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Common fixed point results for α-ψ-contractions on a metric space endowed with graph
Journal of Inequalities and Applications volume 2014, Article number: 136 (2014)
Abstract
Abdeljawad (Fixed Point Theory Appl., 2013:19) introduced the concept of α-admissible for a pair of mappings. More recently Salimi et al. [Fixed Point Theory Appl., 2013:151] modified the notion of α-ψ-contractive mappings. In this paper we introduce the concept of an α-admissible map with respect to η and modify the α-ψ-contractive condition for a pair of mappings and establish common fixed point results for two, three, and four mappings in a closed ball in complete dislocated metric spaces. As an application, we derive some new common fixed point theorems for ψ-graphic contractions defined on dislocated metric space endowed with a graph as well as preordered dislocated metric space. Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature.
MSC: 46S40, 47H10, 54H25.
1 Introduction and preliminaries
Fixed point results of mappings satisfying certain contractive condition on the entire domain has been at the center of rigorous research activities, for example, see [1–33]. From application point of view the situation is not yet completely satisfactory because it frequently happens that a mapping T is a contraction not on the entire space X but merely on a subset Y of X. Recently Arshad et al. [8] proved a result concerning the existence of fixed points of a mapping satisfying a contractive condition on closed ball in a complete dislocated metric space (see also [9, 14, 15, 25, 33]). The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [5, 17, 29]).
The existence of fixed points of α-ψ-contractive and α-admissible mappings in complete metric spaces has been studied by several researchers (see [18–20] and references therein). In this paper we discuss common fixed point results for α-ψ-contractive type mappings in a closed ball in complete dislocated metric space. Our results improve several well known recent conventional results in [2, 8, 31]. We also derive some new common fixed point theorems for ψ-graphic contractions as well as ordered contractions on preordered metric space. We give examples which show how these results can be used when the corresponding results cannot.
Consistent with [2, 7, 8, 17, 31], the following definitions and results will be needed in the sequel.
Definition 1.1 [17]
Let X be a non-empty set and let be a function, called a dislocated metric (or simply -metric) if the following conditions hold for any :
-
(i)
if , then ;
-
(ii)
;
-
(iii)
.
The pair is then called a dislocated metric space. It is clear that if , then from (i), . But if , may not be 0.
Definition 1.2 [17]
A sequence in a -metric space is called a Cauchy sequence if given , there corresponds such that for all we have .
Definition 1.3 [17]
A sequence in -metric space converges with respect to if there exists such that as . In this case, x is called a limit of and we write .
Definition 1.4 [17]
A -metric space is called complete if every Cauchy sequence in X converges to a point in X.
Definition 1.5 Let X be a non-empty set and . A point is called point of coincidence of T and f if there exists a point such that , here x is called coincidence point of T and f. The mappings T, f are said to be weakly compatible if they commute at their coincidence point (i.e., whenever ).
We require the following lemmas for subsequent use.
Lemma 1.6 [8]
Let X be a non-empty set and be a function. Then there exists such that and is one-to-one.
Lemma 1.7 [7]
Let X be a non-empty set and the mappings have a unique point of coincidence v in X. If and are weakly compatible, then S, T, f have a unique common fixed point.
Let Ψ denote the family of all nondecreasing functions such that for all , where is the n th iterate of ψ.
Lemma 1.8 [31]
If , then for all .
Definition 1.9 [2]
Let and . We say that the pair is α-admissible if such that , then we have and .
Definition 1.10 [31]
Let and two functions. We say that T is α-admissible mapping with respect to η if such that , then we have . Note that if we take , then T is called an α-admissible mapping [32].
2 Common fixed point results in dislocated metric space
We first extend the concept of α-η-admissibility for the pair of mappings.
Definition 2.1 Let and two functions. We say that the pair is α-admissible with respect to η if such that , then we have and . Also, if we take , then the pair is called α-admissible, if we take, , then we say that the pair is η-subadmissible mapping. If we take , then we obtain Definition 1 of Abdeljawad [2]. Also, if we take , we obtain Definition 1.10.
Theorem 2.2 Let be a complete dislocated metric space and be two mappings. Suppose there exist two functions, such that the pair is α-admissible with respect to η. For , , and , assume that
and
Suppose that the following assertions hold:
-
(i)
;
-
(ii)
for any sequence in such that for all and as then for all .
Then there exists a point in such that .
Proof Let in X be such that and . Continuing this process, we construct a sequence of points in X such that
By assumption and the pair is α-admissible with respect to η, we have, from which we deduce that which also implies that . Continuing in this way we obtain for all . First, we show that for all . Using inequality (2), we have
It follows that
Let for some . If , where then using inequality (1), we obtain
Thus we have
If , then as where (), we obtain,
Thus from inequality (3) and (4), we have
Now,
Thus . Hence for all . Now inequality (5) can be written as
Fix and let such that . Let with , then by using the triangle inequality, we obtain
Thus we proved that is a Cauchy sequence in . As every closed ball in a complete dislocated metric space is complete, so there exists such that . Also
On the other hand, from (ii), we have
Now using the triangle inequality, together with (1) and (8), we get
Letting and by using inequality (7), we obtain . Hence . Similarly by using
we obtain , that is, . Hence S and T have a common fixed point in . □
If for all in Theorem 2.2, we obtain the following result.
Corollary 2.3 Let be a complete dislocated metric space and , and be an arbitrary point in . Suppose there exists such that the pair is α-admissible. For , assume that
and
Suppose that the following assertions hold:
-
(i)
;
-
(ii)
for any sequence in such that for all and as then for all .
Then there exists a point in such that .
If for all in Theorem 2.2, we obtain following result.
Corollary 2.4 Let be a complete dislocated metric space and be two mappings. Suppose there exists such that the pair is η-subadmissible. For and , assume that
and
Suppose that the following assertions hold:
-
(i)
;
-
(ii)
for any sequence in such that for all and as then for all .
Then there exists a point in such that .
Corollary 2.5 (Theorem 2.2 of [32])
Let be a complete metric space and be an α-admissible mapping. Assume that for ,
holds for all . Also, suppose that the following assertions hold:
-
(i)
there exists such that ;
-
(ii)
for any sequence in X with for all and as , we have for all .
Then S has a fixed point.
Theorem 2.6 On adding the condition ‘if is any common fixed point in of S and T, x be any fixed point of S or T in , then ’ to the hypotheses of Theorem 2.2, S and T have a unique common fixed point and .
Proof Assume that be another fixed point of T in , then, by assumption, ,
A contradiction to the fact that for each , . So . Hence T has no fixed point other than . Similarly, S has no fixed point other than . Now, , then
This implies that
□
Example 2.7 Let and be defined by . Then is complete dislocated metric space (see [8]). Let be defined by
and
Considering, , , and . Now . Also,
Also if , then
Then the contractive condition does not hold on X. Also, if , then
Therefore, all the conditions of Corollary 2.3 are satisfied and S and T have a common fixed point 0.
Now we apply our Theorem 2.6 to obtain unique common fixed point of three mappings on a closed ball in complete dislocated metric space.
Theorem 2.8 Let be a dislocated metric space, such that , and . Suppose there exist two functions, α-admissible with respect to η and such that
and
Suppose that
-
(i)
the pair and f are α-admissible with respect to η;
-
(ii)
;
-
(iii)
if is a sequence in such that for all n and as then for all ;
-
(iv)
if fx is any point in such that and fy be any point in such that or , then ;
-
(v)
fX is complete subspace of X and and are weakly compatible.
Then S, T, and f have a unique common fixed point fz in . Moreover, .
Proof By Lemma 1.6, there exists such that and is one-to-one. Now since , we define two mappings by and , respectively. Since f is one-to-one on E, then g, h are well defined. Now . Then . Let , choose a point in fX such that and let . Continuing this process and having chosen in fX such that
As f is α-admissible then implies
Also if is α-admissible then implies
This implies that the pair is α-admissible. As . Continuing this process, we have . Following similar arguments to those of Theorem 2.2, . Also by inequality (10).
Note that for and . Then by using inequality (9), we have
As fX is a complete space, all conditions of Theorem 2.6 are satisfied, we deduce that there exists a unique common fixed point of g and h. Now or . Thus fz is the point of coincidence of S, T and f. Let be another point of coincidence of f, S and T then there exists such that , which implies that . A contradiction as is a unique common fixed point of g and h. Hence . Thus S, T and f have a unique point of coincidence . Now since and are weakly compatible, by Lemma 1.7 fz is a unique common fixed point of S, T, and f. □
Similarly, we can apply our Theorem 2.6 to obtain unique common fixed point and point of coincidence of four mappings in complete dislocated metric space. One can easily obtain conclusion by using the technique given in the proof of Theorem 2.8 [8].
Theorem 2.9 Let be a dislocated metric space and S, T, g and f be self-mappings on X such that , and . Suppose there exist two functions is α-admissible with respect to η and such that
and
Suppose that
-
(i)
the pairs and are α-admissible with respect to η;
-
(ii)
;
-
(iii)
if is a sequence in such that for all n and as then for all n;
-
(iv)
if is any point in such that and be any point in such that or , then ;
-
(v)
fX is complete subspace of Xand and are weakly compatible.
Then S, T, f, and g have a unique common fixed point fz in .
A partial metric version of Theorem 2.2 is given below.
Theorem 2.10 Let be a complete partial metric space, be two maps, and . Suppose there exist two functions, such that be α-admissible with respect to η and . Assume that
and
Suppose that the following assertions hold:
-
(i)
;
-
(ii)
for any sequence in such that for all and as then for all .
Then there exists a point in such that .
3 Fixed point results for graphic contractions in dislocated metric spaces
Consistent with Jachymski [24], let be a dislocated metric space and Δ denotes the diagonal of the Cartesian product . Consider a directed graph G such that the set of its vertices coincides with X, and the set of its edges contains all loops, i.e., . We assume G has no parallel edges, so we can identify G with the pair . Moreover, we may treat G as a weighted graph (see [24]) by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length m () is a sequence of vertices such that , and for . A graph G is connected if there is a path between any two vertices. G is weakly connected if is connected (see for details [1, 11, 21, 24]).
Definition 3.1 [24]
We say that a mapping is a Banach G-contraction or simply G-contraction if T preserves the edges of G, i.e.,
and T decreases the weights of the edges of G in the following way:
Now we extend the concept of G-contraction for the pair of maps as follows.
Definition 3.2 Let be a dislocated metric space endowed with a graph G and be self-mappings. Assume that for , and following conditions hold:
Then the mappings are called a ψ-graphic contractive mappings. If for some , then we say are G-contractive mappings.
Theorem 3.3 Let be a complete dislocated metric space endowed with a graph G and be ψ-graphic contractive mappings and . Suppose that the following assertions hold:
-
(i)
and for all ;
-
(ii)
if is a sequence in such that for all and as , then for all .
Then S and T have a common fixed point.
Proof Define, by At first we prove that the mappings are α-admissible. Let with , then . As are ψ-graphic contractive mappings, we have and . That is, and . Thus S, T are α-admissible mappings. From (i) there exists such that . That is, .
If with , then . Now, since S, T are ψ-graphic contractive mappings, . That is,
Let with as and for all . Then for all and as . So by (ii) we have for all . That is, . Hence, all conditions of Corollary 2.3 are satisfied and S and T have a common fixed point.
Theorem 3.2(2o) [24] and Theorem 2.3(2) [12] are extended to ψ-graphic contractive pair defined on a dislocated metric space as follows. □
Theorem 3.4 Let be a complete dislocated metric space endowed with a graph G and be ψ-graphic contractive mappings and . Suppose that the following assertions hold:
-
(i)
and for all ;
(iis) and imply for all , that is, is a quasi-order [24]and if is a sequence in such that for all and as , then there is a subsequence with for all .
Then S, T have a common fixed point.
Proof Condition (iis) implies that of (ii) in Theorem 3.3 (see Remark 3.1 [24]). Now the conclusion follows from Theorem 3.3. □
Corollary 3.5 Let be a complete dislocated metric space endowed with a graph G and be two mappings and . Suppose that the following assertions hold:
-
(i)
are G-contractive mappings;
-
(ii)
and ;
-
(iii)
if is a sequence in such that for all and as , then for all .
Then S and T have a common fixed point.
Corollary 3.6 Let be a complete dislocated metric space endowed with a graph G and be a mapping and . Suppose that the following assertions hold:
-
(i)
S is Banach G-contraction on ;
-
(ii)
and ;
-
(iii)
if is a sequence in such that for all and as , then for all .
Then S has a fixed point.
Corollary 3.7 Let be a complete dislocated metric space endowed with a graph G and be a mapping. Suppose that the following assertions hold:
-
(i)
S is Banach G-contraction on X and there is such that ;
-
(ii)
if is a sequence in X such that for all and as , then for all .
Then S has a fixed point.
The study of existence of fixed points in partially ordered sets has been initiated by Ran and Reurings [28] with applications to matrix equations. Agarwal, et al. [3, 4], Bhaskar and Lakshmikantham [10], Ciric et al. [13] and Hussain et al. [22, 23] presented some new results for nonlinear contractions in partially ordered metric spaces and noted that their theorems can be used to investigate a large class of problems. Roldán et al. [30] and Harandi et al. [6] proved some results in preordered metric spaces which is a generalization of partially ordered metric spaces. Here as an application of our results we deduce some new common fixed point results in preordered dislocated metric spaces.
Recall that if is a preordered set and is such that for , with implies , then the mapping T is said to be nondecreasing. If for , with implies and , then the pair is called jointly nondecreasing.
Let X be a non-empty set. Then is called a preordered dislocated metric space if is a dislocated metric on X and ⪯ is a preorder on X. Let be a preordered dislocated metric space. Define the graph G by
For this graph, the first condition in Definition 3.2 means S, T are jointly nondecreasing with respect to this order. From Theorems 3.3-Corollary 3.7 we derive the following important results in preordered dislocated metric spaces.
Theorem 3.8 Let be a preordered complete dislocated metric space and let the pair of self-maps of X be jointly nondecreasing and . Suppose that the following assertions hold:
-
(i)
for all , with ;
-
(ii)
and for all ;
-
(iii)
if is a nondecreasing sequence in such that as , then for all .
Then S and T have a common fixed point.
Corollary 3.9 Let be a preordered complete dislocated metric space and let the pair of self-maps of X be jointly nondecreasing and . Suppose that the following assertions hold:
-
(i)
there exists such that for all with ;
-
(ii)
and ;
-
(iii)
if is a nondecreasing sequence in such that as , then for all .
Then S and T have a common fixed point.
Corollary 3.10 Let be a preordered complete dislocated metric space and let the pair of self-maps of X be jointly nondecreasing. Suppose that the following assertions hold:
-
(i)
there exists such that for all with ;
-
(ii)
;
-
(iii)
if is a nondecreasing sequence in X such that as , then for all .
Then S and T have a common fixed point.
Corollary 3.11 Let be a preordered complete dislocated metric space and be a nondecreasing map and . Suppose that the following assertions hold:
-
(i)
there exists such that for all with ;
-
(ii)
and ;
-
(iii)
if is a nondecreasing sequence in such that as , then for all .
Then S has a fixed point.
Corollary 3.12 Let be a preordered complete dislocated metric space and be a nondecreasing map. Suppose that the following assertions hold:
-
(i)
there exists such that for all with ;
-
(ii)
there exists such that ;
-
(iii)
if is a nondecreasing sequence in X such that as , then for all .
Then S has a fixed point.
Corollary 3.13 [27]
Let be a preordered complete metric space and be a nondecreasing mapping such that
for all with where . Suppose that the following assertions hold:
-
(i)
there exists such that ;
-
(ii)
if is a sequence in X such that for all and as , then for all .
Then S has a fixed point.
Remark 3.14 We can similarly obtain partial metric and preordered partial metric versions of all results proved here which provide new results in the literature.
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.
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Hussain, N., Arshad, M., Shoaib, A. et al. Common fixed point results for α-ψ-contractions on a metric space endowed with graph. J Inequal Appl 2014, 136 (2014). https://doi.org/10.1186/1029-242X-2014-136
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DOI: https://doi.org/10.1186/1029-242X-2014-136