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; multi-valued mappings
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 multi-valued mappings are, for instance, [1-5]. Coincidence and common fixed points of nonlinear hybrid contractions (i.e., contractions involving single-valued and multi-valued mappings) have been recently studied by many authors. To mention some of the achievements, we cite, for example, [6-12].
The concept of commutativity of single-valued mappings  was extended in  to the setting of a single-valued mapping and a multi-valued mapping on a metric space. This concept of commutativity has been further generalized by different authors, viz weakly commuting , compatible , weakly compatible . It is interesting to note that in all the results obtained so far concerning common fixed points of hybrid mappings the (single-valued and multi-valued) mappings under consideration satisfy either the commutativity condition or one of its generalizations (see, for instance, [6-10]). In this note, we show the existence of fixed points of hybrid contractions which do not satisfy any of the commutativity conditions or its above-mentioned generalizations. Our result extends and improves several well-known 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 [19-21] and generalized contractive hybrid pairs are considered in . Finally, in , fixed point results are proved in topological vector space valued cone metric spaces (with nonnormal cones).
For a metric space , let and denote respectively the hyper-space of non-empty closed bounded and non-empty closed subsets of X, where H is the Hausdorff metric induced by d. For and , we shall use the following notations:
We recall some definitions.
The mappings f and T are said to be weakly commuting on X if
Definition 5 The mappings f and T are said to be coincidentally commuting if they commute at their coincidence points.
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.
We recall the following lemma.
We remark that the above-mentioned lemma has been used in [8,9] and  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 integral-type contractivity condition. In contrast to , we avoid the complete character of the base space X, and we introduce hybrid mappings. With respect to the study in , we consider here occasionally coincidentally idempotent mappings.
which trivially implies that
Suppose also that
ThenTandfhave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof In view of (1) and Nadler’s remark in , 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  and using that , we can choose such that and for a certain . The continuation of this process allows to construct the two above-mentioned sequences and inductively.
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,
Hence, for n large enough, we have
Making n tend to +∞ in the previous inequality, we have
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 T-regular with respect to f. However, this property can be deduced directly from the fact that
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
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 .
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 .
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
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).
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
Theorem 19In Theorem 9, we can assume, instead of condition (2), one of the inequalities
Similarly, in Corollary 10, we can consider one of the contractivity conditions
Proof It follows from the inequality
hence, for instance,
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.
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.
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.
(ii) fandThave a coincidence point.
Further, if fandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Also, we have
Now, we recall that
for every n, which implies that
Following this procedure, we prove that
We check that the right-hand 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.
Now we state some fixed point theorems for Kannan-type multi-valued mappings which extend and generalize the corresponding results of Shiau et al. and Beg and Azam [6,27]. A proper blend of the proof of Theorem 9 and those of [, Th. 6, Th. 7, Th. 8 respectively] and [, Theorems 3.1, 3.2, 3.3] will complete the proof.
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 asymptoticallyT-regular sequencewith respect tofinY. Ifis complete or
thenfandThave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof By hypotheses,
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
Now, suppose that is complete. Note that is closed and bounded for every . Take fixed. By the results in , we can affirm that for every , there exists such that .
In this case,
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.
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 ,
This implies that
The following result extends [, Theorem 3.3].
Remark 28 Note that condition (23) in Theorem 27 cannot be replaced by
Remark 29 In Theorem 27, condition (23) can be replaced by the following:
The authors declare that they have no competing interests.
Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.
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 MTM2010-15314, and co-financed by EC fund FEDER.
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