Abstract
The aim of this paper is to introduce two classes of generalized α-ψ-contractive type mappings of integral type and to analyze the existence of fixed points for these mappings in complete metric spaces. Our results are improved versions of a multitude of relevant fixed point theorems of the existing literature.
MSC:54H25, 47H10, 54E50.
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1 Introduction and preliminaries
Recently, Samet et al. [1] introduced a very interesting notion of α-ψ-contractions via α-admissible mappings. In this paper, the authors [1] proved the existence and uniqueness of a fixed point for such a class of mappings in the context of complete metric spaces. Furthermore, the famous Banach [2] fixed point result was observed as a consequence of their main results. Following this initial paper, several authors have published new fixed point results by modifying, improving and generalizing the notion of α-ψ-contractions in various abstract spaces; see, e.g., [3–8]. Very recently, Shahi et al. [9] gave the integral version of α-ψ-contractive type mappings and proved some related fixed point theorems. As a consequence of the main results of this paper [9], the well-known integral contraction theorem of Branciari [10] and hence the celebrated Banach contraction principle were obtained.
In the present work, we introduce two classes of generalized α-ψ-contractive type mappings of integral type inspired by the report of Karapınar and Samet [7]. Also, we analyze the existence and uniqueness of fixed points for such mappings in complete metric spaces. Our results generalize, improve and extend not only the results derived by Shahi et al. [9], Samet et al. [1] and Branciari [10] but also various other related results in the literature. Moreover, from our fixed point theorems, we will derive several fixed point results on metric spaces endowed with a partial order.
We recall some necessary definitions and basic results from the literature. Throughout the paper, let ℕ denote the set of all nonnegative integers.
Berzig and Rus [4] introduced the following definition.
Definition 1.1 (see [4])
Let . We say that α is N-transitive (on X) if
for all .
In particular, we say that α is transitive if it is 1-transitive, i.e.,
As consequences of Definition 1.1, we obtain the following remarks.
Remark 1.1 (see [4])
-
(1)
Any function is 0-transitive.
-
(2)
If α is N transitive, then it is kN-transitive for all .
-
(3)
If α is transitive, then it is N-transitive for all .
-
(4)
If α is N-transitive, then it is not necessarily transitive for all .
Let Ψ be a family of functions satisfying the following conditions:
-
(1)
ψ is nondecreasing.
-
(2)
for all , where is the n th iterate of ψ.
In the literature, such mappings are called in two different ways: (c)-comparison functions in some sources (see, e.g., [11]), and Bianchini-Grandolfi gauge functions in some others (see, e.g., [12–14]).
It can be easily verified that if ψ is a (c)-comparison function, then for any .
Define such that φ is nonnegative, Lebesgue integrable and satisfies
Shahi et al. in [9] introduced the following new concept of α-ψ-contractive type mappings of integral type.
Definition 1.2 Let be a metric space and be a given mapping. We say that T is an α-ψ-contractive mapping of integral type if there exist two functions and such that for each ,
where .
In what follows, we recollect the main results of Shahi et al. [9].
Theorem 1.1 [9]
Let be a complete metric space and be a transitive mapping. Suppose that is an α-ψ-contractive mapping of integral type and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Theorem 1.2 [9]
Let be a complete metric space and be a transitive mapping. Suppose that is an α-ψ-contractive mapping of integral type and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then T has a fixed point, that is, there exists such that .
Notice that in the theorems above, the authors proved only the existence of a fixed point. To guarantee the uniqueness of the fixed point, they needed the following condition.
(U): For all , there exists such that and , where denotes the set of fixed points of T.
2 Main results
In this section, we present our main results. First, we introduce two classes of generalized α-ψ-contractive type mappings of integral type in the following way.
Definition 2.1 Let be a metric space and be a given mapping. We say that T is a generalized α-ψ-contractive mapping of integral type I if there exist two functions and such that for each ,
where and .
Definition 2.2 Let be a metric space and be a given mapping. We say that T is a generalized α-ψ-contractive mapping of integral type II if there exist two functions and such that for each ,
where and .
Remark 2.1 It is evident that if is an α-ψ-contractive mapping of integral type, then T is a generalized α-ψ-contractive mapping of integral types I and II.
The following is the first main result of this manuscript.
Theorem 2.1 Let be a complete metric space and be a transitive mapping. Suppose that is a generalized α-ψ-contractive mapping of integral type I and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Proof Let be an arbitrary point of X such that . We construct an iterative sequence in X in the following way:
If for some , then, obviously, is a fixed point of T and the proof is completed. Hence, from now on, we suppose that for all n. Due to the fact that T is α-admissible, we find that
Iteratively, we obtain that
for all .
By applying inequality (3) with and and using (5), we deduce that
where
By utilizing (7) and regarding the properties of the function ψ, we derive from (6) that
Notice that the case
is impossible due to the property for all . By using mathematical induction, we get, for all ,
where .
Letting in (9) and taking the property of ψ on the account, we find that
which, from (1), implies that
We shall prove that is a Cauchy sequence. Suppose, on the contrary, that there exist an and subsequences and such that with
Due to the definition of , we have that
By elementary evaluation, (11), we find that
In view of (11), (12) and the triangular inequality, we deduce that
Letting in the inequality above, we conclude that
Owing to the transitivity of α, we infer from (5) that
Regarding inequality (3) and by using (16), we obtain
In view of (12) and using the triangular inequality, we get
Therefore, using (11), we infer that
Now, from (3), (12), (13), (14), (15), (16) and (19), it then follows that
which is a contradiction. This implies that is a Cauchy sequence in . Due to the completeness of , there exists such that as . The continuity of T yields that as , that is, as . By the uniqueness of the limit, we obtain . Therefore, z is a fixed point of T. □
Theorem 2.2 Let be a complete metric space and be a transitive mapping. Suppose that is a generalized α-ψ-contractive mapping of integral type I and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k;
-
(iv)
ψ is continuous for all .
Then T has a fixed point, that is, there exists such that .
Proof From the proof of Theorem 2.1, we infer that the sequence defined by for all converges to . We obtain, from hypothesis (iii) and (3), that there exists a subsequence of such that for all k. Now, applying inequality (3), we get, for all k,
On the other hand, we have
Recall from the proof of Theorem 2.1 that the sequence converges to . Consequently, as , the limit of the terms , , tends to 0. Thus, by letting in (22), we get that
Assume that . In view of (23) and for k large enough, we get , which implies from (21) that
Letting in (24) and by using (23), assumption (iv), together with the property of , we derive that
which is a contradiction. Thus, we have , that is, . □
One can easily deduce the following result from Theorem 2.1.
Theorem 2.3 Let be a complete metric space and be a transitive mapping. Suppose that is a generalized α-ψ-contractive mapping of integral type II and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
In the next theorem, we exclude the continuity hypothesis of T in Theorem 2.3.
Theorem 2.4 Let be a complete metric space and be a transitive mapping. Suppose that is a generalized α-ψ-contractive mapping of integral type II and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then T has a fixed point, that is, there exists such that .
Proof From the proof of Theorem 2.3, we infer that the sequence defined by for all converges to . We obtain, from hypothesis (iii) and (3), that there exists a subsequence of such that for all k. Now, applying inequality (4), we get, for all k,
On the other hand, we have
Letting in the above equality, we get that
Assume that . In view of (28) and for k large enough, we get , which implies from (26) that
Letting in (29) and using (28), we obtain that
which is a contradiction. Thus, we have , that is, . □
Remark 2.2 Notice that in Theorem 2.2, the continuity of ψ is assumed as an extra condition. Despite Remark 2.1, Theorem 2.4 can be derived from Theorem 2.2 due to the additional assumption on ψ.
In order to ensure the uniqueness of a fixed point of a generalized α-ψ-contractive mapping of integral type II, we need an additional condition (U) defined in the previous section.
Theorem 2.5 If the condition (U) is added to the hypotheses of Theorem 2.1, then the fixed point u of T is unique.
Proof We shall show the uniqueness of a fixed point of T by reductio ad absurdum. Suppose, on the contrary, that v is another fixed point of T with . From the hypothesis (U), we obtain that there exists such that
Using the α-admissible property of T, we get from (31) for all
Consider the sequence in X by for all and . From (32), for all n, we infer that
On the other hand, we have
Due to the monotone property of ψ and using the above inequality, we infer from (33) that
for all n. Let us examine the possibilities for the inequality above. For simplicity, let
If , then due to the properties of the function ψ, we get
which is a contradiction. If , then
thereby implying that
for all . Letting in the above inequality, we obtain that
which from (1) implies that
Let us analyze the last case: . Regarding the properties of ϕ and the triangle inequality, we have
Notice that if , then, as in the analysis of the first case, we get a contradiction. Hence,
and hence we easily deduce that
for each n. Consequently, we find that
for all . Letting in the above inequality, we obtain that
which from (1) implies that
Similarly, we can show that
From equations (41) and (42), we obtain that . Therefore, we have proved that u is the unique fixed point of T. □
The following result can be easily deduced from Theorem 2.5 due to Remark 2.1.
Theorem 2.6 Adding the condition (U) to the hypotheses of Theorem 2.3 (resp. Theorem 2.4), one obtains that u is the unique fixed point of T.
3 Consequences
In this section, we shall list some existing results in the literature that can be deduced easily from our Theorem 2.6.
3.1 Standard fixed point theorems
Theorem 1.1 and Theorem 1.2 are immediate consequences of our main results Theorem 2.1 and Theorem 2.3 where .
Corollary 3.1 (see Karapınar and Samet [7])
Let be a complete metric space and be a transitive mapping. Suppose that is a generalized α-ψ-contractive mapping and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Proof It is sufficient to take for all in Theorem 2.3. □
If one replaces for all in Theorem 1.1, the following fixed point theorem is observed.
Corollary 3.2 (see Samet et al. [1])
Let be a complete metric space and be an α-ψ-contractive mapping satisfying the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
If we take = 1 for all and for in Theorem 1.1, we derive the following result.
Corollary 3.3 (see Branciari [10])
Let be a complete metric space, , and let be a mapping such that for each ,
where . Then T has a unique fixed point such that for each , .
The following corollary is concluded from Corollary 3.1 by taking for all .
Corollary 3.4 (see Karapınar and Samet [7])
Let be a complete metric space and be a given mapping. Suppose that there exists a function such that
for all . Then T has a unique fixed point.
By taking for in Corollary 3.4, we get the next result.
Corollary 3.5 (see Ćirić [15])
Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
for all . Then T has a unique fixed point.
Corollary 3.6 (see Hardy and Rogers [16])
Let be a complete metric space and be a given mapping. Suppose that there exist constants with such that
for all . Then T has a unique fixed point.
For the proof of the above corollary, it is sufficient to chose in Corollary 3.5.
The next two results are obvious consequences of Corollary 3.5.
Corollary 3.7 (see Kannan [17])
Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
for all . Then T has a unique fixed point.
Corollary 3.8 (see Chatterjea [18])
Let be a complete metric space and be a given mapping. Suppose that there exists a constant such that
for all . Then T has a unique fixed point.
By taking in Corollary 3.3, we obtain the following corollary.
Corollary 3.9 (Rhoades and Abbas [19])
Let T be a self-map of a complete metric space satisfying
for all and , where . Then T has a unique fixed point .
Corollary 3.10 (Berinde [20])
Let be a complete metric space and be a given mapping. Suppose that there exists a function such that
for all . Then T has a unique fixed point.
Proof Let for all and for all in Theorem 1.1. Then all the conditions of Theorem 1.1 are satisfied and the proof is completed. □
It is evident that we have the celebrated result of Banach.
Corollary 3.11 (Banach [2])
Let be a complete metric space and be a given mapping satisfying
where . Then T has a unique fixed point.
3.2 Fixed point theorems on ordered metric spaces
Recently, there have been so many interesting developments in the field of existence of a fixed point in partially ordered sets. This idea was initiated by Ran and Reurings [21] where they extended the Banach contraction principle in partially ordered sets with some application to a matrix equation. Later, many remarkable results have been obtained in this direction (see, for example, [22–29] and the references cited therein). In this section, we will establish various fixed point results on a metric space endowed with a partial order. For this, we require the following concepts.
Definition 3.1 Let be a partially ordered set and be a given mapping. We say that T is nondecreasing with respect to ⪯ if
Definition 3.2 Let be a partially ordered set. A sequence is said to be nondecreasing with respect to ⪯ if for all n.
Definition 3.3 [7]
Let be a partially ordered set and d be a metric on X. We say that is regular if for every nondecreasing sequence such that as , there exists a subsequence of such that for all k.
Now, we have the following result.
Corollary 3.12 Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exist functions and such that for all with , we have
where . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Proof Consider the mapping by
Clearly, α is transitive. In view of the definition of α, we infer that T is an α-ψ-contractive mapping of integral type, that is,
for all . From condition (i), we have . Now, we proceed to show that T is α-admissible. For this, let for all . Moreover, owing to the monotone property of T, we have, for all ,
Thus, T is α-admissible. Now, if T is continuous, we obtain the existence of a fixed point from Theorem 2.3. Now, assume that is regular. Suppose that is a sequence in X such that for all n and as . Due to the fact that the space is regular, there exists a subsequence of such that for all k. Owing to the definition of α, we get that for all k. In this case, we get the existence of a fixed point from Theorem 2.4. Now, we have to show the uniqueness of the fixed point. For this, let . By hypothesis, there exists such that and , which implies from the definition of α that and . Therefore, we obtain the uniqueness of the fixed point from Theorem 2.6. □
We can now easily derive the following results from Corollary 3.12.
Corollary 3.13 (Shahi et al. [9])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function such that for all with , we have
where . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Corollary 3.14 (Karapınar and Samet [7])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Proof By taking for all in Corollary 3.12, we get the proof of this corollary. □
Corollary 3.15 (Karapınar and Samet [7])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Proof By taking for all in Corollary 3.13, we get the proof of this corollary. □
Corollary 3.16 (Shahi et al. [9])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function such that for all with , we have
where . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Proof By taking for all and some in Corollary 3.13, we get the proof of this corollary. □
Corollary 3.17 (Ran and Reurings [21], Nieto and Rodriguez-Lopez [29])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Proof Taking for all in Corollary 3.16, we get the proof of this corollary. □
Corollary 3.18 (see Karapınar and Samet [7])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Corollary 3.19 (see Karapınar and Samet [7])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exist constants with such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Corollary 3.20 (see Karapınar and Samet [7])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
Corollary 3.21 (see Karapınar and Samet [7])
Let be a partially ordered set and d be a metric on X such that is complete. Let be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant such that
for all with . Suppose also that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point. Moreover, if for all there exists such that and , we have uniqueness of the fixed point.
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The authors are grateful to the reviewers for their careful reviews and useful comments. The first author was supported by the Research Center, College of Science, King Saud University.
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Karapınar, E., Shahi, P. & Tas, K. Generalized α-ψ-contractive type mappings of integral type and related fixed point theorems. J Inequal Appl 2014, 160 (2014). https://doi.org/10.1186/1029-242X-2014-160
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DOI: https://doi.org/10.1186/1029-242X-2014-160