Common Fixed Point Theorems for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.4652pt" version="1.1" viewBox="-0.0657574 -11.3994 10.5867 11.4652" width="10.5867pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g></svg>-</span>Kannan–Suzuki Type Mappings in TVS-Valued Cone Metric Space with Some Applications

Wangwe, Lucas; Kumar, Santosh

doi:https://doi.org/10.1155/2022/6504663

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Special Issue

Fixed Points and Computational Optimization

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 6504663 | https://doi.org/10.1155/2022/6504663

Common Fixed Point Theorems for -Kannan–Suzuki Type Mappings in TVS-Valued Cone Metric Space with Some Applications

Lucas Wangwe¹and Santosh Kumar¹

Academic Editor: Sun Young Cho

Received12 Jan 2022

Accepted23 Feb 2022

Published19 Apr 2022

Abstract

This research paper generalizes and extends various fixed-point results that demonstrate common fixed-point theorems for -Kannan–Suzuki type mappings in TVS-valued cone metric spaces. The results are supported using interpretative exemplifications and applications that include nonlinear fractional as well as two-point periodic ordinary differential equations.

1. Introduction

The fixed-point theory is at the foundation of nonlinear analysis, which is a prominent research area of mathematics. Fixed point theory is, in fact, a simple, powerful, and useful tool for nonlinear analysis. It also has fruitful applications in mathematics and in various scientific domains, including physics, chemistry, computer science, etc. As a result, this theory has attracted a large number of researchers who are guiding the theory’s growth in various areas.

In 1922, Banach [1] established a fixed point theorem in 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. In addition to an acceptable contraction condition, the metrical common fixed-point theorems usually include constraints on commutativity, continuity, completeness, and appropriate containment of ranges of detailed maps. The goal of researchers in this field is to weaken one or more of these conditions. The use of weak conditions of commutativity is to improve common fixed point theorems in analysis. Connell [2] provided an example of a noncomplete metric space , but every contraction on it has a fixed point. Kannan [3] proposed an alternative contractive condition that was not the same as the Banach contraction condition. Also, Subrahmanyam [4] proved the converse of Banach fixed-point theorem using Kannan mapping. Furthermore, to evaluate a fixed point for a stringent type Kannan contraction, the assumption of continuity of the mapping and the compactness requirement on metric space are necessary.

In 2007, Huang and Zhang [5] generalised the Banach fixed point theorem by introducing the structure of cone metric by substituting real numbers with an ordered Banach space and establishing a convergence criterion for sequences in a cone metric space. In normal cone metric space, Huang and Zhang [5] proved some fixed-point theorems for Kannan type contractive conditions; nevertheless, Rezapour and Hamlbarani [6] neglected this idea in some results by Huang. For normal and nonnormal cones in cone metric spaces, several authors have examined fixed point theorems and common fixed-point theorems for self-mappings. We refer to the reader [7–10] and the references therein. By relaxing the normalcy criteria set by Huang and Zhang [5], Beg et al. [11], investigated common fixed points for a pair of maps on topological vector space (TVS) valued cone metric spaces in 2009. They demonstrated that the class of TVS-valued cone metric spaces is larger than the class of cone metric spaces, used in [12–16] and the references therein. Recently, Hu and Gu [17] proved some fixed point theorems of -contractive mappings in Menger PSM-spaces. For a class of contractive mappings, Reich and Alexander [18] generalised fixed points and convergence results. In Hausdroff TVS, Ram and Lai [19] presented the existence results on generalised strong operator equilibrium problems. In TVS-Cone Metric Spaces, Dubey and Mishra [20] demonstrated some fixed-point results of single-valued mapping for -distance. Using some facts about topological vector space, Tas [21] constructed a new notion of a TVS cone - metric space. Lee [22] introduces chain recurrent set, trapping region, attracting set and repelling set for a flow on a TVS-cone metric space. By using generalised metric spaces, Ge and Yang [23] proved a common generalisation of TVS-cone metric spaces, partial metric spaces and -metric spaces, and a unified approach is proposed for some fixed point results. Later, Suzuki [24] and Rida et al. [25] gave a generalisation of the Banach contraction principle that characterises metric completeness.

Wardowski [26] used a new sort of contraction called -contraction to give an intriguing generalisation of the Banach fixed point theorem. Many scholars have used his method to build new fixed-point theorems since then. The associated results and references can be found in [27–30] and the references therein. Piri and Kumam [28], extended Wardowski’s [26] results in 2014 by introducing the notion of -Suzuki contraction and obtained some intriguing results utilising the Secelean [29] concept. In the complete -metric spaces, Alsulami et al. [31] demonstrated fixed points of generalised -Suzuki type Contractions. Budhia et al. [32] proved an extension of almost- and -Suzuki contractions with graph and demonstrated some applications to fractional calculus whereas Chandok et al. [33] formulated some fixed point results for the generalised -Suzuki type contractions in -metric spaces. Derouiche and Ramou [34] proved new fixed-point results for -contractions of Suzuki Hardy-Rogers type in -metric spaces and provided some applications. Beg et al. [11] proposed a fixed point of orthogonal -Suzuki contraction mapping on 0-complete -metric spaces with some applications. Mani et al. [35] introduced generalised orthogonal -contraction and orthogonal -Suzuki contraction mappings and proved some fixed point theorems for a self-mapping in orthogonal metric space. Vujakovic and Radenovic [36] introduced certain fixed point results for -contraction of Piri-Kumam-Dung-type mappings in metric spaces.

In 2019, Goswami et al. [27] introduced -contractive type mappings in -metric spaces and proved some fixed point results with suitable examples. Recently, Batra et al. [37] noticed in their subsequent analysis that the definition introduced by Goswami et al. [27] is not meaningful in general. Therefore, they provided suitable examples to support their opinion on this definition. Also, due to these reasons, Batra et al. [37] presented -contraction and Kannan mapping concepts for defining -Kannan mappings, which is, in a true sense, a generalisation of Kannan mappings.

This paper aims to extend and generalise the results due to Batra et al. [37], Filipovic et al. [38], Morales and Rojas [9], Rahimi et al. [39] and Wangwe and Kumar [40] using a pair of two self-mappings in -Kannan–Suzuki type mapping in TVS-valued cone metric space, where we consider a map to be sequentially convergent, one to one and continuous. By doing so, we will extend several other results of the same setting in the literature. Finally, we will provide some applications to the nonlinear Riemann–Liouville fractional differential equation and nonlinear Volterra-integral differential equation.

2. Preliminaries

The definitions, lemmas, and theorems will help us prove our main points in the upcoming sections.

In 1968, Kannan [3] developed a 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 [3]). Let be a self mapping on a complete metric space such thatfor all and . Then, possesses a unique fixed point and for any the iterate sequence converges to .

Equation (1) is equivalent tofor some .

Definition 1 (see [11]). Let be always a topological space and a subset of . Then, is called a cone if the following hold:(i) is a nonempty, closed and (ii) for all and nonnegative real number (iii)For given cone . If the interior of , is nonempty we say that is solid. If is solid cone, then is a component of , and in this case we use the notation to indicate that . Note that if and , then for all .

The following axioms satisfy TVS-valued cone complete metric space.

Definition 2 (see [11]). Let be a nonempty set and the mapping , satisfies the following:(i), for all and if and only if (ii), for all (iii), for all Then, is called a cone metric on , and is called topological vector space valued cone metric space.

Example 1 (see [12, 9, 41]). Let , and such that , where . Then, is a TVS-valued cone metric space.

The following definition is due to Beg et al. [11] in TVS-valued cone metric space.

Definition 3 (see [8]). Let be a topological vector space valued cone metric space, and let and be a sequence in . Then,(i) converges to whenever for every with there is a natural number such that for all . We denote this by(ii) is Cauchy sequence whenever for every with , there is a natural number such that for all .(iii) is called topological vector space valued cone metric space if every Cauchy sequence is convergent.

Definition 4 (see [42]). 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.

Definition 5 (see [43]). 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.

The extended version of sequentially convergent mappings in TVS-valued cone metric space is given as follows.

Definition 6 (see [9]). Let be a cone metric space, is a solid cone and . Then(i) is said to be continuous if for all ,(ii) is said to be sequentially convergent if we have, for every sequence , if is convergent, then also is convergent,(iii) is said to be subsequentially convergent if we have, for every sequence and is convergent, implies has a convergent subsequence.

In 2011, Filipovic et al. [38] generalised Theorem 3.1 and Theorem 3.5 from [9] by using the sequentially convergent mappings in cone metric space and considered to be a solid cone. They proved results on two self mappings as follows.

Definition 7 (see [38]). Let be a cone metric space and two mappings. A mapping is said to be - Hardy-Rogers contraction if there exists with such that for all .

Theorem 2 (see [38]). Let be a complete cone metric space and a solid cone, in addition let be a one-to-one, continuous mappings and a -hardy-Rogers contraction. Then,(i)For every the sequence is Cauchy.(ii)There is such that (iii) is sequentially convergent, then has a convergent, subsequence.(iv)There is a unique such that (v)If is sequentially convergent, then for each the iterate sequence converges to

Theorem 3 (see [38]). Let be a complete cone metric space and a solid cone, in addition let be a one-to-one, continuous mappings and such that and thatholds for some and for all . Then has property .

Remark 1 (see [44]). Let denote the fixed point set of a map . A map has property if for each . We shall say that a pair of maps and has property if for each .

Secelean [29] proved the following lemma.

Lemma 1 (see [29]). Let be an increasing function and be a sequence of positive real numbers. Then the following holds:(a)If , then (b)If and , then

Let be the set of all functions defined as , which satisfies the following conditions: (F1) is strictly increasing i.e., for all such that (F2″) there is a sequence of positive real numbers such that or (F3″) is continuous on

The following function belongs to :(i)(ii)(iii)

Definition 8 (see [28]). Let be a metric space. A mapping is said to be an -Suzuki contraction if there exists , such that for all with where .

In 2014, Piri and Kumam [28] established a generalisation of Banach contraction principle, which is as follows:

Theorem 4 (see [28]). Let be a complete metric space and be a -Suzuki contraction. Then has a unique fixed point and for every a sequence is convergent to .

Remark 2 (see [28]). We denote by the set of all functions satisfying -suzuki type contraction condition due to [28, 29] and let denote by the set of all functions satisfying -contraction condition by Wardowski [26], then(i)(ii)(iii)

For more details on -Suzuki contraction mapping, one can see [31–33] and the references therein.

Motivated by Batra et al. [37], we use the following notations: Let be a nonempty set and denotes the metric space with metric . Let the cardinality of a set is denoted by and is set of all fixed points of a mapping .

Batra et al. [37] gave a new generalisation family of contraction called -Kannan mapping and introduced the following definition:

Definition 9 (see [37]). Let be a mapping satisfying . 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. [37].

Remark 3 (see [37]). By properties of , it follows that every -Kannan mapping on a metric space , satisfies following condition:for every .

Furthermore, it is concluded that . Let be a self map of a metric space . is said to be a Picard operator (PO) if has unique fixed point and for all .

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

We recall the following examples from Batra et al. [37] of such functions which satisfies :

Example 2 (see [37]). Let be defined as . Then clearly, are satisfied by . In fact holds for every for all with .
Thus, if is a Kannan mapping with constant satisfyingfor every , then it also satisfies (8) and (11) with . In fact, whenever , then from (12), we get or .

The following lemma introduced by Batra et al. [37].

Lemma 2 (see [37]). Let be a metric space and be a -Kannan mapping. Then, as for all .

Batra et al. [37] introduced a -Kannan mapping using the properties by Subrahmanyam [4] which is an extension of Goswami et al. [27] and Wardowski [26] results. They proved the following result.

Theorem 5 (see [37]). Let be a complete metric space and suppose is a -Kannan mapping, then is a Picard operator (PO).

Using the following definitions, we introduce some fundamental properties for a fixed point and common fixed point theorems.

Definition 10 (see [45]). Let be a pair of self-mappings on a metric space . Then 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 of and .

Definition 11 (see [46]). Let be self-mappings of a nonempty set . A point is coincidence point of and if . The set of coincidence point of and is denoted by .

Definition 12 (see [46, 47]). Let be a pair of self-mappings on a metric space . Then, the pair is said to be as follows:(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 .

3. Main Results

To prove this section’s main results, we commerce by obtaining a more general version of Definition 8 and 9 using a pair of two self mappings in -Kannan–Suzuki type mapping setting. We denotes as a TVS-valued cone metric space.

Definition 13. Let be a mapping satisfying . A pair of two self mapping is said to be an -Kannan–Suzuki type mapping if the following holds: (FKS1) (FKS2) there exists such thatfor all , with and .

Following remark is motivated by the work of Batra et al. [37] given as follows.

Remark 4. By properties of , it follows that every -Kannan–Suzuki type mapping on a TVS-valued cone metric space , satisfies the following condition:for every .

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

Example 3. Let be defined as . Then clearly, are satisfied by . In fact holds for every . Moreover, condition (14) takes the form:for all with .
Thus, if is a Kannan mapping with constant satisfyingfor every . Then it also satisfies (16) and (14) with .

Example 4. Let be defined as . Then, are satisfied by . Condition (14) takes the form:for all with .

Example 5. Let be defined as . Then, are satisfied by . Condition (14) takes the form:for all with .

We prove the following lemma which is an extension of Lemma 2.

Lemma 3. Let be a complete TVS-valued cone metric space and be an -Kannan–Suzuki type mapping. Then,for all .

Proof. Suppose that is an arbitrary point in . If for some , then sequence converges in , and hence the sequence for all .
Assume that for any . Then, by (14) with , we getBy Remark 4, we obtainUsing (22) in (21), as results yields toLetting in (23), we getwhich is a contradiction. Hence, as .

Motivated by Batra et al. [37] and Filipovic et al. [38], we give a proof of an extended version of Theorem 2, 4, and 5 in -Kannan–Suzuki type mappings with a pair of two self-mappings in complete TVS-valued cone metric space.

Theorem 6. Let be a complete TVS-valued cone metric space and a solid cone, in addition let be a one-to-one, continuous mappings and a --Kannan–Suzuki type contraction. Then,(i)For every the sequence is convergent(ii)There is such that (iii) is sequentially convergent, then has a convergent, subsequence(iv)There is a unique such that (v)If is sequentially convergent, then for each the iterate sequence converges to

Proof. By (i), we prove that is a Cauchy sequence. Let be arbitrary. If for some , then sequence converges in and hence the sequence for all . Suppose that for any . Then, by (14), Lemma 3 with , we getFrom Remark 4, we haveUsing (26) in (25), as results yields toLetting in (27), we getwhich is a contradiction. Hence, as . Thus, converges.
Since is sequentially convergent, using , we prove that the iterate of a sequence converge to a fixed . To see this, suppose be an arbitrary point in . Let the sequence be defined by and , for . Thus, we haveequivalent toLet and , using inequality (14), we obtainUsing Remark 4 and the increasing property of , we deduceand hence,By , this implies thatConsequently, we getsoBy induction and (36), we deduceLetting in (37), we find thatConsequently, using Lemma 1 and property of results inThus, there exists such thatwhich is a contradiction. Hence, we haveTherefore, we have as . Denote , for all and , for -Kannan–Suzuki type mappings.
By (39), we prove that is a Cauchy sequence since is complete. Consider such that . Assume on the contrary that there exists and sequences and such thatUsing (iii) of Definition 2, we getFrom (39) and the above inequality, we haveFrom , (44), and (14), we getEquivalently,which is a contradiction. So, for every in . Hence, is a Cauchy sequence in . The completeness of ensures the existence of such thatBy (47) and Definition 6, it follows that as . By sequential continuity of and , we haveSince is a complete metric space, there exists such thatNow, we prove that is a fixed point of . Thus, by (iii) of Definition 2 and , we haveBy Remark 4, it implies thatApplying (51) in (50), we obtainLetting and using in above inequality, we getwhich is a contradiction. Hence, .
Next, we prove that is a unique fixed point of . Assume on contrary that there exists such that or . Let and is a fixed point of . Using Remark 4 and (14), it follows that or which is a contradiction. Thus, is a unique fixed point of .
Moreover, is a subsequentially convergent, has a convergent subsequence, there exists and such thatDue to the continuity of , it implies thatBy (49), we conclude thatUsing Remark 4 and , we getThus, using (iii) of Definition 2, we haveBy Remark 4,Using (57) and (59) in (58), we obtainSuppose thatLetting and using Definition 3, (61), and (62) in (60), we obtainwhich followsSince is one to one and continuous, . So, has a fixed point. As is sequentially convergent, we conclude that converges to the fixed point of .

Next, we prove our second main results by extending Theorem 3 using an -Kannan–Suzuki type mapping in TVS-valued cone metric space.

Theorem 7. Let be a complete TVS-valued cone metric space and a solid cone. In addition, let be a one-to-one, continuous and sequentially mappings and such that , and thatholds for some and for all . Then has property .

Proof. By Remark 1, let for some . If , that is is a unique fixed point of and . Hence, the proof completed. On contrary, we suppose . Let and such that and using (65), we getConsequently, we getRepeating the same argument several times, we finally obtainBy following similar procedure as the proof of Theorem 6, we can conclude that , i.e., . Since is one to one and sequentially convergent, then , which is a contradiction. Hence, .

We give an example where generalised Kannan mapping will not be applicable. However, - Kannan Suzuki type mapping is applicable.

Example 6. Consider the sequence and be an Euclidean metric on . Then is a TVS-valued cone complete metric space. Let the mapping be determined as follows:for . Let there exist , so that, for all condition (1) is satisfied although is not true for every . That is a contradiction; hence, Kannan’s theorem cannot be applicable.
The mapping be determined asFor all , is continuous, one to one, and subsequentially convergent.
We consider a sequence in and assume that is sequentially compact in complete TVS-valued cone metric space. By assumption is sequentially compact with we can cover the space with finitely many balls of radius 1; then one of them contains many for ; i.e., there is a ball of radius 1 so that there is a subsequence of whose members all belongs to . We denote this subsequence by ; thus, all belongs to .
Similar by sequentially compactness conditions with , we can find a subsequence of and a ball of radius so that all belongs to . Continuing this way, we obtain for any a subsequence of and a ball of radius so that all belongs to .
Now, let , . Then, we show that is a -Kannan–Suzuki type mapping in TVS-valued cone metric space with respect to and . By using and , we haveTo see this, we now calculate for .Applying (72), (73), and (74) in (71) becomes

Thus the inequality (71) and all conditions imposed in Theorem 6 are satisfied. Hence, and has unique fixed point that is in , where is a solid cone.

4. Some Applications

Two applications of the theorem stated in the previous part will be presented in this section.

4.1. Existence of a Solution for Nonlinear Riemann–Liouville Type Fractional Differential Equation

As a convolution mapping, the nonlinear fractional differential equation is equally and identically utilized in several applications in the domains of science, engineering, and mathematics.(i)In image processing: convolutional filtering is used in many essential algorithms in digital image processing, such as edge detection and related procedures. An out-of-focus photograph is created by convolutioning a crisp image with a lens function in optics. This is referred to as bokeh in photography. For example, applying blurring to a picture in image processing software.(ii)In digital data processing: Savitzky–Golay smoothing filters are used for analyzing spectroscopic data in analytical chemistry. This can boost the signal-to-noise ratio while reducing spectral distortion along with a convolution in statistics that is weighted in moving average.(iii)In acoustics: reverberation is the convolution of the original sound with echoes from objects surrounding the sound source. Convolution is a technique for mapping the impulse response of a physical room to a digital audio stream in digital signal processing. The imposition of a spectral or rhythmic structure on a sound is known as convolution in electronic music. This envelope or structure is frequently derived from a different sound. Filtering one signal via the other is called convolution of two signals.(iv)In electrical engineering: the output of a linear time-invariant (LTI) system is obtained by the convolution of one function (the input signal) with a second function (the impulse response). At any one time, the output is the sum of all previous input function values, with the most recent values often having the most influence (expressed as a multiplicative factor). This component is provided by the impulse response function as a function of the time since each input value happened.(v)In physics: a convolution operation can be found whenever there is a linear system with a “superposition principle.” For example, in spectroscopy, line widening owing to the Doppler effect produces a Gaussian spectral line form on its own, whereas collision broadening produces a Lorentzian line shape. The Line form is a convolution of Gaussian and Lorentzian, which is a Voigt function, when both effects are active. The measured fluorescence in time-resolved fluorescence spectroscopy is a sum of exponential decays from each delta pulse, and the excitation signal may be considered as a chain of delta pulses.(vi)In computational fluid dynamics: the convolution process is used in the large eddy simulation (LES) turbulence model to reduce the range of length scales required in computing, lowering the computational cost.(vii)In probability theory: the convolution of the distributions of two independent random variables is the probability distribution of the sum of their distributions.(viii)In kernel density estimation: a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian.(ix)In radiotherapy: in the handling of planning systems, the convolution-superposition algorithm is used in the majority of recent computation codes.

The above applications of a convolution show that the fractional derivative as convolution has multiple purposes. It can represent memory, like in the instance of elasticity theory. It may be understood as a filter, with the Caputo and Caputo–Fabrizio types in particular being viewed as a filter of the local derivative with power and exponent functions (one can see in [48]).

The purpose of this section is to provide an application of Theorem 6 to find a common solution of a nonlinear fractional differential equation, where we can apply -Kannan–Suzuki type mappings in complete TVS-valued cone metric spaces.

Here, we investigates the Riemann–Liouville derivative fractional integral of order . This form of fractional derivative for a continuous function denoted by , is given bywhere denotes the integer part of the real number and , provided that the right hand side is pointwise defined on . (see [49–54]). Also, the Riemann–Liouville fractional integral of order is given byfor . The notation stands for largest integer not greater than . If , then , for and if , then for .

The following nonlinear fractional differential equation with integral boundary valued conditions is inspired by Kilbas et al. [55], Cabada and Hamdi [56], and Cabada and Wang [50].where denotes the Riemann–Liouville fractional derivative of order and is a continuous function.

We recall the following lemmas from Bai, and Lü [57].

Lemma 4. Let . If we assume , then the fractional differential equation:has, , as unique solution.

Since for all . From Lemma 4, we deduce the following lemma.

Lemma 5. Assume that , with fractional derivative of order that belongs to . Then,for some , , as unique solution.

The unique solution of (78) is given by

Recall that the Green function related to the problem (78) is given by

Consider the space , be endowed with the ordering if for all and define by , .

This space defines the metric such that and . Then, is a TVS-valued cone metric space. A function is a unique solution of the fractional differential integral equation (82) if and only if is a solution of the nonlinear fractional differential equation (78).

Now, we prove the following theorem.

Theorem 8. Suppose the following condition hold:(i) for all (ii) for all (iii) is sequentially continuous(iv)there exists a continuous function , such that for all and , such that Then, the fractional differential Equation has a common solution as a fixed point .

Proof. Let us define a map byfor , then is sequentially continuous. This implies that and possess 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 (iv), we haveThis implies thatSince , we haveThus, for each , we haveTaking logarithms on both sides of (91) using and the property of , we getequivalent towhereUsing (94) in (93) and applying -Kannan–Suzuki type conditions leads toFor , satisfies -Kannan–Suzuki type mapping. Therefore, is a contraction mapping on . Since all the conditions of Theorem 8 are satisfied. Therefore, there exists a common fixed point of and , that is, is a solution to fractional nonlinear differential equation (78).

4.2. The Existence of Coincidence Solution for Nonlinear Volterra-Integral Equations

This section investigates the coincidence solution for nonlinear Volterra-integral equations in the setting of TVS-valued cone metric spaces. Nieto [58] initiated the study of the existing solution of an ordinary differential equation. Since then, several authors utilized his ideas to find the solution of ordinary differential equations. In detail, one can see the literature in [55, 59–62] and the references therein.

Integral equation methods help solve many problems of the applied fields like mathematical economics and optimal control theory because this problem is often reduced to integral equations.

Integral equations appear in several forms. However, in this section, we are interested with the integral equation, namely, Volterra integral-differential equation which is of the form

The following integral equation inspired by [12, 63–66].where are given functions and is unknown function to be found.

Let be the class of continuous functions from the set to the set . We denote and . We denote that , and

Denote be the space of functions satisfyingwhere is a positive constant, that is,for constant . Let be a Banach space. Define a norm in the space by

Define the mapping byandfor . The coincidence fixed point of and is also a solution of the Volterra integral-differential equation (97).

Now we prove the results by establishing the existence solution of a coincidence fixed point for a pair of self mappings:

Theorem 9. Suppose the following conditions holds:(i)For the continuous functions , we have where and .(ii)There exists a nonnegative constant such that and for all .

Then, the nonlinear Volterra-integral equation (97) has a unique solution in which is the coincidence fixed point of equations (101) and (102).

Proof. Let be two operators such that and . Now we verify that the two operators are contractive maps in . Let . On contrary we claim that and are not contractive maps in . From equations (101) and (102), using condition (i) and (ii) of Theorem 9, we havewhich is a contradiction. Hence is a common fixed of and , also a solution to integral (97).
From (105), since and using of Definition 13, wherewe haveUsing by taking natural logarithms in both sides of (107), we getBy (106), we obtain a -Kannan–Suzuki contraction as defined in Definition 13. Thus, all conditions imposed in Theorem 6 and Theorem 9 are satisfied. Hence, is a common fixed point of and in .

5. Conclusion

The novelty of this study to fixed point theory is the fixed point result given in Theorem 6. This theorem provides the common fixed points conditions for a pair of two self mappings in TVS-valued cone metric spaces. This paper extended and generalised the results due to Batra et al. [37], Filipovic et al. [38], Morales and Rojas [9], Rahimi et al. [39], and Wangwe and Kumar [40] using a pair of two self-mappings in -Kannan–Suzuki type mapping in TVS-valued cone metric space, where we consider a map to be sequentially convergent, one to one and continuous. By doing so, we extended several other results of the same setting in the literature. These results have some applications in many areas of applied mathematics, especially in nonlinear Riemann–Liouville fractional differential equation and nonlinear Volterra-integral differential equation.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. In addition, all authors read and approved the final manuscript.

References

S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
E. H. Connell, “Properties of fixed point spaces,” Proceedings of the American Mathematical Society, vol. 10, no. 6, pp. 974–979, 1959.
View at: Publisher Site | Google Scholar
R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968.
View at: Google Scholar
P. V. Subrahmanyam, “Completeness and fixed-points,” Monatshefte für Mathematik, vol. 80, no. 4, pp. 325–330, 1975.
View at: Publisher Site | Google Scholar
L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.
View at: Publisher Site | Google Scholar
S. Rezapour and R. Hamlbarani, “Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 719–724, 2008.
View at: Publisher Site | Google Scholar
M. Abbas and B. E. Rhoades, “Fixed and periodic point results in cone metric spaces,” Applied Mathematics Letters, vol. 22, no. 4, pp. 511–515, 2009.
View at: Publisher Site | Google Scholar
I. Beg, A. Azam, and M. Arshad, “Common fixed points for maps on topological vector space valued cone metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 560264, 8 pages, 2009.
View at: Publisher Site | Google Scholar
J. R. Morales and E. Rojas, “Cone metric spaces and fixed point theorems of T-Kannan contractive mappings,” International Journal of Mathematics and Analysis, vol. 4, no. 4, pp. 175–184, 2010.
View at: Google Scholar
F. Vetro and S. Radenovic, “Some results of Perov type in rectangular cone metric spaces,” Journal of Fixed Point Theory and Applications, vol. 20, p. 16, 2018.
View at: Publisher Site | Google Scholar
I. Beg, G. Mani, and A. J. Gnanaprakasam, “Fixed point of orthogonal F-suzuki contraction mapping on O-complete b-metric spaces with applications,” Journal of Function Spaces, vol. 2021, Article ID 6692112, 12 pages, 2021.
View at: Publisher Site | Google Scholar
A. Azam and I. Beg, “Kannan type mapping in TVS-valued cone metric spaces and their application to Urysohn integral equations,” Sarajevo Journal of Mathematics, vol. 9, no. 22, pp. 243–255, 2013.
View at: Publisher Site | Google Scholar
M. Đorđević, D. Đorić, Z. Kadelburg, S. Radenović, and D. Spasić, “Fixed point results under c-distance in tvs-cone metric spaces,” Fixed Point Theory and Applications, vol. 2011, no. 1, 2011.
View at: Publisher Site | Google Scholar
Z. Kadelburg, S. Radenovic, and V. Rakocevic, “Topological vector spaces valued cone metric spaces and fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, Article ID 170253, p. 18, 2010.
View at: Publisher Site | Google Scholar
Z. Kadelburg and S. Radenovic, “Coupled fixed point results under TVS-cone metric and W-cone-distance,” Advanced Fixed Point Theory, vol. 2, no. 1, pp. 29–46, 2012.
View at: Google Scholar
S. Radenovic and B. E. Rhoades, “Fixed point theorem for two non-self mappings in cone metric spaces,” Computers & Mathematics with Applications, vol. 57, pp. 1701–1707, 2009.
View at: Google Scholar
P. Hu and F. Gu, “Some fixed point theorems of λ-contractive mappings in menger PSM-spaces,” Journal of Nonlinear Functional Analysis, vol. 2020, no. 2020, p. 33, 2020.
View at: Google Scholar
S. Reich and J. Z. Alexander, “Fixed points and convergence results for a class of contractive mappings,” Journal of Nonlinear and Variational Analysis, vol. 5, no. 2021, pp. 665–671, 2021.
View at: Google Scholar
T. Ram and P. Lal, “Existence results on generalized strong operator equilibrium problems in Hausdorff TVS,” Communications in Optimization Theory, vol. 2021, p. 14, 2021.
View at: Google Scholar
A. Dubey and U. Mishra, “Some fixed point results of single-valued mapping for c-distance in tvs-cone metric spaces,” Filomat, vol. 30, no. 11, pp. 2925–2934, 2016.
View at: Publisher Site | Google Scholar
N. Tas, “On the topological equivalence of some generalized metric spaces,” Journal of Linear and Topological Algebra, vol. 9, no. 1, pp. 67–74, 2020.
View at: Google Scholar
K. B. Lee, “The chain recurrent set on compact TVS-cone metric spaces,” Journal of the Chungcheong Mathematical Society, vol. 33, no. 1, pp. 157–163, 2020.
View at: Google Scholar
X. Ge, S. Yang, and S. Yang, “Some fixed point results on generalized metric spaces,” AIMS Mathematics, vol. 6, no. 2, pp. 1769–1780, 2021.
View at: Publisher Site | Google Scholar
T. Suzuki, “A generalised Banach contraction principle that characterises metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
View at: Publisher Site | Google Scholar
O. Rida, C. Karim, and M. El Miloudi, “Related Suzuki-type fixed point theorems in ordered metric space,” Fixed Point Theory and Applications, vol. 1, pp. 1–26, 2020.
View at: Publisher Site | Google Scholar
D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory and Applications, vol. 1, pp. 1–6, 2012.
View at: Publisher Site | Google Scholar
N. Goswami, N. Haokip, and V. N. Mishra, “F-contractive type mappings in b-metric spaces and some related fixed point results,” Fixed Point Theory and Applications, vol. 2019, no. 1, pp. 1–13, 2019.
View at: Publisher Site | Google Scholar
H. Piri and P. Kumam, “Some fixed point theorems concerning F-contraction in complete metric spaces,” Fixed Point Theory and Applications, vol. 2014, no. 1, p. 210, 2014.
View at: Publisher Site | Google Scholar
N.-A. Secelean, “Iterated function systems consisting of F-contractions,” Fixed Point Theory and Applications, vol. 2013, no. 1, pp. 1–3, 2013.
View at: Publisher Site | Google Scholar
D. Wardowski and N. Van Dung, “Fixed points of F-weak contractions on complete metric spaces,” Demonstratio Mathematica, vol. 47, no. 1, pp. 146–155, 2014.
View at: Publisher Site | Google Scholar
H. H. Alsulami, H. Piri, and H. Piri, “Fixed points of GeneralizedF-suzuki type contraction in complete b-metric spaces,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 969726, 8 pages, 2015.
View at: Publisher Site | Google Scholar
L. B. Budhia, P. Kumam, J. Martínez-Moreno, and D. Gopal, “Extensions of almost-F and F-Suzuki contractions with graph and some applications to fractional calculus,” Fixed Point Theory and Applications, vol. 1, pp. 1–14, 2016.
View at: Publisher Site | Google Scholar
S. Chandok, H. Huang, and S. Radenović, “Some fixed point results for the generalised F-suzuki type contractions in b-metric spaces,” Proceedings of the American Mathematical Society, vol. 11, no. 1, pp. 81–89, 2018.
View at: Google Scholar
D. Derouiche and H. Ramoul, “New fixed point results for F-contractions of Hardy-Rogers type in b-metric spaces with applications,” Journal of Fixed Point Theory and Applications, vol. 22, no. 4, pp. 1–44, 2020.
View at: Publisher Site | Google Scholar
G. Mani, A. J. Gnanaprakasam, L. N. Mishra, and V. N. Mishra, “Fixed point theorems for orthogonal F-suzuki contraction mappings on O-complete metric space with an applications,” Malaya Journal of Matematik, vol. 9, no. 1, pp. 369–377, 2021.
View at: Publisher Site | Google Scholar
J. Z. Vujakovic and S. N. Radenovic, “On some F-contraction of Piri-Kumam-Dung-type mappings in metric spaces,” Journal of Nonlinear and Variational Analysis, vol. 68, no. 4, pp. 697–714, 2020.
View at: Google Scholar
R. Batra, R. Gupta, and P. Sahni, “A new extension of Kannan contractions and related fixed point results,” The Journal of Analysis, vol. 337, no. 1, pp. 1–6, 2020.
View at: Publisher Site | Google Scholar
M. Filipovic, L. Paunovic, S. Radenovic, and M. Rajović, “Remarks on ”Cone metric spaces and fixed point theorems of T-Kannan and T-Chatterjea contractive mappings,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1467–1472, 2011.
View at: Google Scholar
H. Rahimi, B. E. Rhoades, S. Radenovic, and S. Rad, “Fixed and periodic point theorems for T-contractions on cone metric spaces,” Filomat, vol. 27, no. 5, pp. 881–888, 2013.
View at: Publisher Site | Google Scholar
L. Wangwe and S. Kumar, “A common fixed point theorem for generalised F -kannan mapping in metric space with applications,” Abstract and Applied Analysis, vol. 2021, Article ID 6619877, 12 pages, 2021.
View at: Publisher Site | Google Scholar
H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Springer, New York, NY, USA, 1971.
J. Munkres, Topology: Pearson New International Edition, Springer, Pearson, Berlin, Germany, 2013.
A. Branciari, “A fixed point theorem of Banach-Caccippoli type on a class of generalised metric spaces,” Publicationes Mathematicae Debrecen, vol. 57, pp. 31–37, 2000.
View at: Google Scholar
G. S. Jeong and B. E. Rhoades, “More maps for which F(T)=F(Tn),” Demonstratio Mathematica, vol. 40, no. 3, pp. 671–680, 2007.
View at: Publisher Site | Google Scholar
G. Jungck, “Compatible mappings and common Fixed points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771–779, 1986.
View at: Publisher Site | Google Scholar
G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261–263, 1976.
View at: Publisher Site | Google Scholar
S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,” Publications de l’Institut Mathématique, vol. 32, no. 46, pp. 149–153, 1982.
View at: Google Scholar
U. Zölzer, DAFX: Digital Audio Effects, Wiley, Hoboken, NJ, USA, 2002.
D. Baleanu, S. Rezapour, and H. Mohammadi, “Some existence results on nonlinear fractional differential equations,” Philosophical Transactions of the Royal Society A, vol. 371, no. 1990, pp. 1–7, 2013.
View at: Google Scholar
A. Cabada and G. Wang, “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 403–411, 2012.
View at: Publisher Site | Google Scholar
J. Henderson and R. Luca, Boundary Valued Problems for System of Differential, Difference and Fractional Equations of Positive Solutions, Elsevier, Amsterdam, Netherlands, 2016.
T. Kanwal, A. Hussain, H. Baghani, and M. de la Sen, “New fixed point theorems in orthogonal F -metric spaces with application to fractional differential equation,” Symmetry, vol. 12, no. 5, p. 832, 2020.
View at: Publisher Site | Google Scholar
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
Y. Zhou, J. Wang, and L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2016.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006.
A. Cabada and Z. Hamdi, “Nonlinear fractional differential equations with integral boundary value conditions,” Applied Mathematics and Computation, vol. 228, pp. 251–257, 2014.
View at: Publisher Site | Google Scholar
Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
View at: Publisher Site | Google Scholar
J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
View at: Publisher Site | Google Scholar
P. Borisut, P. Kumam, V. Gupta, and N. Mani, “Generalized (ψ, α, β)-weak contractions for initial value problems,” Mathematics, vol. 7, no. 3, p. 266, 2019.
View at: Publisher Site | Google Scholar
V. Gupta, W. Shatanawi, and N. Mani, “Fixed point theorems for (ψ, β)-geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations,” Journal of Fixed Point Theory and Applications, vol. 19, no. 2, pp. 1251–1267, 2017.
View at: Publisher Site | Google Scholar
J. Harjani and K. Sadarangani, “Generalised contractions in partially ordered metric spaces and applications to ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1188–1197, 2010.
View at: Publisher Site | Google Scholar
F. Yan, Y. Su, and Q. Feng, “A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations,” Journal of Fixed Point Theory and Applications, vol. 152, 2012.
View at: Publisher Site | Google Scholar
M. Abbas, A. Latif, and Y. I. Suleiman, “Fixed points for cyclic R-contractions and solution of nonlinear Volterra integro-differential equations,” Fixed Point Theory and Applications, vol. 1, pp. 1–9, 2016.
View at: Google Scholar
C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, UK, vol. 148, 1991.
H. K. Nashine, R. Pathak, P. S. Somvanshi, S. Pantelic, and P. Kumam, “Solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic (, )-contraction,” Advances in Difference Equations, vol. 2013, no. 1, p. 106, 2013.
View at: Publisher Site | Google Scholar
B. G. Pachpatte, “On a nonlinear Volterra integral equation in two variables,” Sarajevo Journals of Mathematics, vol. 6, no. 18, pp. 59–73, 2010.
View at: Google Scholar

Copyright

Copyright © 2022 Lucas Wangwe and Santosh Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

428

Downloads

484

Citations