https://doi.org/10.5831/HMJ.2017.39.2.161
SOME NEW COMMON FIXED POINTS OF GENERALIZED RATIONAL CONTRACTIVE MAPPINGS IN DISLOCATED METRIC SPACES WITH
APPLICATION
Sami Ullah Khan∗, Muhammad Arshad, Tahair Rasham and Abdullah Shoaib
Abstract. The objective of this manuscript is to continue the study of fixed point theory in dislocated metric spaces, introduced by Hitzler et al. [12]. Concretely, we apply the concept of dislocated metric spaces and obtain theorems asserting the existence of com- mon fixed points for a pair of mappings satisfying new generalized rational contractions in such spaces.
1. Introduction
Fixed point theory has an important role in non linear analysis. In this field the first important and significant result was proved by Banach [7] for the contraction mapping in a complete metric space. After that, huge number of fixed point theorems have been established by various authors and they made different generalizations of the Banach’s result.
The notion of metric spaces introduced by Frechet [10], is one of the helpful topic in Analysis. The study of metric spaces expressed the most important role to many fields both in pure and applied science such as biology, medicine, physics and computer science (see [14], [24] ). Some generalizations of the notion of a metric space have been proposed by some authors, such as, rectangular metric spaces, semi metric spaces, pseudo metric spaces, probabilistic metric spaces, fuzzy metric spaces, quasi metric spaces, quasi semi metric spaces, D-metric spaces, and cone
Received November 14, 2016. Accepted April 11, 2017.
2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25.
Key words and phrases. Common fixed point, Dislocated metric, Contractive type mappings
Research funding can be written here.
*Corresponding author
metric spaces (see [ [2], [9], [11], [19], [20]]). Branciari [8] introduced the notion of a generalized metric space replacing the triangle inequality by a rectangular type inequality. He then extended Banach’s contraction principle in such spaces.
In 1994, S. G. Matthews [17] intoduced the concept of partial met- ric spaces and obtained various fixed point theorems. In particular, he established the precise relationship between partial metric spaces and quasi-metric spaces, and proved a partial metric generalization of Ba- nach’s contraction mapping theorem. Hitzler and Seda [12] introduced the concept of dislocated topologies and named their corresponding gen- eralized metric a dislocated metric. They have also established a fixed point theorem incomplete dislocated metric spaces to generalize the cele- brated Banach contraction principle. The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [13]).
2. Definitions and Relevant results
Definition 2.1. [12] Let X be a nonempty set. A mapping dl : X × X → [0, ∞) is called a dislocated metric (or simply dl-metric) if the following conditions hold. for any j, k, l ∈ X :
1. If dl(j, k) = 0 , then j = k;
2. dl(j, k) = dl(k, j);
3. dl(j, k) ≤ dl(j, l) + dl(l, k).
Then dl is called a dislocated metric on X, and the pair (X, dl) is called dislocated metric space or dl metric space.
Example 2.2. If X = R+∪ {0}, then dl(j, k) = j + k defines a dislocated metric on X.
Definition 2.3. [12] A sequence {jn} in dl−metric space is called Cauchy sequence if for given ε > 0, there corresponds n0∈ N such that for all n, m ≥ n0 , we have dl(jm, jn) < ε.
Definition 2.4. [12] A sequence {jn} in dl−metric space converges with respect to dlif there exists j ∈ X such that dl(jn, j) → 0 as n → ∞.
In this case, j is called limit of {jn} and we write jn→ j.
Every metric space is a dislocated metric, but the converse may not be true.
Definition 2.5. Let X = R and dl : X × X → [0, ∞) defined by dl(j, k) = |j| + |k| for all j, k ∈ X.
Note that dl is a dislocated metric, but not a metric since dl(1, 1) = 2 > 0.
Definition 2.6. [12] A dl−metric space (X, dl) is called complete if every Cauchy sequence in X converges to a point in X.
Definition 2.7. Let X = [0, 1] and dl(j, k) = max{j, k}. Then the pair (X, dl) is dislocated metric space, but it is not a metric space.
Definition 2.8. [12] Let (X, dl) be a dislocated metric space. A mapping T : X → X is called contraction if there exists 0 ≤ λ < 1 such that
d(T (j) , T (k)) ≤ λd(j, k), for all j, k ∈ X with j 6= k.
Then T has a unique fixed point in X.
The purpose of this paper is to prove common fixed point theorems for generalized rational contractions on dislocated metric spaces. We provide an example and an application to a system of integral equations to validate our results.
3. The Results
In this section we will prove the existance of common fixed points of two self mappings involving rational expressions in dislocated metric space.
Theorem 3.1. Let (X, d) be a complete dislocated metric space and let the mappings S, T : X → X satisfy:
dl(Sj, T k) ≤ a1 dl(j, k) + a2
dl(j, Sj) .dl(k, T k) dl(j, k) + a3
dl(j, T k) .dl(k, Sj) dl(j, k) + a4
dl(j, Sj) dl(k, T k) dl(j, T k) + dl(j, k) + dl(k, Sj) (1)
for all j, k ∈ X, where a1, a2, a3, a4 are nonnegative reals with a1+ a2+ a3+ a4< 1. Then S, T have a unique common fixed point.
Proof: Let j0 be an arbitrary point in X and define j1 = Sj0 and j2 = T j1such that
dl(j1, j2) = dl(Sj0, T j1) .
Then
dl(j1, j2) ≤ a1dl(j0, j1) + a2dl(j0, Sj0) .dl(j1, T j1) dl(j0, j1) + a3dl(j0, T j1) .dl(j1, Sj0)
dl(j0, j1)
+ a4 dl(j0, Sj0) dl(j1, T j1)
dl(j0, T j1) + dl(j0, j1) + dl(j1, Sj0),
≤ a1dl(j0, j1) + a2dl(j0, j1) .dl(j1, j2) dl(j0, j1) + a3dl(j0, j2) .dl(j1, j1)
dl(j0, j1)
+ a4 dl(j0, j1) dl(j1, j2)
dl(j0, j2) + dl(j0, j1) + dl(j1, j1),
≤ a1dl(j0, j1) + a2dl(j1, j2) + a4 dl(j0, j1) dl(j1, j2) dl(j0, j2) + dl(j0, j1). As (owing to triangular inequality),
dl(j1, j2) < a1dl(j0, j1) + a2dl(j1, j2) + a4
dl(j0, j1) dl(j1, j2) dl(j0, j2) + dl(j0, j1), where
dl(j1, j2) ≤ dl(j1, j0) + dl(j0, j2) . Hence
dl(j1, j2) < a1+ a4 1 − a2
|dl(j0, j1)| ,
< λdl(j0, j1) , where λ = a1−a1+a4
2 . Similarly, by repeating the same process for dl(j2, j3) = dl(T j1, Sj2) = dl(Sj2, T j1)
we get
|dl(j2, j3)| < λ2|dl(j0, j1)| . Consequently, we get
|dl(j2n+1, j2n+2)| < λdl(j2n, j2n+1)
< λ2dl(j2n−1, j2n)
< λ2n+1dl(j0, j1) .
Hence for any m > n,
dl(jn, jm) < dl(jn, jn+1) + dl(jn+1, jn+2) + · · · · +dl(jm−1, jm) ,
< λn+ λn+1+ · · · · +λm−1 dl(j0, j1) ,
< λn
1 − λdl(j0, j1) , and
dl(jn, jm) < λn
1 − λdl(j0, j1) .
−→ 0, as m, n −→ ∞
This implies that {jn} is a Cauchy sequence. Since X is complete, there exist u ∈ X such that jn −→ u. It fallows that u = Su, otherwise d (u, Su) = z > 0 and we would then have
dl(u, Su) ≤ dl(u, j2n+2) + dl(j2n+2, Su) dl(u, Su) ≤ dl(u, j2n+2) + dl(T j2n+1, Su) dl(u, Su) ≤ dl(u, j2n+2) + dl(Su, T j2n+1)
dl(u, Su) ≤ dl(u, j2n+2) + a1dl(u, j2n+1) + a2dl(u, Su) .dl(j2n+1, T j2n+1) dl(u, j2n+1)
+ a3dl(u, T j2n+1) .dl(j2n+1, Su) dl(u, j2n+1)
+ a4 dl(u, Su) dl(j2n+1, T j2n+1)
dl(u, T j2n+1) + dl(u, j2n+1) + dl(j2n+1, Su),
≤ dl(u, j2n+2) + a1dl(u, j2n+1) + a2dl(u, Su) .dl(j2n+1, j2n+2) dl(u, j2n+1)
+ a3dl(u, j2n+2) .dl(j2n+1, Su) dl(u, j2n+1)
+ a4 dl(u, Su) dl(j2n+1, j2n+2)
dl(u, j2n+2) + dl(u, j2n+1) + dl(j2n+1, Su). This implies that
z ≤dl(u, j2n+2) + a1|dl(u, j2n+1)| + a2z.dl(j2n+1, j2n+2) dl(u, j2n+1) + a3dl(u, j2n+2) .dl(j2n+1, Su)
dl(u, j2n+1)
+ a4 zdl(j2n+1, j2n+2)
dl(u, j2n+2) + dl(u, j2n+1) + dl(j2n+1, Su),
which on making n → ∞, gives rise dl(u, Su) = 0 a contradiction so that u = Su. Similarly, one can show that u = T u and its uniqueness.
Example 3.2. Let X = [0, 1] be a dislocate metirc space dl : X × X → X defined by
dl(j, k) = j 2 +k
2 Let S : X → X defined by
Sj = 2j 3
, j ∈ X.
And T : X → X defined by T k = 4k
3
, k ∈ X.
Now
dl(Sj, Sk) = j 3 +2k
3 . Take j = 12 and k = 13 then we have
dl(Sj, Sk) = j 3+ 2k
3 = 1 6 +2
9 = 5
18 = 0.2070.
Now by using the contractive condition we have dl(Sj, T k) ≤ a1 dl(j, k) + a2dl(j, Sj) .dl(k, T k)
dl(j, k) + a3dl(j, T k) .dl(k, Sj) dl(j, k) +a4 dl(j, Sj) dl(k, T k)
dl(j, T k) + dl(j, k) + dl(k, Sj).
As given that a1+ a2+ a3+ a4 < 1. choose a1 = 13, a2 = 14 a3 = 15 and a4 = 17, then clearly a1+ a2+ a3+ a4 < 1. Now putting j = 12 and k = 13,then
5
18 ≤ 1 3(1
4 +1 6) +1
4
(14 +16).(16 +29) (14 +16) +1
5
(14 +29) + (16 +16) (14+ 16) +1
7
(14 +16).(16 +29) (14 +29) + (14+ 16) + (16 +16) 0.207 ≤ 0.1387 + 0.0972 + 0.2266 + 0.009044 0.207 ≤ 0.4715
Hence, all contractive conditions of theorem (3.1) are satisfied.
Corollary 3.3. Let (X, dl) be a complete dislocated metric space and let the mappings S, T : X → X satisfy:
dl(Sj, T k) ≤ a1 dl(j, k) + a2dl(j, Sj) .dl(k, T k) dl(j, k) + a3 dl(j, Sj) dl(k, T k)
dl(j, T k) + dl(j, k) + dl(k, Sj)
for all j, k ∈ X, where a1, a2, a3are nonnegative reals with a1+ a2+ a3 <
1. Then S, T have a unique common fixed point.
Proof : By putting a3 = 0 in Theorem 3.1, we get the required result.
Theorem 3.4. Let (X, dl) be a complete dislocated metric space and let the mappings S, T : X → X satisfy:
dl(S (j) , T (k)) ≤ a dl(j, k) + bdl(j, S (j)) dl(k, T (k)) 1 + dl(j, k)
for all j, k ∈ X, where a, b are nonnegative reals with a + b < 1. Then S, T have a unique common fixed point.
Proof: Let j0 be an arbitrary point in X and define j1= S (j0) and j2 = T (j1) such that
dl(j1, j2) = dl(S (j0) , T (j1)) Then
≤ adl(j0, j1) + bdl(j0, S (j0)) dl(j1, T (j1)) 1 + dl(j0, j1) ,
≤ adl(j0, j1) + bdl(j0, j1) dl(j1, j2) 1 + dl(j0, j1) ,
≤ adl(j0, j1) + bdl(j1, j2)
dl(j0, j1) 1 + dl(j0, j1)
,
≤ adl(j0, j1) + bdl(j1, j2) . This implies that
dl(j1, j2) ≤
a 1 − b
dl(j0, j1)
≤ λdl(j0, j1) (2)
Where
a 1−b
= λ Similarly,
dl(j2, j3) = dl(j3, j2) = dl(S (j2) , T (j1))
dl(S (j2) , T (j1)) ≤ adl(j2, j1) + bdl(j2, S (j2)) dl(j1, T (j1)) 1 + dl(j2, j1) ,
≤ adl(j2, j1) + bdl(j2, j3) dl(j1, j2) 1 + dl(j2, j1) ,
≤ adl(j2, j1) + bdl(j2, j3)
dl(j1, j2) 1 + dl(j2, j1)
, dl(j2, j3) ≤ adl(j1, j2) + bdl(j2, j3)
This implies that
dl(j2, j3) ≤
a 1 − b
dl(j1, j2) ,
≤ λ.λdl(j0, j1) ,
≤ λ2dl(j0, j1) . Consequently, we get
dl(jn, jn+1) ≤ λdl(jn−1, jn)
≤ λ2dl(jn−2, jn−1) ...
≤ λndl(j0, j1) .
To prove that {jn} is a Cauchy sequence, we have for any m > n, dl(jn, jm) ≤ dl(jn, jn+1) + dl(jn+1, jn+2) + · · · + dl(jm−1, jm) ,
≤ λndl(j0, j1) + λn+1dl(j0, j1) + · · · + λm−1dl(j0, j1) ,
≤ λn+ λn+1+ · · · + λm−1 dl(j0, j1) ,
≤
λn 1 − λ
dl(j0, j1) ,
−→ 0 as m, n −→ ∞.
Hence {jn} is a Cauchy sequence. Since X is complete, so for any u ∈ X such that jn −→ u and suppose θ = dl(u, Su) . Therefore we
have
dl(u, Su) ≤ dl(u, j2n+2) + dl(j2n+2, Su)
= dl(u, j2n+2) + dl(T (j2n+1) , Su)
= dl(u, j2n+2) + dl(Su, T (j2n+1))
≤ dl(j2n+2, u) + adl(u, j2n+1) + bdl(u, Su) dl(j2n+1, T (j2n+1)) 1 + dl(u, j2n+1)
≤ dl(j2n+2, u) + adl(u, j2n+1) + bdl(u, Su) dl(j2n+1, j2n+2) 1 + dl(u, j2n+1) θ ≤ dl(u, j2n+2) + adl(u, j2n+1) + bθ + dl(j2n+1, j2n+2)
1 + dl(u, j2n+1) letting n → ∞ , and jn−→ u we get,
(1 − b) θ ≤ 0 (1 − b) 6= 0
θ = dl(u, Su) = 0.
which implies that u = Su. It fallows similarly that u = T u. Now, we show that S and T have a unique common fixed point. For this, assume that v in X is a second common fixed point of S and T. Then
dl(u, v) = dl(Su, T v)
≤ a dl(u, v) + bdl(u, Su) dl(v, T v) 1 + dl(u, v)
≤ a dl(u, v).
This implies that
(1 − a) dl(u, v) ≤ 0 1 − a 6= 0 dl(u, v) = 0.
This implies that u = v, completing the proof of the theorem.
As an application of Theorems (3.1) and (3.4) we prove the following theorem for two finite families of mappings.
Theorem 3.5. If {Tp}m1 and {Sp}n1 are two finite pair wise commut- ing finite families of self mappings defined on complete dislocated metric space (X, dl) such that the mappings T and S satisfy the conditions of theorems (3.1) and (3.4), then the component maps of the two families {Tp}m1 and {Sp}n1 have a unique common fixed point.
Proof. In view of theorems (3.1) and (3.4), one can infer that T and S have a unique common fixed point q i.e. T q = Sq = q. Now we are required to show that q is common fixed point of all the components maps of both the families. In view of pairwise commutativity of families of {Tp}m1 and {Sp}n1 , (for every 1 ≤ i ≤ m ) we can write
Tiq = TiSq = STiq and Tiq = TiT q = T Tiq
which shows that Tiq (for every i ) is also a common fixed point of T and S. By using the uniqueness of common fixed point, we can write Tiq = q (for every i ) which shows that q is the common fixed point of the family {Tp}m1 . Using the foregoing arguments, one can also shows that (for every 1 ≤ i ≤ n ) Siq = q.
This completes the proof of the theorem.
By setting {Sp}n1 = Γ and {Tp}m1 = Ω in theorems (3.1) and (3.4), we derive the following common fixed point theorems involving iterates of mappings.
Corollary 3.6. If Γ and Ω are two commuting self mappings defined on a complete dislocated metric space (X, dl) satisfying the condition :
dl(Γmj, Ωnk) ≤ a1 dl(j, k) + a2dl(j, Γmj) .dl(k, Ωnk) dl(j, k)
+a3dl(j, Ωnk) .dl(k, Γmj) dl(j, k)
+a4 dl(j, Γmj) dl(k, Ωnk) dl(j, Ωnk) + dl(j, k) + dl(k, Γmj)
for all j, k ∈ X, where a1, a2, a3are nonnegative reals with a1+a2+2a3 <
1. Then Γ, Ω have a unique common fixed point.
Corollary 3.7. If Γ and Ω are two commuting self mappings defined on a complete dislocated metric space (X, dl) satisfying the condition :
dl(Γmj, Ωnk) ≤ a dl(j, k) + bdl(j, Γmj) dl(k, Ωnk) 1 + dl(j, k)
for all j, k ∈ X, where a, b, are nonnegative reals with a + b < 1. Then Γ, Ω have a unique common fixed point.
4. Existence of a common solution for a system of integral equations
In this section, we show that theorem 3.4 can be applied to the exis- tance of a common solution of the system of the integral equations.
Theorem 4.1. Let X = C ([a, b], R) , where b > a ≥ 0 and dl : X × X → R be defined by
dl(j, k) = max
t∈[a,b]
kj (t) − k (t)k∞p
1 + a2ecot−1a. Consider the following system of integral equations:
j (t) = Z b
a
k1(t, r, j (r)) dr + g (t) , j (t) =
Z b a
k2(t, r, j (r)) dr + h (t) , (3)
where, X = C [a, b] , t ∈ [a, b] ⊂ R and j, g, h ∈ X..
Suppose that k1, k2 : [a, b] × [a, b] × R → R are continuous and such that
(4.2) Fj(t) =
Z b a
k1(t, r, j (r)) dr and
(4.3) Gj(t) =
Z b a
k2(t, r, j (r)) dr
for all j ∈ X and for all t ∈ [a, b] .Then the existence of a solution to (4.1) is equaivalent to the existance of common fixed point of S and T.
Let us consider
kFj(t) − Gk(t) + g (t) − h (t)k∞p
1 + a2ecot−1a≤ A (j, k) (t)+B (j, k) (t) where
A (j, k) (t) = kj (t) − k (t)k∞p
1 + a2ecot−1a and
B (j, k) (t)
= kFj(t) + g (t) − j (t)k∞kGk(t) + h (t) − y (t)k∞ 1 + d (j, k)
p1 + a2ecot−1a Then the system of integral Equations (4.2) and (4.3) have a unique common solution.
Proof: It is easily to check that (X, dl) is a dislocated metric space.
Define two mappings S ,T : X ×X → X by Sj = Fj+g and Tj = Gj+h.
Then
d (S (j) , T (k)) = max
t∈[a,b]kFj(t) − Gk(t) + g (t) − h (t)k∞p
1 + a2ecot−1a d (j, S (j)) = max
t∈[a,b]
kFj(t) + g (t) − j (t)k∞p
1 + a2ecot−1a and
d (k, T (k)) = max
t∈[a,b]kGk(t) + h (t) − k (t)k∞p
1 + a2ecot−1a Thus by theorem 3.4, we get S and T have a common fixed point.
Thus there exists a unique point v ∈ X such that v = Sv = T v. Now, we have
j = S (j) = Fj+ g and
j = T (j) = Gj + h that is
j (t) = Z b
a
k1(t, r, j (r)) dr + g (t) and
j (t) = Z b
a
k2(t, r, j (r)) dr + h (t)
Therefore, we can conclude that the system of integral equations (4.1) have a unique common fixed point.
Conflict of Interests
The authors declare that they have no competing interests.
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Sami Ullah Khan
Department of Mathematics, International Islamic University, H-10, Is- lamabad - 44000, Pakistan.
Department of Mathematics, Gomal University D. I. Khan, KPK, Pak- istan.
E-mail: [email protected] Muhammad Arshad
Department of Mathematics, International Islamic University, H-10, Is- lamabad - 44000, Pakistan.
E-mail: marshad− [email protected] Tahair Rasham
Department of Mathematics, International Islamic University, H-10, Is- lamabad - 44000, Pakistan.
E-mail: tahir [email protected] Abdullah Shoaib
Department of Mathematics, Ripha International University, Islamabad, Pakistan.
E-mail: [email protected]
