### 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; multi-valued mappings### Introduction

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 [13] was extended in [14] 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 [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 (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 [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
*d*. For

and

We recall some definitions.

**Definition 1** Mappings *f* and *T* are said to be *commuting* at a point
*f* and *T* are said to be *commuting* on *X* if

**Definition 2** Mappings *f* and *T* are said to be *weakly commuting at a point*

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
*X* such that

**Definition 4** The mappings *f* and *T* are said to be *f*-*weak compatible* if

(i)

(ii)

whenever
*X* such that

Let
*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
*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

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]

*Let*
*and*
*be**f*-*weak compatible*. *If*
*for some*
*and*
*for all**x*, *y**in**Y*, *where*
*then*

We remark that the above-mentioned 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 integral-type
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 9***Let**Y**be 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*

*and*

*Suppose also that*

*where*
*is such that*

*and*

*Then**T**and**f**have a coincidence point*. *Further*, *if**f**and**T**are occasionally coincidentally idempotent*, *then**f**and**T**have a common fixed point*.

*Proof* In view of (1) and Nadler’s remark in [24], given the point
*Y* and
*X* such that, for each

Indeed, since

We claim that

where

Suppose that

where

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

which tends to zero as

Since
*u*. Let

Suppose that

where

Here, we have used that
*Tw* closed, and

Hence, for *n* large enough, we have

Making *n* tend to +∞ in the previous inequality, we have

and, therefore, since
*w* is a coincidence point for *T* and *f*.

Although this fact is not relevant to the proof, we note that

Indeed,

where

therefore

Then
*ψ*, we get
*n* and, therefore,
*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

where

If

which is a contradiction. Hence
*i.e.*, *fw* is a common fixed point of *f* and *T*. □

Let Φ denote the family of maps *ϕ* from the set

**Corollary 10***Let**Y**be an arbitrary non*-*empty set*,
*be a metric space*,
*and*
*be such that*

*for all**x*, *y**in**Y*, *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*
*and**q**determined by* (11). *Then**T**and**f**have a coincidence point*. *Further*, *if**f**and**T**are occasionally coincidentally idempotent*, *then**f**and**T**have 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
*γ*. Indeed, using that

and

**Remark 12** Assuming (8), condition (9) is trivially valid if
*γ* the identity mapping. Moreover, if

**Remark 13** According to Remark 12, for
*ψ* satisfying (4), an admissible function *γ* can be obtained by taking

provided that

**Example 14** Taking *ψ* as the constant function

so that we must choose *γ* as a nonnegative function satisfying that

**Example 15** A simple calculation provides that, for the function

**Example 16** Now, we choose

which is equivalent to
*γ* satisfying that

Of course,

**Example 17** Take

that is,

Now, for each
*z*.

It is easy to prove that for
*τ* is strictly negative on

Moreover,

The following example shows that Theorem 9 is a proper generalization of the fixed point results in [7-10].

**Example 18** Let
*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
*f* and *T* are not coincidentally idempotent, but
*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
*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 [7-9] and [10] cannot be applied to these mappings *f* and *T*.

**Theorem 19***In Theorem *9, *we can assume*, *instead of condition* (2), *one of the inequalities*

*or*

*where*
*and*

*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

Of course, the function

Note that, taking

**Corollary 20***Let 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*
*Then**f**and**T**have a coincidence point*. *Further*, *if **f**and**T**are occasionally coincidentally idempotent*, *then**f**and**T**have a common fixed point*.

**Corollary 21***Let 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*
*Then**f**and**T**have a coincidence point*. *Further*, *if **f**and**T**are occasionally coincidentally idempotent*, *then**f**and**T**have a common fixed point*.

Let

(

This property obviously holds if *η* is continuous since *η* attains its maximum (less than 1) on each compact

**Definition 22** A sequence
*T*-regular 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 23***Let**Y**be 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 that**Tx**is a compact set for every*

*If*
*is complete*, *then*

(i) *there exists an asymptotically**T*-*regular sequence*
*with respect to**f**in**Y*,

(ii) *f**and**T**have a coincidence point*.

*Further*, *if **f**and**T**are occasionally coincidentally idempotent*, *then**f**and**T**have a common fixed point*.

*Proof* For some
*Y*, let
*Y* such that

Using (1), we can choose

hence

Note that, in the previous inequalities, we have used that
*T*-regular with respect to *f*.

By induction, we construct a sequence
*Y* and
*n*,

and

Also, we have

It follows that the sequence
*t*. Clearly
*η*, there will exist

For this

which contradicts the assumption that
*i.e.*,
*T*-regular with respect to *f*.

We claim that
*ψ*, we get

Now, we recall that

for every *n*, which implies that

Following this procedure, we prove that

Therefore,

We check that the right-hand side in the last inequality tends to 0 as
*n*, *m*). Indeed, we check that
*η*. Given

In consequence, for

and this expression is bounded independently of *m*, *n*.

Hence
*p* in

Letting

Now, if *f* and *T* are occasionally coincidentally idempotent, then

Thus,
*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.*[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 24***Let**Y**be an arbitrary non*-*empty set*,
*be a metric space*,
*and*
*be such that* (1) *holds and*

*for all*
*where*
*are bounded on bounded sets*, *r**is 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 asymptotically**T*-*regular sequence*
*with respect to**f**in**Y*. *If*
*is complete or*

*then**f**and**T**have a coincidence point*. *Further*, *if**f**and**T**are occasionally coincidentally idempotent*, *then**f**and**T**have a common fixed point*.

*Proof* By hypotheses,

Since
*T*-regular with respect to *f* in *Y*, then

If

where the number of terms containing
*r*, and therefore fixed. Calculating the limit as

Therefore,

and, by the properties of *ψ*, we get
*u* is a coincidence point.

Now, suppose that

Given

By the hypothesis on
*n*, hence

and taking the limit as

In this case,

and
*f* and *T* are coincidentally idempotent, then

Since

Therefore

obtaining
*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
*X* and

**Theorem 26***In addition to the hypotheses of Theorem *24, *suppose that*
*is compact for all*
*If*
*is a cluster point of*
*then**z**is a coincidence point of**f**and**T*.

*Proof* Let
*u* is obtained in the proof of Theorem 24. Note that, for every

hence

In consequence,

Using that there exists a subsequence
*fz*, the properties of

then, taking the limit when

This implies that

and, by the properties of
*ψ*, we deduce that
*z* is a coincidence point of *f* and *T*. □

The following result extends [[10], Theorem 3.3].

**Theorem 27***Let**Y**be an arbitrary non*-*empty set*,
*be a metric space*,
*and*
*be such that* (1) *and* (21) *hold*, *where*
*are bounded on bounded sets and such that*