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

Authors: Lucas Wangwe; Santosh Kumar
Journal: Journal of Mathematics
 (2022)

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## Abstract

This research paper generalizes and extends various fixed-point results that demonstrate common fixed-point theorems forF-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.


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## Body

## 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 X is a complete metric space and G:X⟶X is a contraction map, i.e., ϱGσ,Gς≤κϱσ,ς for all σ,ς∈X and κ∈[0,1), then G has a unique fixed point or Gσ=σ 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 X, 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 λ-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 S- metric space. Lee [22] introduces chain recurrent set, trapping region, attracting set and repelling set for a flow f 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 b-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 F-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 F-Suzuki contraction and obtained some intriguing results utilising the Secelean [29] concept. In the complete b-metric spaces, Alsulami et al. [31] demonstrated fixed points of generalised F-Suzuki type Contractions. Budhia et al. [32] proved an extension of almost-F and F-Suzuki contractions with graph and demonstrated some applications to fractional calculus whereas Chandok et al. [33] formulated some fixed point results for the generalised F-Suzuki type contractions in b-metric spaces. Derouiche and Ramou [34] proved new fixed-point results for F-contractions of Suzuki Hardy-Rogers type in b-metric spaces and provided some applications. Beg et al. [11] proposed a fixed point of orthogonal F-Suzuki contraction mapping on 0-complete b-metric spaces with some applications. Mani et al. [35] introduced generalised orthogonal F-contraction and orthogonal F-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 F-contraction of Piri-Kumam-Dung-type mappings in metric spaces.In 2019, Goswami et al. [27] introduced F-contractive type mappings in b-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 F-contraction and Kannan mapping concepts for defining F-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 F-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]).
LetG:X⟶X be a self mapping on a complete metric space X,ϱ such that(1)ϱGσ,Gς≤κϱσ,Gσ+ϱς,Gς,for all σ,ς∈X and 0≤κ≤1/2. Then, G possesses a unique fixed point σ∗∈X and for any σ∈X the iterate sequence Gnσ converges to σ∗.Equation (1) is equivalent to(2)ϱGσ,Gς≤κ2ϱσ,Gσ+ϱς,Gς,for some κ∈0,1.Definition 1 (see [11]).
LetE,τ be always a topological space and P a subset of E. Then, P is called a cone if the following hold:(i)
P is a nonempty, closed and P≠0;(ii)
λσ+μς∈P for all σ,ς∈P and nonnegative real number λ,μ;(iii)
P∩−P=0.
For given coneP⊆E. If the interior of PintP, is nonempty we say that P is solid. If P is solid cone, then P is a component of P, and in this case we use the notation σ≪ς to indicate that ς−σ∈intP. Note that if σ≪ς and ς≤υ, then σ≪υ for all σ,ς,υ∈intP.The following axioms satisfy TVS-valued cone complete metric space.Definition 2 (see [11]).
LetX be a nonempty set and the mapping ϱ:X×X⟶E, satisfies the following:(i)
0≤ϱσ,ς, for all σ,ς∈X and ϱσ,ς=0 if and only if σ=ς;(ii)
ϱσ,ς=ϱς,σ, for all σ,ς∈X;(iii)
ϱσ,ς≤ϱσ,υ+ϱυ,ς, for all σ,ς,υ∈X.
Then,ϱ is called a cone metric on X, and X,ϱ is called topological vector space valued cone metric space.Example 1 (see [12, 9, 41]).
LetE=C0,1,ℝ2,P=σ,ς∈E|σ,ς≥0⊂ℝ2, X=ℝ and ϱ:X×X⟶E such that ϱσ,ς=σ−ςψt, where ψt=et. Then, X,Σ 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]).
LetX,ϱ be a topological vector space valued cone metric space, and let x∈X and σnn≥1 be a sequence in X. Then,(i)
σnn≥1 converges to X whenever for every c∈E with 0≪c there is a natural number ℕ such that ϱσn,σ≪c for all n≥ℕ. We denote this by(3)limn⟶∞σn=σ⇔σn⟶σ.(ii)
σnn≥1 is Cauchy sequence whenever for every c∈E with 0≪c, there is a natural number ℕ such that ϱσn,σm≪c for all n,m≥ℕ.(iii)
X,ϱ is called topological vector space valued cone metric space if every Cauchy sequence is convergent.Definition 4 (see [42]).
LetX be a topological space. If σn is a sequence of points of X, and if n1<n2<…<ni<… is an increasing sequence of positive integers, then the sequence ςi defined by setting ςi=σni is called a subsequence of the sequence σn. The space X is said to be sequentially compact if every sequence of points of X has a convergent subsequence.Definition 5 (see [43]).
LetX,d be a metric space. A mapping G:X⟶X is said to be sequentially convergent if we have, for every sequence ςn, if Gςn is convergence then ςn also is convergence. G is said to be subsequentially convergent if we have, for every sequence ςn, if Gςn is convergence then ςn 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]).
LetX,ϱ be a cone metric space, P is a solid cone and G:X⟶X. Then(i)
G is said to be continuous if(4)limn⟶∞σn=σ⇒limn⟶∞Gσn=Gσ,for allσn∈X,(ii)
G is said to be sequentially convergent if we have, for every sequence ςn, if Gςn is convergent, then ςn also is convergent,(iii)
G is said to be subsequentially convergent if we have, for every sequence ςn and Gςn is convergent, implies ςn 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 P to be a solid cone. They proved results on two self mappings as follows.Definition 7 (see [38]).
LetX,ϱ be a cone metric space and T,f:X⟶X two mappings. A mapping f is said to be T- Hardy-Rogers contraction if there exists ai≥0,i=1,…,5 with ∑i=15ai≤1 such that for all σ,ς∈X.(5)ϱTfσ,Tfς≤a1ϱTσ,Tς+a2ϱTσ,Tfσ+a3ϱTς,Tfς+a4ϱTσ,Tfς+a5ϱTς,Tfσ.Theorem 2 (see [38]).
LetX,ϱ be a complete cone metric space and P a solid cone, in addition let T:X⟶X be a one-to-one, continuous mappings and f:X⟶X a T-hardy-Rogers contraction. Then,(i)
For everyσ0∈X the sequence Tfnσ0 is Cauchy.(ii)
There isvσ0∈X such that limn⟶∞Tfnσ0=vσ0.(iii)
T is sequentially convergent, then fnσ0 has a convergent, subsequence.(iv)
There is a uniqueuσ0∈X such that fuσ0=uσ0.(v)
IfT is sequentially convergent, then for each σ0∈X the iterate sequence fnσ0 converges to uσ0.Theorem 3 (see [38]).
LetX,ϱ be a complete cone metric space and P a solid cone, in addition let T:X⟶X be a one-to-one, continuous mappings and f:X⟶X such that Ff≠∅ and that(6)ϱTfσ,Tf2σ≼λϱTσ,Tfσ,holds for some λ∈0,1 and for all σ∈X,σ≠fσ. Then f has property P.Remark 1 (see [44]).
LetFT denote the fixed point set of a map T. A map T has property P if FT=FTn for each n∈ℕ. We shall say that a pair of maps T and f has property Q if FT∩Ff=FTn∩Ffn for each n∈ℕ.Secelean [29] proved the following lemma.Lemma 1 (see [29]).
LetF:ℝ+⟶ℝ be an increasing function and xn be a sequence of positive real numbers. Then the following holds:(a)
Iflimn⟶∞Fαn=−∞, then limn⟶∞αn=0,(b)
IfinfF=−∞ and limn⟶∞αn=0, then limn⟶∞Fαn=−∞,LetF be the set of all functions defined as F:ℝ+⟶ℝ, which satisfies the following conditions:(F1)F is strictly increasing i.e., for all α,β∈ℝ+ such that α<β⇒Fα<Fβ(F2″) there is a sequenceαnn∈ℕ of positive real numbers such that limn⟶∞Fαn=−∞ or infF=−∞(F3″)F is continuous on 0,∞The following functionF:ℝ+⟶ℝ belongs to F:(i)
F1z=lnz(ii)
F2z=−1/z(iii)
F3z=−1/z+zDefinition 8 (see [28]).
LetX,ϱ be a metric space. A mapping G:X⟶X is said to be an F-Suzuki contraction if there exists τ>0, such that for all σ,ς∈X with Gσ≠Gς(7)12ϱσ,Gσ<ϱσ,ς⇒τ+FϱGσ,Gς≤Fϱσ,ς,where F∈F.In 2014, Piri and Kumam [28] established a generalisation of Banach contraction principle, which is as follows:Theorem 4 (see [28]).
LetX,ϱ be a complete metric space and G:X⟶X be a F-Suzuki contraction. Then G has a unique fixed point σ∗∈X and for every σ0∈X a sequence Gnσ0n∈ℕ is convergent to σ∗.Remark 2 (see [28]).
We denote byF the set of all functions satisfying F-suzuki type contraction condition due to [28, 29] and let denote by ℱ the set of all functions satisfying F-contraction condition by Wardowski [26], then(i)
ℱ⊈F(ii)
F⊈ℱ(iii)
ℱ∩F≠∅For more details onF-Suzuki contraction mapping, one can see [31–33] and the references therein.Motivated by Batra et al. [37], we use the following notations: Let X be a nonempty set and X,ϱ denotes the metric space with metric ϱ. Let the cardinality of a set A is denoted by cardA and FixG is set of all fixed points of a mapping G.Batra et al. [37] gave a new generalisation family of contraction called F-Kannan mapping and introduced the following definition:Definition 9 (see [37]).
LetF be a mapping satisfying F1−F3. A mapping G:X⟶X is said to be an F-Kannan mapping if the following holds:(K1)(8)Gσ≠Gς⇒Gσ≠σorGς≠ς.(K2)∃Υ>0 such that(9)ϒ+FϱGσ,Gς≤Fϱσ,Gσ+ϱς,Gς2,for allσ,ς∈X, with Gσ≠Gς.The remark presented below is due to Batra et al. [37].Remark 3 (see [37]).
By properties ofF, it follows that every F-Kannan mapping T on a metric space X,ϱ, satisfies following condition:(10)ϱGσ,Gς≤ϱσ,Gσ+ϱς,Gς2,for every σ,ς∈X.Furthermore, it is concluded thatCardFixG≤1. Let G be a self map of a metric space X,ϱ. G is said to be a Picard operator (PO) if G has unique fixed point σ∗ and limn⟶∞Gnσ=σ∗ for all σ∈X.Then the family of all functionsF:ℝ+⟶ℝ satisfying the condition F1−F3 is denoted by ℱ.We recall the following examples from Batra et al. [37] of such functions F:ℝ+⟶ℝ which satisfies F1−F3:Example 2 (see [37]).
LetF1:ℝ+⟶ℝ be defined as F1z=lnz. Then clearly, F1−F3 are satisfied by F1z. In fact F3 holds for every k∈0,1(11)ϱGσ,Gς≤e−ϒϱσ,Gσ+ϱς,Gς2,for all σ,ς∈X with Gσ≠Gς.
Thus, ifG:X⟶X is a Kannan mapping with constant κ∈0,1 satisfying(12)ϱGσ,Gς≤κϱσ,Gσ+ϱς,Gς2,for every σ,ς∈X, then it also satisfies (8) and (11) with ϒ=ln1/κ. In fact, whenever Gσ≠Gς, then from (12), we get Gσ≠σ or Gς≠ς.The following lemma introduced by Batra et al. [37].Lemma 2 (see [37]).
LetX,ϱ be a metric space and G:X⟶X be a F-Kannan mapping. Then, ϱGnσ,Gn+1σ⟶0 as n⟶∞ for all σ∈X.Batra et al. [37] introduced a F-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]).
LetX,ϱ be a complete metric space and suppose G:X⟶X is a F-Kannan mapping, then G 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]).
LetG,f be a pair of self-mappings on a metric space X,ϱ. Then coincidence point of the pair G,f is a point σ∈X such that Gσ = fσ=σ∗, then σ∗ is called coincidence point of the pair G,f. If σ∗=σ, then σ is said to be a common fixed point of f and G.Definition 11 (see [46]).
LetG,f be self-mappings of a nonempty set X. A point σ∈X is coincidence point of G and f if t=Gσ=fσ. The set of coincidence point of G and f is denoted by CG,f.Definition 12 (see [46, 47]).
LetT,f be a pair of self-mappings on a metric space X,ϱ. Then, the pair T,f is said to be as follows:(i)
Commuting if, for allσ∈X, Gfσ = fGσ,(ii)
Weakly commuting if, for allϱGfσ,fGσ≤ϱGσ,fσ,(iii)
Compatible iflimn⟶∞ϱGfσn,fGσn=0, whenever σn is a sequence in X such that limn⟶∞Gσn=limn⟶∞fσn=t,(iv)
Weakly compatible if, for allGfσ = fGσ, for every coincidence point σ∈X.


## 3. Main Results

To prove this section’s main results, we commerce by obtaining a more general version of Definition8 and 9 using a pair of two self mappings in F-Kannan–Suzuki type mapping setting. We denotes X,ϱ as a TVS-valued cone metric space.Definition 13.
LetF be a mapping satisfying F1−F3. A pair of two self mapping G,f:X⟶X is said to be an F-Kannan–Suzuki type mapping if the following holds:(FKS1)(13)Gfσ≠Gfς⇒Gfσ≠σorGfς≠ς.(FKS2) there existsϑ>0 such that(14)12ϱσ,Gσ<ϱσ,σ⇒ϑ+FϱGfσ,Gfς≤FϱGσ,Gfσ+ϱGς,Gfς2,for all σ,ς∈X, with Gfσ≠Gfς and F∈F.Following remark is motivated by the work of Batra et al. [37] given as follows.Remark 4.
By properties ofF, it follows that every F-Kannan–Suzuki type mapping G on a TVS-valued cone metric space X,ϱ, satisfies the following condition:(15)ϱGfσ,Gfς)≤ϱGσ,Gfσ+ϱGς,Gfς2,for every σ,ς∈X.We give the following examples in the context of a pair of two self mappings:Example 3.
LetF1:ℝ+⟶ℝ be defined as F1z=lnz. Then clearly, F1−F3 are satisfied by F1z. In fact F3 holds for every κ∈0,1. Moreover, condition (14) takes the form:(16)ϱGfσ,Gfς)≤e−ϑϱGσ,Gfσ+ϱGς,Gfς2,for all σ,ς∈X with Gfσ≠Gfς.
Thus, ifG,f:X⟶X is a Kannan mapping with constant κ∈0,1 satisfying(17)ϱ(Gfσ,Gfς))≤κϱ(Gσ,Gfσ)+ϱ(Gς,Gfς)2.for every σ,ς∈X. Then it also satisfies (16) and (14) with ϑ=ln1/κ.Example 4.
LetF2:ℝ+⟶ℝ be defined as F2z=−1/z,z>0. Then, F1−F3 are satisfied by F2z. Condition (14) takes the form:(18)ϱGσ,Gfσ+ϱGς,Gfς2≤ϱGfσ,Gfς1−ϑϱGfσ,Gfς,for all σ,ς∈X with Gfσ≠Gfς.Example 5.
LetF3:ℝ+⟶ℝ be defined as F3z=−1/z,z>0. Then, F1−F3 are satisfied by F3z. Condition (14) takes the form:(19)ϱGσ,Gfσ+ϱGς,Gfς2≤ϱGfσ,GfςϱGσ,Gfσ+ϱGς,Gfς/22−1ϱGfσ,Gfς+ϑϱGfσ,Gfς2−1,for all σ,ς∈X with Gfσ≠Gfς.We prove the following lemma which is an extension of Lemma2.Lemma 3.
LetX,ϱ be a complete TVS-valued cone metric space and G,f:X⟶X be an F-Kannan–Suzuki type mapping. Then,(20)ϱGfnσ0,Gfn+1σ0⟶0asn⟶∞,for all σ∈X.Proof.
Suppose thatσ0 is an arbitrary point in X. If Gfnx0=Gfn+1σ0 for some n∈ℕ, then sequence σnn∈ℕ converges in X, and hence the sequence ϱGfnσ0,Gfn+1σ0⟶0asn⟶∞ for all σ∈X.
Assume thatGfnσ0≠Gfn+1σ0 for any n∈ℕ. Then, by (14) with ϑ>0, we get(21)12ϱσ0,Gσ0<ϱσ0,Gσ0⇒ϑ+FϱGfnσ0,Gfn+1σ0≤FϱGfn−1σ0,Gfnσ0+ϱGfnσ0,Gfn+1σ02.
By Remark4, we obtain(22)ϱGfnσ0,Gfn+1σ0≤ϱGfn−1σ0,Gfnσ0+ϱGfnσ0,Gfn+1σ02.
Using (22) in (21), as results yields to(23)ϑ+FϱGfnσ0,Gfn+1σ0≤FϱGfnσ0,Gfn+1σ0.
Lettingn⟶∞ in (23), we get(24)ϑ+0≤0,ϑ≤0,which is a contradiction. Hence, ϱGfnσ0,Gfn+1σ0⟶0 as n⟶∞.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 F-Kannan–Suzuki type mappings with a pair of two self-mappings in complete TVS-valued cone metric space.Theorem 6.
LetX,ϱ be a complete TVS-valued cone metric space and P a solid cone, in addition let G:X⟶X be a one-to-one, continuous mappings and f:X⟶X a G-F-Kannan–Suzuki type contraction. Then,(i)
For everyσ0∈X the sequence Gfnσ0 is convergent(ii)
There isv∗∈X such that limn⟶∞Gfnσ0=v∗(iii)
G is sequentially convergent, then fnσ0 has a convergent, subsequence(iv)
There is a uniqueu∗∈X such that fu∗=u∗(v)
IfG is sequentially convergent, then for each σ0∈X the iterate sequence fnσ0 converges to u∗Proof.
By (i), we prove thatGfnσ0 is a Cauchy sequence. Let σ0∈X be arbitrary. If Gfnσ0=Gfn+1σ0 for some n∈ℕ, then sequence σnn∈ℕ converges in X and hence the sequence ϱGfnσ0,Gfn+1σ0⟶0asn⟶∞ for all σ∈X. Suppose that Gfnσ0≠Gfn+1σ0 for any n∈ℕ. Then, by (14), Lemma 3 with ϑ>0, we get(25)12ϱσn,Gσn<ϱσn,Gσn⇒ϑ+FϱGfnσ0,Gfn+1σ0≤FϱGfn−1σ0,Gfnσ0+ϱGfnσ0,Gfn+1σ02.
From Remark4, we have(26)ϱGfnσ0,Gfn+1σ0≤ϱGfn−1σ0,Gfnx0+ϱGfnσ0,Gfn+1σ02,2ϱGfnσ0,Gfn+1σ0≤ϱGfn−1σ0,Gfnσ0+ϱGfnσ0,Gfn+1σ0,ϱGfnσ0,Gfn+1σ0≤ϱGfn−1σ0,Gfnσ0.
Using (26) in (25), as results yields to(27)ϑ+FϱGfnσ0,Gfn+1σ0≤FϱGfn−1σ0,Gfnσ0+ϱGfnσ0,Gfn+1σ02,ϑ+FϱGfnσ0,Gfn+1σ0≤F2ϱGfn−1σ0,Gfnσ02,ϑ+FϱGfnσ0,Gfn+1σ0≤FϱGfn−1σ0,Gfnσ0.
Lettingn⟶∞ in (27), we get(28)ϑ+0≤0,ϑ≤0,which is a contradiction. Hence, ϱGfnσ0,Gfn+1σ0⟶0 as n⟶∞. Thus, Gfnσ0 converges.
SinceG is sequentially convergent, using v, we prove that the iterate of a sequence fnσ0 converge to a fixed u∈X. To see this, suppose σ0∈X be an arbitrary point in X. Let the sequence σnn≥1 be defined by σn+1=fσn=fn+1σ0=ffnσ0 and σn=fσn−1=fnσ0=ffn−1σ0, for n≥1∈ℕ. Thus, we have(29)ϱσn,σn+1≤ϱfσn−1,fσn=ϱfnσ0,fn+1σ0=ϱffn−1σ0,ffnσ0.equivalent to(30)ϱGσn,Gσn+1≤ϱGfσn−1,Gfσn=ϱGfnσ0,Gfn+1σ0=ϱGffn−1σ0,Tffnσ0.
Letσ=fn−1σ0 and ς=fnσ0, using inequality (14), we obtain(31)12ϱσn,Gσn<ϱσn,Gσn⇒ϑ+F(ϱGffn−1σ0,Gffnσ0≤FϱGfn−1σ0,Gffn−1σ0+ϱGfnσ0,Gffnσ02,FϱGσn,Gσn+1≤FϱGσn−1,Gσn+ϱGσn,Gσn+12−ϑ.
Using Remark4 and the increasing property of F, we deduce(32)ϱGffn−1σ0,Gffnσ0≤ϱGfn−1σ0,Gffn−1σ0+ϱGfnσ0,Gffnσ02,ϱGσn,Gσn+1<ϱGσn−1,Gσn+ϱGσn,Gσn+12,and hence,(33)2ϱGσn,Gσn+1−ϱGσn,Gσn+1<ϱGσn−1,Gσn,ϱGσn,Gσn+1<ϱGσn−1,Gσn.
ByF1, this implies that(34)FϱGσn,Gσn+1<FϱGσn−1,Gσn.
Consequently, we get(35)ϑ+FϱGσn,Gσn+1≤FϱGσn−1,Gσn,so(36)FϱGσn,Gσn+1≤FϱGσn−1,Gσn−ϑ.
By induction and (36), we deduce(37)FϱGσn+1,Gσn+2≤FϱGσn−1,Gσn−2ϑ,FϱGσn+2,Gσn+3≤FϱGσn−1,Gσn−3ϑ,⇒FϱGσn,Gσn+1≤FϱGσn−1,Gσn−nϑ.
Lettingn⟶∞ in (37), we find that(38)limn⟶∞FϱGσn,Gσn+1=−∞.
Consequently, using Lemma1 and property F2″ of F results in(39)limn⟶∞ϱGσn,Gσn+1=0.
Thus, there existsn∈ℕ such that(40)ϱGσn,Gσn+1<ϱGσn,G2σn<cϱσn,Gσn<ϱσn,Gσn,which is a contradiction. Hence, we have(41)limn⟶∞ϱσn,Gσn=0.
Therefore, we haveϱGσn,Gσn+1⟶0 as n⟶∞. Denote αn=ϱGσn,Gσn+1=0, for all n≥1 and n∈ℕ, for F-Kannan–Suzuki type mappings.
By (39), we prove that Gσn is a Cauchy sequence since X,ϱ is complete. Consider n,m∈N such that m>n. Assume on the contrary that there exists c>0 and sequences pnn≥1∞ and qnn≥1∞ such that(42)pn>qn>n,ϱGσpn,Gσqn≥c,ϱGσpn−1,Gσqn≤c,∀n∈ℕ.
Using (iii) of Definition2, we get(43)ϱGσpn,Gσqn≤ϱGσpn,Gσpn−1+ϱGσpn−1,Gσqn≤ϱGσpn,Gσpn−1+c.
From (39) and the above inequality, we have(44)limn⟶∞ϱGσpn,Gσqn=c.
FromF3″, (44), and (14), we get(45)ϑ+FϱGσpn,Gσqn≤FϱGσpn−1,Gσpn+ϱGσpn,Gσqn2.
Equivalently,(46)ϑ+Fc≤Fc,ϑ≤0,which is a contradiction. So, Gσn=Gσm for every m≥n in X. Hence, Gσn is a Cauchy sequence in X. The completeness of X ensures the existence of u∗∈X such that(47)ϱGu∗,u∗=limn,m⟶∞ϱGσn,Gσm=0=limn⟶∞ϱGσn,u∗=0.
By (47) and Definition 6, it follows that Gσn+1⟶u∗ as n⟶∞. By sequential continuity of f and G, we have(48)u∗=limn⟶∞fnσ0=limn⟶∞σn=limn⟶∞σn+1=limn⟶∞fσn=fu∗.u∗=limn⟶∞Gfnσ0=limn⟶∞Gσn=limn⟶∞Gσn+1=limu⟶∞G2σn=Gu∗.
SinceX is a complete metric space, there exists u∗∈X such that(49)limn⟶∞Gσn=Gu∗=u∗.
Now, we prove thatu∗ is a fixed point of G. Thus, by (iii) of Definition 2 and ϱu∗,Gu∗≥0, we have(50)ϱu∗,Gu∗≤ϱu∗,Gσn+1+ϱGσn+1,Gu∗.
By Remark4, it implies that(51)ϱGσn+1,Gu∗≤ϱGσn,Gσn+1+ϱGσn+1,Gu∗2.
Applying (51) in (50), we obtain(52)ϱu∗,Gu∗≤ϱu∗,Gσn+1+ϱGσn,Gσn+1+ϱGσn+1,Gu∗2.
Lettingn⟶∞ and using in above inequality, we get(53)ϱu∗,Gu∗≤ϱu∗,Gu∗+ϱu∗,Gu∗+ϱGu∗,Gu∗2,ϱu∗,Gu∗≤ϱu∗,Gu∗+ϱu∗,Gu∗2,ϱu∗,Gu∗≤2ϱu∗,Gu∗+ϱu∗,Gu∗2,2ϱu∗,Gu∗≤2ϱu∗,Gu∗+ϱu∗,Gu∗,2ϱu∗,Gu∗−2ϱu∗,Gu∗≤ϱu∗,Gu∗,0≤ϱu∗,Gu∗,which is a contradiction. Hence, Gu∗=u∗.
Next, we prove thatu∗ is a unique fixed point of G. Assume on contrary that there exists v∗∈intP such that u∗≠v∗ or Gu∗≠Gv∗. Let Gσn⟶v∗ and v∗ is a fixed point of G. Using Remark 4 and (14), it follows that u∗=v∗ or Gu∗=Gv∗ which is a contradiction. Thus, u∗ is a unique fixed point of G.
Moreover,G is a subsequentially convergent, fnσ0 has a convergent subsequence, there exists σ∗∈X and fnkx0k=1∞ such that(54)limk⟶∞fnkx0=v∗.
Due to the continuity ofG, it implies that(55)limk⟶∞Gfnkσ0=Gv∗.
By (49), we conclude that(56)Gv∗=u∗.
Using Remark4 and λ=1/2, we get(57)ϱGffnk−1σ0,Tffnkσ0≤λ(ϱGfnk−1σ0,Gffnk−1σ0+ϱGfnkσ0,Gffnkσ0,≤λ(ϱGfnk−1σ0,Gffnk−1σ0+ϱGffnk−1σ0,Gffnkσ0,≤λϱGfnk−1σ0,Gffnk−1σ0+λϱGffnk−1σ0,Gffnkσ0,1−λϱGffnk−1σ0,Gffnkσ0≤λϱGfnk−1σ0,Gffnk−1σ0,ϱGffnk−1σ0,Gffnkσ0≤λ1−λϱGfnk−1σ0,Gffnk−1σ0.
Thus, using (iii) of Definition2, we have(58)ϱGfv∗,Gv∗≤ϱGfv∗,Gfnk+1σ0+ϱGfnk+1σ0,Gv∗.
By Remark4,(59)ϱGfv∗,Gfnk+1σ0=ϱGfv∗,Gffnkσ0≤λϱGv∗,Gfv∗+ϱGffnk−1σ0,Gffnkσ0.
Using (57) and (59) in (58), we obtain(60)ϱGfv∗,Gv∗≤λϱGv∗,Gfv∗+ϱGffnk−1σ0,Gffnkσ0+ϱGfnk+1σ0,Gv∗,≤λϱGv∗,Gfv∗+λ1−λϱGfnk−1σ0,Gffnk−1σ0+ϱGfnk+1σ0,Gv∗≤λϱGv∗,Gfv∗+λλ1−λnk−1ϱGfnk−1σ0,Gffnk−1σ0+ϱGfnk+1σ0,Gv∗,≤λ1−λλ1−λnk−1ϱGfnk−1σ0,Gffnk−1σ0+11−λϱGfnk+1σ0,Gv∗,≤λ1−λnkϱGfnk−1σ0,Gffnk−1σ0+11−λϱGfnk+1σ0,Gfnkσ0.
Suppose that(61)λ1−λnkϱGfnk−1σ0,Gffnk−1σ0=c2.(62)λ1−λϱGfnk−1σ0,Gffnk−1σ0=c2.
Lettingk⟶∞ and using Definition 3, (61), and (62) in (60), we obtain(63)ϱGfv∗,Gv∗≤c2+c2,which follows(64)ϱGfv∗,Gv∗≤c.
SinceG is one to one and continuous, fv∗=v∗. So, f has a fixed point. As Gfnσ0 is sequentially convergent, we conclude that Gfnσ0 converges to the fixed point of f.Next, we prove our second main results by extending Theorem3 using an F-Kannan–Suzuki type mapping in TVS-valued cone metric space.Theorem 7.
LetX,ϱ be a complete TVS-valued cone metric space and P a solid cone. In addition, let G:X⟶X be a one-to-one, continuous and sequentially mappings and f:X⟶X such that Ff≠∅, ϑ>0 and that(65)12ϱσ,Gσ<ϱσ,ς⇒ϑ+FϱGfσ,Gf2σ≤FϱGσ,Gfσ,holds for some λ∈0,1 and for all σ∈X,σ≠fσ. Then f has property Q.Proof.
By Remark1, let u∈FGn∩Ffn for some n∈ℕ. If u=fu, that is u is a unique fixed point of G and f. Hence, the proof completed. On contrary, we suppose u≠fu. Let σ=u=fn−1u and ς=fu=ffn−1u such that fn−1≠ffn−1 and using (65), we get(66)12ϱu,Gu<ϱu,fu,ϱu,Gu<2ϱu,fu,⇒ϑ+FϱGffn−1u,Gf2fn−1u≤FϱGfn−1u,Gffn−1u,ϑ+FϱGffn−1u,Gffnu≤FϱGfn−1u,TGfnu.
Consequently, we get(67)FϱGffn−1u,Gffnu≤FϱGfn−1u,Gfnu−ϑ.
Repeating the same argument several times, we finally obtain(68)FϱGffn−1u,Gffnu≤FϱGfn−1u,Gfnu−nϑ.
By following similar procedure as the proof of Theorem6, we can conclude that ϱGu,Gfu=c, i.e., Gu=Gfu. Since G is one to one and sequentially convergent, then u=fu, which is a contradiction. Hence, u∈FGn∩Ffn.We give an example where generalised Kannan mapping will not be applicable. However,F- Kannan Suzuki type mapping is applicable.Example 6.
Consider the sequenceX=0,1∪1/2,1/3,1/4,… and d be an Euclidean metric on X. Then X,ϱ is a TVS-valued cone complete metric space. Let the mapping f:X⟶X be determined as follows:(69)f0=0,f1/i=1i+1,for n≥2. Let there exist λ∈[0,(1/2)), so that, for all σ,ς∈X condition (1) is satisfied although is not true for every λ>0. That is a contradiction; hence, Kannan’s theorem cannot be applicable.
The mappingG:X⟶X be determined as(70)G0=0,G1/i=12i.
For alli≥2, G is continuous, one to one, and subsequentially convergent.
We consider a sequenceσi in X and assume that X is sequentially compact in complete TVS-valued cone metric space. By assumption X is sequentially compact with ϵ=1 we can cover the space X with finitely many balls of radius 1; then one of them contains many σi for i≥2; i.e., there is a ball B1 of radius 1 so that there is a subsequence of σi whose members all belongs to B1. We denote this subsequence by σi; thus, all σi belongs to B1.
Similar by sequentially compactness conditions withϵ=1/2, we can find a subsequence σi2 of σi1 and a ball B2 of radius 1/2 so that all σi2 belongs to B2. Continuing this way, we obtain for any k≥2 a subsequence σik of σik−1 and a ball Bk of radius 2−k so that all σik belongs to Bk.
Now, leti,j∈ℕ, j>i. Then, we show that f,G is a F-Kannan–Suzuki type mapping in TVS-valued cone metric space with respect to F2z=−1/z and ϑ≥0. By using FKS2 and F2z, we have(71)12ϱσ,Gσ<ϱσ,ς⇒ϱGσ,Gfσ+ϱGς,Gfς2≤ϱGfσ,Gfς1−ϑϱGfσ,Gfς.
To see this, we now calculateϱfσ,Gς for σ=1/i,ς=1/j,i≥1.(72)ϱGσ,Gfσ=ϱT1/i,Gf1/i≤12i−121/i+1et.(73)ϱGfσ,Gfς=ϱGf1/i,Gf1/j≤21/i+1−21/j+1et.(74)ϱGς,Gfς=ϱG1/j,Gf1/j≤21/j−21/j+1et.
Applying (72), (73), and (74) in (71) becomes(75)12ϱσ,Gσ<ϱσ,ς,12ϱ1/i,G1/i<ϱ1/i,1/j,ϱ1/i,G1/j<2ϱ1/i,1/j,1i−12jet<21i−1jet,2i−i2i.iet<2j−iijet,⇒ϱG1/i,Gf1/i+ϱG1/j,Gf1/j2≤ϱGf1/i,Gf1/j1−ϑϱGf1/i,Gf1/j,⇒1/2i−1/21/i+1et+21/j−21/j+1et2≤21/i+1−21/j+1et1−ϑ21/i+1−21/j+1et.Thus the inequality (71) and all conditions imposed in Theorem 6 are satisfied. Hence, G and f has unique fixed point that is v∗=0 in P⊆E∈X, where P 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 Theorem6 to find a common solution of a nonlinear fractional differential equation, where we can apply F-Kannan–Suzuki type mappings in complete TVS-valued cone metric spaces.Here, we investigates the Riemann–Liouville derivative fractional integral of orderα>0. This form of fractional derivative for a continuous function g:[0,∞)⟶ℝ denoted by Daαf, is given by(76)D0+αgt=ddtn−1I0+αgt=1Γn−αddtn∫0tt−sn−α−1gsds,where α denotes the integer part of the real number α and n=α+1, provided that the right hand side is pointwise defined on 0,∞. (see [49–54]). Also, the Riemann–Liouville fractional integral of order α is given by(77)I0αgt=1Γα∫att−sα−1gsds,for α>0. The notation α stands for largest integer not greater than α. If α=m∈ℕ, then D0+mgt=gmt, for t>0 and if α=0, then D0+0gt=gt for t>0.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].(78)D0+ασt+gt,σt=0,0<t<1,σ0=σ′0,σ′1=λ∫01σsds,0<λ<1,where D0+α denotes the Riemann–Liouville fractional derivative of order α and g:0,1⟶X is a continuous function.We recall the following lemmas from Bai, and Lü [57].Lemma 4.
Letα>0. If we assume σ∈C0,1∩L0,1, then the fractional differential equation:(79)D0+ασt=0,has(80)σt=C1tα−1+C2tα−2+⋯+CNtα−N,Ci∈ℝ, i=1,2,…,N, as unique solution.SinceD0+αI0+ασt=σ for all σ∈C0,1∩L0,1. From Lemma 4, we deduce the following lemma.Lemma 5.
Assume thatσ∈C0,1∩L0,1, with fractional derivative of order α>0 that belongs to σ∈C0,1∩L0,1. Then,(81)I0+αD0+ασt=σt+C1tα−1+C2tα−2+⋯+CNtα−N,for some Ci∈ℝ, i=1,2,…,N, as unique solution.The unique solution of (78) is given by(82)σt=∫atGt,sgs,usds.Recall that the Green function related to the problem (78) is given by(83)Gft,s=tα−11−sα−1α−λ+λs−α−λt−sα−1α−λΓα,0≤s≤t≤1,tα−11−sα−1α−λ+λsα−λΓα,0≤t≤s≤1.Consider the spaceX=C0,1,ℝn, E=C0,1 be endowed with the ordering σ≤ς if σt≤ςt for all t∈C0,1 and define P∈E by P=σ,ς∈E:σt,ςt≥0⊂ℝ2, X=ℝ.This space defines the metricϱ:X×X⟶E such that(84)ϱσ,ς=supt∈0,1σt−ςtψt,∀σ,ς∈X and ψt=et. Then, X,ϱ is a TVS-valued cone metric space. A function σ∈C0,1,X is a unique solution of the fractional differential integral equation (82) if and only if σ=u∗ is a solution of the nonlinear fractional differential equation (78).Now, we prove the following theorem.Theorem 8.
Suppose the following condition hold:(i)
Gft,s∈C0,1×0,1,X≥0 for all t,s∈0,1(ii)
∫01Gft,s≤γs for all t,s∈0,1(iii)
g∈C0,1×X,X is sequentially continuous(iv)
there exists a continuous functiong:0,1×ℝ⟶ℝ+, such that(85)gt,σs−gt,ςs≤e−ϑγsσs−ςs,for allt∈0,1 and ϑ>0, such that(86)γs=tα−1αλ+αα+1−α+1αtα+λtααα+1α−λΓα.Then, the fractional differential Equation4.1 has a common solution as a fixed point σ∗∈C0,1,X.Proof.
Let us define a mapG,f:P⟶E by(87)Gfσt=∫01Gft,sgs,σsds,for t∈0,1, then Gfnσ0 is sequentially continuous. This implies that f∈Gfnσ0 and fnσ0 possess a fixed point u∗∈Gf. To prove the existence of fixed point of Gf, we prove that Gf is sequentially and contraction. To show Gf is sequentially continuous, let Gfσ≠Gfς, for all σ,ς∈0,1. By condition (iv), we have(88)Gfσ−Gfς=∫01Gft,sgs,σsds−∫01Gft,sgs,ςsds,≤∫01Gft,sgs,σs−gs,ςsds,≤∫0ttα−11−sα−1α−λ+λs−α−λt−sα−1α−λΓαds+∫t1tα−11−sα−1α−λ+λsα−λΓαdse−ϑσs−ςset,≤tα−1αλ+αα+1−α+1αtα+λtααα+1α−λΓαe−ϑσs−ςs.
This implies that(89)Gfσ,Gfς≤e−ϑγsσ−ςet.
Sinceγs<1, we have(90)Gfσ,Gfς≤e−ϑσ−ςet.
Thus, for eachσ,ς∈X, we have(91)ϱGfσ,Gfς≤e−ϑMσ,ς.
Taking logarithms on both sides of (91) using F1z=lnz and the property of F, we get(92)lnϱGfσ,Gfς≤lne−ϑMσ,ςequivalent to(93)ϑ+FϱGfσ,Gfς≤FMσ,ς,where(94)Mσ,ς=ϱGσ,Gfσ+ϱGς,Gfς2.
Using (94) in (93) and applying F-Kannan–Suzuki type conditions leads to(95)12ϱσ,Gσ<ϱσ,ς⇒ϑ+FϱGfσ,Gfς≤FϱGσ,Gfσ+ϱGς,Gfς2.
Forγ∈[0,1), ϑ>0 satisfies F-Kannan–Suzuki type mapping. Therefore, Gf is a contraction mapping on X. Since all the conditions of Theorem 8 are satisfied. Therefore, there exists u∗∈C0,1 a common fixed point of G and f, that is, u∗ 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(96)unt,σ=ft,σ+∫aσKσ,t,utdt,whereun=dnudσn.The following integral equation inspired by [12, 63–66].(97)uσ,ς=lσ,ς+∫0σgσ,ς,ε,uε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τdεdν,where l,g,h are given functions and u is unknown function to be found.LetCG,f be the class of continuous functions from the set G to the set f. We denote E=ℝ+×ℝ+,E1=lσ,ς,s:0≤s≤σ≤∞,ς∈ℝ+ and E2=lσ,ς,s,t:0≤s≤σ≤∞,0≤t≤ς≤∞. We denote that l∈CE,ℝ, g∈CE1×ℝ,ℝ and h∈CE2×ℝ,ℝDenoteX be the space of functions z∈Cℝ+×ℝ+,ℝ satisfying(98)zσ,t=Oeλσ+ς,where λ is a positive constant, that is,(99)zσ,ς≤M0eλσ+ς,for constant M0>0. Let X,. be a Banach space. Define a norm in the space X by(100)zX=supσ,ς∈Ezσ,ςe−λσ+ς.Define the mappingG,f:X×X⟶[0,∞) by(101)Gfnuσ,ς=lσ,ς+∫0σgσ,ς,ε,uε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τdεdν,and(102)Gfnvσ,ς=lσ,ς+∫0σgσ,ς,ε,vε,ydε+∫0σ∫0ςhσ,ς,ν,τ,vν,τ)dεdν,for u,v∈X. The coincidence fixed point of Gfnu and Gfnv 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 functionsg,h∈X, we have(103)gσ,ς,ε,uε,ς−gσ,ς,ε,vε,ς≤γ1σ,ς,εu−v,hσ,ς,ν,τ,uν,τ−hσ,ς,ν,τ,vν,τ≤γ2σ,ς,ν,τu−v,whereγ1∈C(E1,[0,∞)) and γ2∈C(E2,[0,∞)).(ii)
There exists a nonnegative constantδ such that δ<1 and(104)∫0σγ1σ,ς,εeλε+ςdε+∫0σ∫0ςγ2σ,ς,ν,τeλν+τdτdν≤δeλσ+ς−ϑ,for allσ,ς,ε,ν,τ∈E1∪E2.Then, the nonlinear Volterra-integral equation (97) has a unique solution in E1∪E2 which is the coincidence fixed point of equations (101) and (102).Proof.
LetG,f:X⟶X be two operators such that Gfv∈X and Gfnv∈X. Now we verify that the two operators are contractive maps in X. Let u,v∈X. On contrary we claim that G and f are not contractive maps in X. From equations (101) and (102), using condition (i) and (ii) of Theorem 9, we have(105)Gfnu−GfnvX=lσ,ς+∫0σgσ,ς,ε,uε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τdτdν−lσ,ς−∫0σgσ,ς,ε,vε,ςdε−∫0σ∫0ςhσ,ς,ν,τ,vν,τdτdν,≤∫0σgσ,ς,ε,uε,ς−gσ,ς,ε,vε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τ−hσ,ς,ν,τ,vν,τdτdν,≤∫0σγ1σ,ς,εeλσ+ςdε+∫0σ∫0ςγ2σ,ς,ν,τeλν+τdτdνu−vX,≤δeλσ+ς−ϑu−vX,≤δeλσ+ς−ϑu−vX,Gfnu−GfnvX≤δe−ϑu−vXeλσ+ς,ϱGfu,Gfv≤δe−ϑMu,v,which is a contradiction. Hence u is a common fixed of G and f, also a solution to integral (97).
From (105), since δ<1 and using FKS2 of Definition 13, where(106)Mu,v=ϱGu,Gfu+ϱGv,Gfv2,we have(107)ϱGfu,Gfv≤e−ϑMu,v.
UsingF1z=lnz by taking natural logarithms in both sides of (107), we get(108)ϑ+ϱGfu,Gfv≤Mu,v.
By (106), we obtain a F-Kannan–Suzuki contraction as defined in Definition 13. Thus, all conditions imposed in Theorem 6 and Theorem 9 are satisfied. Hence, u∗ is a common fixed point of G and f in X.


## 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 Theorem6 to find a common solution of a nonlinear fractional differential equation, where we can apply F-Kannan–Suzuki type mappings in complete TVS-valued cone metric spaces.Here, we investigates the Riemann–Liouville derivative fractional integral of orderα>0. This form of fractional derivative for a continuous function g:[0,∞)⟶ℝ denoted by Daαf, is given by(76)D0+αgt=ddtn−1I0+αgt=1Γn−αddtn∫0tt−sn−α−1gsds,where α denotes the integer part of the real number α and n=α+1, provided that the right hand side is pointwise defined on 0,∞. (see [49–54]). Also, the Riemann–Liouville fractional integral of order α is given by(77)I0αgt=1Γα∫att−sα−1gsds,for α>0. The notation α stands for largest integer not greater than α. If α=m∈ℕ, then D0+mgt=gmt, for t>0 and if α=0, then D0+0gt=gt for t>0.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].(78)D0+ασt+gt,σt=0,0<t<1,σ0=σ′0,σ′1=λ∫01σsds,0<λ<1,where D0+α denotes the Riemann–Liouville fractional derivative of order α and g:0,1⟶X is a continuous function.We recall the following lemmas from Bai, and Lü [57].Lemma 4.
Letα>0. If we assume σ∈C0,1∩L0,1, then the fractional differential equation:(79)D0+ασt=0,has(80)σt=C1tα−1+C2tα−2+⋯+CNtα−N,Ci∈ℝ, i=1,2,…,N, as unique solution.SinceD0+αI0+ασt=σ for all σ∈C0,1∩L0,1. From Lemma 4, we deduce the following lemma.Lemma 5.
Assume thatσ∈C0,1∩L0,1, with fractional derivative of order α>0 that belongs to σ∈C0,1∩L0,1. Then,(81)I0+αD0+ασt=σt+C1tα−1+C2tα−2+⋯+CNtα−N,for some Ci∈ℝ, i=1,2,…,N, as unique solution.The unique solution of (78) is given by(82)σt=∫atGt,sgs,usds.Recall that the Green function related to the problem (78) is given by(83)Gft,s=tα−11−sα−1α−λ+λs−α−λt−sα−1α−λΓα,0≤s≤t≤1,tα−11−sα−1α−λ+λsα−λΓα,0≤t≤s≤1.Consider the spaceX=C0,1,ℝn, E=C0,1 be endowed with the ordering σ≤ς if σt≤ςt for all t∈C0,1 and define P∈E by P=σ,ς∈E:σt,ςt≥0⊂ℝ2, X=ℝ.This space defines the metricϱ:X×X⟶E such that(84)ϱσ,ς=supt∈0,1σt−ςtψt,∀σ,ς∈X and ψt=et. Then, X,ϱ is a TVS-valued cone metric space. A function σ∈C0,1,X is a unique solution of the fractional differential integral equation (82) if and only if σ=u∗ is a solution of the nonlinear fractional differential equation (78).Now, we prove the following theorem.Theorem 8.
Suppose the following condition hold:(i)
Gft,s∈C0,1×0,1,X≥0 for all t,s∈0,1(ii)
∫01Gft,s≤γs for all t,s∈0,1(iii)
g∈C0,1×X,X is sequentially continuous(iv)
there exists a continuous functiong:0,1×ℝ⟶ℝ+, such that(85)gt,σs−gt,ςs≤e−ϑγsσs−ςs,for allt∈0,1 and ϑ>0, such that(86)γs=tα−1αλ+αα+1−α+1αtα+λtααα+1α−λΓα.Then, the fractional differential Equation4.1 has a common solution as a fixed point σ∗∈C0,1,X.Proof.
Let us define a mapG,f:P⟶E by(87)Gfσt=∫01Gft,sgs,σsds,for t∈0,1, then Gfnσ0 is sequentially continuous. This implies that f∈Gfnσ0 and fnσ0 possess a fixed point u∗∈Gf. To prove the existence of fixed point of Gf, we prove that Gf is sequentially and contraction. To show Gf is sequentially continuous, let Gfσ≠Gfς, for all σ,ς∈0,1. By condition (iv), we have(88)Gfσ−Gfς=∫01Gft,sgs,σsds−∫01Gft,sgs,ςsds,≤∫01Gft,sgs,σs−gs,ςsds,≤∫0ttα−11−sα−1α−λ+λs−α−λt−sα−1α−λΓαds+∫t1tα−11−sα−1α−λ+λsα−λΓαdse−ϑσs−ςset,≤tα−1αλ+αα+1−α+1αtα+λtααα+1α−λΓαe−ϑσs−ςs.
This implies that(89)Gfσ,Gfς≤e−ϑγsσ−ςet.
Sinceγs<1, we have(90)Gfσ,Gfς≤e−ϑσ−ςet.
Thus, for eachσ,ς∈X, we have(91)ϱGfσ,Gfς≤e−ϑMσ,ς.
Taking logarithms on both sides of (91) using F1z=lnz and the property of F, we get(92)lnϱGfσ,Gfς≤lne−ϑMσ,ςequivalent to(93)ϑ+FϱGfσ,Gfς≤FMσ,ς,where(94)Mσ,ς=ϱGσ,Gfσ+ϱGς,Gfς2.
Using (94) in (93) and applying F-Kannan–Suzuki type conditions leads to(95)12ϱσ,Gσ<ϱσ,ς⇒ϑ+FϱGfσ,Gfς≤FϱGσ,Gfσ+ϱGς,Gfς2.
Forγ∈[0,1), ϑ>0 satisfies F-Kannan–Suzuki type mapping. Therefore, Gf is a contraction mapping on X. Since all the conditions of Theorem 8 are satisfied. Therefore, there exists u∗∈C0,1 a common fixed point of G and f, that is, u∗ 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(96)unt,σ=ft,σ+∫aσKσ,t,utdt,whereun=dnudσn.The following integral equation inspired by [12, 63–66].(97)uσ,ς=lσ,ς+∫0σgσ,ς,ε,uε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τdεdν,where l,g,h are given functions and u is unknown function to be found.LetCG,f be the class of continuous functions from the set G to the set f. We denote E=ℝ+×ℝ+,E1=lσ,ς,s:0≤s≤σ≤∞,ς∈ℝ+ and E2=lσ,ς,s,t:0≤s≤σ≤∞,0≤t≤ς≤∞. We denote that l∈CE,ℝ, g∈CE1×ℝ,ℝ and h∈CE2×ℝ,ℝDenoteX be the space of functions z∈Cℝ+×ℝ+,ℝ satisfying(98)zσ,t=Oeλσ+ς,where λ is a positive constant, that is,(99)zσ,ς≤M0eλσ+ς,for constant M0>0. Let X,. be a Banach space. Define a norm in the space X by(100)zX=supσ,ς∈Ezσ,ςe−λσ+ς.Define the mappingG,f:X×X⟶[0,∞) by(101)Gfnuσ,ς=lσ,ς+∫0σgσ,ς,ε,uε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τdεdν,and(102)Gfnvσ,ς=lσ,ς+∫0σgσ,ς,ε,vε,ydε+∫0σ∫0ςhσ,ς,ν,τ,vν,τ)dεdν,for u,v∈X. The coincidence fixed point of Gfnu and Gfnv 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 functionsg,h∈X, we have(103)gσ,ς,ε,uε,ς−gσ,ς,ε,vε,ς≤γ1σ,ς,εu−v,hσ,ς,ν,τ,uν,τ−hσ,ς,ν,τ,vν,τ≤γ2σ,ς,ν,τu−v,whereγ1∈C(E1,[0,∞)) and γ2∈C(E2,[0,∞)).(ii)
There exists a nonnegative constantδ such that δ<1 and(104)∫0σγ1σ,ς,εeλε+ςdε+∫0σ∫0ςγ2σ,ς,ν,τeλν+τdτdν≤δeλσ+ς−ϑ,for allσ,ς,ε,ν,τ∈E1∪E2.Then, the nonlinear Volterra-integral equation (97) has a unique solution in E1∪E2 which is the coincidence fixed point of equations (101) and (102).Proof.
LetG,f:X⟶X be two operators such that Gfv∈X and Gfnv∈X. Now we verify that the two operators are contractive maps in X. Let u,v∈X. On contrary we claim that G and f are not contractive maps in X. From equations (101) and (102), using condition (i) and (ii) of Theorem 9, we have(105)Gfnu−GfnvX=lσ,ς+∫0σgσ,ς,ε,uε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τdτdν−lσ,ς−∫0σgσ,ς,ε,vε,ςdε−∫0σ∫0ςhσ,ς,ν,τ,vν,τdτdν,≤∫0σgσ,ς,ε,uε,ς−gσ,ς,ε,vε,ςdε+∫0σ∫0ςhσ,ς,ν,τ,uν,τ−hσ,ς,ν,τ,vν,τdτdν,≤∫0σγ1σ,ς,εeλσ+ςdε+∫0σ∫0ςγ2σ,ς,ν,τeλν+τdτdνu−vX,≤δeλσ+ς−ϑu−vX,≤δeλσ+ς−ϑu−vX,Gfnu−GfnvX≤δe−ϑu−vXeλσ+ς,ϱGfu,Gfv≤δe−ϑMu,v,which is a contradiction. Hence u is a common fixed of G and f, also a solution to integral (97).
From (105), since δ<1 and using FKS2 of Definition 13, where(106)Mu,v=ϱGu,Gfu+ϱGv,Gfv2,we have(107)ϱGfu,Gfv≤e−ϑMu,v.
UsingF1z=lnz by taking natural logarithms in both sides of (107), we get(108)ϑ+ϱGfu,Gfv≤Mu,v.
By (106), we obtain a F-Kannan–Suzuki contraction as defined in Definition 13. Thus, all conditions imposed in Theorem 6 and Theorem 9 are satisfied. Hence, u∗ is a common fixed point of G and f in X.


## 5. Conclusion

The novelty of this study to fixed point theory is the fixed point result given in Theorem6. 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 F-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.



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