#### Abstract

This paper is aimed at proving a common fixed point theorem for -Kannan mappings in metric spaces with an application to integral equations. The main result of the paper will extend and generalise the recent existing fixed point results in the literature. We also provided illustrative examples and some applications to integral equation, nonlinear fractional differential equation, and ordinary differential equation for damped forced oscillations to support the results.

#### 1. Introduction and Preliminaries

In 1922, Banach [1] established a fixed point theorem in a metric space which states that if is a complete metric space and is a contraction map, i.e., for all and , then has a unique fixed point or has a unique solution. Since then, researchers are finding the ways to determine the fixed points of the maps by changing one or more conditions such as contractive condition, continuity of the maps, and completeness of the space etc. Kannan [2] gave an alternative contractive condition which was different from the Banach contraction condition. In 1968, Kannan [2] used this new contractive condition and proved the following theorem for self-mappings in complete metric spaces as a result of a generalisation of the Banach fixed point theorem.

Theorem 1 (see [2]). *Let be a complete metric space and a self-mapping be a mapping such that
for all and . Then, has a unique fixed point and for any the sequence of iterate converges to .*

An equivalent form of (1), for some .

In 1959, Connell [3] gave an example of a metric space which is not complete and every contraction on has a fixed point. Also, Subrahmanyam [4] proved the converse of the Banach fixed point theorem using Kannan mapping. Moreover, the assumption of continuity of the mapping and the compactness condition on metric space is required for the existence of a fixed point for a strict type Kannan contraction.

In 2000, Branciari [5] introduced a class of generalised metric spaces by replacing triangular inequality to similar ones which involve four or more points instead of three and improved the Banach contraction mapping principle. This motivated several researchers to prove fixed point results in such spaces. For more details on the fixed point theory of a generalised metric space, we refer to the reader [6–11]. In 2008, Azam and Arshad [12] using the concept of Branciari [5] investigated fixed points for the mappings given by Kannan [2] by applying the rectangular property in a generalised metric space.

We will require the following definitions and preliminary results to prove our results.

*Definition 2 (see [12]). *Let be a nonempty set. Suppose that the mapping satisfies
(i), for all and if and only if (ii), for all (iii), for all and for all distinct points [rectangular property]

Then, is called a generalised metric and is a generalised metric space.

*Definition 3 (see [5]). *Let be a metric space. A mapping is said to be sequentially convergent if we have, for every sequence , if is convergence then also is convergence. is said to be subsequentially convergent if we have, for every sequence , if is convergence then has a convergent subsequence.

*Definition 4 (see [13]). *Let be a topological space. If is a sequence of points of , and if is an increasing sequence of positive integers, then the sequence defined by setting is called a subsequence of the sequence . The space is said to be sequentially compact if every sequence of points of has a convergent subsequence.

For more details on the sequentially convergent property, one can see [14, 15].

In 2011, Moradi and Alimohammadi [16] generalised Kannan’s results, by using the sequentially convergent mappings and rectangular property in a metric space. Since then, several researchers involved in investigations of Kannan’s contraction mappings using a rectangular property in different spaces. For more details, one can see [12, 17–20] and the references therein. Furthermore, Morandi and Alimohammadi [16] investigated and extended Kannan’s mapping [2] by using the concept due to Branciari [5]. They proved results on two self-mappings as follows.

Theorem 5 (see [16]). *Let be a complete metric space and be mappings such that is continuous, one-to-one, and subsequentially convergent. If and
then has a unique fixed point. Also, if is sequentially convergent then for every , the sequence of iterates converges to this fixed point.*

Wardowski [21] gave an interesting generalisation of the Banach fixed point theorem using a different type of contraction called -contraction. Since then, many researchers following his approach to construct new fixed point theorems for which one can see [22–27] and the references therein.

Wardowski [21] gave the following definitions.

Let be a function defined as , which satisfies the following conditions:

(F1) is strictly increasing, i.e., for all such that ,

(F2) For each sequence of positive numbers if and only if

(F3) There exists such that

*Definition 6 (see [21]). *A mapping is said to be a -contraction if there exists , such that

In 2012, Wardowski [21] introduced a generalization of Banach contraction principle, which is as follows:

Theorem 7 (see [21]). *Let be a complete metric space and be a -contraction. Then, has a unique fixed point and for every a sequence is convergent to .*

In 2019, Goswami et al. [22] defined -contractive type mappings in -metric spaces and proved some fixed point results with suitable examples. Recently, Batra et al. [28] noticed that the definition introduced by Goswami et al. [22] is not meaningful in general. They provided suitable examples to support their opinion on this definition. Therefore, Batra et al. [28] gave the notions of -contraction and Kannan mapping to define a new class of contractions called -Kannan mappings which is in true sense a generalization of Kannan mappings.

Motivated by Batra et al. [28], we use the following notations: Let be a nonempty set and denotes the metric space with metric . Define the cardinality of a set by and denotes the set of all fixed points of a mapping .

Batra et al. [28] gave a new generalization family of contraction called -Kannan mapping and introduced the following definition.

*Definition 8 (see [28]). *Let be a mapping satisfying (F1)–(F3). A mapping is said to be an -Kannan mapping if the following holds:

(K1)
(K2) such that
for all with .

The remark presented below is due to Batra et al. [28].

*Remark 9 (see [28]). *By properties of , it follows that every -Kannan mapping on a metric space satisfies the following condition:
for every .

Further, it is concluded that . Let be a self-map of a metric space . We say that is a Picard operator (PO) if has unique fixed point and for all .

Then, the family of all functions satisfying the condition is denoted by .

One can use the following examples in Batra et al. [28] of such functions which satisfy .

*Example 10 (see [28]). *Let be defined as . Then, clearly, (F1)-(F3) are satisfied by . In fact, (F3) holds for every ,
for all with .

Thus, if is a Kannan mapping with constant satisfying
for every .

*Example 11 (see [28]). *Let be defined as . Then, (F1)-(F3) are satisfied by .
for all with .

The following lemma was proved by Batra et al. [28]

Lemma 12 (see [28]). *Let be a metric space and be an F-Kannan mapping. Then, as for all .*

Batra et al. [28] introduced an *F*-Kannan mapping using the properties by Subrahmanyam [4] which is an extension of Goswami et al. [22] and Wardowski [21] results. They proved the following result.

Theorem 13 (see [28]). *Let be a complete metric space and suppose is an F-Kannan mapping, then is a Picard operator (PO).*

In 2017, Gopal et al. [29] specified the fundamental properties for a fixed point theorem which ensures the existence of a common fixed point for suitable assumptions. Those assumptions are sufficient and include conditions of commutativity, containment of ranges of mappings, continuity of at least one mapping or weaker notion, contractive, and all essential common fixed point theorem attempts to obtain or soften required values of one or more such conditions.

*Definition 14 (see [30]). *Let be a pair of self-mappings on a metric space . Then, the coincidence point of the pair is a point such that , then is called coincidence point of the pair . If , then is said to be a common fixed point.

*Definition 15 (see [31]). *Let be self-mappings of a nonempty set . A point is the coincidence point of and if . The set of coincidence point of and is denoted by .

*Definition 16 (see [31, 32]). *Let be a pair of self-mappings on a metric space . Then, the pair () is said to be
(i)Commuting if, for all , ,(ii)Weakly commuting if, for all ,(iii)Compatible if , whenever is a sequence in such that ,(iv)Weakly compatible if, for all , for every coincidence point .

This paper is aimed at extending and generalising the results due to Batra et al. [28], and Morandi and Alimohammadi [16] using a pair of two self-mappings in -Kannan mapping. Doing so, we will be able to extend several other results of the same setting in the literature. We will provide some applications of the theorem to the integral equation, nonlinear fractional differential equation, and ordinary differential equation for damped forced oscillations to support the results.

#### 2. Main Results

We present the main result of this paper by assuming a map to be sequentially convergent with a pair of two self-mappings in -Kannan mappings. We shall start with the extension of Definition 8 using a pair of two self-mappings in the -Kannan mapping setting.

*Definition 17. *Let be a mapping satisfying (F1)–(F3). A pair of two self-mapping is said to be an -Kannan mapping if the following holds:

(FK1)
(FK2) There exists such that
for all with .

By following Batra et al. [28], we presented the remark as below.

*Remark 18. *By properties of , it follows that every -Kannan mapping on a metric space satisfies the following condition:
for every .

We give the following examples in the context of a pair of two mappings:

*Example 19. *Let be defined as . Then, clearly, (F1)–(F3) are satisfied by . In fact, (F3) holds for every . Moreover, condition (13) above takes the form:
for all with .

Thus, if is a Kannan mapping with constant satisfying for every . Then, it also satisfies (15) and (13) with .

*Example 20. *Let be defined as . Then, (F1)–(F3) are satisfied by . Condition (13) above takes the form:
for all with .

We prove the following lemma which will be useful in proving of the main theorem.

Lemma 21. *Let be a metric space and be an -Kannan mapping. Then,
for all .*

*Proof. *Let be arbitrary. If for some , then sequence converges in and hence the sequence for all .

Assume that for any . Then, by (13) with , we get
From Remark 18, we have
Using (20) in (19), as results yield to
Letting in (21), we get
which is a contradiction. Hence, as .

Motivated by Morandi and Alimohammadi [16], we prove the extended version of Theorem 5 using -Kannan mappings with a pair of two self-mappings in metric space.

Theorem 22. *Let be a complete metric space and be an F-Kannan mapping such that is continuous, injection, and subsequentially convergent. If , and
then has a unique fixed point. Also, if is subsequentially convergent then for every the sequence of iterates converges to this fixed point.*

*Proof. *Assume be an arbitrary point in . Let the sequence be defined by and , for .

Using inequality (13), we obtain
Since is strictly increasing, by using Remark 9, we deduce
and hence
By (F1), this implies that
Consequently, we get
so,
Similarly, we obtain
which implies that
Applying (29) in (31), we obtain
by (F1).

Using induction and (29), we deduce
Letting in (33) and using property (F2) of results in
By Lemma 21, we have as . Denote , for all and , for -Kannan mappings.

Using condition of the function there exists such that
From (33), for every , we have
On taking limit as in (36), we get
From (39), there exist such that , for all , which follows that
Therefore, converges.

By (40), we prove that is a Cauchy sequence since is complete. Consider such that ,
This shows that the series converges, which implies that
So, for every in . Hence, is a Cauchy sequence in . The completeness of ensures the existence of such that
By (43), it follows that as . By continuity of and , we have
Since is a complete metric space, there exists such that
Now, we prove that is a fixed point of . Thus, by (iii) of Definition 2, we have
By Remark 18, it implies that
Applying (47) in (46), we obtain
Letting and using Lemma 21 in above inequality, we get
That is, .

Next, we prove that is a unique fixed point of . Assume the contrary, i.e, there exists such that . Let and is a fixed point of . Using Remark 18 and Lemma 21, it follows that which is a contradiction. Thus, is a PO on .

Moreover, is a subsequentially convergent, has a convergent subsequence, and there exists and so that . Since is continuous and
Due to the continuity of , it implies that
By (45), we conclude that
Using Remark 18 and (ii) of Definition 2, we get
Thus, using equation (13) and (iii) of Definition 2, we have
As is sequentially increasing, this implies that
By Lemma 21 when for any and (13), we obtain
Using (56) and (57) in (55), we obtain
which follows
Letting in (59), we obtain
Since is injection, . So, has a fixed point. As is sequentially convergent, we conclude that converges to the fixed point of . Implying that and , then, there exists a point such that , that is, is a common fixed point of and . which satisfies all fundamental property of Definition 16.

Dasgupta et al. [33] and Moradi and Alimohammadi [16], in their work, considered an example in which Kannan Theorem is not applicable. At the same time, generalised Kannan mappings imply the existence of a fixed point for the considered mapping. In this work, we use one more example of this type which satisfies -Kannan mapping.

*Example 23. *Consider the sequence and be a Euclidean metric on . Then, is a complete metric space. Let the mapping be determined as follows:
for . Let there exists , so that for all condition (1) is satisfied although is not true for every , which is a contradiction, hence, Kannan’s theorem cannot be applicable.

The mapping be determined as

For all , is continuous, injection, and subsequentially convergent.

Now, let , . Then, we prove that is an -Kannan mapping with respect to and .

By using (KF2) with , we note that (23) becomes

To see this, we now calculate for .

Applying (64),(65), and (66) in (63) becomes

By Theorem 22, has unique fixed point that is .

#### 3. Some Applications

In this section, we will provide three applications of the theorem proved in the previous section.

##### 3.1. Existence of a Solution for an Integral Equation

In this section, as inspired by Nashine et al. [34], we establish the existence of a solution for the following Volterra type integral equation: This problem is equivalent to the integral equation where and are continuous functions. Let be the space of all continuous functions defined on . Notice that endowed with metric is a complete metric space and can be equipped with the partial order given by , or for all . It was shown by Nieto and Rodrigurz-Lopez [35] that is regular. For more applications to nonlinear integral equations, one can see [36–38] and the references therein.

Now, we define a mapping by

If is a fixed point of , then is a solution of (68).

We prove our results, by establishing the existence of a common fixed point for a pair of self-mappings.

Theorem 24. *Let be self-maps of a metric space such that the following condition holds
*(1)*For all and , we have
*(2)*There exists two functions with constants such that, for all , we have
*(3)*For
*

Then, integral equation (68) has a solution

*Proof. *From (a), for all , we have
Therefore, for all .

By using condition (2) of Theorem 24, we obtain
Thus, for all . Hence, is a common fixed point of and , also a solution to integral equation. Then, the integral equation (68) has a solution .

##### 3.2. Existence of a Solution for Nonlinear Fractional Differential Equation

The purpose of this section is to provide an application of Theorem 22 to get a common solution of a nonlinear fractional differential equation, where we can apply -Kannan mappings in metric spaces.

Here, we investigates the Caputo derivative with the fractional order of the nonlinear fractional differential equation. This form of fractional derivative for a continuous function is given as where denotes the integer part of the real number (see [37, 39]). Also, the Riemann-Liouville fractional integral of order is given by

The Caputo fractional differential equation has several applications in mathematics, i.e., in image processing, digital data processing, electrical signal, acoustics, physics, and probability theory (one can see in [40]). The following nonlinear fractional differential equation is inspired by Kilbas et al. [41], Baleanu et al. [39], Budhia et al. [42], and Kanwal et al. [37]: where denotes the Caputo fractional derivative of order and is a continuous function.

Consider the space of the continuous function defined on . Suppose that is a Banach space, and Let be the Banach space of all continuous functions from into with the norm for (one can see in [43]).

This space defines the metric by

. This is a complete metric space.

Nonlinear fractional equation (78) can be written as

A function is a solution of the fractional differential integral equation (80) if and only if is a solution of the nonlinear fractional differential equation (78).

Now, we prove the following theorem.

Theorem 25. *Suppose the following condition hold:
*(i)* is sequentially continuous*(ii)*There exists a continuous function , such that
for all and for all such that and a constant with a constant , such that
*

Then, fractional differential equation (78) has a common solution as a fixed point .

*Proof. *Let us define by
for , then is sequentially continuous. Suppose that
this implies that and posses a fixed point . To prove the existence of fixed point of , we prove that is sequentially and contraction. To show is sequentially continuous, let , for all . By condition (ii), we have