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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 259768, 12 pages
Discussion on Generalized-(αψ, β)-Contractive Mappings via Generalized Altering Distance Function and Related Fixed Point Theorems
1Tunis College of Sciences and Techniques, Tunis University, 5 Avenue Taha Hussein, Tunis, Tunisia
2Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
3Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
4Department of Mathematics, University of Jaén, Campus las Lagunillas s/n, 23071 Jaén, Spain
Received 5 August 2013; Accepted 24 October 2013; Published 24 February 2014
Academic Editor: Yong Zhou
Copyright © 2014 Maher Berzig et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We extend the notion of (αψ, β)-contractive mapping, a very recent concept by Berzig and Karapinar. This allows us to consider contractive conditions that generalize a wide range of nonexpansive mappings in the setting of metric spaces provided with binary relations that are not necessarily neither partial orders nor preorders. Thus, using this kind of contractive mappings, we show some related fixed point theorems that improve some well known recent results and can be applied in a variety of contexts.
1. Introduction and Preliminaries
After the appearance of the pioneering Banach contractive mapping principle and due to its possible applications, fixed point theory has become one of the most useful branches of nonlinear analysis, with applications to very different settings, including, among others, resolution of all kind of equations (differential, integral, matrix, etc.), image recovery, convex minimization and split feasibility, and equilibrium problems.
In the last decades, fixed point theorems in partially ordered metric spaces have attracted much attention, especially after the works of Ran and Reurings , Nieto and Rodríguez-López , Bhaskar and Lakshmikantham , Berinde and Borcut [4, 5], Karapınar [6, 7], Berzig and Samet , and Karapınar et al. [9–11], among others. Their results have been extended to contractivity conditions in which altering distance functions (a notion introduced by Khan et al. ) play an important role. Very recently, Alghamdi and Karapınar  used a similar notion in -metric spaces, and Berzig and Karapinar  also considered a more general kind of contractivity conditions using a pair of generalized altering distance functions.
In this paper, by introducing the notion of generalized-()-contractive mappings, we collect, improve, and generalize some existing results on this topic in the literature.
Now, we recollect some basic definitions and useful results for the sake of completeness of the paper. First, we recollect the concept of altering distance function as follows.
Definition 1 (Khan et al. ). A function is called an altering distance function if the following properties are satisfied:(i) is continuous and nondecreasing;(ii) if and only if .
In what follows, we state the definition of -preserving mapping which plays crucial roles in the setting of main results.
Definition 2 (see, e.g., ). Let be a set and be a binary relation on . We say that is -preserving mapping if
Throughout the paper, let denote the set of all nonnegative integers, and let be the set of all real numbers.
Example 3 (see, e.g., ). Let and a function be defined as . Define by
Define the first binary relation by if and only if , and define the second binary relation by if and only if . Then, we obtain easily that is simultaneously -preserving and -preserving.
Definition 4 (see ). Let . We say that is –transitive on if
The following remark is a consequence of the previous definition.
Remark 5 (see ). Let . We have the following. (1)If is transitive, then it is -transitive for all .(2)If is -transitive, then it is -transitive for all .
Definition 6 (see ). Let be a metric space and , be two binary relations on . We say that is -regular if for every sequence in such that as and there exists a subsequence such that
Definition 7. We say that a subset of is -directed if for all , there exists such that
Definition 8. Let be a mapping. We say that a subset of is -directed with respect to if for all , there exists such that
Remark 9. A subset of is an -directed subset if, and only if, it is an -directed subset with respect to the identity mapping .
We recall the notion of a pair of generalized altering distance as follows.
Definition 10. We say that the pair of functions is a pair of generalized altering distance (where ) if the following hypotheses hold: (a1) is continuous;(a2) is nondecreasing;(a3).
Definition 11 (see ). Let be a metric space and be a given mapping. We say that is an -contractive mapping if there exists a pair of generalized altering distance functions and two mappings such that
2. Main Results
Firstly, we present two technical properties that will be very useful in the proof of our main result.
Lemma 12. If is a pair of generalized altering distance functions and are such that , then one, and only one, of the following conditions holds:
Proof. Firstly, notice that both possibilities are not compatible. Suppose that . Since is nondecreasing and , so and . Defining for all , we have that . By (a3), .
Lemma 13. Let be a sequence in a metric space .(1)If is not Cauchy, then there exists and two subsequences and verifying that, for all , Furthermore, .(2)In addition to this, if also verifies , then
Proof. The first part is well-known as can be chosen to be the lowest integer that does not verify , then . The first part of the second item can be proved as follows. For all , Therefore, taking limit as in we deduce that . To prove the second part of the second item, we proceed by induction methodology on . If , it follows from item . Suppose that (12) holds for some . On the one hand, and on the other hand, Joining both inequalities, Taking limit as and using (12) and , we conclude that
Next we introduce the notion of generalized--contractive mappings which is an extension of Definition 11.
Definition 14. Let be a metric space and let be a given mapping. We say that is a generalized--contractive mapping if there exists a pair of generalized altering distance functions and two mappings such that
where is given by one of the following cases:(i), + ;(ii);(iii);(iv);(v),
for all .
In the sequel, the binary relations and are defined as follows.
Definition 15. Let be a set, and are two mappings. We define two binary relations and on by
Now we are ready to study the existence and the uniqueness of fixed points.
2.1. Existence of Fixed Points
We may now state our first main result.
Theorem 16. Let be a complete metric space and and be an ()-contractive mapping of type I satisfying the following conditions: (i) and are -transitive;(ii) is -preserving and -preserving;(iii)there exists such that and ;(iv) is continuous.
Then, has a fixed point; that is, there exists such that .
Proof. Let such that for . Define the sequence in by
If for some , then is a fixed point . Assume that ; that is,
From (ii) and (iii), we have
By induction, from (ii) it follows that
Substituting and in (19), we obtain
So, by (24) it follows that
(the last equality follows from ). By Lemma 12, either or , but the second case is impossible by (22). Then, we get
From (29), is monotone decreasing and, consequently, there exists such that
Notice that (26) and (29) imply that, for all ,
Letting in (31) and taking into account that is continuous, we obtain that the sequence has finite limit and
which implies that . Then, by (a3), we get
Next we show that is a Cauchy sequence reasoning by contradiction. If is not Cauchy, Lemma 13 assures us that there exists and two subsequences and verifying that, for all , and also Since , consider, for all , the Euclidean division , whose quotient will be denoted by (then ) and whose rest will be denoted by as follows: (36) Notice that and are convenient integer numbers such that and . Hence, can only take a finite quantity of integer numbers, in the interval . Therefore, there exist subsequences of and (also verifying (34) and (35)) such that is constant (it does not depend on ). In order not to complicate the notation, we will suppose that where is constant.
Let define for all . Taking into account item of Remark 5, (24), and (i), we obtain for all . Furthermore, by (35), Following the same technique as in Lemma 13, we also deduce that Apply the contractivity condition (19) to and , and we get Now, using (39), we get where, by (22), for all , Lemma 12 shows that . Furthermore, by (22), (35), and (41), Taking limit as in we deduce that . By (a3), , which contradicts (45) and the fact that . This contradiction implies that is a Cauchy sequence.
Since is a complete metric space, then there exists such that as . From the continuity of , it follows that as . Due to the uniqueness of the limit, we derive that ; that is, is a fixed point of .
Proof. Following the lines of the proof of Theorem 16, we get that is a Cauchy sequence. Since is a complete metric space, then there exists such that . Furthermore, the sequence satisfies (24); that is, Now, since is -regular, then there exists a subsequence of such that , that is, , and , that is, , for all . By setting and in (19), we obtain, for all , that is, where We prove that reasoning by contradiction. If , then , for all . By Lemma 12, Futhermore, By (49), for all , Using the continuity of and letting in the above inequality, we get By (a3) and (52), which contradicts that . This contradiction concludes that is a fixed point of .
Taking into account that the same proofs of the above theorems can be followed point by point to demonstrate the next result.
The uniqueness of the fixed point is studied in the following result.
Proof. Assume that is of type III; that is, (the other cases are similar). Suppose that and are any two fixed points of . Since is -directed, there exists such that , , , and ; that is,
Define for all . We claim that and . Hence, by the unicity of the limit, we will conclude that . Therefore, it is only necessary to prove that .
Indeed, since is -preserving for , from (57), we get that and, proceeding by induction, we have Using (59) and (19), we deduce that where (the last equality holds because ). By Lemma 12, either or . If , then . The second case yields to . In any case, we deduce that Since is a bounded below, nonincreasing sequence, there exists such that . By (61), for all . By the continuity of and taking limit as , we deduce that . Using (a3), we have ; that is, . This finishes the proof.
Now, we derive a particular condition which ensures the uniqueness of the fixed point for the mappings of type I, II, III, IV, or V as follows:(C):if are such that , then either or .
For instance, if and we consider and for all , then and verify condition (C).
Theorem 20. Adding to the hypotheses of Theorem 16 (resp., Theorem 17) that is -directed and is of type I, II, III, IV, or V, we obtain the unicity of the fixed point of whenever condition (C) is satisfied.
Proof. Following the lines of the proof of Theorem 19, we will prove that . Since is -directed with respect to , there exists such that the sequence converges (to some ) and also , , , and ; that is, Now we will prove that . By induction, we have that and for all . Substituting and in (19), we get that is, where Now from inequality (67) and the condition (C), it follows that, for all , If there is some such that , the proof is finished (because ). On the contrary, assume that for all . If , then which is a contradiction. Hence, necessarily, for all , and then Thus, we deduce that is a nonincreasing, bounded below sequence, so there exists such that . Therefore, By (67), for all . By the continuity of and taking limit as , we deduce that . Using (a3), we have ; that is, . This completes the proof.
Proof. Following the lines of the proof of Theorem 19, we will prove that . Since is -directed with respect to , there exists such that the sequence converges (to some ) and also , , , and ; that is, Now we will prove that . By induction, we have and , for all . Substituting and in (19), we get where Notice that Taking into account that, for all , and taking limit as , we deduce that the sequence has finite limit and so . By (a3), we conclude that ; that is, .
Very recently, a mapping satisfying contraction on metric spaces endowed with a binary relation has been introduced by Samet and Turinici in ; therefore, this work has been extended and improved in [14, 18]. In this section, using our main results, we derive some consequences on metric spaces endowed with -transitive binary relation, as on metric spaces endowed with a partial order. Furthermore, we establish a fixed point results for cyclic mappings.
3.1. Fixed Point Results on Metric Spaces Endowed with -Transitive Binary Relation
In this section, we establish a fixed point theorem on metric space endowed with -transitive binary relation . Therefore, we denote by if is -related to .
Definition 22. We say that is -regular if for every sequence in such that , and there exists a subsequence such that
Definition 23. We say that a subset of is -directed if for every , there exists such that and .
Corollary 24. Let be a nonempty set endowed with a binary relation . Suppose that there is a metric on such that is complete. Let satisfy the -weakly -contractive conditions; that is,
where , are altering distance functions and is given by Definition 14. Suppose also that the following conditions hold:(i) is -transitive ();(ii) is a -preserving mapping;(iii)there exists such that ;(iv) is continuous or is -regular.
Then has a fixed point. Moreover, if we suppose that is -directed with respect to or , then we have the uniqueness of the fixed point.
Proof. In view to link this theorem to the main result, we define the mapping by
and we define the mapping by
where for are defined by (a) if ;(b) if ;(c) if ;(d) if ;(e) if .
In case is neither -related nor -related to , the functions and are well defined, since and .
We can verify easily that and are -transitive.
Next, we claim that is a -contractive mapping. Indeed, in case , we get easily and in case is neither -related nor -related to , we have hence, our claim holds.
Moreover, since is -preserving, we get and similarly, we have Thus, is -preserving for . Now, if condition (iii) is satisfied; that is, is continuous, the existence of a fixed point follows from Theorem 16. Suppose now that the is -regular; hence, let be a nondecreasing sequence in such that ; that is, and , for all . Suppose also that as . Since is -regular, there exists a subsequence such that for all . This implies from the definition of and that and , for all , which implies that for and for all . In this case, the existence of a fixed point follows from Theorem 17.
To show the uniqueness, suppose that is -directed with respect to (resp., ); that is, for all , there exists a such that and (resp., with being a convergent sequence), which implies from the definition of and that and ; that is, is -directed with respect to (resp., ). Hence, Theorem 20 or 19 (resp., Theorem 21) gives us the uniqueness of this fixed point.
3.2. Fixed Point Results in Partially Ordered Metric Spaces
We start by defining the binary relations for and the concept of -directed.
Definition 25. Let be a partially ordered set. (1)We define two binary relations and on by (2)We say that is -directed if every have a common upper bound; that is, there exists such that and .
The following definition is useful later.
Definition 26. Let be a partially ordered set and be a metric on . We say that is -regular if for every nondecreasing sequence in such that , there exists a subsequence such that for all .
Notice that, by the transitivity condition of , in such a case, we have for all .
Corollary 27. Let be a partially ordered set and be a metric on such that is complete. Suppose that the mapping is weakly contractive; that is,
where and are altering distance functions and is given by Definition 14. Suppose also that the following conditions hold:(i) is a nondecreasing mapping;(ii)there exists with ;(iii) is continuous or is -regular.
Then has a fixed point. Moreover, if is -directed with respect to or , we have the uniqueness of the fixed point.
Proof. The proof follows immediately from the previous proof, since is a binary, 1-transitive relation.
3.3. Fixed Point Results for Cyclic Contractive Mappings
The main result of Kirk et al. in  is as follows.
Theorem 28 (see ). For , let be a nonempty closed subset of a complete metric space and let be a given mapping. Suppose that the following conditions hold: (i)