado, 2007;Sedghi, Turkoghlu, & Shobe, 2007;Sharma, 2002;Vasuki, 1998;Vasuki & Veeramani, 2003).The role of topology in logic programming has come to be recognized in recent years.In particular topological methods are employed in order to obtain fixed point semantics for logic programs.In classical approach to logic programming semantics in which positive or definite positive programs are considered (those in which negation does not occur) Knaster -Tarski fixed point theorem can be applied to obtain a least fixed point of an operator called the single step or immediate consequence operator.However when the syntax is enhanced in the sense that negation is allowed, the approach using Knaster -Tarski theorem does not work.In such cases the Banach contraction mapping theorem for complete metric spaces is an alternative to Knaster -Tarski fixed point theorem.However topological spaces which arise in the area of denotational semantics are often not Hausdorff and so spaces which are weaker than metric spaces in a topological sense had to be cosidered.Motivated by this fact Hitzler and Seda (Hitzler & Seda, 2000) introduced the concept of dislocated metric space and studied the dislocated topologies which is a generalisation of the conventinal topologies and can be thought of as underlying the notion of dislocated metrics.They also proved a generalized version of Banach contraction mapping theorem which was applied to obtain fixed point semantics for logic programs.Later George and Khan (Reny George & Khan, 2005) introduced the concept of dislocated fuzzy metric spaces and studied the associated topologies.Alaca (Alaca, 2010) introduced the concept of Dislocated Fuzzy Quasi Metric Space (DFqM − S pace)in the sense of Kramosil and Michalek as well as George and Veeramani and discussed the topologies associated with it which is conventional in nature.In this paper we have discussed the dislocated fuzzy topologies associated with a DFqM − S pace and also proved a common fixed point theorem of Presic type which extends and generalises the well known Banach contraction principle in a Fuzzy metric space and also fuzzyfies other known results.

Preliminaries
Let (X, d) be a metric space and T : X → X and f : X → X be mappings.Let C(T, f ) denote the set of all coincidence points of the mappings f and T , that is C(T, f ) = {u : f u = T u}.
Definition 2.1 The mappings f and T are said to be weakly compatible if and only if they commute at their coincidence points.
Remark 2.2 Clearly mappings f and T are not weakly compatible if and only if there exist some u ∈ C(T, f ), such that f T u T f u.Obviously, if C(T, f ) = ϕ then f and T are weakly compatible.
Definition 2.3 The mappings f and T are said to be occasinally weakly compatible (owc) if and only if they commute at some coincidence point of f and T , i.e. f T u = T f u for some u ∈ C(T, f ).
Remark 2.4 Occasionally weakly compatible mappings requires the set of coincidence points to be non empty.In other words if C(T, f ) = ϕ then mappings f and T cease to be owc.Hence owc mappings cannot be considered as a generalisation of weakly compatible mappings.Hence we put it in the following way.
Definition 2.5 The mappings f and T are said to be occasionally weakly compatible (owc) if and only if f T u = T f u for some u ∈ C(T, f ) whenever C(T, f ) ϕ.
Definition 2.6 f is said to be coincidentally idempotent with respect to T if and only if f is idempotent at the coincidence points of f and T .
Definition 2.7 The mapping f is said to be occasionally coincidentally idempotent (oci) if and only if and A, B ∈ P(X), where P(X) is the power set of X.If x ≻ A we read it as x is below A. Definition 2.9 (Hitzler & Seda 2000) Let X be a set, ≻ be a d-membership relation on X.For each x ∈ X, let Ux be the collection of all subsets of X satisfying the following conditions: Then (Ux, ≻) is called a d-neighborhood system for x and each U ∈ U x is called a d-neighborhood of x.
Remark 2.10 If U = {U x : x ∈ X} then (X, U, ≻) is called a dislocated topological space or a d-topological space.
Remark 2.11 Note that points can have empty d-neighborhoods and the above definition is exactly the definition of a topological neighbourhood system if the relation ≻ is replaced by the membership relation ∈.

Dislocated Fuzzy Quasi Metric Space and Topologies
In this section we will define Dislocated Fuzzy Quasi Metric Space and discuss the topologies associated with it.Definition 3.2 Let X be any non empty set, * be a continuous t-norm and M : X 2 ×[0, ∞) → [0, 1] be a fuzzy set.Consider the following conditions: For all x, y, z ∈ X and t, s ∈ [0, ∞) (Kramosil & Michalec 1975).If M satisfies conditions F M1 and F M3 to F M6 then we say that (X, M, * ) is a Dislocated Fuzzy Metric Space in the sense of Kramosil and Michalek (in short D satisfies conditions F M1 and F M3 to F M5 and F M7 then we say that (X, M, * ) is a Dislocated Fuzzy Metric Space in the sense of George and Veeramani (in short D GV F M − S pace) (Reny George & Khan 2005).If M satisfies conditions F M1, F M3, F M5 and F M6 then we say that (X, M, * ) is a Dislocated Fuzzy Quasi Metric Space in the sense of Kramosil and Michalek (in short D K M FqM − S pace) (Alaca 2010).If M : X 2 × (0, ∞) → [0, 1] satisfies conditions F M3, F M5 and F M7 then we say that (X, M, * ) is a Dislocated Fuzzy Quasi Metric Space in the sense of George and Veeramani (in short D GV FqM − S pace) (Alaca 2010).
Obviously each D GV FqM − S pace can be cosidered as a D K M FqM − S pace by defining M(x, y, 0) = 0 for all x, y ∈ X (see Gregory & Romuguera 2004) We define an open ball with centre x and radius r (0 r, t) and its not necessary that x ∈ B(x, r, t) for all x ∈ X.
Proof: (N1) Clearly if U ∈ U x then x ≻ U and vice versa.
Remark 3.6 The above construction yields the usual dislocated fuzzy topology associated with a DFqM − S pace and consequently we may define a dislocated topology T D on X as T D = {A ⊂ X : x ≻ A for some x ∈ X and ≻ is as defined in Proposition 3.
Definition 3.9 A sequence x n in a DFqM − S pace (X, M, * ) is said to be bi-convergent to a point x ∈ X if and only if Lim n→∞ M ‡ (x n , x, t) = 1 for all t > 0. In this case we say that limit of the sequence x n is x.Proof: Let x n be a sequence in X and suppose u and v are two limits of x n .Then we have Proposition 3.16 Let (X, M, * ) be a DFqM−S pace(DF M−S pace) and x n be a sequence in X.If sequence x n bi-converges (converges) to x ∈ X then M(x, x, t) = 1 for all t > 0.

Common Fixed Point Theorems
Let (X, M, * ) be a DFqM − S pace, T :  Definition 4.3 f is said to be coincidentally idempotent with respect to T if and only if f is idempotent at the coincidence points of f and T .
Definition 4.4 The mapping f is said to be occasionally coincidentally idempotent (oci)with respect to T if and only if f f u = f u for some u ∈ C(T, f ) whenever C(T, f ) ϕ.

Now we present our main results as follows:
Theorem 4.5 Let (X, M, * ) be a DF M − S pace, k a positive integer, f : X −→ X and T : X k −→ X be mappings, such that where x 1 , x 2 , ...x k+1 are arbitrary elemants in X, 0 < q < 1 2 and t ∈ [0, ∞) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided the pair ( f, T ) are weakly compatible.
Theorem 4.6 Let (X, M, * ) be a DF M − S pace, k a positive integer, f : X −→ X and T : X k −→ X be mappings, such that where x 1 , x 2 , ..., x k+1 are arbitrary elemants in X, 0 < q < 1 and t ∈ [0, ∞) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided either f is oci with respect to T and the pair ( f, T ) are weakly compatible or f is coincidentally idempotent with respect to T and the pair ( f, T ) are occasionally weakly compatible.
Proof: Proceeding as in the proof of previous theorem, we can show that C(T, f ) ϕ. Now suppose f is oci with respect to T and the pair ( f, T ) are weakly compatible.Then there will exist z ∈ C( f, T ) such that f f z = f z and also f (T (z, z, ..., z)) = T ( f z, f z, ..., f z).

Thus we have
is a common fixed point of f and T .The proof follows on the same lines in the other case also.
Theorem 4.7 Let (X, M, * ) be a DFqM − S pace, k a positive integer, f : X −→ X and T : where x 1 , x 2 , ...x k+1 are arbitrary elemants in X, 0 < q < 1 2 and t ∈ [0, ∞) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided the pair ( f, T ) are weakly compatible.
Proof: Proceeding as in Theorem 4.5 we see that M(y n , y n+p , t) → 1, as n → ∞ and so < y n > is a L-Cauchy sequence in f (X) and since f (X) is L-complete, there will exist z in f (X) such that lim n→∞ y n = z.Rest of the proof goes on the same lines as that of Theorem 4.5.
Theorem 4.8 Let (X, M, * ) be a DFqM − S pace, k a positive integer, f : X −→ X and T : X k −→ X be mappings, such that T (X k ) ⊆ f (X) ( 13) where x 1 , x 2 , ..., x k+1 are arbitrary elemants in X, 0 < q < 1 and t ∈ [0, ∞) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided either f is oci with respect to T and the pair ( f, T ) are weakly compatible or f is coincidentally idempotent with respect to T and the pair ( f, T ) are occasionally weakly compatible.
Proof: The proof follows on the same line as in the previous theorem and Theorem 4.6.
Theorem 4.9 Let (X, M, * ) be a DFqM − S pace, k a positive integer, f : X −→ X and T : X k −→ X be mappings, such that T (X k ) ⊆ f (X) ( 16) where x 1 , x 2 , ...x k+1 are arbitrary elemants in X, 0 < q < 1 2 and t ∈ [0, ∞) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided the pair ( f, T ) are weakly compatible.
Proof: Let x 1 , x 2 , ..., x k be arbitrary elemants in X.By ( 16) we define sequence < y n > in f (X) as follows: qt).By the method of mathematical induction we will prove that where Clearly from the definition of K, we see that (4.4) is true for n = 1, 2, ..., k.Let the k inequalities ) 2 , be the induction hypothesis.Then we have Thus inductive proof of ( 20 Hence < y n > is a R-Cauchy sequence in f (X) and since f (X) is R-complete, there will exist z in f (X) such that lim n→∞ y n = z.Let z = f (u) for some u ∈ X.Then we have Similarly it can be shown that M( f u, T (u, u, ..., u), t) = 1.Hence f u = T (u, u, ..., u), and so C(T, f ) ϕ. Now suppose the pair ( f, T ) are weakly compatible.Then we have 2 3 ) ⋆ ... ⋆ M(T (u, u, ..., f u), T (u, u, ..., u), qt 2 k−1 ) By (4.17), Proposition 3.14 and 3.15 we get ). Repeating the above process n times we get Theorem 4.10 Let (X, M, * ) be a DFqM − S pace, k a positive integer, f : X −→ X and T : where x 1 , x 2 , ..., x k+1 are arbitrary elemants in X, 0 < q < 1 2 and t ∈ [0, ∞) lim t→∞ M(x, y, t) = 1 for all x, y ∈ X. (24) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided either f is oci with respect to T and the pair ( f, T ) are weakly compatible or f is coincidentally idempotent with respect to T and the pair ( f, T ) are occasionally weakly compatible.
Taking k = 1 in the Theorems 4.7 and 4.9, we get the following.
Corollary 4.11 Let (X, M, * ) be a DFqM − S pace, f : X −→ X and T : X −→ X be mappings, such that T (X) ⊆ f (X) ( 25) for all x, y ∈ X, 0 < q < 1 2 and t ∈ [0, ∞) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided the pair ( f, T ) are weakly compatible.
Taking k = 1 in the Theorems 4.8 and 4.10, we get the following.
Corollary 4.12 Let (X, M, * ) be a DFqM − S pace, f : X −→ X and T : X −→ X be mappings, such that T (X) ⊆ f (X) (29) M(T x, T y, qt) ≥ M( f x, f y, t), (30) where x, y ∈ X , 0 < q < 1 and t ∈ [0, ∞) f(X) is L-complete or R-complete (31) Then f and T has a coincidence point, i.e.C( f, T ) ϕ.Further f and T has a common fixed point provided either f is oci with respect to T and the pair ( f, T ) are weakly compatible or f is coincidentally idempotent with respect to T and the pair ( f, T ) are occasionally weakly compatible.
If we take f to be the identity mapping in the above corollaries, we get the following.
is an abelian monoid with unit one such that, for all a, b, c, d in [0, 1], a * b ≥ c * d whenever a ≥ c and b ≥ d.
. Hereafter by a Dislocated Fuzzy Quasi Metric Space (DFqM − S pace) we mean a D GV FqM − S pace or a D K M FqM − S pace.Definition 3.4 Let (X, M, * ) be a DFqM − S pace.We define a left open ball (L-open ball) with centre x and radius r 5. The topology T D is called the Dislocated Fuzzy topology associated with the DFqM − S pace (X, M, * ).Clearly open balls in a DFqM −S pace is open in the dislocated topology T D .Collection of open balls in a DFqM −S pace does not in general yield a conventional topology.Proceeding on the same lines as in the proof of above proposition we have the following.Proposition 3.7 Let (X, M, * ) be a DFqM − S pace.Define a d-membership relation ≻ L by x ≻ L A if and only if there exists r denote the set of all coincidence points of the mappings f and T , that isC(T, f ) = {z : f z = T (z, z, ..., z)}.Definition 4.1 The mappings f and T in a DFqM − S pace (DF M − S pace) are said to be weakly compatible if and only
Definition 3.11 A sequence x n in a DFqM−S pace (X, M, * ) is said to be bi-Cauchy if and only if Lim n→∞ M ‡ (x n , x n+p , t) = 1 for all t > 0, p > 0. Definition 3.12 A DFqM − S pace is said to be Left (Right) complete if and only if every Left (Right) Cauchy sequence in it is bi-convergent.Definition 3.13 A DFqM − S pace is said to be bi-complete if and only if every bi-Cauchy sequence in it is bi-convergent.Remark 3.14 Clearly a sequence x n in a DFqM − S pace (X, M, * ) is bi-Cauchy sequence if and only if sequence x n is a Cauchy sequence in the DF M − S pace (X, M ‡ , * ).A DFqM − S pace (X, M, * ) is bi-Complete if and only if the DF M − S pace (X, M ‡ , * ) is complete.Proposition 3.15 Limit of a sequence in a DFqM − S pace (X, M, * ) is unique.
Definition 3.10 A sequence x n in a DFqM − S pace (X, M, * ) is said to be Left (Right) Cauchy sequence if and only if Lim n→∞ M(x n , x n+p , t) = 1 (Lim n→∞ M(x n+p , x n , t) = 1) for all t > 0, p > 0.
) is complete.