Abstract
In this note, some coincidence and common fixed points of nonlinear hybrid mappings have been obtained under certain noncommutativity conditions of mappings. Our results improve several known results in the field of hybrid fixed point theory.
MSC: 54H25, 47H10, 54C60.
Keywords:
coincidence point; fixed point; occasionally coincidentally idempotent; multivalued mappingsIntroduction
As a generalization of the Banach fixed point theorem, Nadler’s contraction principle has lead to an excellent fixed point result in the area of nonlinear analysis. Some other works focused on fixed point results for multivalued mappings are, for instance, [15]. Coincidence and common fixed points of nonlinear hybrid contractions (i.e., contractions involving singlevalued and multivalued mappings) have been recently studied by many authors. To mention some of the achievements, we cite, for example, [612].
The concept of commutativity of singlevalued mappings [13] was extended in [14] to the setting of a singlevalued mapping and a multivalued mapping on a metric space. This concept of commutativity has been further generalized by different authors, viz weakly commuting [15], compatible [16], weakly compatible [8]. It is interesting to note that in all the results obtained so far concerning common fixed points of hybrid mappings the (singlevalued and multivalued) mappings under consideration satisfy either the commutativity condition or one of its generalizations (see, for instance, [610]). In this note, we show the existence of fixed points of hybrid contractions which do not satisfy any of the commutativity conditions or its abovementioned generalizations. Our result extends and improves several wellknown results in the field of hybrid fixed point theory. Some other recent related references are [17,18], where common fixed point theorems for hybrid mappings on a symmetric space are proved under the assumptions of weak compatibility and occasional weak compatibility. Some analogous results for the case of contractivity conditions of integral type are presented in [1921] and generalized contractive hybrid pairs are considered in [22]. Finally, in [23], fixed point results are proved in topological vector space valued cone metric spaces (with nonnormal cones).
Preliminaries
For a metric space , let and denote respectively the hyperspace of nonempty closed bounded and nonempty closed subsets of X, where H is the Hausdorff metric induced by d. For and , we shall use the following notations:
and
We recall some definitions.
Definition 1 Mappings f and T are said to be commuting at a point if . The mappings f and T are said to be commuting on X if for all .
Definition 2 Mappings f and T are said to be weakly commuting at a point if
The mappings f and T are said to be weakly commuting on X if
Definition 3 The mappings f and T are said to be compatible if for all and , whenever is a sequence in X such that and , as .
Definition 4 The mappings f and T are said to be fweak compatible if for all and the following limits exist and satisfy the inequalities:
whenever is a sequence in X such that and as .
Let denote the set of all coincidence points of the mappings f and T, that is, .
Definition 5 The mappings f and T are said to be coincidentally commuting if they commute at their coincidence points.
Definition 6 Mappings f and T are said to be coincidentally idempotent if for every , that is, if f is idempotent at the coincidence points of f and T.
Definition 7 Mappings f and T are said to be occasionally coincidentally idempotent (or, in brief, oci) if for some .
It should be remarked that coincidentally idempotent pairs of mappings are occasionally coincidentally idempotent, but the converse is not necessarily true as shown in Example 18 of this note.
Main results
We recall the following lemma.
Lemma 8[8]
Letandbefweak compatible. Iffor someandfor allx, yinY, where, , then.
We remark that the abovementioned lemma has been used in [8,9] and [10] to prove the existence of fixed points of hybrid mappings. However, we have noticed some typos in its original statement which have been rectified in the above statement without altering the proof.
Next, we prove a fixed point result for hybrid mappings under a general integraltype contractivity condition. In contrast to [20], we avoid the complete character of the base space X, and we introduce hybrid mappings. With respect to the study in [21], we consider here occasionally coincidentally idempotent mappings.
Theorem 9LetYbe an arbitrary nonempty set, be a metric space, andbe such that
is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that
which trivially implies that
and
Suppose also that
and
ThenTandfhave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof In view of (1) and Nadler’s remark in [24], given the point , we can construct two sequences in Y and in X such that, for each ,
Indeed, since , there exists such that . Besides, given , by Nadler’s remark in [24] and using that , we can choose such that and for a certain . The continuation of this process allows to construct the two abovementioned sequences and inductively.
We claim that is a Cauchy sequence. Using the inequality in (2) and also property (7), which is trivially valid for , it follows, for , that
where
Suppose that
so that
where we have also used (6) (a consequence of (4)), (7), (8) and (9). The previous inequalities imply that
which is a contradiction. In consequence,
where , by hypothesis, and hence is a Cauchy sequence in . This is clear from the following inequality, valid for , ,
Since is complete, then the sequence has a limit in , say u. Let and prove that .
Suppose that , then, by (2), we have
where
Here, we have used that , as , , as , due to and Tw closed, and
Hence, for n large enough, we have
Making n tend to +∞ in the previous inequality, we have
and, therefore, since and , we get , which is a contradiction. Hence , that is, w is a coincidence point for T and f.
Although this fact is not relevant to the proof, we note that since
Indeed,
where
therefore
Then and, by the properties of ψ, we get as . From the definition of , we deduce that for every n and, therefore, , so that is asymptotically Tregular with respect to f. However, this property can be deduced directly from the fact that
Now, if f and T are occasionally coincidentally idempotent, then for some . Then we have
where
If , then from inequality (10) and using (5) (which is guaranteed by (4)), we have that
which is a contradiction. Hence . Thus we have and , i.e., fw is a common fixed point of f and T. □
Let Φ denote the family of maps ϕ from the set of nonnegative real numbers to itself such that
Corollary 10LetYbe an arbitrary nonempty set, be a metric space, andbe such that,
for allx, yinY, where (satisfying (11) for a certain),
is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Suppose also that (7), (8) and (9) hold for a certainandqdetermined by (11). ThenTandfhave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof It is a consequence of Theorem 9 since (11) and (12) imply that
Remark 11 The condition
implies the validity of hypothesis (7) in Theorem 9 for the particular case of γ the identity mapping. Moreover, for , hypotheses (8) and (9) are trivially satisfied for this choice of γ. Indeed, using that , we get
and
Remark 12 Assuming (8), condition (9) is trivially valid if for every or, equivalently, for every , that is, for every . Note that this last condition is trivially valid for γ the identity mapping. Moreover, if for every , then for every and, therefore, if , then , obtaining (8) if .
Remark 13 According to Remark 12, for fixed and ψ satisfying (4), an admissible function γ can be obtained by taking
Example 14 Taking ψ as the constant function , , in the statement of Theorem 9, condition (7) is reduced to
so that we must choose γ as a nonnegative function satisfying that for (obviously, since ) in order to guarantee conditions (7), (8) and (9).
Example 15 A simple calculation provides that, for the function , , condition (7) is written as for and, therefore, in this case condition (8) is never fulfilled. If we take , , for and fixed, then (7) implies that for .
Example 16 Now, we choose , , where and are fixed. Note that the case has already been studied in Example 14. In this case , condition (7) is reduced to
which is equivalent to for . Note that this inequality implies, for , that . If we add the hypothesis for , then we guarantee the validity of conditions (8) and (9) due to Remark 12. Hence, we can take any nonnegative function γ satisfying that
Of course, and are valid choices.
Example 17 Take , . Condition (7) is equivalent to
that is,
Now, for each fixed, we calculate , which is obviously positive, and we check that its value is equal to z.
It is easy to prove that for fixed, the function is decreasing on . Indeed, the sign of its derivative coincides with the sign of the function and also with the sign of for . Now, the function τ is strictly negative on since and for .
Moreover, for each ; in consequence, for every . Therefore, if for every , then (7) follows. Note also that if , then . Finally, for , if we take such that for , we deduce the validity of (7), (8) and (9).
The following example shows that Theorem 9 is a proper generalization of the fixed point results in [710].
Example 18 Let be endowed with the Euclidean metric, let and be defined by and . Let be defined by for all . Then mappings f and T are not commuting and also do not satisfy any of its generalizations, viz weakly commuting, compatibility, weak compatibility. Also the mappings f and T are not coincidentally commuting. Note that , but and so f and T are not coincidentally idempotent, but and thus f and T are occasionally coincidentally idempotent. For all x and y in X, we have
Note that these inequalities are valid if
which is satisfied taking, for instance, the constant function . On the other hand, γ is chosen as the identity map and it satisfies (8) and (9).
Note that 0 is a common fixed point of f and T. We remark that the results of [79] and [10] cannot be applied to these mappings f and T.
Theorem 19In Theorem 9, we can assume, instead of condition (2), one of the inequalities
or
Similarly, in Corollary 10, we can consider one of the contractivity conditions
or
where, and (satisfying (11) for a certain) and the conclusion follows.
Proof It follows from the inequality
and the nonnegative character of a, b and ψ. Indeed, ,
hence, for instance,
Note that, in cases (16) and (17), it is not necessary to assume the nondecreasing character of the function ϕ since, using that , we deduce (14) and (15), respectively. □
Of course, the function is admissible in the results of this paper.
Note that, taking and in the inequalities of Theorem 19, we obtain the corresponding contractivity conditions of Theorem 9 and Corollary 10. On the other hand, taking and in Theorem 19, we have the following results, which are also corollaries of Theorem 9.
Corollary 20Let Y be an arbitrary nonempty set, be a metric space, andbe such that conditions (1), (3) hold and
whereandis a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Assume also that (7), (8) and (9) are fulfilled for a certain. ThenfandThave a coincidence point. Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Corollary 21Let Y be an arbitrary nonempty set, be a metric space, andbe such that conditions (1), (3) hold and
where (satisfying (11) for) andis a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Assume also that (7), (8) and (9) are fulfilled for a certain. ThenfandThave a coincidence point. Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Let be a function having the following property (see, for instance, [6,25]):
() For , there exist , such that implies .
This property obviously holds if η is continuous since η attains its maximum (less than 1) on each compact .
Definition 22 A sequence is said to be asymptotically Tregular with respect to f if .
The following theorem is related to the main results of Hu [[25], Theorem 2], Jungck [14], Kaneko [26], Nadler [[24], Theorem 5] and Beg and Azam [[6], Theorem 5.4 and Corollary 5.5].
Theorem 23LetYbe an arbitrary nonempty set, be a metric space, andbe such that condition (1) holds and
for all, wheresatisfies () andis nonincreasing.
Suppose also thatTxis a compact set for every.
(i) there exists an asymptoticallyTregular sequencewith respect tofinY,
(ii) fandThave a coincidence point.
Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof For some in Y, let and choose in Y such that . Then, by (20), we have
Using (1), we can choose such that and satisfying that
hence
Note that, in the previous inequalities, we have used that . If , then and is asymptotically Tregular with respect to f.
By induction, we construct a sequence in Y and in such that, for every n,
Also, we have
It follows that the sequence is decreasing and converges to its greatest lower bound, say t. Clearly . If , then by the property () of η, there will exist and such that
For this , there exists such that , whenever . Hence , whenever . Let . Then for , we have
which contradicts the assumption that . Thus ; i.e., as . Hence the sequence is asymptotically Tregular with respect to f.
We claim that is a Cauchy sequence. Let with , then, by the nonincreasing character of ψ, we get
Now, we recall that
for every n, which implies that
Following this procedure, we prove that
Therefore,
We check that the righthand side in the last inequality tends to 0 as . Since as , it suffices to show that is bounded (uniformly on n, m). Indeed, we check that is bounded for any sequence with nonnegative terms and tending to 0 as , using the property () of the function η. Given , by (), there exist , such that implies . Since , given , there exists such that, for every , we have . This implies that for every .
and this expression is bounded independently of m, n.
Hence is a Cauchy sequence in . Since is complete, converges to some p in . Let . Then . Next, we have
Letting , we get . Thus we have . Hence .
Now, if f and T are occasionally coincidentally idempotent, then for some . Then we have
Thus, . It follows that . Hence, fw is a common fixed point of T and f. □
Now we state some fixed point theorems for Kannantype multivalued mappings which extend and generalize the corresponding results of Shiau et al.[10] and Beg and Azam [6,27]. A proper blend of the proof of Theorem 9 and those of [[10], Th. 6, Th. 7, Th. 8 respectively] and [[9], Theorems 3.1, 3.2, 3.3] will complete the proof.
Theorem 24LetYbe an arbitrary nonempty set, be a metric space, andbe such that (1) holds and
for all, where () are bounded on bounded sets, ris some fixed positive real number andis a Lebesgue measurable mapping which is summable on each compact interval andfor each. Suppose that there exists an asymptoticallyTregular sequencewith respect tofinY. Ifis complete or
thenfandThave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof By hypotheses,
Since is asymptotically Tregular with respect to f in Y, then and are bounded sequences and , , as . This provides the property as , so that is a Cauchy sequence in .
If is complete, there exists such that as . Let be such that . Then
where the number of terms containing is a finite number depending on r, and therefore fixed. Calculating the limit as and taking into account that the length of the intervals in the last integral tends to zero, we get
Therefore,
and, by the properties of ψ, we get , which implies that and u is a coincidence point.
Now, suppose that is complete. Note that is closed and bounded for every . Take fixed. By the results in [24], we can affirm that for every , there exists such that .
Given , we choose and, for this fixed, we choose such that . Then
By the hypothesis on and the Cauchy character of , we deduce that is a Cauchy sequence. Since is complete, there exists such that . By hypotheses, for every n, hence
and taking the limit as , we get
In this case,
and , which implies that . Now, if f and T are coincidentally idempotent, then for some . Hence
Therefore
obtaining and . Since , we deduce that and . In consequence, and fw is a common fixed point of T and f. □
Remark 25 In the statement of Theorem 24, condition (22) can be replaced by the more general one
To complete the proof with this more general hypothesis, take into account that for , is a closed set in X and . Using that is complete, we deduce that is complete. Hence is a sequence in and it is a Cauchy sequence in . Therefore, there exists such that as . Note also that is a closed set in the complete space , then is complete and, therefore, a closed set, then . Once we have proved that as in , the proof follows analogously.
Theorem 26In addition to the hypotheses of Theorem 24, suppose thatis compact for all. Ifis a cluster point of, thenzis a coincidence point offandT.
Proof Let be such that , this is possible since is compact. It is obvious that a cluster point of is a cluster point of . Let be a cluster point of and , then we check that , where u is obtained in the proof of Theorem 24. Note that, for every ,
hence
In consequence,
Using that there exists a subsequence converging to fz, the properties of and the inequality
then, taking the limit when , we get and . To prove that , using that , we get
This implies that
and, by the properties of and ψ, we deduce that , which proves that z is a coincidence point of f and T. □
The following result extends [[10], Theorem 3.3].
Theorem 27LetYbe an arbitrary nonempty set, be a metric space, andbe such that (1) and (21) hold, where () are bounded on bounded sets and such that
ris some fixed positive real number andis a Lebesgue measurable mapping which is summable on each compact interval, andfor each. Suppose that
Ifis complete oris complete, thenfandThave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof Using Theorem 24, it suffices to prove that there exists an asymptotically Tregular sequence with respect to f in Y. Let and take in Y such that for every . Then
Hence,
or also, using the hypothesis on and ,
The properties of ψ imply that for every , and is nonincreasing and bounded below. Therefore it is convergent to the infimum, that is,
and is asymptotically Tregular with respect to f in Y. □
Remark 28 Note that condition (23) in Theorem 27 cannot be replaced by
since the infimum taking the sequence could be positive (we calculate the infimum in a smaller set).
Remark 29 In Theorem 27, condition (23) can be replaced by the following:
Indeed, since
then
Remark 30 In Theorem 27, if we are able to obtain a sequence with an infinite number of terms which are different, then we can relax condition (23) to the following:
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
Dedicated to Professor Jean Mawhin, on the occasion of his seventieth birthday.
We thank the editor, the anonymous referees and also Professor Stojan Radenović for their helpful comments and suggestions. This research was partially supported by the University Grants Commission, New Delhi, India; Ministerio de Economía y Competitividad, project MTM201015314, and cofinanced by EC fund FEDER.
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