The aim of this work is to extend the notion of weakly isotone increasing mappings to multivalued and present common endpoint theorems for -weakly isotone increasing multivalued mappings satisfying generalized -weak contractive as well as almost contractive inequalities in complete partially ordered metric spaces. Examples are given in support of the new results obtained.
MSC: 47H10, 54H25, 54H10.
Keywords:endpoint; common endpoint; partial ordering; multivalued mapping; control function; weak contractive inequalities; weakly isotone increasing mappings
1 Introduction and preliminaries
The Banach contraction principle  is a remarkable result in the metric fixed point theory. Over the years, it has been generalized in different directions and spaces by several mathematicians. In 1997, Alber and Guerre-Delabriere  introduced the concept of weak contraction in the following way.
Using the concept of weak contractiveness, they succeeded in establishing the existence of fixed points for such mappings in Hilbert spaces. Later on, Rhoades  proved that the results in  are also valid in complete metric spaces. He also proved the following fixed point theorem which is a generalization of the Banach contraction principle.
Weak contractive inequalities of the above type have been used to establish fixed point results in a number of subsequent works, some of which are noted in [4,5]. Since then, fixed point theory for single-valued as well as for multivalued weakly contractive type mappings was studied by many authors. Fixed point theorems for multivalued mappings are quite useful in Control theory and have been frequently used in solving problems in Economics and Game theory.
The development of a geometric fixed point theory for multifunctions was initiated by Nadler  in 1969. He used the concept of a Hausdorff metric ℋ to establish the multivalued contraction principle containing the Banach contraction principle as a special case as follows.
Since then, this discipline has been developed further, and many profound concepts and results have been established with considerable generality; see, for example, the works of Itoh and Takahashi , Mizoguchi and Takahashi , Hussain and Abbas , and references cited therein. Very recently, results on common fixed points for a pair of multivalued operators have been obtained by applying various types of contractive conditions; we refer the reader to [10-14]. In some cases, multivalued mapping defined on a nonempty set assumes a compact value for each x in . There are situations when, for each x in , is assumed to be a closed and bounded subset of . To prove the existence of a fixed point of such mappings, it is essential for mappings to satisfy certain contractive conditions which may involve the Hausdorff metric.
The Fixed Point Theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [, Theorem 2.1] who presented its applications to matrix equations. Subsequently, Nieto and Rodríguez-López  extended the result of  for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed point theorems in ordered metric spaces. In , Nashine et al. extended the results in  by using -weakly isotone increasing mappings and relaxing other conditions without taking into account any commutativity condition. Beg and Butt  studied set-valued mappings and proved common fixed point results for mappings satisfying implicit relation in a partially ordered metric space. Recently, Amini  proved endpoint theorems for multivalued mappings in a metric space. More recently, Choudhury and Metiya  as well as Nashine and Kadelburg  also proved fixed point theorems for multivalued mappings in the framework of a partially ordered metric space.
We will use the following terminology.
Definition 1.5 ()
The purpose of this paper is to prove the existence of a common endpoint for a pair of -weakly isotone increasing multivalued mappings under a generalized -weakly contractive condition and under a variant of so-called almost contractive conditions of Berinde  without using the continuity of any map and any commutativity condition in a complete ordered metric space. Our results generalize the results of Abbas and Ðorić , Choudhury and Metiya  and Hussain et al.  for more general contractive and weakly contractive conditions for a pair of weakly isotone increasing multivalued mappings. They also extend the results of Babu et al., Berinde , Choudhury et al. and Ćirić et al. from single-valued mappings in metric spaces to multivalued mappings in ordered metric spaces. Also, the results on common fixed points of weakly isotone increasing mappings in  are modified to the results on common endpoints of -weakly isotone increasing mappings under suitable conditions. Examples are presented to show the usage of the results and, in particular, that the order can be crucial.
In this section, we prove common endpoint theorems for a pair of weakly isotone increasing multivalued mappings under a generalized -weak contractive condition. In order to formulate the results, we extend to multivalued mappings the notion of weakly isotone increasing mappings given by Vetro [, Definition 4.2].
Definition 2.2 ()
The main result of this section is as follows.
which implies that
Now, from (2.2) and from the triangle inequality for δ, we have
From (2.5) and (2.6), it follows that
It, furthermore, implies that
a contradiction. So, we have
and since φ is lower semicontinuous, we have
From (2.8) and (2.4), it follows that
We proceed by negation and suppose that the inequality in (2.9) is not true. That is, there exists for which we can find nonnegative integer sequences and such that is the smallest element of the sequence such that for each ,
This means that
From (2.10) and the triangle inequality for δ, we have
We note that
Using (2.8) and (2.11), we get
using (2.8) and (2.12), we get
Also, from (2.2), (2.8) and (2.12), we have
whereφ, ψare as in Theorem 2.3 and
whereφ, ψare as in Theorem 2.3 and
Remark 2.7 In [, Corollary 2.5], it was proved that if
and hence has a unique fixed point. If the condition (2.18) fails, it is possible to find examples of mappings with more than one fixed point (cf.).
We illustrate the results of this section with two simple examples. The first one shows how a multivalued variant (Corollary 2.5) can be used. The other shows that (in the single-valued case) the use of order can be crucial.
Let again be given by , . It is again easy to show that in the cases , , as well as , , the condition (2.17) of Corollary 2.6 is satisfied, and it follows that has a fixed point A. However, for (incomparable) points , , the condition (2.17) is not satisfied, and so the respective result in the metric space without order cannot be applied to reach the conclusion. Indeed, in this case, , ,
3 Common endpoint for almost contractive conditions
Proof First of all, we show that if or has an endpoint, then it is a common endpoint of and . Indeed, let z be an endpoint of and assume . If we use the inequality (3.1) for and the properties of φ, we have
By (3.2), we have
On the other hand, by the triangular inequality, we have
Thus, we have
which implies that
By (3.5), we have
If we set
Hence, by (3.7) and (3.9), we deduce
From (3.3) and (3.10), it follows that
Now, we prove that is a Cauchy sequence. To this end, it is sufficient to verify that is a Cauchy sequence. Suppose, on the contrary, that it is not. Then there exists an such that for each even integer 2k there are even integers , with such that
From (3.12), (3.13) and the triangular inequality, we have
Hence, by (3.10), it follows that
Now, from the triangular inequality, we have
On the other hand, we have
On the other hand, we have
It follows that
Now, using (3.10), (3.14), (3.15), (3.17) and the continuity of φ, we get
(i) The condition
implies the condition (3.1).
(ii) The condition (3.19) is equivalent to the condition (3.1) if we suppose that φ is a non-decreasing function.
(iii) From Theorem 3.1 we can derive a corollary involving the condition (3.19).
(iv) Under the hypothesis that φ is a non-decreasing function, we can state many other corollaries using the equivalences established by Jachymski in  for single-valued mappings.
First, we check that is -weakly isotone increasing. Suppose that and . Then implies that and so . This means that for any , we have for all . Similarly, one can prove that for each , we have for all .
Hence, the condition (3.1) is satisfied.
To conclude this section, we provide a sufficient condition to ensure the uniqueness of the endpoint in Theorem 3.1,
Theorem 3.5Adding to the hypotheses of Theorem 3.1 the condition
The authors declare they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
The first and fourth authors gratefully acknowledge the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University during this research. The third author is thankful to the Ministry of Science and Technological Development of Serbia.
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