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Some new fixed point results in partial ordered metric spaces via admissible mappings
Fixed Point Theory and Applications volume 2014, Article number: 117 (2014)
Abstract
The purpose of this paper is to discuss the existence of fixed points for new classes of mappings defined on an ordered metric space. The obtained results generalize and improve some fixed point results in the literature. Some examples show the usefulness of our results.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
Over the last decades, the fixed point theory has become increasingly useful in the study of nonlinear phenomena. In fact, the fixed point theorems and techniques have been developed in pure and applied analysis, topology and geometry. It is well known that a fundamental result of this theory is Banach’s contraction principle [1]. Consequently, in the last 50 years, it has been extensively studied and generalized to many settings; see for example [2–14].
In 2008, Dutta and Choudhury proved the following theorem.
Theorem 1.1 (See [15])
Let be a complete metric space and be such that
where are continuous, non-decreasing, and if and only if . Then f has a unique fixed point .
Note that the above theorem remains true if the hypothesis on ϕ is replaced by ϕ is lower semi-continuous and if and only if (see e.g. [16, 17]).
Eslamian and Abkar stated the following theorem as a generalization of Theorem 1.1.
Theorem 1.2 Let be a complete metric space and be such that
where are such that ψ is continuous and non-decreasing, α is continuous and β is lower semi-continuous,
Then f has a unique fixed point .
Aydi et al. [18] proved that Theorem 1.2 is a consequence of Theorem 1.1
On the other hand, Ran and Reurings [19] initiate the fixed point theory in the metric spaces equipped with a partial order relation. Let X be a nonempty set equipped with a partial order relation ⪯ such that the function is a metric on X, then the triple is called a partially ordered metric space. Two elements are comparable if either or . We write if but . A sequence in X is said to be non-decreasing with respect to ⪯ if for all . A mapping is said to be non-decreasing with respect to ⪯ if implies . In further discussion, if there is no confusion, for the mappings on X and sequences in X, we use the phrase ‘non-decreasing’ instead ‘non-decreasing with respect to ⪯’.
Harjani and Sadarangani [20] extended Theorem 1.1 in the framework of partially ordered metric spaces in the following way.
Theorem 1.3 Let be a partially ordered complete metric space. Let be a continuous non-decreasing mapping such that
where are continuous and non-decreasing and if and only if . If there exists such that , then f has a fixed point .
Choudhury and Kundu [21] generalized Theorems 1.2 and 1.3 as follows.
Theorem 1.4 Let be a partially ordered complete metric space. Let be a non-decreasing mapping such that
where are such that ψ is continuous and non-decreasing, α is continuous, β is lower semi-continuous,
If there exists such that , then f has a unique fixed point .
Aydi et al. [18] proved that Theorem 1.4 is a consequence of Theorem 1.3.
Karapinar and Salimi [22] proved the following theorem as a generalization of Theorems 1.2 and 1.3 where the approach of Aydi et al. [18] cannot be modified for it.
Theorem 1.5 Let be an ordered metric space such that is complete and let be a non-decreasing self mappings. Assume that there exist , , and such that
and
for all comparable where
and
Suppose that either
-
(a)
f is continuous, or
-
(b)
if a non-decreasing sequence is such that , then for all .
If there exists such that , then f has a fixed point.
On the other hand, in 2012, Samet et al. [23] introduced the concepts of α-ψ-contractive and α-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. More recently, Salimi et al. [24] modified the notions of α-ψ-contractive and α-admissible mappings and established fixed point theorems which are proper generalizations of the recent results in [22, 23]. For more on α-admissible mappings, see [25–27] and the references therein.
Samet et al. [23] defined the notion of α-admissible mappings as follows.
Definition 1.6 Let T be a self-mapping on X and be a function. We say that T is an α-admissible mapping if
In [23] the authors consider the family Ψ of non-decreasing functions such that for each , where is the n th iterate of ψ and give the following theorem.
Theorem 1.7 Let be a complete metric space and T be an α-admissible mapping. Assume that
for all , where . Also, suppose that the following assertions hold:
-
(i)
there exists such that ,
-
(ii)
either T is continuous or for any sequence in X with for all and as , we have for all .
Then T has a fixed point.
Recently, Hussain et al. [28] obtained the following Geraghty type [29] fixed point theorems via α-admissible mappings.
Theorem 1.8 Let be a complete metric space and be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies and
for all where . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that and for all n, then .
If there exists such that , then f has a fixed point.
Theorem 1.9 Let be a complete metric space and be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies and
for all where . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that and for all n, then .
If there exists such that , then f has a fixed point.
Theorem 1.10 Let be a metric space such that is complete and be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that and for all n, then .
If there exists such that , then f has a fixed point.
For more details on α-admissible mappings and related fixed point results we refer the reader to [30–32].
More recently, Salimi et al. [24] modified and generalized the notions of α-ψ-contractive mappings and α-admissible mappings by the following ways.
Definition 1.11 [24]
Let T be a self-mapping on X and be two functions. We say that T is an α-admissible mapping with respect to η if
Note that if we take then this definition reduces to Definition 1.6. Also, if we take, then we say that T is an η-subadmissible mapping.
The following result properly contains Theorem 1.7, and Theorems 2.3 and 2.4 of [22].
Theorem 1.12 [24]
Let be a complete metric space and T be an α-admissible mapping with respect to η. Assume that
where and
Also, suppose that the following assertions hold:
-
(i)
there exists such that ,
-
(ii)
either T is continuous or for any sequence in X with for all and as , we have for all .
Then T has a fixed point.
For more details on modified α-ψ-contractive mappings and related fixed point results we refer the reader to [33, 34].
2 Main results
In this section, motivated by the work of Hussain et al. [28] and Salimi et al. [24] we state and prove the following fixed point results in the setting of partially ordered metric spaces.
Theorem 2.1 Let be a partially ordered metric space such that is complete. Assume and be two mappings such that f is a non-decreasing and γ-admissible mapping. Assume that there exist , , and such that
and
for all comparable where . Suppose that either
-
(i)
f is continuous, or
-
(ii)
if a non-decreasing sequence is such that as , , and for all n, then , , and for all .
If there exists such that , , and , then f has a fixed point.
Proof Let . We define an iterative sequence in the following way:
Since f is non-decreasing and , we have
and hence is a non-decreasing sequence. Let . Since f is a γ-admissible mapping and , we deduce that . By continuing this process, we get for all . Also, assume . Similarly we get for all . If for some , then the point is the desired fixed point of f which completes the proof. Hence, we suppose that , that is, for all n. Hence, (2.3) implies
We want to show that the sequence is non-increasing sequence of reals. Suppose, to the contrary, that there exists some such that
Since ψ is non-decreasing, we obtain
Taking and in (2.2) we derive
Hence
for all . Now, by taking and in (2.7) and applying (2.6) we have
which contradicts (2.12). Therefore, we conclude that holds for all . Hence is a non-increasing sequence of positive real numbers. Thus, there exists such that . We shall show that by method of reductio ad absurdum. For this purpose, we assume that . By (2.7) together with the properties of α, β, ψ we have
which is a contradiction. Hence
We shall show that the sequence is a Cauchy sequence. Suppose that it is not. Then there are and sequences and such that for all positive integers k with
Additionally, corresponding to , we may choose such that it is the smallest integer satisfying (2.9) and . Thus,
Now, for all we have
So
Again, we have
and
By taking the limit as in the above inequalities and applying (2.8) and (2.10), we deduce
Now, from (2.2) with and we have
Then
Taking the lim inf as in the above inequality, we have
So we have
which contradicts the fact that for all . Hence
that is, the sequence is a Cauchy sequence. Since is complete, then there exists such that as . Suppose that (i) holds. Then
Hence, is a fixed point of f. Suppose that (ii) holds, that is, , , and for all . We claim that is a fixed point of f, that is, . Suppose, to the contrary, that . Due to condition (2.2), we have
Taking the lim inf as in the above inequality, we obtain
which is a contradiction. Hence and so, . □
Example 2.2 Let be endowed with the usual metric for all and be defined by
Define also , , , and by
We prove that Theorem 2.1 can be applied to f. But Theorem 1.5 cannot be applied.
Clearly, is a complete metric space. We show that f is a γ-admissible mapping. Let . If then . On the other hand, for all we have . It follows that . Thus the assertion holds. Because of the above arguments, . Now, if is a sequence in X such that for all and as , then and hence . This implies that . Also and for all . Let . Then . Indeed, if or . So, or . That is, which is a contradiction. Without any loss of generality we assume that . We get
Then the condition of Theorem 2.1 holds and f has a fixed point. Let and , then
That is, the contractive condition of Theorem 1.5 does not hold for this example.
Corollary 2.3 Let be a partially ordered metric space such that is complete. Assume and be two mappings such that f is a non-decreasing γ-admissible mapping. Assume that there exist , , and such that
Suppose that either
-
(i)
f is continuous, or
-
(ii)
if a non-decreasing sequence is such that as , , and for all n, then , , and for all ,
-
(iii)
for all comparable where .
If there exists such that , , and , then f has a fixed point.
Proof Let . Then from (iii) we have
That is,
Hence, all conditions of Theorem 2.1 hold and f has a fixed point. □
Now, we prove our second main result as follows.
Theorem 2.4 Let be a partially ordered metric space such that is complete. Assume and are two mappings such that f is a non-decreasing γ-admissible mapping. Assume that there exist , , and such that
and
for all comparable . Suppose that either
-
(i)
f is continuous, or
-
(ii)
if a non-decreasing sequence is such that as , , and for all n, then , , and for all .
If there exists such that , , and , then f has a fixed point.
Proof Let . We define an iterative sequence in the following way:
From Theorem 2.1 we know that is a non-decreasing sequence, and for all . Also, similarly, we suppose that for all n. We shall show that the sequence is non-increasing sequence of reals. Assume that there exists some such that
Hence
Taking and in (2.13) and applying (2.6) we get
Hence
for all . Now, by taking and in (2.15) and using (2.14) we deduce
which is a contradiction. Then holds for all and so there exists such that . Reviewing the proof of Theorem 2.1 we can show that . Now, suppose, to the contrary, that is not a Cauchy sequence. Then there exist and sequences and such that for all positive integers k with
and
By (2.13) with and we have
and so
By taking the lim inf as in the above inequality, we have
Therefore
which is a contradiction. Hence,
Then is a Cauchy sequence. Since is complete, there exists such that as . Let (i) hold. Then
So, is a fixed point of f. Now, we assume that (ii) holds, that is, , , and for all . We claim that is a fixed point of f, equivalently, . Suppose, to the contrary, that . From (2.13), we have
Taking the lim inf as in the above inequality, we obtain
which is a contradiction. Then and hence, . □
Example 2.5 Let X and d be as in Example 2.2. Define by
Define also γ, ψ, α, and β as in Example 2.2. We shall show that Theorem 2.4 can be applied to f, but Theorem 1.5 cannot be applied. Proceeding as in the proof of Example 2.2 f is a γ-admissible mapping, , and if is a sequence in X such that for all and as , then . Let . Then . Assume . We get
Then the condition of Theorem 2.4 holds and so f has a fixed point. Let and , then
Hence, the condition of Theorem 1.5 does not hold for this example.
Corollary 2.6 Let be a partially ordered metric space such that is complete. Assume and are two mappings such that f is a non-decreasing γ-admissible mapping. Assume that there exist , , and such that
and
for all comparable . Suppose that either
-
(i)
f is continuous, or
-
(ii)
if a non-decreasing sequence is such that as , , and for all n, then , , and for all .
If there exists such that , , and , then f has a fixed point.
Theorem 2.7 Let be a partially ordered metric space such that is complete. Assume that and are two mappings such that f is a non-decreasing γ-admissible mapping. Assume that there exist , , and such that
and
for all comparable . Suppose that either
-
(i)
f is continuous, or
-
(ii)
if a non-decreasing sequence is such that as , , and for all n, then , , and for all .
If there exists such that , , and , then f has a fixed point.
Proof Let . We define an iterative sequence in the following way:
From Theorem 2.1 we know that is a non-decreasing sequence, , and for all . Also, similarly, we suppose that for all n. We shall show that the sequence is non-increasing. Assume that there exists some such that
Hence
Taking and in (2.18) and applying (2.19) we get
Hence
for all . Now, by taking and in (2.20) and using (2.19) we deduce
which is a contradiction. Then holds for all and so there exists such that . Proceeding as in the proof of Theorem 2.1 we conclude that . Now, suppose, to the contrary that is not a Cauchy sequence. Then there exist and sequences and such that for all positive integers k with
and
By (2.18) with and we have
and so
Taking the lim inf as in the above inequality, we have
Therefore
which is a contradiction. Hence
that is, is a Cauchy sequence. Since is complete, there exists such that as . Let (i) hold. Then
So, is a fixed point of f. Now, we assume that (ii) holds, that is, , , and for all . We claim that is a fixed point of f, or equivalently, . Suppose, to the contrary, that . From (2.18), we have
Taking the lim inf as in the above inequality, we obtain
which is a contradiction. Then , and hence . □
Example 2.8 Let X and d be as in Example 2.2. Define by
Define also γ, ψ, α, and β as in Example 2.2. We shall show that Theorem 2.7 can be applied for f, but Theorem 1.5 cannot be applied. Reviewing the proof of Example 2.2, f is a γ-admissible mapping, and if is a sequence in X such that for all and as , then . Let . Then . Assume . We get
Then the condition of Theorem 2.7 holds and f has a fixed point. Clearly, the condition of Theorem 1.5 does not hold for this example.
Corollary 2.9 Let be a partially ordered metric space such that is complete. Assume and are two mappings such that f is a non-decreasing γ-admissible mapping. Assume that there exist , , and such that
and
for all comparable . Suppose that either
-
(i)
f is continuous, or
-
(ii)
if a non-decreasing sequence is such that as , , and for all n, then , , and for all .
If there exists such that , , and , then f has a fixed point.
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Acknowledgements
The authors are thankful to the referees for their valuable comments on this paper. Wei Long acknowledges support from the Research Project of Jiangxi Normal University (2012-114).
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Long, W., Khaleghizadeh, S., Salimi, P. et al. Some new fixed point results in partial ordered metric spaces via admissible mappings. Fixed Point Theory Appl 2014, 117 (2014). https://doi.org/10.1186/1687-1812-2014-117
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DOI: https://doi.org/10.1186/1687-1812-2014-117