Abstract
Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) introduced α-ψ-contractive mappings and proved some fixed point results for these mappings. More recently Salimi et al. (Fixed Point Theory Appl. 2013:151, 2013) modified the notion of α-ψ-contractive mappings and established certain fixed point theorems. Here, we continue to utilize these modified notions for single-valued Geraghty and Meir-Keeler-type contractions, as well as multi-valued contractive mappings. Presented theorems provide main results of Hussain et al. (J. Inequal. Appl. 2013:114, 2013), Karapinar et al. (Fixed Point Theory Appl. 2013:34, 2013) and Asl et al. (Fixed Point Theory Appl. 2012:212, 2012) as corollaries. Moreover, some examples are given here to illustrate the usability of the obtained results.
MSC:46N40, 47H10, 54H25, 46T99.
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Dedication
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
1 Introduction and preliminaries
In metric fixed point theory, the contractive conditions on underlying functions play an important role for finding solution of fixed point problems. Banach contraction principle is a remarkable result in metric fixed point theory. Over the years, it has been generalized in different directions by several mathematicians (see [1–21]). In 2012, Samet et al. [15] introduced the concepts of α-ψ-contractive and α-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Afterwards, Karapinar and Samet [13] generalized these notions to obtain fixed point results. More recently, Salimi et al. [14] 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 [13, 15]. Here, we continue to utilize these modified notions for single-valued Geraghty and Meir-Keeler-type contractions, as well as multivalued contractive mappings. Presented theorems provide main results of Hussain et al. [9], Karapinar et al. [11] and Asl et al. [12] as corollaries. Moreover, some examples are given here to illustrate the usability of the obtained results.
Denote with Ψ the family of nondecreasing functions such that for all , where is the n th iterate of ψ.
The following lemma is obvious.
Lemma 1.1 If , then for all .
Samet et al. [15] defined the notion of α-admissible mappings as follows.
Definition 1.1 Let T be a self-mapping on X, and let be a function. We say that T is an α-admissible mapping if
Theorem 1.1 [15]
Let be a complete metric space, and let T be an α-admissible mapping. Assume that
for all , where . Also, suppose that
-
(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.
Very recently Salimi et al. [14] modified the notions of α-admissible and α-ψ-contractive mappings as follows.
Definition 1.2 [14]
Let T be a self-mapping on X, and let 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 1.1. Also, if we take , then we say that T is η-subadmissible mapping.
The following result properly contains Theorem 1.1 and Theorems 2.3 and 2.4 of [13].
Theorem 1.2 [14]
Let be a complete metric space, and let 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.
2 Modified α-η-Geraghty type contractions
Our first main result of this section is concerning α-η-Geraghty-type [4] contractions.
Theorem 2.1 Let be a complete metric space, and let be an α-admissible mapping with respect to η. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Proof Let such that . Define a sequence in X by for all . If for some , then is a fixed point for f, and the result is proved. Hence, we suppose that for all . Since f is an α-admissible mapping with respect to η and , we deduce that . By continuing this process, we get for all . Then,
Now from (2.1), we have
Now, if for some , then
Also, if for some , then
That is, for all , we have
Hence,
for all , which implies that . It follows that the sequence is decreasing. Thus, there exists such that . We shall prove that . From (2.2), we have
which implies that . Regarding the property of the function β, we conclude that
Next, we shall prove that is a Cauchy sequence. Suppose, to the contrary, that is not a Cauchy sequence. Then there is and sequences and such that for all positive integers k, we have
By the triangle inequality, we derive that
. Taking the limit as in the inequality above, and regarding the limit in (2.3), we get
Again, by the triangle inequality, we find that
and
Taking the limit in inequality above as , together with (2.3) and (2.4), we deduce that
Now, since
then from (2.1), (2.4) and (2.5), we have
Hence,
Letting in the inequality above, we get
That is, , which is a contradiction. Hence is a Cauchy sequence. Since X is complete, then there is such that . First, we suppose that f is continuous. Since f is continuous, then we have
So z is a fixed point of f. Next, we suppose that (b) holds. Then, , and so, . Now by (2.1), we have
and hence
Letting in the inequality above, we get , that is, . □
If in Theorem 2.1 we take, , then we have the following corollary.
Corollary 2.1 Let be a complete metric space, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.2 Let be a complete metric space, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all , where . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.3 Let be a complete metric space, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.4 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 that and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Further, if in Theorem 2.1 we take , then we have the following corollary.
Corollary 2.5 Let be a complete metric space, and let be a η-subadmissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.6 Let be a complete metric space, and let be a η-subadmissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all , where . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.7 Let be a complete metric space, and let be a η-subadmissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.8 Let be a metric space such that is complete, and let be η-subadmissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
From Corollary 2.1, we can deduce the following corollary.
Corollary 2.9 Let be a complete metric space, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Also, from the corollary above, we can deduce the following corollaries.
Corollary 2.10 (Theorem 4 of [9])
Let be a complete metric space, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all , where . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.11 (Theorem 6 of [9])
Let be a complete metric space, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Corollary 2.12 (Theorem 8 of [9])
Let be a metric space such that is complete, and let be an α-admissible mapping. Assume that there exists a function such that for any bounded sequence of positive reals, implies that and
for all . Suppose that either
-
(a)
f is continuous, or
-
(b)
if is a sequence in X such that , for all n, then .
If there exists such that , then f has a fixed point.
Example 2.1 Let be endowed with the usual metric for all , and let be defined by
Define also and by
We prove that Corollary 2.9 can be applied to f, but Corollaries 2.10, 2.11 and 2.12 (Theorem 4, 6 and 8 of [9]) cannot be applied to f.
Clearly, is a complete metric space. We show that f is an α-admissible mapping. Let with , then . On the other hand, for all , we have . It follows that . Hence, the assertion holds. Also, . Now, if is a sequence in X such that for all and as , then , and hence . This implies that for all .
Let . Then . We get,
That is,
then the conditions of Corollary 2.1 hold, and f has a fixed point.
Let , , and let , then
That is, Corollary 2.10 (Theorem 4 of [9]) cannot be applied for this example.
Let, , and let , then
That is, Corollary 2.11 (Theorem 6 of [9]) cannot be applied for this example.
Let, , and let , then
That is, Corollary 2.12 (Theorem 8 of [9]) cannot be applied for this example.
3 Modified α-ψ-Meir-Keeler contractive mappings
Recently, Karapinar et al. [11] introduced the notion of a triangular α-admissible mapping as follows.
Definition 3.1 [11]
Let , and let . We say that f is a triangular α-admissible mapping if
-
(T1)
implies that , ;
-
(T2)
implies that .
Lemma 3.1 [11]
Let f be a triangular α-admissible mapping. Assume that there exists such that . Define sequence by . Then
Denote with Ψ the family of nondecreasing functions continuous at such that
-
if and only if ,
-
.
Definition 3.2 [11]
Let be a metric space, and let . Suppose that is a triangular α-admissible mapping satisfying the following condition:
for each there exists such that
for all . Then f is called an α-ψ-Meir-Keeler contractive mapping.
Now, we modify Definition 3.2 as follows.
Definition 3.3 Let be a metric space, and let . Suppose that is a triangular α-admissible mapping satisfying the following condition:
for each there exists such that
for all with . Then f is called a modified α-ψ-Meir-Keeler contractive mapping.
Remark 3.1 Let f be a modified α-ψ-Meir-Keeler contractive mapping. Then
for all when and . Also, if and , then , i.e.,
Theorem 3.1 Let be a complete metric space. Suppose that f is a continuous modified α-ψ-Meir-Keeler contractive mapping, and that there exists such that , then f has a fixed point.
Proof Let and define a sequence by for all . If for some , then, obviously, f has a fixed point. Hence, we suppose that
for all . We have for all . Now, define . By Remark 3.1, we deduce that for all . By applying Lemma 3.1 for
we have
Hence, the sequence is decreasing in , and so, it is convergent to . We will show that . Suppose, to the contrary, that . Note that
Let . Then by hypothesis, there exists a such that (3.2) holds. On the other hand, by the definition of ε, there exists such that
Now by (3.2), we have
which is a contradiction. Hence , that is, . Now, by the continuity of ψ at , we have . For given , by the hypothesis, there exists a such that (3.2) holds. Without loss of generality, we assume that . Since , then there exists such that
We will prove that for any fixed ,
holds. Note that (3.6) holds for by (3.5). Suppose that condition (3.2) is satisfied for some . For , by (3.5), we get
If , then by (3.2), we get
and hence (3.6) holds.
If , by Remark 3.1, we get
Consequently, (3.6) holds for . Hence, for all and , which means
Hence is a Cauchy sequence. Since is complete, there exists such that as . Now, since f is continuous, then
that is, f has a fixed point. □
Corollary 3.1 (Theorem 10 of [11])
Let be a complete metric space. Suppose that f is a continuous α-ψ-Meir-Keeler contractive mapping, and that there exists such that , then f has a fixed point.
Proof Let , where . Then by and Definition 3.2, we deduce that . On the other hand, since , then we have
That is, conditions of Theorem 3.1 hold, and f has a fixed point. □
Theorem 3.2 Let be a complete metric space, and let f be a modified α-ψ-Meir-Keeler contractive mapping. If the following conditions hold:
-
(i)
there exists such that ,
-
(ii)
if is a sequence in X such that for all n, and as , then for all n.
Then f has a fixed point.
Proof Following the proof of Theorem 3.1, we say that for all , and that there exist such that as . Hence, from (ii) . By Remark 3.1, we have
By taking limit as , in the inequality above, we get , that is, . Hence . □
Corollary 3.2 (Theorem 11 of [11])
Let be a complete metric space, and let f be a α-ψ-Meir-Keeler contractive mapping. If the following conditions hold:
-
(i)
there exists such that ,
-
(ii)
if is a sequence in X such that for all n, and as , then for all n.
Then f has a fixed point.
Example 3.1 Let , and let be a metric on X. Define by
and . Clearly, is a complete metric space. We show that f is a triangular α-admissible mapping. Let , if , then . On the other hand, for all , we have and . It follows that . Also, if and , then , and hence, . Thus the assertion holds by the same arguments. Notice that .
Now, if is a sequence in X such that for all , and as , then , and hence . This implies that for all . Let , then . Without loss of generality, take . Then
Clearly, by taking , the condition (3.2) holds. Hence, conditions of Theorem 3.2 hold, and f has a fixed point. But if and
where and . Then
That is, Corollary 3.2 (Theorem 11 of [11]) cannot be applied for this example.
Denote with the family of strictly nondecreasing functions continuous at such that
-
if and only if ,
-
.
Definition 3.4 [11]
Let be a metric space, and let . Suppose that is a triangular α-admissible mapping satisfying the following condition:
for each , there exists such that
for all , where
Then f is called a generalized α--Meir-Keeler contractive mapping.
Definition 3.5 Let be a metric space, and let . Suppose that is a triangular α-admissible mapping satisfying the following condition:
for each there exists such that
for all , where and
Then f is called a modified generalized α--Meir-Keeler contractive mapping.
Remark 3.2 Let f be a modified generalized α--Meir-Keeler contractive mapping. Then
for all , where when . Also, if and , then , which implies that , i.e.,
Proposition 3.1 Let be a metric space, and let be a modified generalized α--Meir-Keeler contractive mapping. If there exists such that , then .
Proof Define a sequence by for all . If for some , then, obviously, the conclusion holds. Hence, we suppose that
for all . Then we have for every . Then by Lemma 3.1 and Remark 3.2, we have
Now, since is strictly nondecreasing, then we get
Hence the case, where
is not possible. Therefore, we deduce that
for all n. That is, is a decreasing sequence in , and it converges to , that is,
Notice that . Let us prove that . Suppose, to the contrary, that . Then . Considering (3.13) together with the assumption that f is a generalized α--Meir-Keeler contractive mapping, for , there exists and a natural number m such that
implies that
Now, since is strictly nondecreasing, then we get
which is a contradiction, since . Then , and so,
□
Theorem 3.3 Let be a complete metric space, and let be an orbitally continuous modified generalized α--Meir-Keeler contractive mapping. If there exist such that , then f has a fixed point.
Proof Define for all . We want to prove that . If this is not so, then there exist and a subsequence of such that
For this , there exists such that implies that . Put and for all . From Proposition 3.1, there exists such that
for all . Let . We get . If , then
which contradicts the assumption (3.14). Therefore, there are values of k such that and . Now if , then
which is a contradiction to (3.15). Hence, there are values of k with such that . Choose the smallest integer k with such that . Thus, , and so,
Now, we can choose a natural number k satisfying such that
Therefore, we obtain
and
Thus, we have
Now, inequalities (3.17)-(3.20) imply that , and so, ; the fact that f is a modified generalized α--Meir-Keeler contractive mapping yields that
Then . We deduce
From (3.16), (3.18) and (3.19), we obtain
which is a contradiction. We obtained that , and so, is a Cauchy sequence. Since X is complete, then there exists such that as . As f is orbitally continuous, so . □
Corollary 3.3 (Theorem 17 of [11])
Let be a complete metric space, and let be an orbitally continuous generalized α--Meir-Keeler contractive mapping. If there exist such that , then f has a fixed point.
Example 3.2 Let , and let be a metric on X. Define by
and ,
Clearly, f is a triangular α-admissible mapping, and it is orbitally continuous. Let , then . Without loss of generality, take . Then
Clearly, by taking , the condition (3.10) holds. Hence, all conditions of Theorem 3.3 are satisfied, and f has a fixed point. But if and
for and , then
and so,
That is, Corollary 3.3 (Theorem 17 of [11]) cannot be applied for this example.
4 Modified α-η-contractive multifunction
Recently, Asl et al. [12] introduced the following notion.
Definition 4.1 Let , and let . We say that T is an -admissible mapping if
where
We generalize this concept as follows.
Definition 4.2 Let be a multifunction, and let be two functions, where η is bounded. We say that T is an -admissible mapping with respect to η if
where
If we take for all , then this definition reduces to Definition 4.1. In case for all , then T is called an -subadmissible mapping.
Notice that Ψ is the family of nondecreasing functions such that for all , where is the n th iterate of ψ.
As an application of our new concept, we develop now a fixed point result for a multifunction, which generalizes Theorem 1.1.
Theorem 4.1 Let be a complete metric space, and let be an -admissible, with respect to η, and closed-valued multifunction on X. Assume that for ,
Also, suppose that the following assertions hold:
-
(i)
there exist and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Proof Let be such that . Since T is an -admissible mapping, then . Therefore, from (4.1), we have
If , then is a fixed point of T. Hence, we assume that . Also, if , then is a fixed point of T. Assume that and . Then we have
and so, by (4.2), we get
This implies that there exists such that
Note that (since ). Also, since , and , then . So . Therefore, from (4.1), we have
Put . Then from (4.3), we have , where . Now, since ψ is strictly increasing, then . Put
and so . If , then is a fixed point of T. Hence, we suppose that . Then
So there exists such that
and then from (4.4), we get
Again, since ψ is strictly increasing, then . Put
So, . If , then is a fixed point of T. Hence, we assume that . Then
and so, there exists such that
Clearly, . Also again, since , and , then , and so, . Then from (4.1), we have
and so, from (4.5), we deduce that
By continuing this process, we obtain a sequence in X such that , , and for all . Now, for all , we can write
Therefore, is a Cauchy sequence. Since is a complete metric space, then there exists such that as . Now, since for all , then , and so, from (4.1), we have
for all . Taking limit as in the inequality above, we get , i.e., . □
If in Theorem 4.1 we take , we have the following corollary.
Corollary 4.1 Let be a complete metric space, and let be an -admissible and closed-valued multifunction on X. Assume that
Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
If in Theorem 4.1 we take , then we have the following result.
Corollary 4.2 Let be a complete metric space, and let be an -subadmissible and closed-valued multifunction on X. Assume that
Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Corollary 4.3 (Theorem 2.1 and 2.3 of [12])
Let be a complete metric space, and let be an -admissible and closed-valued multifunction on X. Assume that
for all . Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Proof Suppose that for . Then by (4.6), we have
That is, conditions of Corollary 4.1 hold, and T has a fixed point. □
Similarly, we can deduce the following corollaries.
Corollary 4.4 Let be a complete metric space, and let be an -admissible and closed-valued multifunction on X. Assume that
for all . Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Corollary 4.5 Let be a complete metric space, and let be an -admissible and closed-valued multifunction on X. Assume that
for all , where . Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Corollary 4.6 Let be a complete metric space, and let be an -subadmissible and closed-valued multifunction on X. Assume that
for all . Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Corollary 4.7 Let be a complete metric space, and let be an -subadmissible and closed-valued multifunction on X. Assume that
for all . Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
Corollary 4.8 Let be a complete metric space, and let be an -admissible and closed-valued multifunction on X. Assume that
for all , where . Also, suppose that the following assertions hold:
-
(i)
there exists and such that ;
-
(ii)
for a sequence converging to and for all , we have for all .
Then T has a fixed point.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and third authors acknowledge with thanks DSR, KAU for the financial support.
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Hussain, N., Salimi, P. & Latif, A. Fixed point results for single and set-valued α-η-ψ-contractive mappings. Fixed Point Theory Appl 2013, 212 (2013). https://doi.org/10.1186/1687-1812-2013-212
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DOI: https://doi.org/10.1186/1687-1812-2013-212