Abstract
Purpose
The purpose of this paper is to study a common fixed point theorem for two pairs of weakly compatible maps in dislocated metric spaces.
Methods
Using familiar techniques, we extend the results of Hitzler and Kang et al. in dislocated metric spaces.
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Introduction
In 1922, S. Banach proved a fixed point theorem for contraction mapping in metric space. Since then a number of fixed point theorems have been proved by different authors, and many generalizations of this theorem have been established. The notion of a dislocated metric (d-metric) space was introduced by Pascal Hitzler in [1] as a part of the study of logic programming semantics. The study of common fixed point mappings in dislocated metric space satisfying certain contractive conditions has been at the center of vigorous research activity, see for example in [2, 3].
In 1996, Jungck [4] introduced the concept of weak compatibility. Since then, many interesting fixed point theorems of compatible and weakly compatible maps under various contractive conditions have been obtained by a number of authors. We proved two common fixed point theorems for four weakly compatible maps.
Preliminaries
For convenience we start with the following definitions, lemmas, and theorems.
Definition 2.1[5] Let X be a non empty set and let d: X × X → [0,∞) be a function satisfying the following conditions:
-
i.
d(x, y) = d(y, x)
-
ii.
d(x, y) = d(y, x) = 0 implies that x = y and
-
iii.
d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.
Then d is called dislocated metric (or simply d-metric) on X.
Definition 2.2[5] A sequence {x n } in a d-metric space (X, d) is called a Cauchy sequence if given ∈ > 0 there corresponds n 0 ∈ N such that for all m n ≥ n 0 , we have d(x m , x n ) < ∈.
Definition 2.3[5] A sequence {x n } in d- metric space converges with respect to d (or in d) if there exists x ∈ X such that d(x n , x) → 0 as n → ∞. In this case, x is called d limit of {x n } and we write x n → x.
Definition 2.4[5] A d- metric space (X d) is called complete if every Cauchy sequence in it is convergent with respect to d.
Definition 2.5[5] Let (X d) be a d- metric space. A map T:X → X is called contraction if there exists a number λ with 0 ≤ λ < 1 such that d(Tx Ty) ≤ λd(x y). We state the following lemmas without proof.
Lemma 2.6[5]Let (X d) be a d-metric space. If T:X→X is a contraction function, then {Tn(x0)} is a Cauchy sequence for x0 ∈ X.
Lemma 2.7[5]The limits in a d-metric space are unique.
Definition 2.8[6] Let A and S be mappings from a metric space (X d) into itself. Then A and S are said to be weakly compatible if they commute at their ‘coincident point’ x, that is, Ax = Sx implies ASx = SAx.
Theorem 2.9[5]Let (X d) be a complete d-metric space and let T:X → X be a contraction mapping, then T has a unique fixed point.
Theorem 2.10[4]Let (X d) be a complete d-metric space and let T:X → X be a contraction mapping, then T has a unique fixed point.
Results and discussion
Theorem 3.1 Let (X,d) be a complete d-metric space. Let A,B,S,T:X→X be continuous mappings satisfying
-
I.
T(X) ⊂ A(X), S(X) ⊂ B(X)
-
II.
The pairs (S,A) and (T,B) are weakly compatible and
-
III.
d(Sx,Ty) ≤ α max{d(Ax,By), d(Ax,Sx), d(By,Ty)}
For all x, y ∈ X, wherethen A,B,S and T have a unique common fixed point.
Proof. Using condition (i), we define sequences {x n } and {y n } in X by the rule
y2n = Bx2n + 1 = Sx2n and y2n + 1 = Ax2n + 2 = Tx2n + 1, n = 0,1,2…, where x0 = x, y0 = y.
If y2n = y2n + 1 is for some n, then Bx2n + 1 = Tx2n + 1. Therefore x2n + 1 is a coincident point of B and T. Also, if y2n + 1 = y2n + 2 for some n, then Ax2n + 2 = Sx2n + 2. Hence, x2n + 2 is a coincident point of S and A.
Assume that y2n ≠ y2n + 1 for all n. Then, we have
Hence, by induction d(y n ,yn + 1) ≤ αnd(y0,y1).
Hence, for any integer n ≥ 1 and q ≥ 1,
Since lim αn = 0, it follows that {y n } is a Cauchy sequence in the complete dislocated metric space (X,d). So there exists z ∈ X such that
Therefore, the subsequences
Since T(X) ⊂ A(X), there exists u ∈ X such that z = Au.
Taking limits as n → ∞ we get,
Again, since S(X) ⊂ B(X), there exists a v ε X such that z = Bv.
We claim that z = Tv, then
so we get z = Tv.
Hence, Su = Au = Tv = Bv = z.
Since the pair (S,A) is weakly compatible, by definition SAu = ASu implies Sz = Az.
Now, we show that z is a fixed point of S in the following:
Since 0 ≤ α <1 and d(Sz,z) ≤ α d (Sz,z) from Equation 1, we gset Sz = z. This implies that Az = Sz = z.
Again, the pair (T,B) are weakly compatible, by definition TBv = BTv implies Tz = Bz.
Now, we show that z is a fixed point of T as:
Hence, we have Az = Bz = Sz = Tz = z. This shows that ‘z’ is a common fixed point of the self mappings A, B, S and T.
Uniqueness. Let u ≠ v be two common fixed points of the mappings A,B,S and T in the following:
Since (X, d) is a dislocated metric space, so we have u = v.
Put A = B = I an identity mapping in above Theorem 3.1 yields Corollary 3.2
Corollary 3.2 Let (X,d) be a complete d-metric space. Let S,T:X → X be continuous mappings satisfying,
where, then S and T have a unique fixed point.
Then S = T in Corollary 3.2 yields Corollary 3.3.
Corollary 3.3 Let (X,d) be a complete d-metric space. Let T:X → X be a continuous mapping satisfying
d(Tx, Ty) ≤ α max{d(x, y), d(x, Tx), d(y, Ty)} for all x, y ∈ X,
where then T have unique common fixed point.
Taking A=T and B=S in Theorem 3.1 yields Corollary 3.4.
Corollary 3.4 Let (X,d) be a complete d-metric space. Let S,T:X → X be continuous mappings satisfying
for all x ,y ∈ X, where, then S and T have unique common fixed point.
Theorem 3.5 Let (X,d) be a complete d-metric space. Let A,B,S,T:X → X be the continuous mapping satisfying,
-
I.
T(X) ⊂ A(X), S(X) ⊂ B(X)
-
II.
The pairs (S,A) and (T,B) are weakly compatible and
-
III.
for all x,y ∈ X where , then A,B,S and T have a unique common fixed point.
Proof. Using condition (i), we define sequences {x n } and {y n } in X by the rule,
y2n = Bx2n + 1 = Sx2n, and y2n + 1 = Ax2n + 2 = Tx2n + 1, n = 0,1,2…,
If y2n = y2n + 1 for some n, then Bx2n + 1 = Tx2n + 1. Therefore, x 2n + 1 is a coincident point of B and T. Also, if y2n + 1 = y2n + 2 for some n, then Ax2n + 2 = Sx2n + 2. Hence, x 2n+2 is a coincident point of S and A.
Assume that y2n ≠ y2n + 1 for all n. Then, we have
This shows that
For every integer q > 0, we have
So, when we get d(y n , yn + q) → 0, this implies that {y n } is a Cauchy sequence in the complete dislocated metric space. So, there exists z ∈ X such that {y n } → z.
Therefore, the subsequences
Since T(X) ⊂ A(X), there exists u ∈ X such that z = Au.
So,
Taking limits as n→∞ we get,
Again, since S(X) ⊂ B(X) there exists v ∈ X, such that z = Bv.
We claim that z=Tv as
so we get z = Tv and hence, Su = Au = Tv = Bv = z.
Since the pair (S,A) is weakly compatible so by definition SAu = ASu implies Sz = Az.
Now, we show that z is the fixed point of S.
Again, the pair (T,B) are weakly compatible, so by definition, TBv = BTv and this also implies Tz=Bz.
Now, we show that z is the fixed point of T.
This shows that z is a common fixed point of the self mappings A,B,S and T.
Uniqueness. Let u ≠ v be two common fixed points of the mappings A,B,S, and T, then we have
Since (X,d) is a dislocated metric space, so we have u = v.
Putting A = B = I an identity mapping in above Theorem 3.5 yields Corollary 3.6.
Corollary 3.6 Let (X,d) be a complete d-metric space. Let S,T:X → X be continuous mappings satisfying,
for all x, y ∈ X wherethen S and T have a unique fixed point. If S = T in Corollary 3.6 yields Corollary 3.7.
Corollary 3.7 Let (X,d) be a complete d-metric space. Let T:X→ X be a continuous mapping satisfying,for all x,y ∈ X,where , then T have unique common fixed point.
Taking A = T and B = S in Theorem 3.5 yields Corollary 3.8. This is the Theorem 2.11 in [7].
Corollary 3.8 Let (X,d) be a complete d-metric space. Let S,T: X → X be continuous mappings satisfying, for all x,y ∈ X,where , then S and T have unique common fixed point.
Conclusion
The results in this study enable others in applying our results when one has to deal with dislocated metric spaces.
References
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IR and PSK formulated the problem and VVK drafted and aligned the manuscript sequentially. The three authors read and approved the final manuscript.
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Kumari, P.S., Kumar, V.V. & Sarma, I.R. Common fixed point theorems on weakly compatible maps on dislocated metric spaces. Math Sci 6, 71 (2012). https://doi.org/10.1186/2251-7456-6-71
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DOI: https://doi.org/10.1186/2251-7456-6-71