# Existence and Monotone Iterative Approximation of Solutions for Neutral Differential Equations with Generalized Fractional Derivatives

Authors: Dussadee Somjaiwang; Parinya Sa Ngiamsunthorn
Journal: Journal of Mathematics
 (2022)

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## Abstract

We study the existence and monotone iterative approximation of mild solutions of fractional-order neutral differential equations involving a generalized fractional derivative of order0<α<1 which can be reduced to Riemann–Liouville or Hadamard fractional derivatives. The existence of mild solutions is obtained via fixed point techniques in a partially ordered space. The approach is constructive and can be applied numerically. In particular, we construct a monotone sequence of functions converging to a solution which is illustrated by a numerical example.


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## Body

## 1. Introduction

Differential equations with fractional order, commonly referred to as fractional differential equations, have important role for describing the dynamics of economic models [1], finance [2], engineering [3], and scientific systems [4–8]. In recent years, various types of the fractional derivatives are defined via fractional integrals and were studied including Riemann–Liouville, Caputo, Hadamard, Caputo–Hadamard, Hilfer, Hilfer–Hadamard, Caputo-type, and Liouville fractional derivatives [9–14].Fractional calculus has triggered several research interests in various aspects including existence and uniqueness of solutions, stability and dynamical analysis, numerical methods for solutions, and modelling. Some authors investigated the nonlocal fractional operator based on a generalized Mittag-Leffler kernel [15, 16]. Dynamics of the fractional differential system is also the main research focus in many applications, for example, HIV modelling [8, 17, 18], COVID-19 modelling [7], and integrator circuit model [19].In many applications, differential equations involve functions and derivatives that contain unknown functions at a shifted time. This class of differential equation is called a delay differential equation and is commonly used to describe systems when the rate of change depends on the stages in the previous times. Delay differential equations have become important for mathematical models of phenomena in physical, engineering, and biomedical sciences. Especially, many authors have studied epidemic models with time delays [20–22].One special type of delay differential equation is the neutral differential equation which has been extensively studied by many researchers. Agarwal and Bahuguna [23] studied the existence and solution of the first-order neutral differential equation of Sobolev type with a nonlocal history condition:(1)ddtBut+ft,ut−τ1+Aut=gt,ut,ut−τ2,0<t≤Thu|−τ,0=ϕ,where τ=maxτ1,τ2,τi>0,i=1,2T<∞,ϕ∈ϱ0:=C−τ,0,X in a real Banach space X. The results were proved by using Schauder’s fixed point theorem. Agarwal et al. [24] studied the existence of fractional neural functional differential equations:(2)DCαxt−gt,xt=ft,xt,t∈t0,+∞,t0≥0xt0=ϕ,where CDα is the Caputo fractional derivative of order 0<α<1. The exisence results of solution of problem (2) were obtained by using Krasnoselskii’s fixed point theorem.In addition, Ren et al. [25] investigated periodic solutions for a second-order neutral differential equation of Rayleigh type:(3)xt−cxt−τt″=ft,x′t+gt,xt−τt+et.Sufficient conditions for the existence periodic solutions were obtained from the fixed point index theorem. In recent years, Boularesy et al. [26] studied the existence and uniqueness of solutions to nonlinear fractional-order neutral differential equation:(4)cD0+αxt−gt,xt−τt=fxt,xt−τt,t∈t0,T,where cD0+α is the standard Caputo fractional derivative of order α∈0,1, by using Krasnoselskii’s fixed point theorem and the Banach contraction principle.Motivated by previous results in the literature and our previous work in [27], we extend problem (4) to study the existence result for mild solutions of fractional-order neutral differential equation with a more general class of fractional derivative:(5)ρDt0+α,βxt−gt,xt−τt=fxt,xt−τt,t∈t0,T,subject to the initial condition(6)xt=ξt,t∈k0,t0,with α∈0,1,ρ>0,0≤β≤1 and γ=α+β1−α, where f∈Ck0,T×ℝ,ℝ,g∈Ck0,T×ℝ,ℝ,k0=inft∈t0,Tt−τt,τ:ℝ+⟶ℝ+ and the initial ξ:k0,t0⟶ℝ. Here, ρDt0+α,β is the standard generalized fractional derivative of order α∈0,1.The main contribution in this paper is that we consider the constructive approach for the existence of solutions for generalized fractional differential equations which can be applied numerically. This generalized fractional derivative can be reduced into Riemann–Liouville fractional derivative, Caputo fractional derivative, and Hadamard fractional derivative as follows. Ifρ⟶1 and β=0, our existence result reduces to the existence of mild solutions with respect to the Riemann–Liouville fractional derivative. On the other hand, if ρ⟶1 and β=1, our result implies the existence of mild solution when the Caputo fractional derivative is considered. In addition, if ρ⟶0+ and β=0, we obtain the existence of mild solutions for the problem with Hadamard fractional derivative. Furthermore, based on the monotone iterative approach, our result gives a constructive method to approximate the mild solutions. By constructing a monotone iterative sequence of functions from upper or lower solutions, we obtain a sequence of functions converging to the mild solutions.This paper can be outlined as follows. We introduce some preliminary background in Section2. We present the existence and monotone iterative approximation result for mild solutions of fractional-order neutral differential equation with delay in the sense of generalized fractional derivative for the Riemann–Liouville and the Hadamard derivatives in Section 3. Finally, in Section 4, we give an example to demonstrate the existence of mild solution of IVP (5).


## 2. Preliminary Results for Neutral Differential Equations with Generalized Fractional Derivatives

Definition 1.
(see [28]). Let a and b be real numbers with 0<a<b<∞. Let Ω=a,b be finite interval on the nonnegative numbers ℝ+. For ρ>0 and 0≤γ<1, we define the space of functions as follows.(1)
Ca,b is the space of continuous functions φ on Ω with the norm(7)‖φ‖C=maxt∈Ω|φt|.(2)
The weighted spaceC1−γ,ρa,b is the set(8)Cγ−1,ρa,b=φ:a,b⟶ℝ:tρ−aρρ1−γφt∈Ca,b,0≤γ<1,equipped with the norm(9)‖φ‖C1−γ,ρ=tρ−aρρ1−γφtC=maxt∈Ωtρ−aρρ1−γφt,andC0,ρa,b=Ca,b.(3)
The weighted spaceC1−γ,ργa,b is the set(10)C1−γ,ργa,b=φ∈C1−γa,b,Da+γ,ρφt∈C1−γa,b,0≤γ<1,andC0,ρ1a,b=Ca,b.Definition 2.
Letα and c be real numbers with α>0 and f∈Xcpa,b, where Xcpa,b denotes the space of Lebesgue mensurable functions. The left-sided generalized factional integral is given by(11)ρIaαft=1Γα∫attρ−sρρα−1fuduu1−ρ,t>a.
It should mentioned that oncep=1, the integral in (11) becomes the Riemann–Liouville fractional integral. In case that one takes the limit as ρ⟶0 in (5), it becomes the Hadamard fractional integral.It can be seen that the operatorρDa+α,β can be expressed as(12)ρDa+α,β=ρIa+β1−αδρρIa+1−γ=ρIa+β1−αρDa+γ,γ=α+β1−α.Definition 3.
Letα be a positive real number such that α∉ℕ and n=α+1, where α is the integer part of α>0. The left-sided generalized fractional derivative is defined by(13)ρDaαft=ϱnΓn−α∫attρ−uρρn−α−1fuduu1−ρ,where ρ>0 and ϱ=t1−ρd/dt.It should be noted that onceρ=1, the derivative in (13) becomes the Riemann–Liouville fractional derivative. Also by taking the limit as ρ⟶0, the derivative in (13)  becomes the Hadamard fractional derivative.Definition 4.
Letα and β be real numbers such that 0<α<1 and 0≤γ<1. If f∈Cγa,b and ρIa+1−αf∈Cγ1a,b, then(14)ρIa+αρDa+αft=ft−ρIa+1−αfaΓαtρ−aρρα−1,for all t∈a,b.Definition 5.
The functionx∈C1−γγk0,T is a mild solution of the initial value problem for the neutral fractional differential equations with generalized fractional derivative (5) if the following integral equation is satisfied:(15)xt=gt,xt−τt+1ΓγΓγ−1∫t0ttρ−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fxs,xs−τsdss1−ρ,t∈t0,T,and xt=ξt for t∈k0,t0, where Lt0=xt0−gt0,xt0−τt0.Consider the Banach spaceC1−γk0,T equipped with partially order relation. For any x1,x2∈C1−γk0,T, we define the order relation x1≤x2 if and only if x1t≤x2t with respect to t on k0,T. This defines a partial ordering on C1−γk0,T. We next outline the preliminary results on partially ordered space. Let X=X,≤¯,‖.‖ be a normed linear space with a partially ordered relation ≤¯. The space X is called regular if for any nondecreasing sequence xnn∈ℕ in X such that xn converges to y as n⟶∞, we have xn≤¯y for all n∈ℕ. In particular, we see that the space C1−γk0,T is regular.Definition 6.
The operatorT:X⟶X is called nondecreasing if for any order x1,x2 in X have relation ≤ preserved under T, that is, x1≤x2 implies Tx1≤Tx2 for all x1,x2∈X.Definition 7.
(see [29]). Let X,≤¯ be a partially ordered set. The operator T:X⟶X is called partially continuous at y∈X if for any ε>0, there is δ>0 such that under the supremum norm, we have ‖x−Ty‖<ε and ‖x−y‖<δ for all x comparable to y in X. T is called partially continuous on X when the operator T is partially continuous at every y in X.Definition 8.
(see [29]). Let X,≤¯ be a partially ordered set. The operator T:X⟶X is called partially bounded if the set TC is bounded for all chains C. In particular, it is called uniformly partially bounded if the set TC is bounded with the same constant for every chain C in X.Definition 9.
(see [29]). Let X,≤¯ be a partially ordered set. The operator T:X⟶X is called partially compact if the set TC is relatively compact in X for every chain C in X.Definition 10.
(see [29]). Let X,≤,‖.‖. The order relations ≤¯ and the metric d induced by the norm ‖.‖ are said to be compatible if the following condition holds: if xnkk∈ℕ is a subsequence of a monotone sequence xnn∈ℕ converging to y, then all of the sequence xnn∈ℕ converges to y.Definition 11.
(see [29]). Let ψ:ℝ+⟶ℝ+ be an upper semi-continuous and nondecreasing function. Then, ψ is called a D−function if ψ0=0.Definition 12.
(see [29]). Let X,≤¯,‖.‖ be a normed space with a partially ordered relation ≤¯. The operator T:X⟶X is called partially nonlinear D-contraction if there is ψ:ℝ+⟶ℝ+ such that(16)‖x1−Tx2‖≤ψ‖x1−x2‖,for all comparable pairs x1,x2∈X with φj<j for all j>0.Theorem 1 (see [30]).
LetX,≤¯,‖.‖ be a compatible and regular partially ordered linear space X. S1:X⟶X and S2:X⟶X are nondecreasing and satisfy(a)
An operatorS1 is partially nonlinear D−contraction.(b)
An operatorS2 is partially continuous and partially compact.(c)
There is a pointx0 in X with x0≺¯S1x0+S2x0.
Then, there exists at least one solutionx∗ of the operator equation S1x+S2x=x. Furthermore, a monotone sequence xnn=0∞ given by(17)xn+1=S1xn+S2xn,n=0,1,…converges monotonically to x∗.To establish the existence result for IVP (5), we consider the following assumptions:(i)
The functionsg:t0,T×ℝ⟶ℝ and f:ℝ×ℝ⟶ℝ are continuous.(ii)
The functiong is nondecreasing in x for each t∈t0,T and x∈ℝ.(iii)
There is a parameterMg>0 such that 0≤|gt,x|≤Mg with respect to t on t0,T and x∈ℝ.(iv)
There exists aD-contraction ϕ such that 0≤gt,x1−gt,x2≤ϕx1−x2 for t∈t0,T, and x1,x2∈ℝ with x1≥x2.(v)
The functionf is nondecreasing in two variables, that is, if x1≤x2 and y1≤y2, fx1,y1≤fx2,y2 for x1,x2,y1,y2∈ℝ.(vi)
There exists a parameterMf>0 such that 0≤|fx,y|≤Mf for x,y∈ℝ.(vii)
There exist functionsu∈Ck0,t0 and u∈C1−γt0,T such that u is a lower solution of IVP (5), that is,(18)ρDt0+α,βut−gt,ut−τt≤fut,ut−τtt∈t0,Tut≤xt,t∈k0,t0.


## 3. Existence and Monotone Iterative Approximation Results

In this section, we present a result on the existence and monotone iterative approximation of a mild solution of fractional-order neutral differential equations involving a generalized fractional derivative.Theorem 2.
Suppose that all seven hypotheses (i)–(vii) are satisfied. Then, IVP (5) has a mild solution x∗:k0,T⟶ℝ in which xt=ξt for t∈k0,t0 and(19)ρDt0+α,βxt−gt,xt−τt∈C1−γt0,Tcan be monotonically approximated by sequence xn,n∈ℕ defined by(20)xn+1t=gt,xnt−τt+1ΓγΓγ−1∫t0ttρ−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fxns,xns−τsdss1−ρ,t∈t0,T,where x0t=ut is the lower solution and Lt0=xt0−gt0,xt0−τt0.Proof. We denoteX=C1−γt0,T for the partially ordered Banach space. A mild solution to IVP (5) can be considered as the following operator equation:(21)S1xt+S2xt=xt,where(22)S1xt=1Γγ−1Γγ∫t0ttρ−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fxs,xs−τsdss1−ρS2xt=gt,xt−τt,for t∈t0,T. We require to show that the operators S1 and S2 satisfy all conditions in Theorem 1 which will be divided into 5 parts.First Step. By Theorem 1, we shall prove that the two operators S1 and S2 are nondecreasing. For all x,y∈X such that x≥y, by consideration of hypothesis (ii), we obtain(23)S1xt=gt,xt−τt≥gt,yt−τt=S1yt.This illustrates that the operatorS1 is a nondecreasing operator in X. For the operator S2, we obtain from hypothesis (v) that(24)S2xt−S2yt=1Γα∫t0ttρ−sρρα−1fxs,xs−τs−fys,ys−τsdss1−ρ≥0,for anyx≥y in X. Clearly, S2 is a nondecreasing operator in X.Second Step. For this step, we will prove that the operator S1 satisfies the following properties:(i)
The operatorS1 is partially bounded on X.(ii)
The operatorS1 is a partially nonlinear D−contraction on X.For this moment, letx∈X. By the property of f in hypothesis (iii), we get(25)|S1xt|=|gt,xt−τt|≤‖S1x‖≤Mg,for allt∈t0,T. Therefore, an operator S1 is partially bounded on X. Then, we show that the operator S1 is D−contraction. For any x,y∈X such that x≥y, we see from assumption (iv) that(26)|S1xt−S1yt|=|gt,xt−τt−gt,yt−τt|≤ϕxt−yt≤ϕ‖x−y‖,for eacht∈k0,T. Hence, we take the supremum norm ‖S1xt−S1yt‖≤ϕ‖x−y‖ for all x,y∈X with x≥y. This implies that S1 is a partially nonlinear D−contraction in the space X.Third Step. We shall show that the operator S2 satisfies condition (b) of Theorem 1, that is, S2 is partially continuous on X. We first show that S2 is pointwise convergent. Let xnn∈ℕ be a sequence in a chain C in X satisfying xn⟶x as n⟶∞. Using the boundedness and continuity of f from assumptions (i) and (vi) together with the dominated convergence theorem, we get(27)limn⟶∞S2xnt=limn⟶∞1ΓγΓγ−1∫t0ttρ−sρργ−2L0dss1−ρtρ−t0ρργ−1+limn⟶∞1Γα∫t0ttρ−sρρα−1fxns,xns−τsdss1−ρ=limn⟶∞1ΓγΓγ−1∫t0ttρ−sρργ−2xt0dss1−ρtρ−t0ρργ−1−limn⟶∞1ΓγΓγ−1∫t0ttρ−sρργ−2gt0,xt0−τt0dss1−ρtρ−t0ρργ−1+limn⟶∞1Γα∫t0ttρ−sρρα−1fxns,xns−τsdss1−ρ=1ΓγΓγ−1∫t0ttρ−sρργ−2limn⟶∞xt0dss1−ρtρ−t0ρργ−1−1ΓγΓγ−1∫t0ttρ−sρργ−2limn⟶∞gt0,xt0−τt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1limn⟶∞fxns,xns−τsdss1−ρ=xt0−gt0,xt0−τt0+1Γα∫t0ttρ−sρρα−1fxs,xs−τsdss1−ρ=S2xt,for eacht∈t0,T. This implies that S2xn converges to S2x pointwise on t0,T. Next, we prove the equicontinuity of S2xnn∈ℕ in X. Let t1,t2∈t0,T with t1<t2. We have(28)|S2xnt2−S2xnt1|≤1ΓγΓγ−1∫t0t1t2ρ−sρργ−2Lt0dss1−ρt2ρ−t0ρργ−1−t1ρ−t0ρργ−1+1ΓγΓγ−1∫t1t2t2ρ−sρργ−2Lt0dss1−ρt2ρ−t0ρργ−1+1ΓγΓγ−1∫t0t1t2ρ−sρργ−2−t1ρ−sρργ−2Lt0dss1−ρt1ρ−t0ρργ−1+1Γα∫t1t2t2ρ−sρρα−1fxns,xns−τsdss1−ρ+1Γa∫t0t1t2ρ−sρρα−1−t1ρ−sρρα−1fxns,xns−τsdss1−ρ≤Lt0ΓγΓγ−1∫t0t1t2ρ−sρργ−2dss1−ρt2ρ−t0ρργ−1−t1ρ−t0ρργ−1+Lt0ΓγΓγ−1∫t1t2t2ρ−sρργ−2dss1−ρt2ρ−t0ρργ−1+Lt0ΓγΓγ−1∫t0t1t2ρ−sρργ−2−t1ρ−sρργ−2dss1−ρt1ρ−t0ρργ−1+MfΓα∫t0t1t1ρ−sρρα−1−t2ρ−sρρα−1dss1−ρ+MfΓα∫t1t2t2ρ−sρρα−1dss1−ρ=Lt0t1−t0γ−1pγ−1ΓγΓγ−1γ−1t2ρ−t0ρργ−1−t1ρ−t0ρργ−1+Lt0t2−t1γ−1ργ−1ΓγΓγ−1γ−1t2ρ−t0ρργ−1+Lt0ΓγΓγ−1∫t0t1t2ρ−sρργ−2−t1ρ−sρργ−2dss1−ρt1ρ−t0ρργ−1+p1−αMfΓα+1t2−t1α+Mfρα−1Γα∫t0t1t2ρ−sρα−1−t1ρ−sρα−1dss1−ρ⟶0,ast2−t1⟶0 uniformly for all n∈ℕ. So, S2xn⟶S2x uniformly. Since S2xn converges to S2x pointwise and uniformly, this implies that operator S2 is partially continuous on X.Fourth Step. We show that S2 is partially compact. Let y∈S2C where C is a chain in X. We have y=S2x for some x∈C. By condition (vi), we get(29)|yt|=|S2xt≤1ΓγΓγ−1∫t0ttp−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+MfΓα∫t0ttρ−sρρα−1dss1−ρ≤Lt0ΓγΓγ−1∫t0ttρ−sρργ−2dss1−ρtρ−t0ρργ−1+MfΓα∫t0ttρ−sρρα−1dss1−ρ≤Lt0t−t0γ−1ργ−1ΓγΓγ−1γ−1tρ−t0ρργ−1+ρ1−αMfΓα+1t−t0α≤Lt0Tγ−1ργ−1ΓγΓγ−1γ−1tρ−t0ρργ−1+ρ1−αMfΓα+1Tα≕K,for allt∈t0,T. By taking the supremum in the last assertion, we get ‖yt‖≤K for all y∈S2C. Clearly, the operator S2 is uniformly bounded on every chain C in X. It remains to prove that S2 is equicontinuous on every chain C in X. Let y∈S2C and take t1,t2∈t0,T with t1<t2. We have(30)|S2xt2−S2xt1|≤1ΓγΓγ−1∫t0t1t2ρ−sρργ−2Lt0dss1−ρt2ρ−t0ρργ−1−t1ρ−t0ρργ−1+1ΓγΓγ−1∫t1t2t2ρ−sρργ−2Lt0dss1−ρt2ρ−t0ρργ−1+1ΓγΓγ−1∫t0t1t2ρ−sρργ−2−t1ρ−sρργ−2Lt0dss1−ρt1ρ−t0ρργ−1+1Γα∫t1t2t2ρ−sρρα−1fxs,xs−τsdss1−ρ+1Γa∫t0t1t2ρ−sρρα−1−t1ρ−sρρα−1fxs,xs−τsdss1−ρ≤Lt0ΓγΓγ−1∫t0t1t2ρ−sρpγ−2dss1−ρt2ρ−t0ρργ−1−t1ρ−t0ρργ−1+Lt0ΓγΓγ−1∫t1t2t2ρ−sρργ−2dss1−ρt2ρ−t0ρργ−1.+Lt0ΓγΓγ−1∫t0t1t2ρ−sρργ−2−t1ρ−sρργ−2dss1−ρt1ρ−t0ρργ−1+MfΓα∫t0t1t1p−sρρα−1−t2ρ−sρρα−1dss1−ρ+MfΓα∫t1t2t2ρ−sρρα−1dss1−p=Lt0t1−t0γ−1ργ−1ΓγΓγ−1γ−1t2ρ−t0ρργ−1−t1ρ−t0ρργ−1+L0t2−t1γ−1ργ−1ΓγΓγ−1γ−1t2ρ−t0ρργ−1+Lt0ΓγΓγ−1∫t0t1t2ρ−sρργ−2−t1ρ−sρργ−2dss1−ρt1ρ−t0ρργ−1+ρ1−αMfΓα+1t2−t1α+Mfρα−1Γα∫t0t1t2ρ−sρα−1−t1p−spα−1dss1−ρ⟶0,ast2−t1⟶0 uniformly for y∈S2C. Thus, we get that S2 is also relatively compact and we can conclude that the operator S2 is partially compact on every chain C in X.Fifth Step. Finally, we prove that there exists an element u∈X which is a lower solution of IVP (5). By hypothesis (vii), there exist functions u∈Ct0,T and u∈C1−γt0,T satisfying(31)ρD0+α,βut−gt,ut−τt≤fut,ut−τt,t∈t0,Tut≤xt,t∈k0,t0.Then, it follows that(32)ut≤gt,ut−τt+1ΓγΓγ−1∫t0ttρ−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fus,us−τsdss1−ρ,whereLt0=xt0−gt0,xt0−τt0 for t∈t0,k. This means that u satisfies u≤S1u+S2u. As a consequence, we conclude that the operator equation S1x+S2x=x has a solution since the two operators S1 and S2 satisfy all conditions in Theorem 1. In addition, the solution of IVP (5) can be monotonically approximated by a sequence xn as n=1,2,….


## 4. Example

Consider the neutral fractional differential equation with generalized fractional derivatives(33)ρDα,βxt−arc tanxt−0.012=tanhxt−12ext−0.01+5,t∈1,2xt=0,t∈0,1.Here,(34)fx,y=tanhx−12ey+5,and(35)gt,x=arc tanx2.Letβ=0; we get α=γ. Clearly, both functions f and g are continuous and nondecreasing. Considering functions f and g in equation (31), we see that(36)0≤|fx,y|≤tanhx−12ey+5≤Mf=6,and(37)0≤|gt,x|=12arctanx≤Mg=π4.Next, we verify that the functiong is a D-contraction with ϕt=1/2t. We have(38)0≤gt,x−gt,y=12arctanx−arctany≤12x−y≤12|x−y|=ϕx−y,for all t∈ℝ. Hence, the functions f and g satisfy assumptions (i)–(vi). Finally, for assumption (vii), we claim that a lower solution of equation (24) is given by ut=0 for all t∈0,2. We see that 0≤xt for all t∈0,1. Let Lt=xt−gt,xt−0.01 for t∈ℝ. It can be seen from(39)gt,ut−τt+1ΓγΓγ−1∫t0ttρ−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fus,us−τsdss1−ρ=gt,ut−0.01+1ΓγΓγ−1∫t0ttρ−sρργ−2L1dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fus,us−0.01dss1−ρ=gt,0+1ΓγΓγ−1∫t0ttρ−sρργ−2−g1,x0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttp−sρρα−1f0,0dss1−ρ<4.5ρ1−αΓα+1,for t∈1,2 that(40)0≤gt,ut−0.01+1ΓγΓγ−1∫0ttρ−sρργ−2L1dss1−ρtρ−t0ρργ−1+1Γα∫0ttρ−sρρα−1fus,us−0.01dss1−ρ,for t∈1,2. Then, we see that all conditions of Theorem 2 are satisfied. Hence, equation (31) has a solution x∗:0,2⟶ℝ in which ρDα,βxt−arctanxt−0.01/2∈C1−γ0,2 can be obtained from the monotone iterative sequence xn,n=0,1,2,… defined by(41)xn+1t=gt,xnt−τt+1ΓγΓγ−1∫t0ttρ−sρργ−2Lt0dss1−ρtρ−t0ρργ−1+1Γα∫t0ttρ−sρρα−1fxns,xns−τsdss1−ρ,t∈1,2xt=0,t∈0,1,with x0t=0 for all t∈1,2.We numerically demonstrate the monotone iterative approximation of solutions corresponding to various fractional derivatives including Riemann–Liouville fractional derivative and Hadamard fractional derivative in Figures1–3.Figure 1
Caseβ=0 and ρ⟶1: the monotone iterative sequence of equation (31) under the Riemann–Liouville fractional derivative for α=0.5.Figure 2
Caseβ=0 and ρ=0.5 with fractional derivative for α=0.5.Figure 3
Caseβ=0 and ρ⟶0+: the monotone iterative sequence of equation (31) under the Hadamard fractional derivative for α=0.5.In Figure1, we numerically compute a monotone iterative increasing sequence xn converging to the mild solution of equation (31) for β=0 and ρ⟶1 which corresponds to the fractional differential equation under the Riemann–Liouville fractional derivative for α=0.5 with time delay τ=0.01. The lower solution is chosen to be x0=0 on t∈0,2. It can be seen that sequence xn increases at t=1.01 and then gradually decreases on t∈1,2 which exhibits the behavior of the mild solution.In Figure2, we numerically compute a monotone iterative increasing sequence xn converging to the mild solution of equation (31) for β=0 and ρ=0.5 which corresponds to the fractional differential equation under the generalized fractional derivative for α=0.5 with time delay τ=0.01. The lower solution is chosen to be x0=0 on t∈0,2. It can be seen that sequence xn increases rapidly at t=1.01 and then becomes constant on t∈1.1,2 which exhibits the behavior of the mild solution.In Figure3, we numerically compute a monotone iterative increasing sequence xn converging to the mild solution of equation (31) for β=0 and ρ⟶0+ which corresponds to the fractional differential equation under the Hadamard fractional derivative for α=0.5 with time delay τ=0.01. Similar to Figure 1, the lower solution is chosen to be x0=0 on t∈0,2. It can be seen that sequence xn increases at t=1.01 and then gradually decreases on t∈1.1,2 which exhibits the behavior of the mild solution.


## 5. Conclusions

We establish the existence and monotone iterative approximation of mild solutions to the neutral fractional differential equations under a generalized fractional derivative which can be applied numerically. The monotone iterative sequence for the mild solution of equation (31) for 0<α<1 admits the following particular cases. If ρ⟶1 and β=0, then we have γ=α and we obtain the existence and approximation of solution under the Riemann–Liouville fractional derivative. If ρ⟶0+ and β=0, then we get the existence and approximation of solution under Hadamard fractional derivative.



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