# A Novel Theoretical Investigation of the Abu-Shady–Kaabar Fractional Derivative as a Modeling Tool for Science and Engineering

Authors: Francisco Martínez; Mohammed K. A. Kaabar
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

A newly proposed generalized formulation of the fractional derivative, known as Abu-Shady–Kaabar fractional derivative, is investigated for solving fractional differential equations in a simple way. Novel results on this generalized definition is proposed and verified, which complete the theory introduced so far. In particular, the chain rule, some important properties derived from the mean value theorem, and the derivation of the inverse function are established in this context. Finally, we apply the results obtained to the derivation of the implicitly defined and parametrically defined functions. Likewise, we study a version of the fixed point theorem forα-differentiable functions. We include some examples that illustrate these applications. The obtained results of our proposed definition can provide a suitable modeling guide to study many problems in mathematical physics, soliton theory, nonlinear science, and engineering.


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## Body

## 1. Introduction

Fractional calculus is theoretically considered as a natural extension of classical differential calculus, which has attracted many researchers, both from a more theoretical point of view and for its diverse applications in sciences and engineering. Thus, from a more theoretical perspective, various definitions of fractional derivatives have been initiated. Fractional definitions try to satisfy the usual properties of the classical derivative; however, the only property inherent in these definitions is the property of linearity. On the contrary, some of the drawbacks that these derivatives present can be located in the following:(i)
The Riemann-Liouville derivative does not satisfyDaα1=0, if α is not a natural number(ii)
Fractional derivative statements do not possess some of the fundamental properties of classical derivatives, such as the product rule, the quotient rule, or the chain rule(iii)
These derived proposals, in general, do not satisfyDαDβf=Dα+βf(iv)
The definition of the Caputo derivative implies that the functionf must be differentiable in the ordinary senseMore information on this definition of fractional derivative can be found in [1, 2].The locally formulated fractional derivative is established through certain quotients of increments. In this sense, Khalil et al. [3] introduced a locally defined derivative, called conformable derivative. Some of the inconveniences that the previous fractional derivatives presented have been successfully solved via this definition. Thus, for example, the aforementioned rules for the derivation of products and quotients of two functions or the chain rule are properties that have been satisfied by the conformable derivative. The physical and geometric meaning of the derivative is studied in [4, 5]. However, in [6], the author shows the disadvantages of using the conformable definition compared to Caputo’s fractional derivative definition, to solve some fractional models.Recently, Abu-Shady and Kaabar [7] introduced a new generalized formulation of the fractional derivative (GFFD) that allows to solve analytically in a simple way some fractional differential equations, whose results agree exactly with those obtained via the Caputo and Riemann-Liouville derivatives. Also, this new definition has advantages compared to the conformable derivative definition. In addition, the study in [7] has been recently extended to study some important special functions in the sense of GFFD which are essential for modeling phenomena [8].The GFFD definition is very important in studying various phenomena in science and engineering due to the powerful applicability of this definition in investigating many fractional differential equations in a very simple direction of obtaining analytical solutions without the need for approximate numerical methods or complicated algorithms like other classical fractional definitions. This definition is a modified version of the conformable definition to overcome all issues and advantages associated with the conformable one.Regarding the geometric behavior of GFFD, by following the previous research study concerning the fractional cords orthogonal trajectories in the sense of conformable definition [5], GFFD can be similarly applied to the same example to interpret its geometrical meaning in more details.One of the limitations of GFFD is that GFFD is locally defined derivative, and some future works are needed to proposed nonlocal formulation of GFFD in order to preserve the nonlocality property of fractional calculus. However, nonlocal definitions come with many associated challenges while working on solving fractional differential equations. Therefore, the future studies will work on overcoming all these challenges.The work is constructed as follows: The GFFD and its main properties are presented in Section2. New results on generalized α-differential functions are proposed in Section 3 to complete the study carried out in [7]. Some interesting applications of the results obtained on generalized α-differentiable functions are presented in Section 4. In particular, illustrative examples of the derivation of implicitly defined functions, of parametrically defined functions and of the application of the fixed point theorem for generalized α-differentiable functions are included. Some conclusions are drawn in Section 5.


## 2. Fundamental Tools

Definition 1 (see [7]).
A given functionf:0,∞⟶R, the GFFD of order ,0<α≤1, of f at t>0 is expressed as
(1)DGFFDft=limε⟶0ft+Γβ/Γβ−α+1εt1−α−ftε,β>−1,β∈R+.
Iff is α-differentiable α−DF in some 0,a, a>0, and limt⟶0+DGFFDft exists, then we have
(2)DGFFDf0=limt⟶0+DGFFDft.Theorem 2 (see [7]).
Let0<α≤1,β>−1,β∈R+ and let f,g be α−DF at a point t>0. Then, we obtain
(i)
DGFFDf+g=aDGFFDf+bDGFFDg,∀a,b∈R(ii)
DGFFDtp=pΓβ/Γβ−α+1tp−α,∀p∈R(iii)
DGFFDψ=0,∀constant functions ft=ψ(iv)
DGFFDfg=fDGFFDg+gDGFFDf(v)
DGFFDf/g=gDGFFDf−fDGFFDg/g2(vi)
If, additionally,f is a differentiable function, then DGFFDft=Γβ/Γβ−α+1t1−αdf/dtt.
The generalizedα-derivative of certain functions using GFFD is expressed as:
(i)
DGFFD1=0(ii)
DGFFDsinkt=kΓβ/Γβ−α+1t1−αcoskt(iii)
DGFFDcoskt=−kΓβ/Γβ−α+1t1−αsinkt(iv)
DGFFDekt=kΓβ/Γβ−α+1t1−αekt
In addition, it is interesting to highlight the generalizedα-derivative of the following functions:
(i)
DGFFDΓβ−α+1/αΓβtα=1(ii)
DGFFDsinΓβ−α+1/αΓβtα=cosΓβ−α+1/αΓβtα(iii)
DGFFDcosΓβ−α+1/αΓβtα=−sinΓβ−α+1/αΓβtα(iv)
DGFFDeΓβ−α+1/αΓβtα=eΓβ−α+1/αΓβtα.Theorem 3 (Rolle’s theorem for generalizedα-differentiable functions) (see [7]).
Leta>0, α∈0,1 and f:a,b⟶R be a given function satisfying
(i)
f is continuous on a,b(ii)
f is generalized α−DF on a,b(iii)
fa=fb
Then,∃c∈a,b, such that DGFFDfc=0.Theorem 4 (Mean value theorem for generalizedα-differentiable functions) (see [7]).
Leta>0, α∈0,1 and f:a,b⟶R be a given function satisfying
(i)
f is continuous on a,b(ii)
f is generalized α−DF on a,b
Then,∃c∈a,b, ∋(3)DGFFDfc=fb−fahbα−aα,where h=Γβ−α+1/αΓβ.


## 3. New Results on Generalizedα-Differentiable Functions

In this section, we establish important results that complete the theory of generalizedα-differentiable functions, introduced in [7].Theorem 5.
If a given functionf:0,∞⟶Risα−DF at t0>0, 0<α≤1, β>−1,β∈R+, then f is continuous at t0.Proof.
Since(4)ft0+Γβ/Γβ−α+1εt01−α=ft0+Γβ/Γβ−α+1εt01−α−ft0εε+ft0,
Then,(5)limε⟶0ft0+Γβ/Γβ−α+1εt01−α=limε⟶0ft0+Γβ/Γβ−α+1εt01−α−ft0εε+ft0=ft0.
Hence,f is continuous at t0Theorem 6 (Chain rule).
Let0<α≤1, β>−1,β∈R+, g generalized α−DF at t>0 and f differentiable at gt then
(6)DGFFDf∘gt=f′gtDGFFDgt.Proof.
We prove the result following a standard limit approach. First, if the functiong is constant in a neighborhood of a>0 then DGFFDf∘gt=0. If g not is constant in a neighborhood of a>0, we can find a t0>0 such that gt1≠gt2 for any t1,t2∈a−t0,a+t0. Now, since g is continuous at a, for ε sufficiently small, we have
(7)DGFFDf∘ga=limε⟶0fga+Γβ/Γβ−α+1εa1−α−fgaε=limϵ⟶0fga+Γβ/Γβ−α+1εa1−α−fgaga+Γβ/Γβ−α+1εa1−α−ga∙ga+Γβ/Γβ−α+1εa1−α−gaε=limϵ⟶0fga+Γβ/Γβ−α+1εa1−α−fgaga+Γβ/Γβ−α+1εa1−α−ga∙limϵ⟶0ga+Γβ/Γβ−α+1εa1−α−gaε.
Making(8)h=ga+ΓβΓβ−α+1εa1−α−ga,in the first factor, so we have
(9)limϵ⟶0fga+Γβ/Γβ−α+1εa1−α−fgaga+Γβ/Γβ−α+1εa1−α−ga=limh⟶0fga+h−fgah,from here
(10)DGFFDf∘ga=limh⟶0fga+ε−fgahlimϵ⟶0ga+Γβ/Γβ−α+1εa1−α−gaε=f′gaDGFFDga.Remark 7.
Using the fact that differentiability implies generalizedα-differentiability and assuming gt>0, Equation (6) can be written as
(11)DGFFDf∘gt=Γβ−α+1Γβgtα−1DGFFDfgtDGFFDgt.Theorem 8 (Extended mean value theorem for generalizedα-differentiable functions) [5].
Leta>0, α∈0,1, and f,g:a,b⟶R be functions satisfying
(i)
f,g are continuous on a,b(ii)
f,g are generalized α−DF on a,b(iii)
DGFFDgt≠0∀t∈a,b(iv)
gb≠ga(v)
DGFFDft and DGFFDgt not annulled simultaneously on a,b
Then,∃c∈a,b, ∋(12)DGFFDfcDGFFDgc=fb−fagb−ga.Proof.
Consider the function(13)Ft=ft−fa−fb−fagb−gagt−ga.
SinceF is continuous on a,b, generalized α−DF on a,b, and Fa=Fb=0, then by Theorem 3, ∃c∈a,b such that DGFFDFc=0. Using the linearity of DGFFD and the fact that the generalized α-derivative of a constant is zero, our result follows.Remark 9.
Observe that Theorem4 is a special case of this theorem for gt=Γβ−α+1/αΓβtαTheorem 10.
Leta>0, α∈0,1 and f:a,b⟶R be a given function satisfying
(i)
f is continuous on a,b(ii)
f is generalized α−DF on a,b
IfDGFFDft=0, for all t∈a,b, then, f is a constant on a,bProof.
SupposeDGFFDft=0 for all t∈a,b. Let t1,t2∈a,b with t1<t2. So, the closed interval t1,t2 is contained in a,b, and the open interval t1,t2 is contained in a,b.
Hence,f is continuous on t1,t2 and α−DF on t1,t2. So, by Theorem 4, there is c∈t1,t2 with
(14)ft2−ft1Γβ−α+1/αΓβt2α−t1α=DGFFDfc=0.
Therefore,ft2−ft1=0 and ft2=ft1
Sincet1 and t2 are arbitrary numbers in a,b with t1<t2, then f is a constant on a,b.Corollary 11 (see [5]).
Leta>0, α∈0,1, and F,G:a,b⟶R be functions such that DGFFDFt=DGFFDGt∀t∈a,b. Then, ∃ a constant C such that
(15)Ft=Gt+C.Proof.
By simply applying the above theorem toHt=Ft−Gt, it can easily be proven.Theorem 12 (see [5]).
Leta>0, α∈0,1, and f:a,b⟶R be a given function satisfying
(i)
f is continuous on a,b(ii)
f is generalized α−DF on a,b
Then, we have(i)
IfDGFFDft>0∀t∈a,b, then f is strictly increasing on a,b(ii)
IfDGFFDft<0∀t∈a,b, then f is strictly decreasing on a,bProof.
Following similar line of argument as given in the Theorem10, there exists c between t1 and t2 with
(16)ft2−ft1Γβ−α+1/αΓβt2α−t1α=DGFFDfc.(1)
IfDGFFDfc>0, then ft2>ft1 for t1<t2. Therefore, f is strictly increasing on a,b, since t1 and t2 are arbitrary number of a,b(2)
IfDGFFDfc<0, then ft2<ft1 for t1<t2. Therefore, f is strictly decreasing on a,b, since t1 and t2 are arbitrary number of a,b.Definition 13.
LetI⊂0,∞ an open interval, α∈0,1, and f:I⟶R be we will say that f∈CαI,R if the f is generalized α−DF on I and generalized α-derivative is continuous on I.Theorem 14.
LetI⊂0,∞ an open interval, α∈0,1, and f:I⟶R be a function of class Cα on the interval I. Suppose fa=b for some a∈I, and DGFFDfa≠0. Then, there is an open neighborhood U of a in which f admits an inverse function f−1 of class Cα on the open neighborhood V=fU of b, and its generalized α-derivative is
(17)DGFFDf−1y=ΓβΓβ−α+12t1−αy1−αDGFFDft,∀y∈V,t=f−1y.Proof.
SinceDGFFDft is continuous in the open interval I, it is a known fact that there exists an open neighborhood U of a in which DGFFDft has a constant sign (the sign of DGFDfa). From Theorem 14, it follows f that is strictly monotonic on U (increasing if DGFFDfa>0 and decreasing if DGFFDfa<0). Therefore, f is continuous and strictly monotonic on U, so there is the inverse function of the one-to-one function f:U⟶V, with V=fU. This inverse f−1:V⟶U is of class Cα and strictly monotonic (in the same sense that f is) on V. Equation (30) can be obtained from the identity ff−1y=y for all y∈V, in which the α-derivative (with respect to y) is calculated, applying the chain rule
(18)Γβ−α+1Γβtα−1DGFFDft∙DGFFDf−1y=ΓβΓβ−α+1y1−α,∀y∈V,t=f−1y.


## 4. Applications

Some interesting applications of the results obtained on generalizedα−DF functions are presented in this section.


### 4.1. Generalizedα-Derivative of an Implicit Function

It is a known fact that an equationFt,y=0 implicitly defines a function y=gt in a certain open interval I, if Ft,gt=0∀t∈I. Suppose that gt and Ft,gt are generalized α−DF functions in an open interval I⊂o.∞, then the derivative DGFFDgt can be found by calculating the generalized α-derivative of Ft,gt, as a compound function, and canceling this derivative calculated.Now, we are going to calculate the derivative of the generalized1/3-differentiable y=gt function at the point t=8, such that g8=1, and it is implicitly defined by the equation
(19)t3−2e6y3−3t3sin1−y3−y3+1=0.Calculating the1/3-derivative in this equation, we obtain
(20)13ΓβΓβ+2/3e6gt3+2t3−2e6gt3gt−2/3DGFFDgt−ΓβΓβ+2/3sin1−gt3+t3cos1−gt3gt−2/3DGFFDgt−13gt−2/3DGFFDgt=0.Takingt=8 and g8=1 in the equation above, we have
(21)13ΓβΓβ+2/3e6+2DGFFDg8−13DGFFDg8=0.Finally, the generalized1/3-derivative is given by
(22)DGFFDg8=−ΓβΓβ+2/3e65.


### 4.2. Generalizedα-Derivative of a Parametrically Defined Function

Lett=tλ, y=yλ be generalized α−DF functions on an open interval I⊂o.∞, with DGFFDtλ≠0∀λ∈I. If t=tλ and y=yλ, define the function y=yλt=yt (where λt is the inverse function of tλ), and then, the generalized α-derivative of this function is given by
(23)DGFFDyt=Γβ−α+1Γβ∙λα−1∙DGFFDyλ∙DGFFDλt=Γβ−α+1Γβ∙λα−1∙DGFFDyλ∙ΓβΓβ−α+12∙λ1−αt1−αDGFFDtλ=ΓβΓβ−α+1DGFFDyλDGFFDtλtλ1−α.Note that the above expression is obtained by applying the chain rule and the derivation formula of the inverse function, both in the generalized sense.Thus, for example, consider the functiony=yt defined parametrically by
(24)tλ=r3λ3++sin3λ3yλ=r1−cos3λ3,∀λ∈0,2π33.,is generalized 1/3-differentiable and its 1/3-derivative is (as a function of t)
(25)DGFFDyt=ΓβΓβ+2/3DGFFDyλDGFFDtλtλ2/3=ΓβΓβ+2/3sin3λ31+cos3λ3r23λ3++sin3λ323=ΓβΓβ+2/3tan3λ32r23λ3++sin3λ323.


### 4.3. Fixed Point Theorem for Generalizedα-Derivative

We present the fixed point theorem for generalizedα-derivative and its respective proof. In addition, we will establish some important results about the iteration of the fixed point in the generalized α-derivative sense. However, we first present some basic concepts and necessary results for the developments that we are going to carry out.Definition 15.
Let the following fixed-point equationt=ft. Here,fis a mapping from X⟶X. We assume that X is endowed with the metric d. A point t∈X which satisfies t=ft is called a fixed point of f.Theorem 16 (Modified mean value theorem for generalizedα-differentiable functions).
Leta>0, α∈0,1 and f:a,b⟶R be a given function satisfying
(i)
f is continuous on a,b(ii)
f is generalized α−DF on a,b
Then,∃c∈a,b, ∋(26)DGFFDfchc1−α=fb−fab−a,where h=Γβ/Γβ−α+1.Proof.
Consider the function(27)gt=ft−fa−fb−fab−at−a.
Then, the functiong satisfies the conditions of Theorem 3. Hence, ∃c∈a,b, ∋Ttαg0Cc=0. Therefore
(28)0=DGFFDgc=DGFFDfc−fb−fab−a∙ΓβΓβ−α+1∙c1−α.
Hence,(29)DGFFDfchc1−α=fb−fab−a.
Now, we establish the fixed point theorem for generalizedα-derivative.Theorem 17.
Leta>0 and α∈0,1. If f∈Ca,b,R and ft∈a,b, ∀t∈a,b, then g has a fixed point at a,b. If also, f is generalized α−DF on a,b and
(30)Γβ−α+1ΓβDGFFDftt1−α≤k<1,∀t∈a,b,then f has a unique fixed point p at a,b.Proof.
Iffa=a or fb=b, the existence of the fixed point is obvious. Suppose that fa≠a and fb≠b, therefore fa>a and fb<b. Let gt=ft−t, clearly continuous on a,b, we have
(31)ga=fa−a>0andgb=fb−b<0.
By the classical intermediate value theorem, then∃p∈a,b∋gp=0; that is
(32)gp=fp−p=0.
Therefore,f has a fixed point at p.
Suppose also that Equation (30) is satisfied and that p and q are fixed points on a,b with p≠q. By Theorem 16, ∃ is a number ξ between p and q, and therefore, in a,b, such that
(33)p−q=fp−fq=Γβ−α+1ΓβDGFDfξξ1−αp−q≤kp−q<p−q,which implies that p−q<p−q which is contradiction, therefore p=q.Remark 18.
To approximate the fixed point of a functionf, we choose an initial approximate value p0, and we obtain the succession pnn=0∞ by taking pn=fpn−1 for each n≥1. If the succession pnn=0∞ of converges p and f is a continuous function, then
(34)p=limn⟶∞pn=limn⟶∞fpn−1=flimn⟶∞pn−1=fp,and a solution of the equation t=ft is obtained. This technique is called iterative technique of the fixed point or functional iteration.The following result provides a first step to determine a procedure that guarantees that the functionf converges a solution of the equation t=ft and that also chooses f correctly in such a way that it makes the convergence as quickly as possible.Theorem 19.
Leta>0 and α∈0.1. If f∈Ca,b,R and ft∈a,b, for all t∈a,b. Also, suppose that f is generalized α−DF on a,b with
(35)Γβ−α+1ΓβDGFFDftt1−α≤k<1,∀t∈a,b.
Ifp0 is any number in a,b, then, the succession defined by
(36)pn=fpn−1,n≥1,converges the unique fixed point p in a,b.Proof.
First, by Theorem17, ∃is a unique fixed point at a,b. On the other hand, since f applies to a,b itself, the sequence pnn=0∞ is defined ∀n≥0 and pn∈a,b∀n. Using Equation (35) and the intermediate value ​​theorem,
(37)pn−p=fpn−1−fp=Γβ−α+1ΓβDGFFDfξξ1−αpn−1−p≤kpn−1−p,where ξ∈a,b. Applying the above equation inductively results
(38)pn−p≤kpn−1−p≤k2pn−2−p≤⋯≤knp0−p.
Sincek<1, it easily follows that
(39)limn⟶∞pn−p≤limn⟶∞knp0−p=0,and pnn=0∞ converges to p.Corollary 20.
Iff satisfies the hypotheses of Theorem 19, then
(40)pn−p≤kn1−kp0−p1,∀n≥1.Proof.
Forn≥1, the procedure used in the proof of Theorem 19 implies that
(41)pn+1−pn=fpn−fpn−1≤kpn−pn−1≤⋯≤knp1−p0.
Hence, form>n≥1(42)pm−pn=pm−pm−1+pm−1−⋯−pn+1+pn+1−pn≤pm−pm−1+pm−1−pm−2+⋯+pn+1−pn≤km−1p1−p0+km−2p1−p0+knp1−p0=kn1+k+k2+⋯+km−n−1p1−p0.
By Theorem19, limn⟶∞pm=p, so
(43)p−pn=limm⟶∞pm−pn≤knp0−p∑i=0∞ki=kn1−kp0−p1.Remark 21.
It is clear that the speed of convergence depends on the factorkn/1−k, and the smaller k can be made, the faster the convergence will be. Convergence can be very slow if k is close to 1.Finally, we present an example that illustrates these last established results.Consider the function:ft=2−t on the interval 1/9,1, and we take α=β=1/2. We can observe that f1/9,1=0.5,0.7937005259840997⊂1/9,1. Also, f is continuous and
(44)Γ1Γ1/2DGFFDftt=−log22t2t≤0.82522692269223647775,∀t∈19,1,so f satisfies the hypotheses of Theorem 17 and has a unique fixed point at 1/9,1. In addition, if we use Corollary 20, we can estimate the number of iterations required to find an approximation of the fixed point with a precision of 10−4. Taking p0=1, to obtain this precision, 54 iterations are required. Also, note that since the generalized 1/2-derivative DGFFDft is negative, the successive approximations oscillate around the fixed point.


## 4.1. Generalizedα-Derivative of an Implicit Function

It is a known fact that an equationFt,y=0 implicitly defines a function y=gt in a certain open interval I, if Ft,gt=0∀t∈I. Suppose that gt and Ft,gt are generalized α−DF functions in an open interval I⊂o.∞, then the derivative DGFFDgt can be found by calculating the generalized α-derivative of Ft,gt, as a compound function, and canceling this derivative calculated.Now, we are going to calculate the derivative of the generalized1/3-differentiable y=gt function at the point t=8, such that g8=1, and it is implicitly defined by the equation
(19)t3−2e6y3−3t3sin1−y3−y3+1=0.Calculating the1/3-derivative in this equation, we obtain
(20)13ΓβΓβ+2/3e6gt3+2t3−2e6gt3gt−2/3DGFFDgt−ΓβΓβ+2/3sin1−gt3+t3cos1−gt3gt−2/3DGFFDgt−13gt−2/3DGFFDgt=0.Takingt=8 and g8=1 in the equation above, we have
(21)13ΓβΓβ+2/3e6+2DGFFDg8−13DGFFDg8=0.Finally, the generalized1/3-derivative is given by
(22)DGFFDg8=−ΓβΓβ+2/3e65.


## 4.2. Generalizedα-Derivative of a Parametrically Defined Function

Lett=tλ, y=yλ be generalized α−DF functions on an open interval I⊂o.∞, with DGFFDtλ≠0∀λ∈I. If t=tλ and y=yλ, define the function y=yλt=yt (where λt is the inverse function of tλ), and then, the generalized α-derivative of this function is given by
(23)DGFFDyt=Γβ−α+1Γβ∙λα−1∙DGFFDyλ∙DGFFDλt=Γβ−α+1Γβ∙λα−1∙DGFFDyλ∙ΓβΓβ−α+12∙λ1−αt1−αDGFFDtλ=ΓβΓβ−α+1DGFFDyλDGFFDtλtλ1−α.Note that the above expression is obtained by applying the chain rule and the derivation formula of the inverse function, both in the generalized sense.Thus, for example, consider the functiony=yt defined parametrically by
(24)tλ=r3λ3++sin3λ3yλ=r1−cos3λ3,∀λ∈0,2π33.,is generalized 1/3-differentiable and its 1/3-derivative is (as a function of t)
(25)DGFFDyt=ΓβΓβ+2/3DGFFDyλDGFFDtλtλ2/3=ΓβΓβ+2/3sin3λ31+cos3λ3r23λ3++sin3λ323=ΓβΓβ+2/3tan3λ32r23λ3++sin3λ323.


## 4.3. Fixed Point Theorem for Generalizedα-Derivative

We present the fixed point theorem for generalizedα-derivative and its respective proof. In addition, we will establish some important results about the iteration of the fixed point in the generalized α-derivative sense. However, we first present some basic concepts and necessary results for the developments that we are going to carry out.Definition 15.
Let the following fixed-point equationt=ft. Here,fis a mapping from X⟶X. We assume that X is endowed with the metric d. A point t∈X which satisfies t=ft is called a fixed point of f.Theorem 16 (Modified mean value theorem for generalizedα-differentiable functions).
Leta>0, α∈0,1 and f:a,b⟶R be a given function satisfying
(i)
f is continuous on a,b(ii)
f is generalized α−DF on a,b
Then,∃c∈a,b, ∋(26)DGFFDfchc1−α=fb−fab−a,where h=Γβ/Γβ−α+1.Proof.
Consider the function(27)gt=ft−fa−fb−fab−at−a.
Then, the functiong satisfies the conditions of Theorem 3. Hence, ∃c∈a,b, ∋Ttαg0Cc=0. Therefore
(28)0=DGFFDgc=DGFFDfc−fb−fab−a∙ΓβΓβ−α+1∙c1−α.
Hence,(29)DGFFDfchc1−α=fb−fab−a.
Now, we establish the fixed point theorem for generalizedα-derivative.Theorem 17.
Leta>0 and α∈0,1. If f∈Ca,b,R and ft∈a,b, ∀t∈a,b, then g has a fixed point at a,b. If also, f is generalized α−DF on a,b and
(30)Γβ−α+1ΓβDGFFDftt1−α≤k<1,∀t∈a,b,then f has a unique fixed point p at a,b.Proof.
Iffa=a or fb=b, the existence of the fixed point is obvious. Suppose that fa≠a and fb≠b, therefore fa>a and fb<b. Let gt=ft−t, clearly continuous on a,b, we have
(31)ga=fa−a>0andgb=fb−b<0.
By the classical intermediate value theorem, then∃p∈a,b∋gp=0; that is
(32)gp=fp−p=0.
Therefore,f has a fixed point at p.
Suppose also that Equation (30) is satisfied and that p and q are fixed points on a,b with p≠q. By Theorem 16, ∃ is a number ξ between p and q, and therefore, in a,b, such that
(33)p−q=fp−fq=Γβ−α+1ΓβDGFDfξξ1−αp−q≤kp−q<p−q,which implies that p−q<p−q which is contradiction, therefore p=q.Remark 18.
To approximate the fixed point of a functionf, we choose an initial approximate value p0, and we obtain the succession pnn=0∞ by taking pn=fpn−1 for each n≥1. If the succession pnn=0∞ of converges p and f is a continuous function, then
(34)p=limn⟶∞pn=limn⟶∞fpn−1=flimn⟶∞pn−1=fp,and a solution of the equation t=ft is obtained. This technique is called iterative technique of the fixed point or functional iteration.The following result provides a first step to determine a procedure that guarantees that the functionf converges a solution of the equation t=ft and that also chooses f correctly in such a way that it makes the convergence as quickly as possible.Theorem 19.
Leta>0 and α∈0.1. If f∈Ca,b,R and ft∈a,b, for all t∈a,b. Also, suppose that f is generalized α−DF on a,b with
(35)Γβ−α+1ΓβDGFFDftt1−α≤k<1,∀t∈a,b.
Ifp0 is any number in a,b, then, the succession defined by
(36)pn=fpn−1,n≥1,converges the unique fixed point p in a,b.Proof.
First, by Theorem17, ∃is a unique fixed point at a,b. On the other hand, since f applies to a,b itself, the sequence pnn=0∞ is defined ∀n≥0 and pn∈a,b∀n. Using Equation (35) and the intermediate value ​​theorem,
(37)pn−p=fpn−1−fp=Γβ−α+1ΓβDGFFDfξξ1−αpn−1−p≤kpn−1−p,where ξ∈a,b. Applying the above equation inductively results
(38)pn−p≤kpn−1−p≤k2pn−2−p≤⋯≤knp0−p.
Sincek<1, it easily follows that
(39)limn⟶∞pn−p≤limn⟶∞knp0−p=0,and pnn=0∞ converges to p.Corollary 20.
Iff satisfies the hypotheses of Theorem 19, then
(40)pn−p≤kn1−kp0−p1,∀n≥1.Proof.
Forn≥1, the procedure used in the proof of Theorem 19 implies that
(41)pn+1−pn=fpn−fpn−1≤kpn−pn−1≤⋯≤knp1−p0.
Hence, form>n≥1(42)pm−pn=pm−pm−1+pm−1−⋯−pn+1+pn+1−pn≤pm−pm−1+pm−1−pm−2+⋯+pn+1−pn≤km−1p1−p0+km−2p1−p0+knp1−p0=kn1+k+k2+⋯+km−n−1p1−p0.
By Theorem19, limn⟶∞pm=p, so
(43)p−pn=limm⟶∞pm−pn≤knp0−p∑i=0∞ki=kn1−kp0−p1.Remark 21.
It is clear that the speed of convergence depends on the factorkn/1−k, and the smaller k can be made, the faster the convergence will be. Convergence can be very slow if k is close to 1.Finally, we present an example that illustrates these last established results.Consider the function:ft=2−t on the interval 1/9,1, and we take α=β=1/2. We can observe that f1/9,1=0.5,0.7937005259840997⊂1/9,1. Also, f is continuous and
(44)Γ1Γ1/2DGFFDftt=−log22t2t≤0.82522692269223647775,∀t∈19,1,so f satisfies the hypotheses of Theorem 17 and has a unique fixed point at 1/9,1. In addition, if we use Corollary 20, we can estimate the number of iterations required to find an approximation of the fixed point with a precision of 10−4. Taking p0=1, to obtain this precision, 54 iterations are required. Also, note that since the generalized 1/2-derivative DGFFDft is negative, the successive approximations oscillate around the fixed point.


## 5. Conclusions

Novel results regarding the Abu-Shady–Kaabar fractional derivative have been investigated in this study which are extensions of the previous research study’s results in [7]. In particular, some important properties of the generalized fractional derivative have been accomplished, such as the chain rule, some consequences of the mean value theorem, and the derivation of the inverse function. It is verifiable with the fact that these newly obtained results are considered as a natural extension of the classical differential calculus. The potential of this new definition of fractional derivative, both from a theoretical point of view and due to its applications, is evident through the developments and illustrative examples included in the previous section. This research can definitely open a new path for more related future works in which the results of classical mathematical analysis are extended in the sense of this new definition of fractional derivative. This definition will be applied further in studying various partial differential equations such as Schrödinger equation and Wazwaz–Benjamin–Bona–Mahony equation to study some solutions that are important in soliton theory and many other interesting research topics. Some specific examples of studies that can be further studied in the sense of GFFD are the Klein–Fock–Gordon equation via the Kudryashov-expansion method [9], the systems of fractional-order partial differential equations via the Laplace optimized decomposition technique [10], and the noninteger fractional-order hepatitis B model [11], by comparing the previous results in the senses of conformable and Caputo definitions with new results using GFFD. Numerical experiments with error analysis including comparison between conformable derivative and our definition including CPU time in the graphical representations in the sense of our proposed definition will be conducted in our future studies. In addition, in our future study, all algorithms and/or pseudo-codes will be provided for the solutions’ steps using one of the common software packages such as MAPLE and MATHEMATICA.



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