Abstract
The aim of this paper is to establish certain new fixed point results for multi-valued as well as single-valued maps satisfying an α-ψ-contractive conditions in complete metric space. As an application, we derive some new fixed point theorems for ψ-graphic contractions defined on a metric space endowed with a graph as well as an ordered metric space. The presented results complement and extend some very recent results proved by Asl et al. (Fixed Point Theory Appl. 2012:212, 2012) and Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) as well as other theorems given by Hussain et al. (Fixed Point Theory Appl. 2013:212, 2013). Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature.
MSC:46S40, 47H10, 54H25.
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1 Introduction
In metric fixed point theory the contractive conditions on underlying functions play an important role for finding solutions of fixed point problems. The Banach contraction principle [1] is a fundamental result in metric fixed point theory. Over the years, it has been generalized in different directions by several mathematicians (see [1–25]). In particular, there has been a number of studies involving altering distance functions which alter the distance between two points in a metric space. In 2012, Samet et al. [25] introduced the concepts of α-ψ-contractive and α-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces.
Denote with Ψ the family of nondecreasing functions such that for all , where is the n th iterate of ψ.
The following lemma is well known.
Lemma 1 If , then the following hold:
-
(i)
converges to 0 as for all ;
-
(ii)
for all ;
-
(iii)
iff .
Samet et al. [25] defined the notion of α-admissible mappings as follows.
Definition 2 Let T be a self-mapping on X and be a function. We say that T is a α-admissible mapping if
Theorem 3 [25]
Let be a complete metric space and T be 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.
Afterwards, Asl et al. [21] generalized these notions by introducing the concepts of -ψ-contractive multifunctions, and of -admissibility, and they obtained some fixed point results for these multifunctions.
Definition 4 [21]
Let be a metric space, be a given closed-valued multifunction. We say that T is called -ψ-contractive multifunction if there exist two functions and such that
for all , where H is the Hausdorff generalized metric, and denotes the family of all nonempty subsets of X.
Definition 5 [21]
Let be a metric space, be a given closed-valued multifunction and . We say that T is called -admissible whenever implies that .
Very recently Hussain et al. [12] modified the notions of -admissible and -ψ-contractive mappings as follows:
Definition 6 Let be a multifunction, be two functions where η is bounded. We say that T is -admissible mapping with respect to η if
where
If for all , then this definition reduces to Definition 5. In the case for all , T is called -subadmissible mapping.
Hussain et al. [12] proved following generalization of the above mentioned results of [21].
Theorem 7 Let be a complete metric space and be a -admissible with respect to η and the 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.
For more details on α-ψ-contractions and fixed point theory, we refer the reader to [3, 6, 10, 13, 14, 22, 23, 26–29].
The aim of this paper is to unify the concepts of α-ψ-contractive type mappings and establish some new fixed point theorems in complete metric spaces for such mappings.
Let be a complete metric space, and . We denote by the open ball with center and radius r and by the closed ball with center and radius r.
The following lemmas of Nadler will be needed in the sequel.
Lemma 8 [19]
Let A and B be nonempty, closed and bounded subsets of a metric space and . Then, for every , there exists such that .
Lemma 9 [4]
Let be a metric space and B be nonempty, closed subsets of X and . Then, for each with and , there exists such that .
2 Main result
The following result, regarding the existence of the fixed point of the mapping satisfying an α-ψ-contractive condition on the closed ball, is very useful in the sense that it requires the contractiveness of the mapping only on the closed ball instead of the whole space.
Theorem 10 Let be a complete metric space and be an -admissible and closed-valued multifunction on X. Assume that for ,
for all and for , there exists such that
for all and . Also suppose that the following assertions hold:
-
(i)
for and ;
-
(ii)
for a sequence in converging to and for all , we have for all .
Then T has a fixed point.
Proof Since and T is -admissible, so . From (2.2), we get
It follows that
If , then
implies that
and we have finished. Assume that . By Lemmas 1 and 8, we take and as . Then
Note that , since
By repeating this process, we can construct a sequence of points in such that , , with
Now, for each with using the triangular inequality, we obtain
Thus we proved that is a Cauchy sequence. Since is closed. So there exists such that as . Now we prove that . Since for all n and T is -admissible with respect to η, so for all n. Then
Taking the limit as in (2.5), we get . Thus . □
Example 11 Let and . Define the multi-valued mapping by
Considering, and , , then and
Clearly T is an α-ψ-contractive mapping with . Now
We prove that all the conditions of our Theorem 10 are satisfied only for . Without loss of generality, we suppose that . The contractive condition of theorem is trivial for the case when . So we suppose that . Then
Put and . Then . Then T has a fixed point 0.
Now we prove that the contractive condition is not satisfied for . We suppose and , then
Similarly we can deduce the following corollaries.
Corollary 12 Let be a complete metric space and be an -admissible and closed-valued multifunction on X. Assume that for , we have
for all and for , there exists such that
for all and . Also suppose that the following assertions hold:
-
(i)
for and ;
-
(ii)
for a sequence in converging to and for all , we have for all .
Then T has a fixed point.
Corollary 13 Let be a complete metric space and be an -admissible and closed-valued multifunction on X. Assume that for , we have
for all and and for , there exists such that
for all and . Also suppose that the following assertions hold:
-
(i)
for and ;
-
(ii)
for a sequence in converging to and for all , we have for all .
Then T has a fixed point.
Theorem 14 Let be a complete metric space and be an -admissible and closed-valued multifunction on X. Assume that for , we have
for all . Also suppose that the following assertions hold:
-
(i)
there exist and with ;
-
(ii)
for a sequence in X converging to and for all , we have for all .
Then T has a fixed point.
Proof Since and T is -admissible, so . If , then we have nothing to prove. Let . If , then is a fixed point of T. Assume that , then from (2.7), we get
If , then . Since for all . Then we get a contradiction. Hence, we obtain . So
Let , then from Lemma 9 we take such that
It is clear that . Put . Then and . Since T is -admissible, so . If , then is fixed point of T. Assume that . Then from (2.7), we get
If , we get contradiction to the fact . Hence we obtain
So . Since , so by Lemma 9 we can find such that
It is clear that . Put . Then and . Since T is -admissible, so . If , then is fixed point of T. Assume that . From (2.7), we have
If . Then we get a contradiction. So . Thus
Since , so by Lemma 9 we can find such that
Continuing in this way, we can generate a sequence in X such that and , and
for all n. Now, for each , we have
This implies that is a Cauchy sequence in X. Since X is complete, there exists such that as . We now show that . Since for all n and T is -admissible, so for all n. Then
and taking the limit as , we get . Thus . □
Example 15 Let and . Define by for all and
Then . Then clearly T is -admissible. Now, for x, y and , it is easy to check that
where , for all . Put and . Then . Then T has fixed point 0.
Corollary 16 Let be a complete metric space and be an -admissible and closed-valued multifunction on X. Assume that for , we have
where
for all . Also suppose that the following assertions hold:
-
(i)
for and ;
-
(ii)
for a sequence in X converging to and for all , we have for all .
Then T has a fixed point.
Corollary 17 Let be a complete metric space and be an -admissible and closed-valued multifunction on X. Assume that for , we have
where
for all and . Also suppose that the following assertions hold:
-
(i)
for and ;
-
(ii)
for a sequence in X converging to and for all , we have for all .
Then T has a fixed point.
If T is single-valued in Theorem 14, we obtain the following fixed point results.
Theorem 18 Let be a complete metric space and be an α-admissible mapping. Assume that for , we have
for all . Also suppose that the following assertions hold:
-
(i)
there exists with ;
-
(ii)
for a sequence in X converging to and for all , we have for all .
Then T has a fixed point.
Corollary 19 Let be a complete metric space and be an α-admissible mapping. Assume that for , we have
where
for all . Also suppose that the following assertions hold:
-
(i)
for some ;
-
(ii)
for a sequence in X converging to and for all , we have for all .
Then T has a fixed point.
Now, we give the following result about a fixed point of self-maps on complete metric spaces.
Theorem 20 Let be a complete metric space, be a mapping, and T be a self-mapping on X such that
for all . Suppose that T is α-admissible and there exist and with . If T is continuous. Then T has a unique fixed point.
Proof Take such that , and define the sequence in X by for all . If for some n, then is a fixed point of T. Assume that for all n. Since T is α-admissible, so it is easy to check that for all natural numbers n. Thus for each natural number n, we have
If , then a contradiction. So we get . Since ψ is nondecreasing, so we have
for all n. It is easy to check that is a Cauchy sequence. Since X is complete, so there exists such that . Further the continuity of T implies that
Therefore is a fixed point of T in X. Now, if there exists another point in X such that , then
a contradiction. Hence is a unique fixed point of T in X. □
Example 21 Let and . Define by whenever , whenever and whenever . Also define the mappings by and
By a routine calculation one can easily show that
for all and is unique fixed point of the mapping T.
3 Fixed point results for graphic contractions
Consistent with Jachymski [15], let be a metric space and Δ denote the diagonal of the Cartesian product . Consider a directed graph G such that the set of its vertices coincides with X, and the set of its edges contains all loops, i.e., . We assume G has no parallel edges, so we can identify G with the pair . Moreover, we may treat G as a weighted graph (see [15]) by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length N () is a sequence of vertices such that , and for . A graph G is connected if there is a path between any two vertices. G is weakly connected if is connected (see for details [7, 9, 13, 15]).
Definition 22 [15]
We say that a mapping is a Banach G-contraction or simply G-contraction if T preserves edges of G, i.e.,
and T decreases weights of edges of G in the following way:
Definition 23 [15]
A mapping is called G-continuous, if given and the sequence
Theorem 24 Let be a complete metric space endowed with a graph G and T be a self-mapping on X. Suppose the following assertions hold:
-
(i)
, ;
-
(ii)
there exists such that ;
-
(iii)
there exists such that
for all where
-
(iv)
if is a sequence in X such that for all and as , then for all .
Then T has a fixed point.
Proof Define, by First we prove that T is an α-admissible mapping. Let, , then . From (i), we have . That is, . Thus T is an α-admissible mapping. From (ii) there exists such that . That is, . If , then and hence . Thus, from (iii) we have . Condition (iv) implies condition (ii) of Theorem 18. Hence, all conditions of Theorem 18 are satisfied and T has a fixed point. □
Corollary 25 Let be a complete metric space endowed with a graph G and T be a self-mapping on X. Suppose the following assertions hold:
-
(i)
T is a Banach G-contraction;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all and as , then for all .
Then T has a fixed point.
As an application of Theorem 20, we obtain;
Theorem 26 Let be a complete metric space endowed with a graph G and T be a self-mapping on X. Suppose the following assertions hold:
-
(i)
, ;
-
(ii)
there exists such that ;
-
(iii)
there exists such that
-
(iv)
T is G-continuous.
Then T has a fixed point.
Let be a partially ordered metric space. Define the graph G by
For this graph, condition (i) in Theorem 24 means T is nondecreasing with respect to this order [8]. From Theorems 24-26 we derive the following important results in partially ordered metric spaces.
Theorem 27 Let be a complete partially ordered metric space and T be a self-mapping on X. Suppose the following assertions hold:
-
(i)
T is nondecreasing map;
-
(ii)
there exists such that ;
-
(iii)
there exists such that
for all where
-
(iv)
if is a sequence in X such that for all and as , then for all .
Then T has a fixed point.
Corollary 28 [20]
Let be a complete partially ordered metric space and be nondecreasing mapping such that
for all with where . Suppose that the following assertions hold:
-
(i)
there exists such that ;
-
(ii)
if is a sequence in X such that for all and as , then for all .
Then T has a fixed point.
Theorem 29 Let be a complete partially ordered metric space and T be a self-mapping on X. Suppose the following assertions hold:
-
(i)
T is nondecreasing map;
-
(ii)
there exists such that ;
-
(iii)
there exists such that
-
(iv)
T is continuous.
Then T has a fixed point.
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.
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Hussain, N., Ahmad, J. & Azam, A. Generalized fixed point theorems for multi-valued α-ψ-contractive mappings. J Inequal Appl 2014, 348 (2014). https://doi.org/10.1186/1029-242X-2014-348
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DOI: https://doi.org/10.1186/1029-242X-2014-348