Abstract
In this paper, we introduce the notion of almost generalized -contractive mappings and we establish some fixed and common fixed point results for this class of mappings in ordered complete b-metric spaces. Our results generalize several well-known comparable results in the literature. Finally, two examples support our results.
MSC:54H25, 47H10, 54E50.
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1 Introduction
A fundamental principle in computer science is iteration. Iterative techniques are used to find roots of equations and solutions of linear and nonlinear systems of equations and differential equations. So, the attractiveness of the fixed point iteration is understandable to a large number of mathematicians.
The Banach contraction principle [1] is a very popular tool for solving problems in nonlinear analysis. Some authors generalized this interesting theorem in different ways (see, e.g., [2–16]).
Berinde in [17, 18] initiated the concept of almost contractions and obtained many interesting fixed point theorems for a Ćirić strong almost contraction.
Now, let us recall the following definition.
Definition 1 [17]
A single-valued mapping is called a Ćirić strong almost contraction if there exist a constant and some such that
for all , where
Babu in [19] introduced the class of mappings which satisfy condition (B).
Definition 2 [19]
Let be a metric space. A mapping is said to satisfy condition (B) if there exist a constant and some such that
for all .
Moreover, Babu in [19] proved the existence of a fixed point for such mappings on complete metric spaces.
Ćirić et al. in [20] introduced the concept of almost generalized contractive condition and they proved some existing results.
Definition 3 [20]
Let be a partially ordered set. Two mappings are said to be strictly weakly increasing if and , for all .
Definition 4 [20]
Let f and g be two self mappings on a metric space . Then they are said to satisfy almost generalized contractive condition, if there exist a constant and some such that
for all .
Ćirić et al. in [20] proved the following theorems.
Theorem 1 Let be a partially ordered set and suppose that there exists a metric d on X such that the metric space is complete. Let be a strictly increasing continuous mapping with respect to ⪯. Suppose that there exist a constant and some such that
for all comparable elements , where
If there exists such that , then f has a fixed point in X.
Theorem 2 Let be a partially ordered set and suppose that there exists a metric d on X such that the metric space is complete. Let be two strictly weakly increasing mappings which satisfy (1.1) with respect to ⪯, for all comparable elements . If either f or g is continuous, then f and g have a common fixed point in X.
Khan et al. [21] introduced the concept of an altering distance function as follows.
Definition 5 [21]
The function is called an altering distance function, if the following properties hold:
-
1.
φ is continuous and non-decreasing.
-
2.
if and only if .
So far, many authors have studied fixed point theorems which are based on altering distance functions (see, e.g., [2, 21–30]).
The concept of a b-metric space was introduced by Czerwik in [31]. After that, several interesting results about the existence of a fixed point for single-valued and multi-valued operators in b-metric spaces have been obtained (see [2, 32–42]). Pacurar [40] proved some results on sequences of almost contractions and fixed points in b-metric spaces. Recently, Hussain and Shah [37] obtained some results on KKM mappings in cone b-metric spaces.
Consistent with [31] and [42], the following definitions and results will be needed in the sequel.
Definition 6 [31]
Let X be a (nonempty) set and be a given real number. A function is a b-metric iff for all , the following conditions hold:
(b1) iff ,
(b2) ,
(b3) .
In this case, the pair is called a b-metric space.
It should be noted that, the class of b-metric spaces is effectively larger than the class of metric spaces, since a b-metric is a metric, when .
Here, we present an example to show that in general, a b-metric need not necessarily be a metric (see also [[42], p.264]):
Example 1 Let be a metric space and , where is a real number. We show that ρ is a b-metric with .
Obviously, conditions (b1) and (b2) of Definition 6 are satisfied.
If , then the convexity of the function () implies
and hence, holds.
Thus, for each ,
So, condition (b3) of Definition 6 is also satisfied and ρ is a b-metric.
Definition 7 [35]
Let be a b-metric space. Then a sequence in X is called:
-
(a)
b-convergent if and only if there exists such that , as . In this case, we write .
-
(b)
b-Cauchy if and only if as .
Proposition 1 (See Remark 2.1 in [35])
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.
Definition 8 [35]
The b-metric space is b-complete if every b-Cauchy sequence in X b-converges.
It should be noted that, in general a b-metric function for is not jointly continuous in all its variables. The following example is an example of a b-metric which is not continuous.
Example 2 (see Example 3 in [36])
Let and let be defined by
Then it is easy to see that for all , we have
Thus, is a b-metric space with . Let for each . Then
that is, , but as .
Aghajani et al. [2] proved the following simple lemma about the b-convergent sequences.
Lemma 1 Let be a b-metric space with , and suppose that and b-converge to x, y, respectively. Then, we have
In particular, if , then, . Moreover, for each we have
In this paper, we introduce the notion of an almost generalized -contractive mapping and we establish some results in complete ordered b-metric spaces, where ψ and φ are altering distance functions. Our results generalize Theorems 1 and 2 and all results in [28] and several comparable results in the literature.
2 Main results
In this section, we define the notion of almost generalized -contractive mapping and prove our new results. In particular, we generalize Theorems 2.1, 2.2 and 2.3 of Ćirić et al. in [20].
Let be an ordered b-metric space and let be a mapping. Set
and
Definition 9 Let be a b-metric space. We say that a mapping is an almost generalized -contractive mapping if there exist and two altering distance functions ψ and φ such that
for all .
Now, let us prove our first result.
Theorem 3 Let be a partially ordered set and suppose that there exists a b-metric d on X such that is a b-complete b-metric space. Let be a non-decreasing continuous mapping with respect to ⪯. Suppose that f satisfies condition (2.1), for all comparable elements . If there exists such that , then f has a fixed point.
Proof Let . Then, we define a sequence in X such that , for all . Since and f is non-decreasing, we have . Again, as and f is non-decreasing, we have . By induction, we have
If , for some , then and hence is a fixed point of f. So, we may assume that , for all . By (2.1), we have
where
and
From (2.2)-(2.4) and the properties of ψ and φ, we get
If
then by (2.5) we have
which gives a contradiction. Thus,
Therefore (2.5) becomes
Since ψ is a non-decreasing mapping, is a non-increasing sequence of positive numbers. So, there exists such that
Letting in (2.6), we get
Therefore, , and hence . Thus, we have
Next, we show that is a b-Cauchy sequence in X. Suppose the contrary, that is, is not a b-Cauchy sequence. Then there exists for which we can find two subsequences and of such that is the smallest index for which
This means that
From (2.8), (2.9) and using the triangular inequality, we get
Using (2.7) and taking the upper limit as , we get
On the other hand, we have
Using (2.7), (2.9) and taking the upper limit as , we get
So, we have
Again, using the triangular inequality, we have
and
Taking the upper limit as in the first and second inequalities above, and using (2.7) and (2.10) we get
Similarly, taking the upper limit as in the third inequality above, and using (2.7) and (2.9), we get
From (2.1), we have
where
and
Taking the upper limit as in (2.14) and (2.15) and using (2.7), (2.10), (2.11) and (2.12), we get
So, we have
and
Similarly, we can obtain
Now, taking the upper limit as in (2.13) and using (2.8), (2.16) and (2.17), we have
which further implies that
so , a contradiction to (2.18). Thus, is a b-Cauchy sequence in X. As X is a b-complete space, there exists such that as , and
Now, suppose that f is continuous. Using the triangular inequality, we get
Letting , we get
So, we have . Thus, u is a fixed point of f. □
Note that the continuity of f in Theorem 3 is not necessary and can be dropped.
Theorem 4 Under the same hypotheses of Theorem 3, without the continuity assumption of f, assume that whenever is a non-decreasing sequence in X such that , , for all , then f has a fixed point in X.
Proof Following similar arguments to those given in Theorem 3, we construct an increasing sequence in X such that , for some . Using the assumption on X, we have , for all . Now, we show that . By (2.1), we have
where
and
Letting in (2.20) and (2.21) and using Lemma 1, we get
and
Similarly, we can obtain
Again, taking the upper limit as in (2.19) and using Lemma 1 and (2.22) we get
Therefore, , equivalently, . Thus, from (2.23) we get and hence u is a fixed point of f. □
Corollary 1 Let be a partially ordered set and suppose that there exists a b-metric d on X such that is a b-complete b-metric space. Let be a non-decreasing continuous mapping with respect to ⪯. Suppose that there exist and such that
for all comparable elements . If there exists such that , then f has a fixed point.
Proof Follows from Theorem 3 by taking and , for all . □
Corollary 2 Under the hypotheses of Corollary 1, without the continuity assumption of f, for any non-decreasing sequence in X such that , let us have for all . Then, f has a fixed point in X.
Let be an ordered b-metric space and let be two mappings. Set
and
Now, we present the following definition.
Definition 10 Let be a partially ordered b-metric space and let ψ and φ be altering distance functions. We say that a mapping is an almost generalized -contractive mapping with respect to a mapping , if there exists such that
for all .
Definition 11 Let be a partially ordered set. Then two mappings are said to be weakly increasing if and , for all .
Theorem 5 Let be a partially ordered set and suppose that there exists a b-metric d on X such that is a b-complete b-metric space, and let be two weakly increasing mappings with respect to ⪯. Suppose that f satisfies 2.24, for all comparable elements . If either f or g is continuous, then f and g have a common fixed point.
Proof Let us divide the proof into two parts as follows.
First part: We prove that u is a fixed point of f if and only if u is a fixed point of g. Suppose that u is a fixed point of f; that is, . As , by (2.24), we have
Thus, we have . Therefore, and hence . Similarly, we can show that if u is a fixed point of g, then u is a fixed point of f.
Second part (construction of a sequence by iterative technique):
Let . We construct a sequence in X such that and , for all non-negative integers. As f and g are weakly increasing with respect to ⪯, we have:
If , for some , then . Thus is a fixed point of f. By the first part, we conclude that is also a fixed point of g.
If , for some , then . Thus, is a fixed point of g. By the first part, we conclude that is also a fixed point of f. Therefore, we assume that , for all . Now, we complete the proof in the following steps.
Step 1: We will prove that
As and are comparable, by (2.24), we have
where
and
So, we have
If
then (2.25) becomes
which gives a contradiction. So,
and hence (2.25) becomes
Similarly, we can show that
By (2.26) and (2.27), we get that is a non-increasing sequence of positive numbers. Hence there is such that
Letting in (2.26), we get
which implies that and hence . So, we have
Step 2: We will prove that is a b-Cauchy sequence. It is sufficient to show that is a b-Cauchy sequence. Suppose the contrary, that is, is not a b-Cauchy sequence. Then there exists , for which we can find two subsequences of positive integers and such that is the smallest index for which
This means that
From (2.29), (2.30) and the triangular inequality, we get
Taking the upper limit in the above inequality and using (2.28), we have
Again, from (2.29) and the triangular inequality, we get
Taking the upper limit in the above inequality and using (2.28), we have
So, we obtain
Similarly, we can obtain
Since and are comparable, using (2.24) we have
where
and
Taking the upper limit in (2.34) and (2.35) and using (2.28) and (2.32), we get
So, we have
and
Similarly, we can obtain
Now, taking the upper limit as in (2.33) and using (2.36), (2.37) and (2.38), we have
which implies that
so , a contradiction to (2.38). Hence is a b-Cauchy sequence in X.
Step 3 (Existence of a common fixed point):
As is a b-Cauchy sequence in X which is a b-complete b-metric space, there exists such that as , and
Now, without any loss of generality, we may assume that f is continuous. Using the triangular inequality, we get
Letting , we get
So, we have . Thus, u is a fixed point of f. By the first part, we conclude that u is also a fixed point of g. □
The continuity of one of the functions f or g in Theorem 5 is not necessary.
Theorem 6 Under the hypotheses of Theorem 5, without the continuity assumption of one of the functions f or g, for any non-decreasing sequence in X such that , let us have , for all . Then, f and g have a common fixed point in X.
Proof Reviewing the proof of Theorem 5, we construct an increasing sequence in X such that , for some . Using the assumption on X, we have , for all . Now, we show that . By (2.24), we have
where
and
Letting in (2.40) and (2.41) and using Lemma 1, we get
and
Similarly, we can obtain
Again, taking the upper limit as in (2.39) and using Lemma 1 and (2.42), we get
Therefore, , equivalently, . Thus, from (2.43) we get and hence u is a fixed point of g. On the other hand, similar to the first part of the proof of Theorem 5, we can show that . Hence, u is a common fixed point of f and g. □
Also, we have the following results.
Corollary 3 Let be a partially ordered set and suppose that there exists a b-metric d on X such that is a b-complete b-metric space. Let be two weakly increasing mappings with respect to ⪯. Suppose that there exist and such that
for all comparable elements . If either f or g is continuous, then f and g have a common fixed point.
Corollary 4 Under the hypotheses of Corollary 3, without the continuity assumption of one of the functions f or g, assume that whenever is a non-decreasing sequence in X such that , then , for all . Then f and g have a common fixed point in X.
Now, in order to support the usability of our results, we present the following examples.
Example 3 Let be equipped with the b-metric for all , where .
Define a relation ⪯ on X by iff , the functions by
and
and the altering distance functions by and , where . Then, we have the following:
-
(1)
is a partially ordered set having the b-metric d, where the b-metric space is b-complete.
-
(2)
f and g are weakly increasing mappings with respect to ⪯.
-
(3)
f and g are continuous.
-
(4)
f is an almost generalized -contractive mapping with respect to g, that is,
for all with and , where
and
Proof The proof of (1) is clear. To prove (2), for each , we know that and . So, and . Hence, and , for each . Therefore, f and g are weakly increasing mappings with respect to ⪯. It is easy to see that f and g are continuous.
To prove (4), let with . So, . Thus, we have the following cases.
Case 1: If , then we have
Now, using the mean value theorem for function , for , we have
that is, we have
for each .
Case 2: If , then we have
Using the mean value theorem for function , for , we have
So, we have
for each . Combining Cases 1 and 2 together, we conclude that f is an almost generalized -contractive mapping with respect to g. Thus, all the hypotheses of Theorem 5 are satisfied and hence f and g have a common fixed point. Indeed, 0 is the unique common fixed point of f and g. □
Remark 1 A subset W of a partially ordered set X is said to be well ordered if every two elements of W are comparable [43]. Note that in Theorems 3 and 4, f has a unique fixed point provided that the fixed points of f are comparable. Also, in Theorems 5 and 6, the set of common fixed points of f and g is well ordered if and only if f and g have one and only one common fixed point.
Example 4 Let be equipped with the following partial order ⪯:
Define b-metric by
It is easy to see that is a b-complete b-metric space.
Define the self-maps f and g by
We see that f and g are weakly increasing mappings with respect to ⪯ and f and g are continuous.
Define by and . One can easily check that f is an almost generalized -contractive mapping with respect to g, with .
Thus, all the conditions of Theorem 5 are satisfied and hence f and g have a common fixed point. Indeed, 0 and 2 are two common fixed points of f and g. Note that the ordered set is not well ordered.
3 Applications
Let Φ denote the set of all functions satisfying the following hypotheses:
-
1.
Every is a Lebesgue integrable function on each compact subset of .
-
2.
For any and any , .
It is an easy matter to check that the mapping defined by
is an altering distance function. Therefore, we have the following results.
Corollary 5 Let be a partially ordered set having a b-metric d such that the b-metric space is b-complete. Let be a non-decreasing continuous mapping with respect to ⪯. Suppose that there exist and such that
for all comparable elements . If there exists such that , then f has a fixed point.
Proof Follows from Theorem 3 by taking and , for all . □
Corollary 6 Let be a partially ordered set having a b-metric d such that the b-metric space is b-complete. Let be two weakly increasing mappings with respect to ⪯. Suppose that there exist and such that
for all comparable elements . If either f or g is continuous, then f and g have a common fixed point.
Proof Follows from Theorem 5 by taking and , for all . □
Finally, let us finish this paper with the following remarks.
Remark 2 Theorem 2.1 of [20] is a special case of Corollary 1.
Remark 3 Theorem 2.2 of [20] is a special case of Corollary 2.
Remark 4 Theorem 2.3, Corollary 2.4 and Corollary 2.5 of [20] are special cases of Corollary 3.
Remark 5 Since a b-metric is a metric when , so our results can be viewed as a generalization and extension of corresponding results in [28] and several other comparable results.
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Rezaei Roshan, J., Parvaneh, V., Sedghi, S. et al. Common fixed points of almost generalized -contractive mappings in ordered b-metric spaces. Fixed Point Theory Appl 2013, 159 (2013). https://doi.org/10.1186/1687-1812-2013-159
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DOI: https://doi.org/10.1186/1687-1812-2013-159