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.
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
for all .
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 f-weak 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.
We recall the following lemma.
Let and bef-weak compatible. If for some and for allx, yinY, where , , then .
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.
Theorem 9LetYbe an arbitrary non-empty set, be a metric space, and be such that
that is, ,
is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that
which trivially implies that
Suppose also that
where is such 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.
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 , hence and
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 , ,
which tends to zero as .
Since is complete, then the sequence has a limit in , say u. Let and prove that .
Suppose that , then, by (2), we have
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
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
Now, if f and T are occasionally coincidentally idempotent, then for some . Then we have
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 non-empty set, be a metric space, and be 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 certain andqdetermined 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
for all x, y in Y and . □
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 .
Remark 13 According to Remark 12, for fixed and ψ satisfying (4), an admissible function γ can be obtained by taking
provided that and for every .
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
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).
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).
Theorem 19In Theorem 9, we can assume, instead of condition (2), one of the inequalities
where , and .
Similarly, in Corollary 10, we can consider one of the contractivity conditions
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 non-empty set, be a metric space, and be such that conditions (1), (3) hold and
where and is 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 non-empty set, be a metric space, and be such that conditions (1), (3) hold and
where (satisfying (11) for ) and is 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.
( ) 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 T-regular with respect to f if .
Theorem 23LetYbe an arbitrary non-empty set, be a metric space, and be such that condition (1) holds and
for all , where satisfies ( ) and is nonincreasing.
Suppose also thatTxis a compact set for every .
If is complete, then
(i) there exists an asymptoticallyT-regular sequence with 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
Note that, in the previous inequalities, we have used that . If , then and is asymptotically T-regular 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 T-regular 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
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 .
In consequence, for , we get
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 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.
Theorem 24LetYbe an arbitrary non-empty set, be a metric space, and be such that (1) holds and
for all , where ( ) are bounded on bounded sets, ris some fixed positive real number and is a Lebesgue measurable mapping which is summable on each compact interval and for each . Suppose that there exists an asymptoticallyT-regular sequence with respect tofinY. If is complete or
thenfandThave a coincidence point. Further, iffandTare occasionally coincidentally idempotent, thenfandThave a common fixed point.
Proof By hypotheses,
Since is asymptotically T-regular 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
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 , 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
Since , we get
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 that is compact for all . If is 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 ,
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 [, Theorem 3.3].
Theorem 27LetYbe an arbitrary non-empty set, be a metric space, and be such that (1) and (21) hold, where (