Abstract
The aim of this work is to modify the notions of α-admissible and α-ψ-contractive mappings and establish new fixed point theorems for such mappings in complete metric spaces. Presented theorems provide main results of Karapinar and Samet (Abstr. Appl. Anal. 2012:793486, 2012) and Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) as direct corollaries. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.
MSC:46N40, 47H10, 54H25, 46T99.
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1 Introduction and preliminaries
Metric fixed point theory has many applications in functional analysis. The contractive conditions on underlying functions play an important role for finding solutions of metric fixed point problems. The 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 [1–25]). In 2012, Samet et al. [24] 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 [19] generalized these notions to obtain fixed point results. The aim of this paper is to modify further the notions of α-ψ-contractive and α-admissible mappings and establish fixed point theorems for such mappings in complete metric spaces. Our results are proper generalizations of the recent results in [19, 24]. Moreover, some examples and applications to integral equations 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 .
Definition 1.1 [24]
Let T be a self-mapping on a metric space and let be a function. We say that T is an α-admissible mapping if
Definition 1.2 [24]
Let T be a self-mapping on a metric space . We say that T is an α-ψ-contractive mapping if there exist two functions and such that
for all .
For the examples of α-admissible and α-ψ-contractive mappings, see [19, 24] and the examples in the next section.
2 Main results
We first modify the concept of α-admissible mapping.
Definition 2.1 Let T be a self-mapping on a metric space 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 an η-subadmissible mapping.
Our first result is the following.
Theorem 2.1 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.
Proof Let be such that . Define a sequence in X by for all . If for some , then is a fixed point for T and the result is proved. Hence, we suppose that for all . Since T is a generalized α-admissible mapping with respect to η and , we deduce that . Continuing this process, we get for all . Now, by (2.1) with , , we get
On the other hand,
which implies
Now, if for some , then
which is a contradiction. Hence, for all , we have
By induction, we have
Fix , there exists such that
Let with . Then, by the triangular inequality, we get
Consequently, . Hence is a Cauchy sequence. Since X is complete, there is such that as . Now, if we suppose that T is continuous, then we have
So, z is a fixed point of T. On the other hand, since
for all and as , we get
for all . Then from (2.1) we have
where
Since , then
By taking limit as in the above inequality, we have
which implies , i.e., . □
By taking in Theorem 2.1, we have the following result.
Corollary 2.1 Let be a complete metric space and let T be an α-admissible mapping. Assume that for ,
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.
By taking in Theorem 2.1, we have the following corollary.
Corollary 2.2 Let be a complete metric space and let T be an η-subadmissible mapping. Assume that for ,
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.
Clearly, Corollary 2.1 implies the following results.
Corollary 2.3 (Theorem 2.1 and Theorem 2.2 of [24])
Let be a complete metric space and let T be an α-admissible mapping. Assume that for ,
holds for all . 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.
Corollary 2.4 (Theorem 2.3 and Theorem 2.4 of [19])
Let be a complete metric space and let T be an α-admissible mapping. Assume that for ,
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.
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 2.1 can be applied to T. But Theorem 2.2 of [24] and Theorem 2.4 of [19] cannot be applied to T.
Clearly, is a complete metric space. We show that T is an α-admissible mapping. Let , if , then . On the other hand, for all we have . It follows that . Hence, the assertion holds. In reason of the above arguments, .
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,
All of the conditions of Corollary 2.1 hold. Hence, T has a fixed point. Let and , then
That is, Theorem 2.2 of [24] cannot be applied to T.
Also, by a similar method, we can show that Theorem 2.4 of [19] cannot be applied to T.
By the following simple example, we show that our results improve the results of Samet et al. [24] and the results of Karapinar and Samet [19].
Example 2.2 Let be endowed with the usual metric for all and let be defined by . Also, define by and by .
Clearly, T is an α-admissible mapping. Also, for all . Hence,
Then the conditions of Corollary 2.1 hold and T has a fixed point. But if we choose and , then
That is, Theorem 2.2 of [24] cannot be applied to T. Similarly, we can show that Theorem 2.4 of [19] cannot be applied to T. Further notice that the Banach contraction principle holds for this example.
Example 2.3 Let be endowed with the usual metric for all and let be defined by
Define also and by
We prove that Corollary 2.2 can be applied to T. But the Banach contraction principle cannot be applied to T.
Clearly, is a complete metric space. We show that T is an η-subadmissible mapping. Let , if , then . On the other hand, for all , we have . It follows that . 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 2.2 hold. Hence, T has a fixed point. Let , and . Then
That is, the Banach contraction principle cannot be applied to T.
From our results, we can deduce the following corollaries.
Corollary 2.5 Let be a complete metric space and let T be an α-admissible mapping. Assume that
holds for all , 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.
Proof Let . Then by (2.2) we have
Then . Hence, the conditions of Corollary 2.1 hold and f has a fixed point. □
Similarly, we have the following corollary.
Corollary 2.6 Let be a complete metric space and let T be an α-admissible mapping. Assume that
hold for all , 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.
Notice that the main theorem of Dutta and Choudhury [9] remains true if ϕ is lower semi-continuous instead of continuous (see, e.g., [1, 8]).
We assume that
and
where if and only if .
Theorem 2.2 Let be a complete metric space and let T be an α-admissible mapping with respect to η. Assume that for 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.
Proof Let such that . Define a sequence in X by for all . If for some , then is a fixed point for T and the result is proved. We suppose that for all . Since T is an α-admissible mapping with respect to η and , we deduce that . By continuing this process, we get for all . Clearly,
Now, by (2.4) with , , we have
which implies
Since ψ is increasing, we get
for all . That is, is a non-increasing sequence of positive real numbers. Then there exists such that . We shall show that . By taking the limit infimum as in (2.5), we have
Hence . That is, . Then
Suppose, to the contrary, that is not a Cauchy sequence. Then there is and sequences and such that for all positive integers k,
Now, for all , we have
Taking limit as in the above inequality and using (2.6), we get
Since
and
then by taking the limit as in the above inequality, and by using (2.6) and (2.7), we deduce that
On the other hand,
Then, by (2.4) with and , we get
By taking limit as in the above inequality and applying (2.7) and (2.8), we obtain
That is, , which is a contradiction. Hence is a Cauchy sequence. Since X is complete, then there is such that . First we assume that T is continuous. Then we deduce
So, z is a fixed point of T. On the other hand, since
for all and as , so
which implies
Now, by (2.4) we get
Passing limit inf as in the above inequality, we have
That is, . □
By taking in Theorem 2.2, we deduce the following corollary.
Corollary 2.7 Let be a complete metric space and let T be an α-admissible mapping. Assume that for 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 .
Then T has a fixed point.
By taking in Theorem 2.2, we deduce the following corollary.
Corollary 2.8 Let be a complete metric space and let T be an η-subadmissible mapping. Assume that for 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 .
Then T has a fixed point.
Example 2.4 Let be endowed with the usual metric
for all , and let be defined by
Define also and by
We prove that Corollary 2.7 can be applied to T, but the main theorem in [9] cannot be applied to T.
By a similar proof to that of Example 2.1, we show that T is an α-admissible mapping. Assume that . Now, if , then and so , which is contradiction. If . Similarly, , which is contradiction. Hence, implies . Therefore, we get
That is,
The conditions of Corollary 2.7 are satisfied. Hence, T has a fixed point. Let and , then
That is, the main theorem in [9] cannot be applied to T.
Example 2.5 Let be endowed with the usual metric for all , and let be defined by
Define also and by
We prove that Corollary 2.8 can be applied to T, but the main theorem in [9] cannot be applied to T.
By a similar proof to that of Example 2.3, we can show that T is an η-subadmissible mapping.
Assume that . Now, if , then , which is a contradiction. Similarly, is a contradiction. Hence, implies . We get
That is,
Then the conditions of Corollary 2.8 hold and T has a fixed point. Let , . Then and , which implies
That is, the main theorem in [9] cannot be applied to T.
In 1984 Khan et al. [20] proved the following theorem.
Theorem 2.3 Let be a complete metric space and let T be a self-mapping on X. Assume that
where and . Then T has a unique fixed point.
Theorem 2.4 Let be a complete metric space and let T be a generalized 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 .
Then T 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 T and the result is proved. Hence, we suppose that for all . Since T is a generalized α-admissible mapping with respect to η and , we deduce that . By continuing this process, we get for all . Clearly,
Now, by (2.9) with , , we have
Since, ψ is increasing, we get
for all . That is, is a non-increasing sequence of positive real numbers. Then there exists such that . We shall show that . By taking the limit as in (2.10), we have
which implies , i.e., . Then
Suppose, to the contrary, that is not a Cauchy sequence. Proceeding as in the proof of Theorem 2.2, there exists such that for all there exist with such that
and
Clearly,
Then, by (2.9) with and , we get
Taking limit as in the above inequality and applying (2.12) and (2.13), we get
and so , which is a contradiction. Hence is a Cauchy sequence. Since X is complete, then there is such that . First, we assume that T is continuous. Then, we deduce
So, z is a fixed point of T. On the other hand, since
for all and as , we get
which implies
Then by (2.12) we deduce
Taking limit as in the above inequality, we have
and then . □
Corollary 2.9 Let be a complete metric space and let T be an α-admissible mapping. 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 .
Then T has a fixed point.
Corollary 2.10 Let be a complete metric space and let T be a generalized 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 .
Then T has a fixed point.
Example 2.6 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 T. But Theorem 2.3 cannot be applied to T.
By a similar proof to that of Example 2.1, we show that T is an α-admissible mapping. Assume that . Now, if , then and so , which is contradiction. If . Similarly, , which is contradiction. Hence, implies . Therefore, we get
That is,
Then the conditions of Corollary 2.9 hold. Hence, T has a fixed point. Let and , then and , and hence
That is, Theorem 2.3 cannot be applied to T.
3 Application to the existence of solutions of integral equations
Integral equations like (3.1) were studied in many papers (see [2, 11] and references therein). In this section, we look for a nonnegative solution to (3.1) in . Let be the set of real continuous functions defined on and let be defined by
for all . Then is a complete metric space.
Consider the integral equation
and let defined by
We assume that
-
(A)
is continuous;
-
(B)
is continuous;
-
(C)
is continuous;
-
(D)
there exist and such that if for , then for every we have
-
(F)
there exists such that ;
-
(G)
if , , then ;
-
(H)
if is a sequence in X such that for all and as , then for all ;
-
(J)
for all and .
Theorem 3.1 Under assumptions (A)-(J), the integral equation (3.1) has a solution in .
Proof Consider the mapping defined by (3.2).
By the condition (D), we deduce
Then
Now, define by
That is, implies
All of the hypotheses of Corollary 2.1 are satisfied, and hence the mapping F has a fixed point that is a solution in of the integral equation (3.1). □
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Acknowledgements
The authors thank the referees for their valuable comments and suggestions. This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second and third authors acknowledge with thanks DSR, KAU for financial support. The first author is thankful for support of Astara Branch, Islamic Azad University, during this research.
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Salimi, P., Latif, A. & Hussain, N. Modified α-ψ-contractive mappings with applications. Fixed Point Theory Appl 2013, 151 (2013). https://doi.org/10.1186/1687-1812-2013-151
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DOI: https://doi.org/10.1186/1687-1812-2013-151