This article intends to initiate the study of Pompeiu–Hausdorff distance induced by an M-metric. The Nadler and Kannan type fixed point theorems for set-valued mappings are also established in the said spaces. Moreover, the discussion is supported with the aid of competent examples and a result on homotopy. This approach improves the current state of art in fixed point theory.
With the introduction of Banach’s contraction principle (BCP), the fixed point theory advanced in various directions. Nadler  obtained the fundamental fixed point result for set-valued mappings using the notion of Pompeiu–Hausdorff metric which is an extension of the BCP. Later on, many fixed point theorists followed the findings of Nadler and contributed significantly to the development of theory (cf. S. Reich [2,3]).
On the other hand, in order to investigate the semantics of data flow networks; Matthews  coined the concept called as partial metric spaces which are used efficiently while building models in computation theory. On the inclusion of partial metric spaces into literature, many fixed point theorems were established in this setting, see [5,6,7,8,9,10,11,12,13,14,15,16]. Recently, Asadi et al.  brought the notion of an M-metric as a real generalization of a partial metric into the literature. They also obtained the M-metric version of the fixed point results of Banach and Kannan. Also, some fixed point theorems have been established in M-metric spaces endowed with a graph, see .
In this work, we introduce the M-Pompeiu–Hausdorff type metric. Furthermore, we extend the fixed point theorems of Nadler and Kannan to M-metric spaces for set-valued mappings. Finally, homotopy results for M-metric spaces are discussed.
The symbols , and represent respectively set of all natural numbers, real numbers and nonnegative real numbers. Let us recall some of the concepts for simplicity in understanding.
().Let X be a nonempty set. Then a partial metric is a function satisfying following conditions:
for all . The pair is called a partial metric space.
The concept of an M-metric  defined in following definition extends and generalize the notion of partial metric.
().Let X be a non empty set. Then an M-metric is a function satisfying the following conditions:
(m) where ;
for all . The pair is called an M-metric space.
().Let us denote , where m is an M-metric on X. Then for every , we have
().Let and as in . Then . Further, if , then .
().Let be a sequence in such that for some , , then
3. -Pompeiu–Hausdorff Type Metric
The concept of a partial Hausdorff metric is defined in [19,20]. Following them we initiate the notion of an M-Pompeiu–Hausdorff type metric induced by an M-metric in this section. Let us begin with the following definition.
A subset A of an M-metric space is called bounded if for all , there exist and such that , that is, .
Let denotes the family of all nonempty, bounded, and closed subsets in . For , define
where and .
Let denote the closure of P with respect to M-metric m. Note that P is closed in if and only if .
Let P be any nonempty set in an M-metric space , then if and only if .
Let , then we have
Since , . Then from Lemma 6, . Therefore, .
For any , and , we have
We rewrite it as
Since b is arbitrary element in Q, we have
Since , we can write above inequality as
As c is arbitrary in R, we have
We rewrite the above inequality as
Again, as a is arbitrary in P, we get
For any following are true
From of Proposition 1, we write .
It follows from (m) of Definition 2.
Using of Proposition 1, we have
In general, for . It can be verified through the following example.
Let and , then clearly is an M-metric space. In view of of Proposition 1, we have
In view of Proposition 2, we call an M-Pompeiu–Hausdorff type metric induced by m.
Let and . Then for every , there is at least one such that
Assume that there exists an such that for all . This implies that
Since , , which is a contradiction. □
Let and . For any , there is at least one such that
Assume that there exists such that for all . This implies that
Thus, , which is a contradiction. □
4. Fixed Point Results
First, we state the Nadler fixed point theorem in the class of M-metric spaces.
Let M-metric space be M-complete and be a multivalued mapping. Suppose there exists such that
for all . Then F admits a fixed point.
Choose and . Clearly, and . Let be arbitrary and . From Lemma 7, for , there exists such that
As , so from (2) we have
Now, from Lemma 7, there exists such that
Continuing in this way, we get a sequence of points in X such that and for ,
By Lemma 5, we have
Also the sequence is M-Cauchy. Thus, M-completeness of X yields existence of such that
Since we have
From (1) and (8), we have
Now, since , . Taking limit as and using (8), we get
As , so we have
Using , we have
Varying limit as and using (8)–(10), we get
Since for every , this implies that
From (12) and (13), we have
Thus, by Lemma 6, □
Let be endowed with m-metric Then is an M-complete M-metric space (as in Example 3). Let be a mapping defined as
We shall show that for , , i.e., (1) holds for all . We have following three possible cases:
Case I: . Then . Here, for ,
Case II: . Then , and . In this case,
Since , . So we get
and . Then one can see that
Case III: . Then , and . In this case,
Since , . So, we get
and . Following Case II, one can easily show that
From above three cases, it is clear that (1) is satisfied for . Thus, all the required conditions of Theorem 1 are satisfied. Hence F admits a fixed point, which is .
Next, we present our fixed point result corresponding to multivalued Kannan contractions in M-metric spaces.
Let M-metric space be M-complete and be a multivalued mapping. Suppose there exists such that
for all . Then F admits a fixed point in X.
Let be arbitrary. Fix an element . We can now choose such that
Again, we can choose such that
Continuing in this way, we get a sequence such that with
Using (14) in (15), we get
Let . Since , we have . So,
Thus, from Lemma 5, we have
Moreover, the sequence is a M-Cauchy. M-completeness of X yields existence of such that
Due to (18), we get
Thus, we have
This implies that
We shall show that . Since
Taking limit as , we get
Suppose , then we have
Taking limit as and using (21), we get which is a contradiction (as ). So
Also, using (20), we have
From (22) and (23), we get
Thus, from Lemma 6, we get □
Let and be defined as
Then is an M-complete M-metric space. Let be a mapping defined as
Then one can easily verify that there exists some λ in such that
Thus F satisfies all the conditions in Theorem 2 and hence it has a fixed point (namely 0) in X.
Let be endowed with m-metric Then is an M-complete M-metric space. We define the mapping as
For and , there does not exist any λ in such that
Thus F does not satisfy (14) in Theorem 2. Evidently, F has no fixed point in X.
5. Homotopy Results in -Metric Spaces
The following result is required in the sequel while proving a homotopy result in M-metric spaces.
Let be a multivalued mapping satisfying (1) for all in M-metric space . If for some , then for .
Let . Then Also
Assume that . We have
Since , it is a contradiction. So for every . □
Let (resp. ) be an open (resp. closed) subset in an M-complete M-metric space such that . Let be a mapping satisfying the following conditions:
for all and each ;
there exists such that for every and all we have
there exists a continuous mapping satisfying
If admits a fixed point in for at least one , then admits a fixed point in for all . Moreover, the fixed point of is unique for any fixed
Consider, the set
Then is nonempty, because has a fixed point in for at least one , that is, there exists such that and as holds, we have
We will show that is both closed and open in . First, we show that it is open.
Let and with . As is open subset of X, for some . Let As is continuous on , there exists such that
Since , by Proposition 3, for every . Keeping this fact in view, we have
Now, using of Proposition 2 and (24), we have
Thus for each fixed , satisfies all the hypotheses of Theorem 1 and so admits a fixed point in . But this fixed point must be in to satisfy . Therefore, and hence is open in .
Next, we show that is closed in . Let be a convergent sequence in to some . We need to show that .
The definition of the set implies that for all , there exists with . Then using , of Proposition 2 and the outcome of Proposition 3, we have
This gives us
Since is continuous and converges to s, varying in the above inequality, we get
As , so
Thus is an M-Cauchy sequence. Using of Definition 3, there exists such that
But , so
Thus, we get . We shall prove . We have
Varying in above inequality, we get
Since , we have
From (25) and (26), we get
Therefore, from Lemma 6, we have . Thus . Hence and is closed in .
As is connected and is both open and closed in it, so . Thus admits a fixed point in for all .
For uniqueness, fix , then there exists such that . Suppose b is another fixed point of , then from we have
a contradiction. Thus, the fixed point of is unique for any □
All authors contributed equally to this paper. All authors have read and approved the final manuscript.
This research received no external funding.
The fifth author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Aydi, H.; Barakat, M.; Felhi, A.; Isik, H. On phi-contraction type couplings in partial metric spaces. J. Math. Anal.2017, 8, 78–89. [Google Scholar]
Ciric, L.; Samet, B.; Aydi, H.; Vetro, C. Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput.2011, 218, 2398–2406. [Google Scholar]
Abodayeh, K.; Mliaki, N.; Abdeljawad, T.; Shatanawi, W. Relation Between Partial Metric Spaces and M-Metric Spaces, Caristi Kirk’s Theorem in M-Metric Type Spaces. J. Math. Anal.2016, 7, 1–12. [Google Scholar]
Ameer, E.; Aydi, H.; Arshad, M.; Alsamir, H.; Noorani, M.S. Hybrid multivalued type contraction mappings in αK-complete partial b-metric spaces and applications. Symmetry2019, 11, 86. [Google Scholar] [CrossRef]
Aydi, H.; Felhi, A.; Karapinar, E.; Sahmim, S. A Nadler-type fixed point theorem in dislocated spaces and applications. Miscolc Math. Notes2018, 19, 111–124. [Google Scholar] [CrossRef]
Karapinar, E.; Shatanawi, W.; Tas, K. Fixed point theorem on partial metric spaces involving rational expressions. Miskolc Math. Notes2013, 14, 135–142. [Google Scholar] [CrossRef]
Shatanawi, W.; Pitea, A. Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl.2013, 2013, 153. [Google Scholar] [CrossRef]
Karapinar, E.; Agarwal, R.P.; Aydi, H. Interpolative Reich-Rus-Ciric type contractions on partial metric spaces. Mathematics2018, 6, 256. [Google Scholar] [CrossRef]
Shatanawi, W.; Postolache, M. Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl.2013, 2013, 54. [Google Scholar] [CrossRef]
Karapinar, E.; Shatanawi, W. On Weakly (C,ψ,ϕ)-Contractive Mappings in Ordered Partial Metric Spaces. Abstr. Appl. Anal.2012, 2012, 495892. [Google Scholar] [CrossRef]
Aydi, H.; Karapinar, E.; Shatanawi, W. Coupled fixed point results for (ψ,φ)-weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl.2011, 62, 4449–4460. [Google Scholar] [CrossRef]
Shatanawi, W.; Nashine, H.K.; Tahat, N. Generalization of some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci.2012, 2012, 686801. [Google Scholar] [CrossRef]
Asadi, M.; Karapinar, E.; Salimi, P. New extension of p-metric spaces with fixed-point results on M-metric spaces. J. Inequal. Appl.2014, 2014, 18. [Google Scholar] [CrossRef]
Souayah, N.; Mlaiki, N.; Mrad, M. The GM-Contraction Principle for Mappings on M-Metric Spaces Endowed With a Graph and Fixed Point Theorems. IEEE Access2018, 6, 25178–25184. [Google Scholar] [CrossRef]
Aydi, H.; Abbas, M.; Vetro, C. Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl.2012, 159, 3234–3242. [Google Scholar] [CrossRef]
Aydi, H.; Abbas, M.; Vetro, C. Common Fixed points for multivalued generalized contractions on partial metric spaces, RACSAM—Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales Serie A Matematicas2014, 108, 483–501. [Google Scholar]