Abstract
In this paper, using the concept of a w-distance on a metric space, we first prove the existence of a fixed point theorem for generalized -contraction multivalued mappings without completeness in metric spaces. Our presented results generalize, extend, and improve the result of Kutbi and Sintunavarat (Abstr. Appl. Anal. 2013:165434, 2013) and various well-known results on the topic in the literature. Also, we give some examples to which the results of Kutbi and Sintunavarat (Abstr. Appl. Anal. 2013:165434, 2013) are not applied, but our results are.
MSC:47H10, 54H25.
Similar content being viewed by others
1 Introduction
In 1996, Kada et al. [1] introduced the concept of w-distance on a metric space, which is a real generalization of a metric. By using this concept, they extended and improved Caristi’s fixed point theorem, Ekland’s variational principle, and Takahashi’s existence theorem from the metric version to a w-distance version. Later, Suzuki and Takahashi [2] using the concept of w-distance to established the fixed point result for multivalued mapping. This result is an improvement of the famous Nadler fixed point theorem.
In 2013, Kutbi [3] improved a useful lemma given in [4] for the w-distance version and established the fixed point results via this lemma. Recently, Kutbi and Sintunavarat [5] introduced the notion of generalized -contraction mapping and proved a fixed point theorem for such a mapping in complete metric spaces via the concept of α-admissible mapping due to Mohammadi et al. [6]. On the other hand, Hussain et al. [7] introduced the concepts of α-complete metric spaces and also established fixed point results in such spaces.
The purpose of this work is to weaken the condition of completeness of the metric space in the result of Kutbi and Sintunavarat [5] by using the concept of α-completeness of the metric space. We also give the example of a nonlinear contraction mapping which is not applied by the results of Kutbi and Sintunavarat [5], but can be applied to our results. The presented results extend and complement recent results of Kutbi and Sintunavarat [5] and many known existence results from the literature.
2 Preliminaries
Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively.
For a metric space , we denote by , , and the collection of nonempty subsets of X, nonempty closed subsets of X and nonempty closed bounded subsets of X, respectively.
For , we define the Hausdorff distance with respect to d by
where . It is well known that is a metric space and is complete if is complete.
Definition 2.1 Let be a metric space and be a multivalued mapping. A point is called a fixed point of T if and the set of fixed points of T is denoted by .
Definition 2.2 ([8])
Let be a metric space and let be a multivalued mapping. T is said to be a contraction if there exists a constant such that for each ,
Definition 2.3 ([1])
Let be a metric space. A function is called a w-distance on X if it satisfies the following conditions for each :
(w1) ;
(w2) a mapping is lower semicontinuous;
(w3) for any , there exists such that and imply .
For a metric space , it is easy to see that the metric d is a w-distance on X. But the converse is not true in the general case (see Examples 2.4 and 2.5). Therefore, the w-distance is a real generalization of the metric.
Example 2.4 Let be a metric space. For a fixed positive real number c, define a function by for all . Then ω is a w-distance on X.
Example 2.5 Let be a normed linear space.
-
1.
A function defined by for all is a w-distance on X.
-
2.
A function defined by for all is a w-distance on X.
Remark 2.6 From Example 2.5, we obtain in general for , and neither of the implications necessarily holds.
Definition 2.7 ([9])
Let be a metric space. The w-distance on X is said to be a -distance if for all .
For more details of other examples and properties of the w-distance, one can refer to [1, 2, 9]. The following lemmas are useful for the main results in this paper.
Lemma 2.8 ([1])
Let be a metric space and be a w-distance on X. Suppose that and are sequences in X and and are sequences in converging to 0. Then the following hold for :
-
1.
if and for any , then ; in particular, if and , then ;
-
2.
if and for any , then converges to z;
-
3.
if for any with , then is a Cauchy sequence;
-
4.
if for any , then is a Cauchy sequence.
Next, we give the definition of some type of mapping. Before giving the next definition, we give the following notation. Let be a metric space and be a w-distance on X. For and , we denote .
Definition 2.9 ([2])
Let be a metric space. The multivalued mapping is said to be a w-contraction if there exist a w-distance on X and such that for any and there is with
Definition 2.10 ([5])
Let be a metric space and be a given mapping. The multivalued mapping is said to be a -contraction if there exist a w-distance on X and such that for any and there is with
Definition 2.11 ([5])
Let be a metric space and be a given mapping. The multivalued mapping is said to be a generalized -contraction if there exist a -distance ω on X and such that for any and there is with
Next, we give the concepts of an α-admissible multivalued mapping and α-completeness of metric spaces.
Definition 2.12 ([6])
Let X be a nonempty set, and be a given mapping. We say that T is an α-admissible whenever, for each and with , we have for all .
Remark 2.13 The concept of α-admissible multivalued mapping is extension of concept of -admissible multivalued mapping due to Asl et al. [10].
Many fixed point results via the concepts of α-admissible mappings occupy a prominent place in many aspects (see [5, 11–17] and references therein).
Definition 2.14 ([7])
Let be a metric space and be a given mapping. The metric space X is said to be α-complete if and only if every Cauchy sequence in X with for all , converges in X.
Example 2.15 Let and define metric by for all . Let A be a closed subset of X. Define by
Clearly, is not a complete metric space, but is an α-complete metric space. Indeed, if is a Cauchy sequence in X such that for all , then for all . Now, since is a complete metric space, there exists such that as .
3 Main results
In this section, we prove a fixed point theorem for generalized -contraction multivalued mappings in α-complete metric space.
Theorem 3.1 Let be a metric space, and be a generalized -contraction multivalued mapping. Suppose that is an α-complete metric space and the following conditions hold:
-
(a)
T is an α-admissible mapping;
-
(b)
there exist and such that ;
-
(c)
if for every with , we have
Then .
Proof We start from and in (b). From the definition of a generalized -contraction of T, we can find such that
Since T is an α-admissible mapping and such that , we have
From (3.1) and (3.2), we obtain
Again, using the definition of a generalized -contraction of T, there exists such that
Since and T is an α-admissible mapping, we get
From (3.3) and (3.4), we have
Continuing this process, we can construct the sequence in X such that ,
and
for all . Now, for each , we have
If for some , then we have and hence . By the property of the w-distance, we get
We find from Lemma 2.8, , and that . This implies that and so is a fixed point of T.
Next, we assume that for all . We obtain from (3.7)
for all .
By repeating (3.8), we get
for all .
For with , we obtain
Since , we get as . By Lemma 2.8, we find that is a Cauchy sequence in X. From (3.5) we know that for all . Using α-completeness of X, we obtain as for some . Since is lower semicontinuous, we have
Finally, we will assume that . By hypothesis, we get
which is a contradiction. Consequently, , that is, z is a fixed point of T as required. This completes the proof. □
Corollary 3.2 (Theorem 3.1 in [5])
Let be a complete metric space, and be a generalized -contraction mapping. Suppose that the following conditions hold:
-
(a)
T is an α-admissible mapping;
-
(b)
there exist and such that ;
-
(c)
if for every with , we have
Then .
Proof We find that the completeness of the metric space implies α-completeness. Therefore, by using Theorem 3.1, we obtain the desired result. □
Theorem 3.3 Let be a complete metric space, and be a -contraction mapping. Suppose that is an α-complete metric space and the following conditions hold:
-
(a)
T is α-admissible mapping;
-
(b)
there exist and such that ;
-
(c)
for every with , we have
Then .
Proof We see that this result can be proven by using a similar method to Theorem 3.1. In order to avoid repetition, the details are omitted. □
Example 3.4 Let and define metric by for all . Define by
Let a multivalued mapping be defined by
Now we show that T is a -contraction multivalued mapping with and w-distance defined by for all . For , let , that is, , we can find such that
Otherwise, it is easy to see that the -contractive condition holds. Therefore, T is a -contraction multivalued mapping.
Clearly, is not a complete metric space and then the main results of Kutbi and Sintunavarat [5] cannot be applied to this case.
Next, we show that our results in this paper can be used for this case. First, we claim that is an α-complete metric space. Let be a Cauchy sequence in X such that for all . So for all . Now, since is a complete metric space, there exists such that as . Consequently, is an α-complete metric space. Also, it is easy to see that T is α-admissible and there exists such that and . Finally, we see that for with , we obtain and hence .
Therefore, all the conditions of Theorem 3.3 are satisfied and so T has a fixed point.
References
Kada O, Susuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44: 381–391.
Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 1996, 8: 371–382.
Kutbi MA: f -Contractive multivalued maps and coincidence points. J. Inequal. Appl. 2013., 2013: Article ID 141
Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83: 261–263. 10.2307/2318216
Kutbi MA, Sintunavarat W: The existence of fixed point theorems via w -distance and α -admissible mappings and applications. Abstr. Appl. Anal. 2013., 2013: Article ID 165434
Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ćirić generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24
Hussain N, Kutbi MA, Salami P: Fixed point in α -complete metric spaces with applications. Abstr. Appl. Anal. 2014., 2014: Article ID 280817
Nadler SB: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475
Du W-S: Fixed point theorems for generalized Hounders metrics. Int. Math. Forum 2008, 3: 1011–1022.
Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212
Agarwal RP, Sintunavarat W, Kumam P:PPF dependent fixed point theorems for an -admissible non-self mapping in the Razumikhin class. Fixed Point Theory Appl. 2013., 2013: Article ID 280
Fathollahi S, Salimi P, Sintunavarat W, Vetro P: On fixed points of α - η - ψ -contractive multifunctions. Wulfenia 2014, 21(2):353–365.
Hussain N, Salimi P, Latif A: Fixed point results for single and set-valued α - η - ψ -contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 212
Kutbi MA, Ahmad J, Azam A: On fixed points of - ψ -contractive multi-valued mappings in cone metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 313782
Latif A, Mongkolkeha C, Sintunavarat W: Fixed point theorems for generalized α - β -weakly contraction mappings in metric spaces and applications. Sci. World J. 2014., 2014: Article ID 784207
Latif A, Gordji ME, Karapınar E, Sintunavarat W:Fixed point results for generalized -Meir Keeler contractive mappings and applications. J. Inequal. Appl. 2014., 2014: Article ID 68
Salimi P, Latif A, Hussain N: Modified α - ψ -contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151
Acknowledgements
The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The second author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript. Moreover, the authors thank the editors and referees for their insightful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kutbi, M.A., Sintunavarat, W. Fixed point theorems for generalized -contraction multivalued mappings in α-complete metric spaces. Fixed Point Theory Appl 2014, 139 (2014). https://doi.org/10.1186/1687-1812-2014-139
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-139