# Fixed-Point Theorems for Rational Interpolative-Type Operators with Applications

Authors: Muhammad Sarwar; Sahibzada Waseem Ahmad; Thabet Abdeljawad
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the current manuscript, two fixed-point theorems for Dass-Gupta and Gupta-Saxena rational interpolative-type operators are studied in the setting of metric spaces. For the authenticity of the presented work, examples and applications to the existence of a solution to the Caputo-Fabrizio fractional derivative and Caputo-Fabrizio fractal-fractional derivative are also discussed.


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## Body

## 1. Introduction and Preliminaries

In [1], Banach proposed his famous postulation, the Banach contraction principle. Banach proposed that a continuous self-operator in the setting of complete metric space possesses a unique fixed point. The contraction condition was generalized in several directions. One of the generalizations was supposed by Kannan [2, 3] in which he dropped the continuity assumption of the operator. As a part of the generalization of the Banach contraction principle, Dass-Gupta [4] and Gupta-Saxena [5] introduced the innovation of rational contraction. In recent times, Karapinar [6] converted the classical Kannan [2] contraction to an interpolative Kannan contraction in order to maximize the rate of convergence of an operator to a unique fixed point. However, Karapinar and Agarwal [7] found a little gap in the article [6] about the assumption of the fixed point being unique. They provided a counterexample to verify that the fixed point need not be unique and invalidate the assumption of a unique fixed point.Invented more than a century ago, fractional calculus has attracted numerous physicists, mathematicians, engineers, and researchers in the field of biological sciences because of its extraordinary involvement in these fields of sciences. Despite the long debate, fractional calculus is still not mature enough, and scientists from every field need to do a lot more in this particular branch of mathematics to solve some of its complicated problems. Fractional calculus is mainly composed of fractional integral equation, fractional differential equations, and fractional integrodifferential equations. From the very first day, scientists are finding ways to solve these types of integral equations cited in the manuscripts which can provide enough insight on these problems and their solution [8–12]; among all the other techniques, one technique is a fixed-point theory; several articles can be found on this topic (see, for instance, [13–17] and references therein).In the fixed-point theory, researchers mostly try to check the existence of a solution to the problem in an underlying set. Because of its complicated nature, it is sometimes impossible to find the exact solution to fractional-type equations in some cases; therefore, frequently, scientists attempt to find the nature of the solution rather than an analytic or exact solution. The particular type of differential equations, namely, Caputo-Fabrizio fractional and Caputo-Fabrizio fractal-fractional differential equations [18–20], can also be used in many problems of the aforesaid fields like heat transfer problem, Fisher reaction diffusion equation, mass spring damper system, modeling of steady heat flow, etc.Motivated by the work done in [6, 7, 21], we have converted the famous rational contractions in [4, 5] to interpolation rational contractions with examples given in each case to verify each theorem. Consequently, the enormous amount of applications of fractional differential equation has led us to contribute the existence of a solution to a couple of fractional differential equations, i.e., the Caputo-Fabrizio fractional differential equation and Caputo-Fabrizio fractal-fractional differential equation. The analysis in the current study indicates that whenever the aforementioned fractional differential equations satisfy certain conditions under specific circumstances, then a contraction theorem guarantees the existence of at least one solution.Theorem 1 (see [1]).
ConsiderM,pto be a complete metric space. LetT:M⟶Mbe a continuous self-operator if there exists a constantδ∈0,1, such that(1)pTf,Tg≤δpf,g.
∀f,g∈M, then T possesses a unique fixed point.Afterwards, a question was raised: what happens to the fixed point when the operator is not continuous? Ultimately, the answer to the question was given by Kannan in [2]. He proposed the following theorem as a modified version of Theorem 1.Theorem 2 (see[2]).
ConsiderM,pto be a complete metric space. LetT:M⟶Mbe as a self-operator if there exists a constantδ∈0,1/2, such that(2)pTf,Tg≤δpf,Tf+pg,Tg.
∀f,g∈M, then T possesses a unique fixed point.Eventually, Theorem1 was generalized to a rational-type contraction by Gupta et al. [4, 5] as follows:Theorem 3 (see [4]).
ConsiderM,pto be a complete metric space. LetT:M⟶Mbe a continuous self-operator if there exists constantsδ1,δ2>0, such thatδ1+δ2<1and(3)pTf,Tg≤δ1pg,Tg1+pf,Tf1+pf,g+δ2pf,g.
∀f,g∈M, then T possesses a unique fixed point.Theorem 4 (see [5]).
ConsiderM,pto be a complete metric space. LetT:M⟶Mbe a self-operator if there exists constantsδ1,δ2,δ3>0, such thatδ1+δ2+δ3<1and(4)pTf,Tg≤δ1pg,Tg1+pf,Tf1+pf,g+δ2pg,Tgpf,Tf1+pf,g+δ3pf,g.
∀ distinct f,g∈M, then T possesses a unique fixed point.


## 2. Main Results

This section provides the extension of the famous rational-type contractions to interpolative rational contraction.Before proceeding to the first result of the current section, consider the definition which can later be used in the proof of the first main result of this section.Definition 5.
SupposeM,pbe a metric space. Furthermore, consider a continuous operatorT:M⟶M. If∃α∈0,1andδ∈0,1, such that(5)pTf,Tg≤δpg,Tgpf,Tfpf,gαpf,g1−α.
∀ distinct f,g∈M/FixM. Then, T is known as an interpolative Dass and Gupta rational-type contraction.It can be analyzed that Definition5 is the conversion of the Dass-Gupta rational contraction in Theorem 3 to an interpolation Dass-Gupta rational contraction.Theorem 6.
In the setting of a complete metric space, the operator for interpolative Dass and Gupta rational-type contraction defined in Definition5 possesses a fixed point.Proof.
Taking an arbitraryf0∈M,p and constructing an iterative sequence fnn∈ℕ as fn=Tfn−1=Tnf0. If fn0=fn0+1 for any n0>0, then fn0 is a fixed point for T, which completes the proof. Consequently, taking fn≠fn+1 for each n≥0 and by replacing f by fn and g by fn−1 in (5), it is deduced that
(6)pfn+1,fn=pTfn,Tfn−1≤δpfn−1,Tfn−1pfn,Tfnpfn,fn−1αpfn,fn−11−α=δpfn−1,fnpfn,fn+1pfn,fn−1αpfn,fn−11−α=δpfn,fn+1αpfn,fn−11−α.
With straightforward calculation, it can be analyzed that(7)pfn,fn+11−α≤δpfn,fn−11−α.
Hence, it can be observed that the sequencepfn,fn−1 is a sequence of nonnegative terms which is nonincreasing. As a consequence, there is a nonnegative constant L such that limn→∞pfn,fn−1=L. It is presumed that L>0. Indeed, from (7), it is obvious that
(8)pfn,fn+1≤δpfn,fn−1≤⋯≤δnpf0,f1.
Lettingn⟶∞ in (8), it can be concluded that L=0.
As a proceeding step, it is proven that the given sequence is a Cauchy sequence. Using the triangle inequality(9)pfn,fn+m≤δpfn,fn+1+⋯+δpfn+m−1,fn+m≤δnpf0,f1+⋯+δn+m−1pf0,f1≤δn1−δpf0,f1.
Lettingn⟶∞ in (9), it is deduced that the sequence fnn∈ℕ is a Cauchy sequence. As stated, M,p is a complete metric space; such an assumption guarantees the existence of a number f∈M such that limn→∞pfn,f=0. At last, it is proven that f is the fixed point of T of the sequence fnn∈ℕ. Suppose that f≠Tf; therefore, pf,Tf>0. Recall that f≠Tf and for each n≥0 and by letting f=fn and g=f in (5), it is determined that
(10)pfn+1,Tf=pTfn,Tf≤δpf,Tfpfn,Tfnpfn,fαpfn,f1−α=δpf,Tfpfn,fn+1pfn,fαpfn,f1−α.
By lettingn⟶∞in ((10)), it is analyzed thatpTf,f=0; thus,f=Tfwhich is, subsequently, a contradiction. Therefore, f=Tf.
The proof is complete.Example 7.
Consider a nonempty setM=0,1,2,3and a distance functionp:M×M⟶0,∞defined byp=f−g∀f,g∈M. It can be analyzed that the metric spaceM,pis a complete metric space. Next, an operatorT:M⟶M, is defined by(11)Tf=1,iff∈0,1,3,2,iff∈2.Now, to check ifT is a Dass and Gupta-type contraction. Consider, for f=1, g=2, δ1=0.7, and δ2=0.1(12)1=pT1,T2>δ1pg,Tg1+pf,Tf1+pf,g+δ2pf,g=δ1p2,21+p1,11+p1,2+δ2p1,2=δ2.Asδ1,δ2∈0,1 along with δ1+δ2=0.8<1, so that T does not satisfy the inequality (3). Therefore, it is not a Dass-Gupta-type contraction in Theorem 3.On the contrary forδ=0.15 and α=1/2. Let f,g∈M/FixM, then f,g∈0,3,3,0, i.e., two cases arise in general:Case 1.
Iff=0 and g=3, then
(13)0=pT0,T3<δp3,T3p0,T0p0,3αp0,31−α=0.212.Case 2.
Iff=3 and g=0, then
(14)0=pT3,T0<δp0,T0p3,T3p3,0αp3,01−α=0.212.Henceforth, the self-operatorT is an interpolative Dass and Gupta-type contraction with the data provided above. The fixed points are 1,2.Next, consider the extension of the Gupta-Saxena-type rational contraction to interpolative contraction. Before proceeding to the theorem, consider the definition.Definition 8.
ConsiderM,pto be a metric space. In addition, consider a continuous operatorT:M⟶M. If∃α,β∈0,1andδ∈0,1, such that(15)pTf,Tg≤δpf,Tfpg,Tgpf,gαpg,Tgpf,Tfpf,gβpf,g1−α−β.
∀ distinct f,g∈M/FixM. Then, T is known as an interpolative Gupta-Saxena rational-type contraction.It can be analyzed that Definition8 is the conversion of Gupta-Saxena rational contraction in Theorem 4 to an interpolation Gupta-Saxena rational contraction.Theorem 9.
In the setting of a complete metric space, the operator for interpolative Gupta-Saxena rational-type contraction defined in Definition8 possesses a fixed point.Proof.
By taking an arbitraryf0∈M,p and constructing an iterative sequence fnn∈ℕ as fn=Tfn−1=Tnf0. If fn0=fn0+1 for any n0>0, then f0 is a fixed point for T, which completes the proof. Consequently, taking fn≠fn+1 for each n≥0 and by replacing f by fn and g by fn−1 in (15), it is deduced that
(16)pfn+1,fn=pTfn,Tfn−1≤δpfn,Tfnpfn−1,Tfn−1pfn,fn−1αpfn−1,Tfn−1pfn,Tfnpfn,fn−1βpfn,fn−11−α−β=δpfn,fn+1pfn−1,fnpfn,fn−1αpfn−1,fnpfn,fn+1pfn,fn−1βpfn,fn−11−α−β=δpfn,fn+1α+βpfn,fn−11−α−β.
With straightforward calculation, it can be analyzed that(17)pfn,fn+11−α−β≤δpfn,fn−11−α−β.
Hence, it can be observed that the sequencepfn,fn−1 is a sequence of nonnegative terms which is nonincreasing. As a consequence, there is a nonnegative constant L such that limn→∞pfn,fn−1=L. It is presumed that L>0. Indeed, from (17), it is obvious that
(18)pfn,fn+1≤δpfn,fn−1≤⋯≤δnpf0,f1.
Lettingn⟶∞ in (18), it can be concluded that L=0.
As a proceeding step, it is proven that the given sequence is a Cauchy sequence. Using the triangle inequality(19)pfn,fn+m≤δpfn,fn+1+⋯+δpfn+m−1,fn+m≤δnpf0,f1+⋯+δn+m−1pf0,f1≤δn1−δpf0,f1.
Lettingn⟶∞ in (19), it is deduced that the sequence fnn∈ℕ is a Cauchy sequence. As stated, M,p is a complete metric space; such an assumption guarantees the existence of a number f0∈M such that limn→∞pfn,f0=0. At last, it is proven that f0 is the fixed point of T of the sequence fnn∈ℕ. Suppose that f≠Tf; therefore, pf,Tf>0. Recall that f≠Tf and for each n≥0 and by letting f=fn and g=f in (15), it is determined that
(20)pfn+1,Tf=pTfn,Tf≤δpfn,Tfnpf,Tfpfn,fαpf,Tfpfn,Tfnpfn,fβpfn,f1−α−β=δpfn,fn+1pf,Tfpfn,fαpf,Tfpfn,fn+1pfn,fβpfn,f1−α−β.
By lettingn⟶∞ in (20), it is analyzed that pTfn,f=0; thus,f=Tf, which is, subsequently, a contradiction. Therefore, f=Tf.
The proof is complete.Example 10.
Consider the distances and operator defined in Example7. In the proceeding example, it is shown thatTis not a Gupta-Saxena-type contraction forδ1=0.3, δ2=0.1, andδ3=0.4. However,Tis an interpolative version of the Gupta-Saxena-type contraction, i.e., forf=0andg=2(21)1=pT1,T2>δ1pg,Tg1+pf,Tf1+pf,g+δ2pg,Tgpf,Tf1+pf,g+δ3pf,g=δ1p2,21+p0,11+p0,2+δ2p2,2p0,11+p0,2+δ3p0,2=2δ3.
Sinceδ1,δ2,δ3∈0,1 along with δ1+δ2+δ3<1 suggests that T does not satisfy inequality (4); therefore, it is not a Gupta-Saxena-type contraction in Theorem 4.On the contrary, forδ=0.15, α=1/3 and β=1/3. Let f,g∈M/FixM, then f,g∈0,3,3,0, i.e., two cases arise in general:Case 1.
Iff=0 and g=3, then
(22)0=pT0,T3<δp0,T0p3,T3p0,3αp3,T3p0,T0p0,3βp0,31−α−β=0.13332.Case 2.
Iff=3 and g=0, then
(23)0=pT3,T0<δp3,T3p0,T0p3,0αp0,T0p3,T3p3,0βp3,01−α−β=0.1333.Henceforth, the self-operatorT is an interpolative version of the Gupta-Saxena-type contraction with the data provided in Example 7. The fixed points are 1,2.


## 3. Applications

The current section provides the existence of a solution to the Caputo-Fabrizio fractional derivative and the Caputo-Fabrizio fractal-fractional derivative. The very first result of this section is the existence of the solution to the Caputo-Fabrizio fractional derivative.The Caputo-Fabrizio fractional derivative of orderγ is defined as
(24)CFDtγAt=Nγ1−γ∫0tA′τexp−γ1−γt−τdτ,with boundary conditionA0=0 where Nγ is a normalization function satisfying N0=N1=1 and a≤t≤τ≤b.ConsiderM=C0,1,R to be the space of continuous functions on the interval 0,1. Furthermore, consider a distance function p:M2⟶0,∞ defined by p=∣f−g∣. Henceforth, the metric space M,p is complete. Now, consider a self-operator T:M⟶M defined by
(25)TA=Nγ1−γ∫0tA′τexp−γ1−γt−τdτ.Then, the fractional derivative (24) will have a solution if the following condition is satisfied:
(26)∣At−Bt∣≤2−γK1−γΠA,B,for allt∈0,1 where α,δ∈0,1, K>0, and ΠA,B=δpB,TBpA,TA/pA,BαpA,B1−α.To begin the proof, consider(27)∣TA−TB∣=∣Nγ1−γ∫0tA′τexp−γ1−γt−τdτ−Nγ1−γ∫0tB′τexp−γ1−γt−τdτ∣≤Nγ1−γexp−γ1−γt−τ∣Aτ0t−Bτ0t∣+γ1−γ∫0texp−γ1−γt−τ∣Aτ−Bτ∣dτ≤Nγ1−γ∣At−Bt∣+Nγ1−γexp−γ1−γt∣A0−B0∣+γ1−γ∫0texp−γ1−γt−τ∣Aτ−Bτ∣dτ.Using inequality (26), the above inequality can be written as
(28)TA−TB≤2−γNγK1−γ1KΠA,B+2−γNγK1−γ1ΠA,B=22−γNγK1−γΠA,B.So,(29)pTA,TB≤δΠA,B,whereδ=22−γNγ/κ1−γ. Consequently, all the propositions of Theorem 6 are satisfied. Therefore, the equation (24) possesses a solution.Next, we discuss the existence of a solution to the Caputo-Fabrizio fractal-fractional derivativeConsider the Caputo-Fabrizio fractal-fractional derivative of orderγ as
(30)aCFFFDtγgt=1−γNγϑtϑ−1gt,yt+γϑNγ∫0tτϑ−1gτ,yτdτ,whereNγ is a normalization function satisfying N0=N1=1.ConsiderM=C0,1,R to be the space of continuous functions on the interval 0,1. Furthermore, consider a distance function p:M2⟶0,∞ defined by p=∣f−g∣. Henceforth, the metric space M,p is complete. Now, consider a self-operator T:M⟶M defined by
(31)Tg=1−γNγϑtϑ−1gt,yt+γϑNγ∫0tτϑ−1gτ,yτdτ.Then, the fractal-fractional derivative (30) will have a solution if the operator (31) satisfies the given conditions ∀t∈0,1(32)g1t,yt−g2t,yt≤Ξg1,g2,(33)1−γϑ−γt≤δNγtϑ−1,where Ξg1,g2=pg1,Tg1pg2,Tg2/pg1,g2αpg2,Tg2pg1,Tg1/pg1,g2βpg1,g21−α−β and δ,α,β∈0,1.Consider,(34)∣Tg1−Tg2∣=∣1−γNγϑtϑ−1g1t,yt+γϑNγ∫0tτϑ−1g1τ,yτdτ−1−γNγϑtϑ−1g2t,yt−γϑNγ∫0tτϑ−1g2τ,yτdτ∣≤1−γNγϑtϑ−1∣g1t,yt−g2t,yt∣+γϑNγ∫0tτϑ−1∣g1τ,yτ−g2τ,yτ∣dτ.With the help of (32), the above inequality can be written as
(35)∣Tg1−Tg2∣≤1−γNγϑtϑ−1Ξg1,g2+γϑNγΞg1,g2∫0tτϑ−1dτ=tϑ−1Nγ1−γϑ−γtΞg1,g2.Finally, using (33) in the above inequality,
(36)pTg1,Tg2≤δΞg1,g2.Subsequently, all the hypotheses of Theorem9 are satisfied. Therefore, Equation (30) has a solution.


## 4. Conclusion

In this study, we converted the famous Dass and Gupta rational contraction and Gupta-Saxena rational contraction to their respective interpolative versions. The motivation for this work is to obtain the fixed point faster than the classical versions of the aforesaid contractions. Further, we studied the existence of a solution to Caputo-Fabrizio fractional differential equation and Caputo-Fabrizio fractal-fractional derivative utilizing the obtained rational interpolation to validate the results.



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