Abstract
In this paper, tripled coincidence points of mappings satisfying some nonlinear contractive conditions in the framework of partially ordered b-metric spaces are obtained. Our results extend the results of Berinde and Borcut (Nonlinear Anal. 74:4889-4897, 2011) and Borcut (Appl. Math. Comput. 218:7339-7346, 2012) from the context of ordered metric spaces to the setting of ordered b-metric spaces. Moreover, some examples of the main result are given. Finally, some tripled coincidence point results for mappings satisfying some contractive conditions of integral type in complete partially ordered b-metric spaces are deduced. Also, an application is given to support our results.
MSC: Primary 47H10; secondary 54H25.
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1 Introduction and preliminaries
Existence of coupled fixed points in partially ordered metric spaces was first investigated in 1987 by Guo and Lakshmikantham [1], and then in [2, 3]. Further results in this direction under weak contraction conditions in different metric spaces were proved in, e.g., [4–6].
Recently, Berinde and Borcut [7] introduced a new concept of a tripled fixed point and obtained some tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. For a survey of tripled fixed point theorems and related topics, we refer the reader to [7–11].
Let be a partially ordered set, and .
-
1.
An element is called a tripled fixed point of f if , and .
-
2.
An element is called a tripled coincidence point of the mappings f and g if , and .
-
3.
An element is called a tripled common fixed point of f and g if , and .
-
4.
We say that f has the mixed g-monotone property if is g-nondecreasing in x, g-nonincreasing in y and g-nondecreasing in z, that is, if for any ,
and
Definition 1.2 [11]
Let be a nonempty set. We say that the mappings and commute if for all .
In [7], Berinde and Borcut proved the following result and formulated it as Theorems 7 and 8.
Theorem 1.3 [7]
Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let be a mapping having the mixed monotone property on X. Assume that there exist constants with for which
for all with , and . Suppose either F is continuous or is regular. If there exist such that , and , then there exist such that , and .
In [10], Borcut and Berinde proved the following result and formulated it as Theorem 4.
Theorem 1.4 Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let and be such that F has the mixed g-monotone property on X. Assume that there exist constants with such that
for all with , and . Suppose that , g is continuous and commutes with F and also suppose either F is continuous or is regular. If there exist such that , and , then there exist such that , and .
Notice that Theorem 1.3 follows from Theorem 1.4 by taking (the identity map).
In [9], Borcut obtained the following.
Theorem 1.5 [[9], Corollary 1]
Let be a partially ordered set and suppose there is a metric d on such that is a complete metric space. Let and be such that f has the g-mixed monotone property. Assume that there exists such that
for all with , and . Suppose , g is continuous and commutes with f and also suppose either
-
(a)
f is continuous, or
-
(b)
has the following properties:
-
(i)
if a non-decreasing sequence , then for all n;
-
(ii)
if a non-increasing sequence , then for all n.
If there exist such that , and , then f and g have a tripled coincidence point.
The concept of a b-metric space was introduced by Czerwik in [12]. Since then, several papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in b-metric spaces (see, e.g., [13–20]).
Consistent with [12] and [20], the following definitions and results will be needed in the sequel.
Definition 1.6 [12]
Let be a (nonempty) set and be a given real number. A function is a b-metric if, for all , the following conditions are satisfied:
(b1) iff ,
(b2) ,
(b3) .
The pair is called a b-metric space.
It should be noted that the class of b-metric spaces is effectively larger than that of metric spaces since a b-metric is a metric when , and there are b-metric spaces which are not metric spaces. Here, we present an easy example of this kind (see also [[20], p.264]).
Example 1.7 [13]
Let be a metric space and , where is a real number. Then ρ is a b-metric with . However, is not necessarily a metric space.
For example, let be the set of real numbers and let be the usual Euclidean metric. Then is a b-metric on ℜ with , but is not a metric on ℜ.
Also, the following example of a b-metric space was given in [19].
Example 1.8 [19]
Let be the set of Lebesgue measurable functions on such that
Define by
As is a metric on , then, from the previous example, D is a b-metric on with .
The purpose of this paper is to obtain some tripled coincidence point theorems for two mappings satisfying a -contractive condition in ordered b-metric spaces. Our results extend, unify and generalize the comparable results in [7, 9, 10] from the context of ordered metric spaces to the setup of ordered b-metric spaces.
We also need the following definitions.
Definition 1.9 [15]
Let be a b-metric space. Then a sequence in is called:
-
(a)
b-convergent if there exists such that as . In this case, we write .
-
(b)
b-Cauchy if as .
Proposition 1.10 (See [[15], Remark 2.1])
In a b-metric space , the following assertions hold:
(p1) A b-convergent sequence has a unique limit.
(p2) Each b-convergent sequence is b-Cauchy.
(p3) In general, a b-metric is not continuous (see also an example in [16]).
Definition 1.11 [15]
Let and be two b-metric spaces.
-
(1)
The space is b-complete if every b-Cauchy sequence in b-converges.
-
(2)
A function is b-continuous at a point if it is b-sequentially continuous at x, that is, whenever is b-convergent to x, is b-convergent to .
Definition 1.12 Let be a b-metric space. Mappings and are called compatible if
and
hold whenever , and are sequences in such that
and
Definition 1.13 Let be a nonempty set. Then is called a partially ordered b-metric space if d is a b-metric on a partially ordered set .
The space is called regular if the following conditions hold:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
The notion of an altering distance function was introduced by Khan et al. [21] as follows.
Definition 1.14 [21]
The function is called an altering distance function if the following properties are satisfied:
-
1.
ψ is continuous and strictly increasing.
-
2.
if and only if .
2 Main results
We use the following simple lemma in proving our main results.
Lemma 2.1 Let be an ordered b-metric space (with the parameter s) and let and .
(a) If a relation ⊑ is defined on by
and a mapping is given by
then is an ordered b-metric space (with the same parameter s). The space is b-complete iff is b-complete.
(b) If the mapping f has the g-mixed monotone property, then the mapping given by
is G-nondecreasing w.r.t. ⊑, i.e.,
where is defined by
(c) If f is continuous from to , then F is continuous in .
(d) If f and g are compatible, then F and G are compatible.
Let be an ordered b-metric space, and . In the rest of this paper, unless otherwise stated, for all , let
and
Now, the main result is presented as follows.
Theorem 2.2 Let be a partially ordered b-metric space with the parameter , and let and be such that . Assume that
for all with , and , or , and , where are altering distance functions and .
Assume also that
-
(1)
f has the mixed g-monotone property;
-
(2)
g is b-continuous and compatible with f.
Also, suppose that either
-
(a)
f is b-continuous and is b-complete, or
-
(b)
is regular and is b-complete.
If there exist such that , and , then f and g have a tripled coincidence point in .
Proof Let D be the b-metric and ⊑ be the partial order on defined in Lemma 2.1. Also, define the mappings by and , as in Lemma 2.1. Then is an ordered b-metric space (with the same parameter s as ) and F is a G-nondecreasing mapping on it such that . Moreover, the contractive condition (1) implies that
holds for all such that GX and GU are ⊑-comparable. Since φ has non-negative values and ψ is strictly increasing, (2) implies that
where for all such that GX and GU are ⊑-comparable. We will prove in the next lemma that under these circumstances, it follows that F and G have a coincidence point which is obviously a tripled coincidence point of f and g. □
The following lemma is an ‘ordered variant’ of the basic result of Czerwik [12] (adapted for two mappings).
Lemma 2.3 Let be a partially ordered b-metric space and let f and g be two self-mappings on . Assume that there exists such that
for all with or . Let the following conditions hold:
-
(i)
f is g-nondecreasing with respect to ⪯ and ;
-
(ii)
there exists such that ;
-
(iii)
f and g are continuous and compatible and is complete, or
(iii′) is regular and one of or is complete.
Then f and g have a coincidence point in .
Proof Because of and (ii), we can define a Jungck sequence by
for all .
It can be proved by induction that for all n. If for some n, then is a coincidence point of f and g. Hence, we suppose that for all n. It can be proved in a standard way (see, e.g., [[18], Lemma 3.1]) that is a Cauchy sequence.
Suppose first that (iii) holds. Then there exists
Further, since f and g are continuous and compatible, we get that
and
We will show that . Indeed, we have
as , and it follows that . It means that f and g have a coincidence point.
In the case (iii′), it follows that
for some . Because of regularity, we have . Applying (4) with and , we have
It follows that when , that is, . Hence, f and g have a coincidence point . □
Let
Taking (the identity mapping on ) in Theorem 2.2, we obtain the following tripled fixed point result.
Corollary 2.4 Let be a b-complete partially ordered b-metric space and let be a mapping having the mixed monotone property. Assume that
for all with , and , or , and , where are altering distance functions and .
Also, suppose that either
-
(a)
f is b-continuous, or
-
(b)
is regular.
If there exist such that , and , then f has a tripled fixed point in .
Taking and for all , in Corollary 2.4, we obtain the following tripled fixed point result.
Corollary 2.5 Let be a b-complete partially ordered b-metric space and let be a mapping having the mixed monotone property. Assume that
for some and all with , and , or , and .
Also, suppose that either
-
(a)
f is b-continuous, or
-
(b)
is regular.
If there exist such that , and , then f has a tripled fixed point in .
Remark 2.6 1. Let in Theorem 2.2,
Then the contractive condition (1) reduces to the following:
which appeared in [8] in the context of G-metric spaces.
Choosing the condition (1) instead of (8), brings at least two new features to the tripled fixed point theory.
-
a.
We obtain more general tripled coincidence point theorems, because when f and g satisfy condition (1), then they also satisfy (8).
-
b.
The technique of the proof is essentially simpler than the one used in [8], that is, we need not use Lemma 14 from [8].
2. We can replace the contractive condition (1) by the following:
where
The following corollary can be deduced from our previously obtained results.
Corollary 2.7 Let be a partially ordered b-complete b-metric space with . Let be a mapping with the mixed monotone property such that
for some and all with , and , or , and . Also, suppose that either
-
(a)
f is b-continuous, or
-
(b)
is regular.
If there exist such that , and , then f has a tripled fixed point in .
Proof If f satisfies (10), then f satisfies (6). Hence, the result follows from Corollary 2.4. □
In Theorem 2.2, if we take , and for all , where , we obtain the following result.
Corollary 2.8 Let be a partially ordered b-complete b-metric space with . Let be a mapping having the mixed monotone property and
for some , and all with , and , or , and . Also, suppose that either
-
(a)
f is b-continuous, or
-
(b)
is regular.
If there exist such that , and , then f has a tripled fixed point in .
Corollary 2.9 Let be a partially ordered b-complete b-metric space with . Let be a mapping with the mixed monotone property such that
for some , and all with , and , or , and . Also, suppose that either
-
(a)
f is b-continuous, or
-
(b)
is regular.
If there exist such that , and , then f has a tripled fixed point in .
Proof If f satisfies (11), then f satisfies the contractive condition of Corollary 2.8. □
In the following theorem, we give a sufficient condition for the uniqueness of the common tripled fixed point (see also [7, 8, 11]).
Theorem 2.10 In addition to the hypotheses of Theorem 2.2, suppose that f and g commute and that for all and , there exists such that is comparable with and . Then f and g have a unique common tripled fixed point.
Proof We shall use the notation as in the proof of Theorem 2.2. It was proved in this theorem that the set of tripled coincidence points of f and g, i.e., the set of coincidence points of F and G in , is nonempty. We shall show that if X and are coincidence points of F and G, that is, and , then .
Choose an element such that is comparable with FX and . Let and choose so that . Then we can inductively define a sequence such that . Since GX and are ⊑-comparable, we may assume that . Using the mathematical induction, it is easy to prove that for all . Applying (1), one obtains that
From the properties of ψ, we deduce that the sequence is non-increasing. Hence, if we proceed as in Theorem 2.2, we can show that
that is, is b-convergent to GX.
Similarly, we can show that is b-convergent to . Since the limit is unique, it follows that .
Since , by commutativity of f and g, we have . Let . Then . Thus, A is another coincidence point of f and g. Then . Therefore, is a tripled common fixed point of f and g.
To prove the uniqueness, assume that P is another common fixed point of F and G. Then and also . Thus, . Hence, the tripled common fixed point is unique. □
3 Examples
The following examples support our results.
Example 3.1 Let be endowed with the usual ordering and the complete b-metric , where . Define and as
Let be defined by and
Now, we have
Analogously, we can show that
and
Thus,
Hence, all of the conditions of Theorem 2.2 are satisfied (with ). Moreover, is a tripled coincidence point of f and g.
Example 3.2 Let be endowed with the usual order and the b-metric with . Consider the mapping given by
and functions defined as and . Take in Corollary 2.4. The contractive condition (6) is satisfied since
It follows that f has a tripled fixed point (which is ).
Note that if instead of the b-metric d we try to use the standard metric (with all other data unchanged), the conclusion cannot be obtained. Indeed, the inequality
does not hold since for , it reduces to .
Example 3.3 Let with the order ⪯ be defined as
Let d be given as
where and . Clearly, is a complete b-metric space with .
Let and be defined as follows:
and and .
Let be two arbitrary altering distance functions.
According to the order defined on and the definition of g, we see that for any element , gx is comparable only with itself.
By a careful computation, it is easy to see that all of the conditions of Theorem 2.2 (case (a)) are satisfied. Finally, Theorem 2.2 guarantees the existence of a tripled coincidence point for f and g, i.e., the point .
4 Applications
In this section, we obtain some tripled coincidence point theorems for a mapping satisfying a contractive condition of integral type in a complete ordered b-metric space.
We denote by Λ the set of all functions verifying the following conditions:
-
(I)
μ is a positive Lebesgue integrable mapping on each compact subset of ;
-
(II)
for all , .
Corollary 4.1 Replace the contractive condition (1) of Theorem 2.2 by the following:
There exists a such that
Let the other conditions of Theorem 2.2 be satisfied. Then f and g have a tripled coincidence point.
Proof Consider the function . Then (13) becomes
Taking and and applying Theorem 2.2, we obtain the proof (it is easy to verify that and are altering distance functions). □
Corollary 4.2 Substitute the contractive condition (1) of Theorem 2.2 by the following:
There exists a such that
Let the other conditions of Theorem 2.2 be satisfied. Then f and g have a tripled coincidence point.
Proof Again, as in Corollary 4.1, define the function . Then (14) reduces to
Now, if we define and and apply Theorem 2.2, then the proof is completed. □
Corollary 4.3 Replace the contractive condition (1) of Theorem 2.2 by the following:
There exists a such that
for altering distance functions , , and . If the other conditions of Theorem 2.2 are satisfied, then f and g have a tripled coincidence point.
Similar to [22], let N be a fixed positive integer. Let be a family of N functions which belong to Λ. For all , we define
We have the following result.
Corollary 4.4 Replace the inequality (1) of Theorem 2.2 by the following condition:
Let the other conditions of Theorem 2.2 be satisfied. Then f and g have a tripled coincidence point.
Proof Consider and . Then the above inequality becomes
Applying Theorem 2.2, we obtain the desired result (it is easy to verify that and are altering distance functions). □
Another consequence of the main theorem is the following result.
Corollary 4.5 Substitute the contractive condition (1) of Theorem 2.2 by the following:
There exist such that
Let the other conditions of Theorem 2.2 be satisfied. Then f and g have a tripled coincidence point.
Proof It is clear that the function is an altering distance function. □
5 Existence of a solution for a system of integral equations
Motivated by the work in [8], we study the existence of solutions for a system of nonlinear integral equations using the results proved in the previous sections.
Consider the integral equations in the following system.
We will consider the system (17) under the following assumptions:
-
(i)
are continuous;
-
(ii)
is continuous;
-
(iii)
is continuous;
-
(iv)
there exists such that for all ,
and
-
(v)
-
(vi)
there exist continuous functions such that
and
We consider the space of continuous functions defined on endowed with the b-metric given by
for all , where and . We endow with the partial order ⪯ given by
for all .
It is known that is regular [23].
Our result is the following.
Theorem 5.1 Under assumptions (i)-(vi), the system (17) has a solution in , where .
Proof As in [8], we consider the operators and defined by
and for all , .
F has the mixed monotone property (see [[8], Theorem 25]).
Let , with , and . Since F has the mixed monotone property, we have
On the other hand,
Note that for all , from (iv) and the fact that for all , , we have
Thus,
Repeating this idea, using the definition of the b-metric d, we get
and
Hence, from the above three inequalities, we have
But from (v), we have
This proves that the operator F satisfies the contractive condition appearing in Corollary 2.8 (with ).
Let α, β, γ be the functions appearing in assumption (vi). Then by (vi) we get
Applying Corollary 2.8, we deduce the existence of such that , and . □
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Acknowledgements
The authors express their gratitude to the referees and Professor Zoran Kadelburg for their helpful suggestions which improved the presentation, in particular the proof of Theorem 2.2.
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JRR, VP and SR have worked together on each section of the paper such as the literature review, results and examples. All authors read and approved the final manuscript.
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Parvaneh, V., Rezaei Roshan, J. & Radenović, S. Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations. Fixed Point Theory Appl 2013, 130 (2013). https://doi.org/10.1186/1687-1812-2013-130
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DOI: https://doi.org/10.1186/1687-1812-2013-130