Abstract
The purpose of this paper is to prove some coincidence and common fixed point theorems for ordered PrešićReich type contractions in ordered metric spaces. Results of this paper generalize and extend several known results from metric spaces into product spaces when the underlying space is an ordered metric space. An example illustrates the case when new results can be applied while old ones cannot.
Keywords:
Prešić type mapping; coincidence point; common fixed point; ordered space1 Introduction and preliminaries
The wellknown Banach contraction mapping principle states that if is a complete metric space and is a selfmapping such that
for all , where , then there exists a unique such that . This point x is called the fixed point of the mapping f. On the other hand, for mappings , Kannan [1] introduced the contractive condition
for all , where is a constant and proved a fixed point theorem using (2) instead of (1). Conditions (1) and (2) are independent, as it was shown by two examples in [2].
Reich [3], for mappings , generalized Banach and Kannan fixed point theorems using the contractive condition
for all , where α, β, γ are nonnegative constants with . An example in [3] shows that condition (3) is a proper generalization of (1) and (2).
In 1965, Prešić [4,5] extended the Banach contraction mapping principle to mappings defined on product spaces and proved the following theorem.
Theorem 1.1Letbe a complete metric space, kbe a positive integer andbe a mapping satisfying the following contractive type condition:
for every, whereare nonnegative constants such that. Then there exists a unique pointsuch that. Moreover, ifare arbitrary points inXand for, , then the sequenceis convergent and.
Note that condition (4) in the case reduces to the wellknown Banach contraction mapping principle. So, Theorem 1.1 is a generalization of the Banach fixed point theorem. Some generalizations and applications of the Prešić theorem can be seen in [419].
The existence of a fixed point in partially ordered sets was investigated by Ran and Reurings [20] and then by Nieto and Lopez [21,22]. Fixed point results in ordered metric spaces were obtained by several authors (see, e.g., [6,18,2332]). The following version of the fixed point theorem was proved, among others, in these papers.
Theorem 1.2 (see [22] and references therein)
Letbe a partially ordered set, and letdbe a metric onXsuch thatis a complete metric space. Letbe a nondecreasing map with respect to ⪯. Suppose that the following conditions hold:
(i) there existssuch thatfor allwith;
(iii) fis continuous.
Pǎcurar [10] introduced the PrešićKannan type contraction and proved some common fixed point theorems for such contractions. Very recently, in [18] (see also [33]) authors introduced the ordered Prešić type contraction and generalized the result of Prešić and proved some fixed point theorems for such mappings. In this paper, we introduce the ordered PrešićReich type contraction and prove some common fixed point theorems for such type of mappings in ordered metric spaces. Our results generalize and extend the results of Prešić [4,5], Pǎcurar [10], Malhotra et al.[18], Luong and Thuan [33], Nieto and López [21] and several known results of metric spaces. An example, which illustrates the case when new results can be applied while old ones cannot, is included.
The following definitions will be needed in the sequel.
Definition 1.3 Let X be a nonempty set, k be a positive integer and be a mapping. If , then is called a fixed point of f.
Definition 1.4 (see [13])
Let X be a nonempty set, k be a positive integer, and be mappings.
(a) An element is said to be a coincidence point of f and g if .
(b) If , then w is called a point of coincidence of f and g.
(c) If , then x is called a common fixed point of f and g.
(d) Mappings f and g are said to be commuting if for all .
(e) Mappings f and g are said to be weakly compatible if they commute at their coincidence points.
Remark that the above definition in the case reduces to the usual definitions of commuting and weakly compatible mappings in the sense of [34] (for details, see the Introduction from [34]).
Definition 1.5 (see [18])
Let a nonempty set X be equipped with a partial order ‘⪯’ such that is a metric space, then is called an ordered metric space. A sequence in X is said to be nondecreasing with respect to ‘⪯’ if . Let k be a positive integer and be a mapping, then f is said to be nondecreasing with respect to ‘⪯’ if for any finite nondecreasing sequence we have . Let be a mapping. f is said to be gnondecreasing with respect to ‘⪯’ if for any finite nondecreasing sequence we have .
Remark 1.6 For , the above definitions reduce to usual definitions of fixed point and nondecreasing mapping in a metric space.
Definition 1.7 Let X be a nonempty set equipped with partial order ‘⪯,’ and let be a mapping. A nonempty subset of X is said to be well ordered if every two elements of are comparable. Elements are called gcomparable if ga and gb are comparable. is called gwell ordered if for all , a and b are gcomparable, i.e., ga and gb are comparable.
Example 1.8 Let , ‘⪯’ be a partial order relation on X defined by . Let and be defined by , , , . Then it is clear that is not well ordered but it is gwell ordered.
Let be an ordered metric space. Let k be a positive integer and be a mapping. f is said to be an ordered Prešić type contraction if
for all with , where are nonnegative constants such that . If (5) is satisfied for all , then f is called a Prešić type contraction.
f is said to be an ordered PrešićKannan type contraction (see [10] for details) if f satisfies following condition:
for all with , where . If (6) is satisfied for all , then f is called a PrešićKannan type contraction.
f is said to be an ordered PrešićReich type contraction (see also [16]) if f satisfies the following condition:
for all with , where , are nonnegative constants such that . If (7) is satisfied for all , then f is called a PrešićReich type contraction.
Note that the PrešićReich type contraction is a generalization of Prešić type and PrešićKannan type contractions. Indeed, for , , a PrešićReich type contraction reduces into a Prešić type contraction and for , , and , , a PrešićReich type contraction reduces into a PrešićKannan type contraction. Also, for , a PrešićReich type contraction reduces into a Reich contraction, so it generalizes the Banach and Kannan contractions.
Now we can state our main results.
2 Main results
Theorem 2.1Letbe an ordered complete metric space. Letkbe a positive integer, andbe two mappings such that, is a closed subset ofXand
for allwith, where, are nonnegative constants such that
Suppose that the following conditions hold:
(II) fisgnondecreasing;
(III) if a nondecreasing sequenceconverges to, thenfor alland.
Thenfandghave a point of coincidence. If, in addition, fandgare weakly compatible, thenfandghave a common fixed point. Moreover, the set of common fixed points offandgisgwell ordered if and only iffandghave a unique common fixed point.
Proof Starting with given , we define a sequence as follows: let , . As , there exists such that . Therefore as , we have , that is, . Again, as f is gnondecreasing and , we have . Choose such that (which is possible since ). So, , that is, . Continuing this process, we obtain
that is,
and for . Thus, is nondecreasing with respect to ‘⪯,’ that is, is gnondecreasing with respect to ‘⪯.’ We shall show that is a Cauchy sequence in . If for any n, then
As , using (8), the above inequality implies that
that is,
since . In view of (9), we have , therefore it follows from the above inequality that , that is, . Similarly, it can be shown that
Therefore is a Cauchy sequence. If for all n, then for any , we have
As is gnondecreasing, using (8), the above inequality implies that
that is,
Let , , , then in view of (9) we have . Therefore
By repeating this process, we obtain
Let and , then it follows from (10) that
As , we have as . Therefore, it follows from the above inequality that . Therefore is a Cauchy sequence. As is closed, there exist , such that
We shall show that v is a point of coincidence of f and g. For any , we obtain
By (III) we have for all , also, as , and . Therefore, using (8) in the above inequality, we obtain
that is,
As and , therefore it follows from the above inequality that
Thus, u is a coincidence point and v is a corresponding point of coincidence of f and g. Suppose, f and g are weakly compatible, then by (11) we have
Again, by (III), ; therefore using (8) and a similar process as several times before, we obtain
Thus v is a common fixed point of f and g. Suppose that the set of common fixed points is gwell ordered. We shall show that the common fixed point is unique. Assume on the contrary that is another common fixed point of f and g, that is, and . As v and are gcomparable, let for example . From (8), it follows that
As , we obtain , that is, , a contradiction. Therefore the common fixed point is unique. For converse, if a common fixed point of f and g is unique, then the set of common fixed points of f and g is singleton, and thus gwell ordered. □
Remark 2.2 Let be an ordered metric space, and let be two mappings. Then f is called an ordered gweak contraction if
for all with , where , , are nonnegative constants such that . If the above inequality is satisfied for all , then f is called a gweak contraction (see [35]). For in Theorem 2.1, we get a fixed point result for an ordered gweak contraction in metric spaces.
The following is a fixed point result for ordered PrešićReich type mappings in metric spaces and can be obtained by taking (that is, the identity mapping of X) in Theorem 2.1.
Corollary 2.3Letbe an ordered complete metric space. Letkbe a positive integer, be a mapping such that the following conditions hold:
(I) fis an ordered PrešićReich type contraction;
(III) fis nondecreasing (with respect to ‘⪯’);
(IV) if a nondecreasing sequenceconverges to, thenfor all.
Thenfhas a fixed point. Moreover, the set of fixed points offis well ordered if and only iffhas a unique fixed point.
The following corollary is a generalization of the result of Prešić in an ordered metric space and can be obtained by taking for in Theorem 2.1.
Corollary 2.4Letbe an ordered complete metric space. Letkbe a positive integer, andbe two mappings such that, is a closed subset ofXand
for allwith, whereare nonnegative constants such that. Suppose that the following conditions hold:
(II) fisgnondecreasing;
(III) if a nondecreasing sequenceconverges to, thenfor alland.
Thenfandghave a point of coincidence. If, in addition, fandgare weakly compatible, thenfandghave a common fixed point. Moreover, the set of common fixed points offandgisgwell ordered if and only iffandghave a unique common fixed point.
The following corollary generalizes the result of Pǎcurar [10] in ordered metric spaces and can be obtained by taking for in Theorem 2.1.
Corollary 2.5Letbe an ordered complete metric space. Letkbe a positive integer, andbe two mappings such that, is a closed subset ofXand
for allwith, whereare nonnegative constants such that. Suppose that the following conditions hold:
(II) fisgnondecreasing;
(III) if a nondecreasing sequenceconverges to, thenfor alland.
Thenfandghave a point of coincidence. If, in addition, fandgare weakly compatible, thenfandghave a common fixed point. Moreover, the set of common fixed points offandgisgwell ordered if and only iffandghave a unique common fixed point.
The following example illustrates that an ordered PrešićReich type contraction may not be an ordered Prešić type or ordered PrešićKannan type or PrešićReich type contraction; moreover, that the fixed point of an ordered PrešićReich type contraction may not be unique (when the set of fixed points of f is not wellordered).
Example 2.6 Let and order relation ‘⪯’ be defined by
and let d be the usual metric on X. Then is an ordered complete metric space. For , define by
Then:
(a) f is not an ordered Prešić type contraction;
(b) f is not an ordered PrešićKannan type contraction;
(c) f is not a PrešićReich type contraction;
(d) f is an ordered PrešićReich type contraction with , .
for all with , where , are nonnegative constants such that . Note that , therefore for , , (14) becomes
But and therefore the above inequality will never hold. Thus f is not an ordered Prešić type contraction.
for all with , where β is a nonnegative constant such that . Note that for all and therefore for , , (15) becomes
But , and therefore the above inequality will never hold. Thus f is not an ordered PrešićKannan type contraction.
where , , , , are nonnegative constants such that . If f is a PrešićReich type contraction, then inequality (16) must be satisfied for all . Note that for , , (16) becomes
But , and therefore the above inequality will never hold. Thus f is not a PrešićReich type contraction.
(d) If f is an ordered PrešićReich type contraction, then inequality (16) must be satisfied for all with . Indeed, we have to check the validity of (16) only for , and . If or , then (16) is satisfied trivially. If or with , that is, , then (16) becomes
which is valid for , . If any one of , , is equal to 1, then with a similar process one can verify the same result. If any two of , , are equal to 1, for example, let , , then (16) becomes
which is valid for , . Similarly, in all possible cases, (16) is satisfied for , . Thus, f is an ordered PrešićReich type contraction. All the conditions of Corollary 2.3 (except the set of fixed points of f is well ordered) are satisfied and the set of fixed points of f is . Note that the set of fixed points of f, that is ℱ, is not well ordered (as ) and fixed point f is not unique. □
Competing interests
All authors of the present paper disclose no actual potential conflict of interests including any financial, personal or other relationships with people or organizations.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
This paper is supported by Grant No. 1740024 from the Ministry of Science and Technical Development of the Republic of Serbia.
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