In this paper two new fixed point results are studied. The first result is a theorem that involves (α −β) type rational singlevalued contractions, in the sense of Geraghty type operators. The second result consists of multivalued modified Hardy Rogers operators, namely the existence of the fixed point, data dependence, local version involving two metrics and homotopy theorems involving two metrics are studied.
Authors
Cristian Daniel Alecsa
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
C.-D. Alecsa, Some fixed point results linked to α –β rational contractions and modified multivalued Hardy-Rogers operators, J. Fixed Point Theory, 2018, 2018:3.
[1] C.D. Agarwal, D. O’Regan, Fixed point theory for generalized contractions on spaces with two metrics, J. Math. Anal. Appl. 248 (2000), no.2, 402-414.
[2] C.D. Agarwal, J.H. Dshalalow, D. O’Regan, Fixed point and homotopy results for generalized contractive maps of Reich type, Appl. Anal. 82 (2003), no.4, 329-350.
[3] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.
[4] P.S. Kumari, D. Panthi, Connecting various type of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings, Fixed Point Theory Appl. 2016 (2016), Article ID 15.22 C.D. ALECSA
[5] T. Lazar, D. O’Regan, A. Petrusel, Fixed points and homotopy results for Ciric-type multivalued operators on a set with two metrics, Bull. Korean Math. Soc. 45 (2008), no.1, 67-73.
[6] A. Oprea, Fixed point theorems for multivalued generalized contractions of rational type in complete metric spaces, Creat. Math. Inform. 23 (2014), 99-106.
[7] N.S. Papageorgiou, S. Hu, Handbook of Multivalued Analysis (vol. I and II), Kluwer Acad. Publ., Dordrecht (1997 and 1999).
[8] L. Paunovic, P. Kaushik, S. Kumar, Some applications with new admissibility contractions in b-metric spaces, J. Nonlinear Sci. Appl. 10 (2017), 4162-4174.
[9] A. Petrusel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn. 59, 169-202.
[10] I.A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219.
[11] I.A. Rus, A. Petrusel, A. Sîntmarian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003), no.8, 1947-1959.
[12] R.J. Shahkoohi, A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 373.
[13] W. Sintunavarat, Generalized Ulam-Hyers stability, well-posedness, and limit shadowing of fixed point problems for α −β−contraction mappings in metric spaces, Sci. World J. 2014(2014), article ID 569174, 7 pages.
[14] F. Zabihi, A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, J. Appl. Math., 2014(2014), Article ID 929821
Paper (preprint) in HTML form
SOME FIXED POINT RESULTS LINKED TO RATIONAL CONTRACTIONS AND MODIFIED MULTIVALUED HARDY-ROGERS OPERATORS
CRISTIAN DANIEL ALECSA 1,2,∗ 1 Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania
2 Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Abstract
In this paper two new fixed point results are studied. The first result is a theorem that involves ( ) type rational singlevalued contractions, in the sense of Geraghty type operators. The second result consists of multivalued modified Hardy Rogers operators, namely the existence of the fixed point, data dependence, local version involving two metrics and homotopy theorems involving two metrics are studied.
Copyright (C) 2018 C.D. Alecsa. 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.
1. Preliminaries for rational Geraghty type mappings
The first idea of the present article is that in the third section we want to prove a theorem based on rational -contractions, so we remind the necessary concepts for this type of operators. For more informations, we let the reader follow [13]. We recall the following crucial concepts.
Definition 1.1. Let be a nonempty set and be a mapping. Then is called -admissible, if it satisfies the following condition for each , with .
Definition 1.2. The mapping is called transitive, if for each , with and , we have .
Moreover, let’s denote by the set of all functions , satisfying .
Also, we recall the definition of contractions, given by Sintunavarat in [13].
Definition 1.3. Let ( ) be a metric space. A mapping is called an -contraction, if there exists and , such that , for each , with .
In [13], the author proved a series of theorems such as : existence of a fixed point assuming that the mapping is continuous, a theorem in which the continuity is dropped and a theorem for the uniqueness of fixed points. Also, this was based on the result of Geraghty [3] from 1973. Moreover, [8] Paunović et. al. extended the result of W. Sintunavarat in the framework of b-metric spaces. Additionally, they have studied fixed points for a given mapping , such that , for each , with , where and , where was the coefficient of the b-metric space ( ).
Furthermore, Zabihi and Razani [14] considered rational type operators and developed some fixed point results in the framework of complete b -metric spaces. In the context of a metric space ( ), a self mapping on was considered, satisfying
, where ,
max and .
Moreover, since we want to define some other type of contractions, we shall recall that the authors in [12] developed new fixed point theorems involving a new type of rational contractive Geraghty mapping in b-metric spaces. This rational Geraghty mapping is introduced as follows, in the case of metric spaces.
Definition 1.4. Let ( ) be a metric space. A mapping is called a rational Geraghty of type , if , for each , where and .
That means that in the third section, we will present a generalized theorem for rational Geraghty mappings of type .
2. Preliminaries for modified multivalued Hardy-Rogers
In this section, we recall some general notions in the framework of multivalued analysis theory. Also, for the following preliminary notions and lemmas (such as: multivalued weakly Picard operators, data dependence of the fixed point set, Haussdorf metric properties) we refer the reader to [9], [10] and [11].
Let ( ) be a metric space and be the family of all nonempty subsets of .
We denote by the family of all nonempty subsets of which are closed, by the family of all nonempty subsets of which are bounded and by the family of all nonempty subsets of which are compact.
Furthermore, we consider the following functionals
We recall some useful results concerning the Haussdorf-Pompeiu generalized functional .
Lemma 2.1. Let and .
Then, for each , there exists such that .
Lemma 2.2. Let be a metric space and .
Suppose that there exists such that :
(i) for each , there exists such that ,
(ii) for each , there exists such that .
Then .
Moreover, if is a nonempty subset of and a multivalued operator, then an element is
(a) a fixed point of if and only if ;
(b) a strict fixed point of if and only if ;
Furthermore, we denote by the set of all fixed points of and by the set of all strict fixed points of .
We also remind the definition of the graphic of a multivalued operator, i.e.
.
Definition 2.3. Let ( ) be a metric space and a multivalued operator. We say that is a multivalued weakly Picard operator (briefly MWP) if for each and for each , there exists a sequence , satisfying the following
(i) ;
(ii) , for each ;
(iii) the sequence is convergent to a fixed point of .
Definition 2.4. Let ( ) be a metric space and an MWP.
Then is called a -weakly Picard operator, with , if there exists a selection of , such that
, for each .
Now we focus our attention to the case of Hardy-Rogers type mappings. In [7], the basic notion of singlevalued Hardy-Rogers contraction appeared.
Definition 2.5. Let be a metric space and be an operator such that there exists with , satisfying
, for each .
In [6], Oprea A. developed a theorem concerning multivalued rational contractions (of Hardy Rogers type).
A multivalued operator is called a multivalued rational type contraction, if satisfies the following condition
, for each .
Oprea has showed that the multivalued rational contractions are MWP-operators and developed theorems for data dependence, fractal theory, Ulam-Hyers stability etc.
In [4], Kumari and Panthi introduced a new type of rational contractions, called modified HardyRogers contractions.
They introduced this as types of cyclic contractions for the case of families of dislocated metric spaces.
We recall the notion of singlevalued contractions in the context of metric spaces, i.e. singlevalued operator that satisfies
Also, regarding Hardy Rogers mappings, our purpose to define the concept of modified HardyRogers contractions under the multivalued case shall be presented in the last section, along with some fixed point results.
3. Some theorems regarding rational Geraghty
In this section we present a generalized theorem for rational Geraghty mappings or type , using the -admissibility conditions by Sintunavarat.
Moreover, we will use the same terminology from the first section.
Definition 3.1. Let ( ) be a metric space.
A mapping is called an -rational Geraghty mapping of type if and only if there exists and , such that , for each , with , where .
Our first main result of this section is the existence theorem for -rational Geraghty mappings of type , using the assumption that is continuous. The techniques used in the theorem’s proof follow the same lines as in the theorems from [13].
Theorem 3.2. Let ( ) be a complete metric space and an rational Geraghty mapping of type I. Also, suppose that the following assumptions hold
(i) is -admissible,
(ii) is transitive,
(iii) there exists , such that ,
(iv) is continuous.
Then, there exists , such that .
Proof. - Let satisfying .
Let’s consider the Picard sequence , for each .
If there exists such that , then is a fixed point and the conclusion holds.
Suppose that for each . So , for each .
From condition (i), we know that is -admissible. Since , then we have that .
Inductively, one can show that , for each .
Now, we estimate
so .
Moreover, we make the following computations :
Since and
,
we get that , for each .
Now, we consider two cases.
(I) If , then we get
.
Since , because , we get the contradiction .
(II) Then, only the second case is valid, i.e. , that is , for each .
So, the sequence is strictly decreasing and nonnegative. It implies that there exists , such that as .
Now, we show that .
Let’s suppose that .
We know that . Taking the limit as , we get that .
Because ,we get that .
But , so . From all this, we find that . This implies that .
Now, because is the maximum between three elements, if it’s limit is 0 , so all the elements have the limit 0 . This means that . This is a contradiction!
•
We now show that is a Cauchy sequence.
By reductio ad absurdum, let’s suppose that ( ) is not Cauchy. Then there exists and there exists and , such that , with and being the smallest index satisfying the following
By triangular inequality, we have that .
Since , taking , it follows that .
Like in [13], since is transitive, we observe that .
Now, we make the follow estimation
So, we get that
.
Furthermore
In the above inequality, taking the limit as , we have that
.
Using the fact that , it follows that , i.e. .
Since is the maximum of three elements and it has the limit 0 , also because , then all of the elements will have the limit 0 , so , which is false; so ( ) is Cauchy.
•
Since is complete with respect to the metric , there exists such that . Because is continuous, we infer that
, so is a fixed point for the Geraghty-type mapping .
Now, also based on [13], we give a theorem where we dropped the continuity of the operator .
Theorem 3.3. Let be a complete metric space and an rational Geraghty mappings of type I.
Let’s suppose that the following assumptions hold
(i) is -admissible,
(ii) is transitive,
(iii) there exists , such that ,
(iv) if ( ) is a sequence satisfying and implies that , for each .
Then, there exists , such that .
Proof. In a similar manner like in the previous proof, we can show that is a Cauchy sequence and therefore there exists , such that when .
From (iv), we have that , for each .
We make the following estimation
So, we have that .
Furthermore, we have that
.
Taking the limit as and using the fact that and that , we get that .
Thus , so the conclusion holds properly.
Finally, we present the theorem for the uniqueness of the fixed point for the Geraghty type operator.
Theorem 3.4. Let’s suppose that all the assumptions from the last theorem are satisfied. Additionally, let’s suppose that one of the following assumptions are valid
(HO) if are two fixed points, then
(H1) for each , there exists such that and .
Then, admits a unique fixed point.
Proof. Let two fixed points for the mapping .
We consider two cases
(H0) We have that .
From the Geraghty condition, it is easy to see that , so the conclusion is true.
(H1) We have that there exists , with and .
Since f is -admissible, by induction, we get that and .
We make the following estimation
So , for each .
Now we show that as .
By reductio ad absurdum, we suppose that .
We know that
.
Since is a fixed point for , then .
Taking the limit as and using the fact that , we get that
. Now, because , it follows that . So, it follows that . This means that .
By the same reasoning as in the last proof, we get that . Furthermore, in a similar way, one can show that as , so .
Remark 3.5. Taking , for each , we get the existence and uniqueness for Geraghty mappings of type as a corollary.
4. Fixed Point Results for Modified Multivalued Hardy-Rogers contractions
In this section we introduce the concept of modified multivalued Hardy-Rogers contractions and then we present some theorems concerning the existence of a fixed point, data dependence, Ulam-Hyers stability. Also, we present a local version involving two metrics and a homotopy theorem.
The first main result of this section is a fixed point theorem for modified multivalued HardyRogers contractions, regarding the existence of fixed points for these types of self-mappings.
Theorem 4.1. Let ( ) be a complete metric space and be a multivalued modified Hardy Rogers operator, i.e.
with all the above coefficients positive.
If , then there exists , such that .
Proof. Let’s consider an arbitrary point and .
Let .
If , then , that means , i.e. .
Let’s suppose that .
For , we can choose , such that .
This means that
So, we have that
Since
,
that is and , because , we get that
.
This means that
.
In a similar manner, for , there exists such that
.
This means that
So, we have that
Like before, since
,
that is and ,
because , we get that
.
This means that
By induction, we infer that , where .
Since is arbitrary taken, we impose the following condition, namely
, which means that .
Equivalently, we can take .
In this way, we can take .
From the hypotheses, we have that , which implies that
, so the definition for is correct.
For to take place, we need the relation .
Now, since , it follows that , which is obviously true.
Now, we show that is a Cauchy sequence, i.e.
.
For each , letting , if follows that ( ) is a Cauchy sequence.
Because the metric is a complete, the sequence ( ) is convergent.
Then exists , such that .
We now show that is a fixed point for the operator . We estimate
Now, we have used the following relations
, for each ,
,
,
because .
Letting , we have that
.
It follows that . The inequality , is satisfied since .
We obtain that . This means that , i.e. .
We have shown that .
Letting , we have .
Letting in the above inequality, we have , so is a MWP operator.
Now we present a theorem concerning the fact that T is a MWP operator.
Theorem 4.2. Let ( ) be a complete metric space and be a multivalued modified Hardy Rogers operator, i.e.
with all the above coefficients positive.
If , then the operator is .
Proof. From the proof of the previous theorem, we have that , with defined as .
Since the sequence ( ) was convergent to a fixed point of , letting and then making , we get that .
Using the triangular inequality, it follows that
From the definition of r , letting , we obtain
So .
But , which is equivaluent to and , which is valid because , so .
Finally, the conclusion holds properly.
The next two theorems which are presented are related to data dependence and Ulam-Hyers stability. For more information about this notions we remind the articles [6], [10] and [11].
Theorem 4.3. Let ( ) be a complete metric space and be two multivalued modified Hardy Rogers operators, i.e.
with all the above coefficients positive.
Let’s suppose that and . Also, suppose that there exists , such that , for each .
Then
.
Proof. Let’s consider . This means that .
Let , i.e. . We denote by .
From the proofs of the previous theorems, we remind that we have shown ,
where , with arbitrary taken as in the previous proofs.
So , with and . This inequality chain is obtained because for , there exists , such that .
Analogous, we have that for , there exists , such that
,
where .
All the above inequalities implies that .
Letting , we get that .
So, the conclusion holds.
Now, the next fixed point theorem involves Ulam-Hyers stability of the fixed point inclusion.
Theorem 4.4. Let be a multivalued modified Hardy Rogers contraction with positive coefficients ( ), with .
Let and , such that .
Then, there exists such that .
Proof. Let and , such that .
Since is compact for the above , it implies that there exists , such that .
From the previous proofs, we have that , with considered above.
Then , that is .
This means that the conclusion is valid under the theorem’s hypotheses.
In the next two theorems and in the last corollary we present local versions involving two metrics and homotopy results with respect to modified multivalued Hardy-Rogers operators. For homotopy-type results we let the reader follow [1] and [2] and [5].
Theorem 4.5. Let ( ) be a complete metric space. Let and .
Let be another metric on and let be a multivalued operator.
Let’s suppose the following assumptions are satisfied
(1) there exists such that , for each
(2) If , then is a closed operator,
If , then ,
(3) for each , we have that is a multivalued modified Hardy Rogers contraction with respect to the metric , i.e.
(4) , with , where all the Hardy-Rogers type coefficients are positive.
Then, we have that there exists , such that .
Proof. From the hypotheses we have that .
Then, for there exists such that
.
This means that . We have that
Then .
Since , then . So there exists such that .
So .
This means that .
In a similar manner, for and in we have that
Then .
Since , then . So there exists such that .
So, applying triangular inequality, we obtain
.
This means that .
So, we have created a sequence , with the following properties :
(i) , for each ,
(ii) , for each ,
(iii) .
It is easy to see that is a Cauchy sequence in ( ).
Using the fact that , for each , it implies that is a Cauchy sequence in ( ).
Because ( ) is a complete metric space, there exists such that .
Furthermore, we have two cases to analyze.
I If , since is a closed operator, then .
II If , we have that
. It follows that
Letting , we get the following inequality
Now, since , as in the proofs of the previous theorems, it follows that . Moreover, because and has closed values, we have that .
Now, the last main result of this section involves a theorem regarding the homotopy of a modified multivalued Hardy-Rogers operator.
Theorem 4.6. Let ( ) be a complete metric space and two metrics on such that there exists , with , for each .
Let an open subset and a closed subset of , such that .
Let’s consider the multivalued operator , which satisfies the following conditions :
(a) , for each and
(b) there exists positive coefficients with as in the previous theorem, such that for each and , we have that
, where
(c) there exists an increasing, continuous function , such that
, for each and
(d) is a closed operator.
Then, we have the following equivalence relation
has a fixed point if and only if has a fixed point.
Proof. Let’s suppose that has a fixed point .
From the assumption (a), we get that .
Let’s denote . Then is nonempty, because .
On the set , we define a partial order relation, i.e.
if and only if and , for each and .
Let , with being a totally ordered subset of Q .
Moreover, denote by .
Now we define the sequence , such that , with .
As in [5], we have that
. This implies that and therefore is a Cauchy sequence with respect to the metric .
Using the fact that is a complete metric space and that there exists , such that for each , we obtain that , with .
Since and is a -closed operator, we have that .
From assumption and therefore .
Since is a totally ordered subset of Q , it follows that , for each , so is an upper bound for .
Using the well known Zorn’s Lemma, admits a maximal element, i.e. .
Now we show that .
Let’s suppose the contrary, i.e. that . We choose and such that , with .
Then
.
Since , it implies that , therefore .
So .
But we know that , so satisfies the assumptions of the previous theorem, therefore for each , there exists satisfying the property that has a fixed point, that is , which implies that .
But . This means that , which is a contraction. So .
For the other implication, we show that has a fixed point by swapping with in the first part of the proof. So, we get the desired result.
When the metric functionals and are identical, we have the following corollary.
Corollary 4.7. Let , with ( ) a complete metric space, open and closed.
Let a closed operator, satisfying the following assumptions
(a) , for each and
(b) be a Hardy Rogers modified multivalued contraction with respect to , for each
(c) , for each and for each , with increasing and continuous.
Then
has a fixed point if and only if has a fixed point.
Conflict of Interests
The authors declare that there is no conflict of interests.
References
[1] C.D. Agarwal, D. O’Regan, Fixed point theory for generalized contractions on spaces with two metrics, J. Math. Anal. Appl. 248 (2000), no.2, 402-414.
[2] C.D. Agarwal, J.H. Dshalalow, D. O’Regan, Fixed point and homotopy results for generalized contractive maps of Reich type, Appl. Anal. 82 (2003), no.4, 329-350.
[3] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.
[4] P.S. Kumari, D. Panthi, Connecting various type of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings, Fixed Point Theory Appl. 2016 (2016), Article ID 15.
[5] T. Lazăr, D. O’Regan, A. Petruşel, Fixed points and homotopy results for Ciric-type multivalued operators on a set with two metrics, Bull. Korean Math. Soc. 45 (2008), no.1, 67-73.
[6] A. Oprea, Fixed point theorems for multivalued generalized contractions of rational type in complete metric spaces, Creat. Math. Inform. 23 (2014), 99-106.
[7] N.S. Papageorgiou, S. Hu, Handbook of Multivalued Analysis (vol. I and II), Kluwer Acad. Publ., Dordrecht (1997 and 1999).
[8] L. Paunović, P. Kaushik, S. Kumar, Some applications with new admissibility contractions in b-metric spaces, J. Nonlinear Sci. Appl. 10 (2017), 4162-4174.
[9] A. Petruşel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn. 59, 169-202.
[10] I.A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219.
[11] I.A. Rus, A. Petruşel, A. Sîntmărian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003), no.8, 1947-1959.
[12] R.J. Shahkoohi, A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 373.
[13] W. Sintunavarat, Generalized Ulam-Hyers stability, well-posedness, and limit shadowing of fixed point problems for -contraction mappings in metric spaces, Sci. World J. 2014(2014), article ID 569174, 7 pages.
[14] F. Zabihi, A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, J. Appl. Math., 2014(2014), Article ID 929821.