# On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

Authors: Dongming Nie; Azmat Ullah Khan Niazi; Bilal Ahmed
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.


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## Body

## 1. Introduction

Fractional differential equations (FDEs) have boosted considerably due to their application in various fields of sciences, such as engineering, chemistry, mechanics, and physics. Recently, fractional ordinary differential equations (ODEs) and partial differential equations (PDEs) have been developed significantly. Indeed, we can also find applications in control, electromagnetism, and electrochemistry (see [1, 2]). For more details in this area, one can also see the monographs of Kilbas et al. [3], Li et al. [4, 5], Liu et al. [6–9], Miller and Ross [10], Podulbny [11], Rehman et al. [12], and the references therein.Positive solution of fractional ODEs have already discussed in [13–15]. However, until now research on existence of positive solution for fractional functional differential equations (FFDEs) is rare. Research in different fields including engineering, physics, and biosciences have proved that numerous system structures explain more exactly with the help of FDEs [16–21] and similarly FDEs with delay [22–27] are more accurate to illustrate the real world problems compared to FDEs without delay. So, in [28], Miller and Ross mentioned that this field has been touched by almost all fields of engineering and science.Idea of Ulam stability started in 1940, when during the talk at Wisconsin University Ulam posed a problem that “When can we assert that an approximate solution of a functional equation can be approximated by a solution of the corresponding equation?” (for more details, see [29]). After one year, Hyers first gave the answer to the Ulam’s question [30] in case of Banach spaces. After that, stability of this type is called the Ulam–Hyers stability. Rassias [31] in 1978 provided an outstanding generalization of the Ulam–Hyers stability of mappings by considering variables.Usually, the discussion of the existence and uniqueness of a solution for nonlinear FDEs normally fixed point theory has been used [16, 32, 33]. Motivated by these and [34–36], in this work, we have discussed existence and uniqueness of solution also after applying some sufficient conditions, obtained positive solution, and at the end established local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability and presented stability results graphically for the class of nonlinear FDEs with delay given by(1)Dc0γzt−gt,zt=ztft,zt,t∈I,zt=ψt≥0,t∈−τ,0,where Dc0γ is the Caputo derivative and 0<γ<1, while ψ∈C and I=0,T where C=C−τ,0;ℝ+ is space of continuous functions, τ is a nonnegative real number also f,g∈I×C,ℝ+ are continuous functions, and zt∈C is defined as ztθ=zt+θ,−τ≤θ≤0.Remaining paper is arranged as follows.Section 2 includes some basic definitions and lemmas used throughout this paper. In Section 3, we have the main results. In Section 4, we establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for problem (1), and in Section 5, simulation of an example is given to show the applicability of our results.


## 2. Preliminaries

LetQ be a cone in a real Banach space X and partial order ≤ introduced by Q in X is as(2)ifv−u∈Qthenu≤v.Definition 1.
Letu,v∈X be the order interval u,v defined as(3)u,v=w∈X:u≤w≤v.Definition 2.
The functionalht,zt is said to be nondecreasing w.r.t. ϕ on I×C such that(4)xϕ≤yϕ,ϕ∈−τ,0,implies that(5)ht,x≤ht,y,for any t,xandt,y∈I×C.Definition 3.
(see [11]). The fractional integral of order γ for a function g with lower limit 0 can be defined as(6)I0γgt=1Γγ∫0tgst−s1−γds,γ>0,t>0,where Γ is the gamma function and right-hand side of upper equality is defined pointwise on ℝ+.Definition 4.
(see [11]). The left Caputo fractional derivative of order γ is given by(7)Dctγft=1Γn−γ∫atfnst−sγ+1−nds,where n=γ+1 (γ stands for the bracket function of γ).Lemma 1.
(Leray Schauder fixed point theorem, see [11]). Let Sr be a nonempty, closed, bounded, and convex subset of Banach space X and Q:Sr⟶Sr is a compact and continuous map; then, Q has a fixed point in Sr.Theorem 1.
IfU⊂Q, where Q is a cone of space of partial order X, F:U⟶X be nondecreasing. If there exists u0,v0∈U such that v0≥u0, u0,v0⊂U, and u0 is lower solution and v0 is an upper solution of y−Fy=0, then y−Fy=0 has minimum solution u∗ and maximum solution v∗ in u0,v0 such that v∗≥u∗, and if anyone of these conditions holds(i)
X is reflexive, F is weak continuous or continuous, and Q is normal(ii)
F is completely continuous and Q is normal(iii)
F is continuous and Q is regular


## 3. Main Results

In this section, we have discussed existence and uniqueness of solution and some conditions for positive solution of equation (1). Consider X=C−τ,T is a Banach space endowed with the supremum norm and cone Q is defined as Q=z∈Q;zt≥0,−τ≤t≤T. From equation (1), we have(8)I0γDc0γzt−gt,zt=I0γztft,zt,and we can easily get that(9)zt=ψ0−g0,ψ+gt,zt+1Γγ∫0tt−τγ−1zτfτ,zτdτ,t∈0,T.Letx⋅:−τ,T⟶ℝ+ be the function defined by(10)xt=ψt,t∈−τ,0,0,t∈I.So,x0=ψ and for every w∈CI,ℝ+ with w0=0, and we defined w¯ as(11)w¯t=0,t∈−τ,0,wt,t∈I.Letx⋅ satisfy equation (9), and we can decompose z. as zt=w¯t+xt,for0≤t≤T so that zt=w¯t+xt, for every 0≤t≤T, where w¯t is such that w¯t<1,∀t∈I and the function w⋅ satisfies(12)w¯t=ψ0−g0,ψ+gt,w¯t+xt+1Γγ∫0tt−sγ−1w¯s+xsfs,w¯s+xsds,t∈I.SetH=z∈Q;z=ψt,t∈−τ,0⊂Q, where Q is a cone and Q⊂X and X is a Banach space with Sup norm w¯t≤supwt;−τ≤t≤T. Let the operator P:H⟶H be defined by(13)Pw¯t=ψ0−g0,ψ+gt,w¯t+xt+1Γγ∫0tt−sγ−1w¯sfs,w¯s+xsds.Before proving main results, we introduce the following conditions:(C1)
Let us takef,g:I×C⟶R+ with g0,ψ=0, where function g is nondecreasing and gt,zt≤L1 and fandg satisfy Lipchitz condition that is, and there exists nonnegative constants c1andc2 such that(14)ft,u−ft,v≤c1u−v,∀t∈I,andu,v∈C,gt,u−gt,v≤c2u−v,∀t∈I,andu,v∈C.(C2)
IfG⊂H be bounded, f is nondecreasing and(15)maxt∈Ift,zt,z∈G=L−1,where L>0. Let H=z∈Q;z=ψt,t∈−τ,0⊂Q. Define the operator P by(16)Pw¯t=ψ0+gt,w¯t+xt+1Γγ∫0tt−sγ−1w¯sfs,w¯s+xsds,t∈I,0,t∈−τ,0.Theorem 2.
Assume thatf and g are nonnegative and continuous; then, operator P:H⟶H is completely continuous.Proof.
As we know thatf and g are nonnegative and continuous, so operator P:H⟶H is continuous. Let G⊂H is bounded, i.e., there exists a constant l>0 such that z≤l,∀z∈G. Here, we have to keep in mind the condition that xt=0whent∈I. If maxft,zt|0≤t≤T,z∈G=L−1, and for all z∈G, we get that(17)Pw¯t≤ψ+gt,wt¯+xt+1Γγ∫0tt−sγ−1w¯sfs,w¯s+xsds≤l+L1+LΓγ∫0tt−sγ−1ds≤l+L1+LTγΓγ+1.
Hence,PG is bounded. Now, we show that PG is equicontinuous, and the proof is divided into three possible steps.Step 1: for everyw¯∈G and t1,t2∈I such that t2>t1, then(18)Pw¯t1−Pw¯t2≤gt1,w¯t1+xt1−gt2,w¯t2+xt2+1Γγ∫0t1t1−sγ−1w¯sfs,w¯s+xsds−1Γγ∫0t2t2−sγ−1w¯sfs,w¯s+xsds≤c2w¯t1−w¯t2+1Γγ∫0t1t1−sγ−1−∫0t2t2−sγ−1w¯sfs,w¯s+xsds≤c2w¯t1−w¯t2+1Γγw¯t‖ft,w¯t+xt∫0t1t1−sγ−1−t2−sγ−1ds−∫t1t2t2−sγ−1ds≤c2w¯t1−w¯t2+LΓγ+1t1γ+t2−t1γ−t2γ+t2−t1γ<c2supwt1−wt2+LΓγ+1t2−t1γ.As t1⟶t2, the right-hand side of above inequality tends to zero.Step 2: for everyw¯∈G, as ψ∈C−τ,0 and t1∈−τ,0 while 0≤t2≤T, there exists δ>0 such that when δ1>t1, ψt1−ψ0<ϵ/2.
Ifδ=minδ1,Γγ+1ϵ−2L1/2L1/γ, then for t2−t1<δ, we get that(19)Pw¯t1−Pw¯t2≤ψt−ψ0+gt2,w¯t2+xt2+1Γγ∫0t2t2−sγ−1w¯sfs,w¯s+xsds≤ϵ2+L1+LΓγ+1t2γ=ϵ2+L1+LΓγ+1δγ≤ϵ2+ϵ2=ϵ.Step 3: for everyw¯∈G, ∀ϵ>0, and t1,t2∈−τ,0 where t2>t1. As ψ is continuous, we have(20)Pw¯t1−Pw¯t2=ψt1−ψt2<ϵ,whenδ1>t2−t1.
Hence,PG is equicontinuous. So, PG¯ is compact by Arzela–Ascoli Theorem.Theorem 3.
Suppose that C1 and C2 hold. Letf,g:I×C⟶ℝ+ and c2+Tγ/Γγ+1c1+L<1. Then, there exists a unique solution for IVP (1) on the interval −τ,T.Proof.
Consider the operatorP:H⟶H defined as(21)Pw¯t=ψ0+gt,w¯t+xt+1Γγ∫0tt−sγ−1w¯sfs,w¯s+xsds.
We will show thatP is a contraction. Let w¯1,w¯2∈CI,ℝ+, then we obtain the following sequence of inequalities:(22)Pw¯1t−Pw¯2t≤gt,w¯1t+xt−gt,w¯2t+xt+1Γγ∫0tt−sγ−1w¯1sfs,w¯1s+xsds−1Γγ∫0tt−sγ−1w¯2sfs,w¯2s+xsds≤c2w¯1t−w¯2t+1Γγ∫0tt−sγ−1w¯1sfs,w¯1s+xs−w¯2sfs,w¯2s+xsds≤c2w¯1t−w¯2t+1Γγ∫0tt−sγ−1w¯1sfs,w¯1s+xs−w¯2sfs,w¯2s+xs≤c2w¯1t−w¯2t+1Γγ∫0tt−sγ−1w¯1sfs,w¯1s+xs−w¯1sfs,w¯2s+xs+w¯1sfs,w¯2s+xs−w¯2sfs,w¯2s+xsds≤c2w¯1t−w¯2t+1Γγ∫0tt−sγ−1w¯1sc1w¯1s−w¯2sds+∫0tt−sγ−1Lsupw1t−w2tds≤c2supw1t−w2t+c1TγΓγ+1supw1t−w2t+LTγΓγ+1supw1t−w2t≤c2+TγΓγ+1c1+Lw¯1−w¯2.
Therefore,(23)Pw¯1t−Pw¯2t≤c2+TγΓγ+1c1+Lw¯1t−w¯2t.
So,P is a contraction, and hence by Banach’s contraction principle P has a unique fixed point.Definition 5.
The functionm∈X is said to be a lower solution of equation (1) (operator P) if(24)Dc0γmt−gt,mt≤mtft,mt,t∈I,mt≤ψt,t∈−τ,0,and the function n∈X is said to be an upper solution of equation (1) if(25)Dc0γnt−gt,nt≥ntft,nt,t∈I,nt≥ψt,t∈−τ,0.
If inequalities are strict, thenmt and nt are strict lower and upper solutions.Theorem 4.
Let(H1)
f and g from I×C to ℝ+ are continuous, and ft,ψ and gt,ψ are nondecreasing in ψ for each t∈I(H2)
m0 is lower and n0 is an upper solution of equation (1), satisfying condition m0t≤n0t, t∈I and m0,n0∈H
Then, equation (1) has at least a positive solution.Proof.
We only take fixed point operatorP. By Theorem 2, P is completely continuous and obviously by equation (16), u0 and v0 are lower and upper solutions of P, respectively. By (H1), w¯1,w¯2∈H,w¯1≤w¯2, we have(26)Pw¯1t=ψ0+gt,w¯1t+xt+1Γγ∫0tt−sγ−1w¯1sfs,w¯1s+xsds≤ψ0+gt,w¯2t+xt+1Γγ∫0tt−sγ−1w¯2sfs,w¯2s+xsds=Pw¯2t,t∈I.
So, the operatorP is nondecreasing. Also, we have(27)Pmt=ψ0+gt,mt+xt+1Γγ∫0tt−sγ−1msfs,msds,Pm0=ψ0+g0,m0+x0≥m0.
Asm∈X is a lower solution, therefore we can say that Pm0≥m0, similarly Pn0≤n0 from the definition of upper solution of P.
Hence,P:m0,n0⟶m0,n0 is a completely continuous operator. Since Q is a normal cone, Theorem 1 implies that P has a fixed point w¯∈m0,n0.(H3)
Here, we suppose that there existsM>0 such that 0<zft,zt≤M,t∈I.Theorem 5.
Assume conditions (H1) and (H3) holds; then, equation (1) has a positive solution.Proof.
Consider the equation(28)Dc0γmt−gt,mt=0,t∈I,mt=ψt,t∈−τ,0.
Obviously, equation (28) has a solution mt=m0−gt0,ψ+gt,mt, which is a lower solution of equation (1). Similarly, consider the equation(29)Dc0γnt−gt,nt=M,t∈I,nt=ψt,t∈−τ,0.
We know thatnt=n0−gt0,ψ+gt,nt+Mtγ/Γγ+1, t∈I is an upper solution of equation (1) and nt≥mt, so the above definition proves that equation (1) has minimum one positive solution.
For getting another result for a positive solution of equation (1), now we take the more general case of (1) and find the existence of a positive solution for the equations:(30)Dc0γνt−gt,νt=Nνt,t∈I.νt=ψt,t∈−τ,0.Theorem 6.
Equation (30) has a least positive solution.Proof.
Equation (30) is equivalent to(31)νt=ν0−g0,ψ+gt,νt+1Γγ∫0tt−sγ−1Nνsds,t∈I,ψt,t∈−τ,0.
LetP:H⟶H be an operator defined as(32)Pνt=ν0+gt,νt+1Γγ∫0tt−sγ−1Nνsds,t∈I,ψt,t∈−τ,0.
P is completely continuous by Theorem 2.
Let us takeBR=νt∈C−τ,T,ℝ+|νt−ν0≤R a bounded, closed, and convex subset of the Banach space C−τ,δ, where δ∈I and R>0. The possible two cases areCase 1: for allν∈BR,t∈I, we obtain(33)Pνt−ν0≤gt,νt+1Γγ∫0tt−sγ−1Nνsds≤L1+NΓγ+1νtγ,t∈I.
Sinceν≤R+ν0, hence(34)Pνt−ν0≤L1+NΓγ+1ν0+Rδγ≤R.
Here,δ<R−L1Γγ+1/Nν0+R1/γ.Case 2: ifPνt=ψt and −τ≤t<0, then(35)Pνt−ν0=ψt−ψ0=νt−ν0≤νt−ν0≤R.
Hence,PBR⊆BR. The Leray–Schauder fixed point theorem guarantees that operator P has at least one fixed point and so equation (30) has at least one positive solution.


## 4. Stability

In this section, we will discuss local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for a class of fractional neutral differential equations. For the caseI=0,T with f,g:I×C⟶ℝ+ are continuous functions on a closed interval or more generally compact sets, then they are bounded so we can replace the supremum by the maximum. In this case, norm is also called the maximum norm. Let X=x∈C−τ,1 be a Banach space with x=maxt∈Ixt, where τ is a nonnegative real number and zt=zt+θ;−τ≤θ≤0. We consider the following differential equation:(36)Dc0γzt−gt,zt=ztft,zt,t∈I,zt=ψt≥0,t∈−τ,0,where Dc0γ is a Caputo derivative with γ∈0,1 and ψ∈C−τ,0,ℝ+. We focus on the following inequalities:(37)Dc0γut−gt,ut−utft,ut≤ϵ,t∈I,(38)Dc0γut−gt,ut−utft,ut≤ψt,t∈I.Definition 6.
(see [36]). Equation (36) is local generalized Ulam–Hyers stable if there exists a positive real number c1 such that, for each positive ϵ and for every solution v¯∈C−τ,T,ℝ of (37), there exists a solution w¯∈C−τ,T,ℝ of (36) with v¯t−w¯t≤c1ϵ,t∈−τ,T.Definition 7.
(see [36]). Equation (36) is local generalized Ulam–Hyers–Rassias stable with respect to ψ if there exists c1ψ>0 such that, for every solution v¯∈C−τ,T,ℝ of (38), there exists a solution w¯∈C−τ,T,ℝ of (36) with v¯t−w¯t≤c1ψψt,t∈−τ,T.Remark 1.
A solution of differential equation is stable (asymptotically stable) if it attracts all other solutions with sufficiently close initial values. On the contrary, in Hyers–Ulam stability, we compare solution of given differential equation with the solution of differential inequality. We say solution of differential equation is stable if it stays close to solution of differential inequality. Hyers–Ulam stability may not imply the asymptotic stability.Remark 2.
(see [37]). A function v¯∈C−τ,T,ℝ is a solution of (37) if and only if there exists h∈CI,ℝ such that(i)
ht≤ϵ,t∈I(ii)
Dc0γut−gt,ut=utft,ut+ht,t∈I
Similarly, a functionv¯∈C−τ,T,ℝ is a solution of (38) if and only if there exists h˜∈CI,ℝ such that(i)
h˜t≤ψt,t∈I(ii)
Dc0γut−gt,ut=utft,ut+h˜t,t∈IRemark 3.
Let0<γ<1 and v¯∈C−τ,T,ℝ be a solution of inequality (37); then, v¯ is a solution of the following inequality:(39)v¯t−v0−gt,vt¯+xt−1Γγ∫0tt−sγ−1v¯sfs,vs¯+xsds≤tγϵΓγ+1.
From Remark2, we have(40)Dc0γut−gt,ut=utft,ut+ht,t∈I,then(41)ut−u0+g0,ψ−gt,ut=1Γγ∫0tt−sγ−1usfs,usds+1Γγ∫0tt−sγ−1hsds.
Therefore,(42)ut=u0−g0,ψ+gt,ut+1Γγ∫0tt−sγ−1usfs,usds+1Γγ∫0tt−sγ−1hsds.
By using the same process as inSection 3, i.e., let x⋅:−τ,T⟶ℝ+ be the function defined by(43)xt=ψt,t∈−τ,0,0,t∈I.
Therefore,x0=ψ and for every v∈CI,ℝ+ with v0=0. We defined v¯ as(44)v¯t=0,t∈−τ,0,vt,t∈I.
Letx⋅ satisfies equation (42). We can decompose u. as ut=v¯t+xt,for0≤t≤T so that ut=v¯t+xt, for every 0≤t≤T, where v¯t is such that v¯t<1,∀t∈I and the function v⋅ satisfies(45)v¯t=v0−g0,ψ+gt,vt¯+xt+1Γγ∫0tt−sγ−1v¯sfs,vs¯+xsds+1Γγ∫0tt−sγ−1hsds.
By following the conditions inSection 3, we get from (42) that(46)v¯t=v0+gt,vt¯+xt+1Γγ∫0tt−sγ−1v¯sfs,vs¯+xsds+1Γγ∫0tt−sγ−1hsds.
Hence,(47)v¯t−v0−gt,vt¯+xt−1Γγ∫0tt−sγ−1v¯sfs,vs¯+xsds≤tγϵΓγ+1.
Ifv¯∈C−τ,T,ℝ is a solution of inequality (38) then v¯ is a solution of the following inequality:(48)v¯t−v0−gt,vt¯+xt−1Γγ∫0tt−sγ−1v¯sfs,vs¯+xsds≤1Γγ∫0tt−sγ−1ψsds,≤λψψt,t∈I.
Before stating stability results, let us take condition (H4) as(H4)
c2+Tγ/Γγ+1c1+L<1Theorem 7.
Suppose thatH4 is true and other two conditions(a)
f,g:I×C⟶ℝ+, h∈CI,ℝ, and ht≤ϵ, h>0.(b)
f and g satisfy Lipchitz conditions:(49)ft,vt¯+xt−ft,wt¯+xt≤c1v¯t−w¯t,gt,vt¯+xt−gt,wt¯+xt‖≤c2v¯t−w¯t,where c1 and c2 are nonnegative real numbers holds, then(i)
Equation (36) has a unique solution(ii)
Equation (36) is local generalized Ulam–Hyers stableProof.
Letv¯∈C−τ,T,ℝ be a unique solution of equation (37). Denote w¯∈C−τ,T,ℝ as the unique solution of the equation(50)Dc0γzt−gt,zt=ztft,zt,t∈I,wt=vt≥0,t∈−τ,0.
Then, we have(51)w¯t=w0+gt,wt¯+xt+1Γγ∫0tt−sγ−1w¯sfs,ws¯+xsds=v0+gt,wt¯+xt+1Γγ∫0tt−sγ−1w¯sfs,ws¯+xsds.
We can see thatv¯t−w¯t=0, for t∈−τ,0. For t∈I, we obtain(52)v¯t−w¯t=v¯t−v0−gt,wt¯+xt−1Γγ∫0tt−sγ−1w¯sfs,ws¯+xsds≤tγϵΓγ+1+gt,vt¯+xt−gt,wt¯+xt+1Γγ∫0tt−sγ−1v¯sfs,vs¯+xs−w¯sfs,ws¯+xsds≤tγϵΓγ+1+c2vt¯−wt¯+c1Γγ∫0tt−sγ−1v¯s−w¯sds+LΓγ∫0tt−sγ−1v¯s−w¯sds.
Fort∈I, by using(53)v¯s−w¯s=max0≤s≤tv¯s−w¯s=max0≤s≤tv¯s+θ−w¯s+θ=maxθ≤s+θ≤t+θv¯s+θ−w¯s+θ≤max−τ≤s1≤tv¯s1−w¯s1wheres+θ=s1,−τ≤θ<0≤max−τ≤s1≤Tv¯s1−w¯s1.
Thus,(54)v¯t−w¯t≤tγϵΓγ+1+λ2max−τ≤t1≤Tv¯t1−w¯t1+c1Γγ∫0tt−sγ−1max−τ≤s1≤Tv¯s1−w¯s1ds+LΓγ∫0tt−sγ−1max−τ≤s1≤Tv¯s1−w¯s1ds≤tγϵΓγ+1+c2v¯t1−w¯t1+c1+LΓγ+1v¯t1−w¯t1Tγ≤TγϵΓγ+1+c2+TγΓγ+1c1+Lv¯t−w¯t.
So,(55)v¯t−w¯t≤c1ϵ,wherec1=Tγ1−c2Γγ+1−c1+LTγ.
Hence, equation (36) is local generalized Ulam–Hyers stable.Theorem 8.
IfH4 is true and(a)
f,g:I×C⟶ℝ+, h˜∈CI,ℝ, and h˜t≤ψt, h>0.(b)
f and g satisfy Lipchitz conditions:(56)ft,vt¯+xt−ft,wt¯+xt≤c1v¯t−w¯t,gt,vt¯+xt−gt,wt¯+xt≤c2v¯t−w¯t,where c1 and c2 are nonnegative real numbers.(c)
The functionψ∈CI,ℝ+ is an increasing function and there exists λψ>0 such that(57)1Γγ∫0tt−sγ−1ψsds≤λψψt,t∈I,holds, then equation (36) is local generalized Ulam–Hyers–Rassias stable with respect to ψ.Proof.
Following the same steps as in Theorem7, we can find the result, i.e., here we will obtain(58)v¯t−w¯t≤λψψt+λ2max−τ≤t1≤Tv¯t1−w¯t1+c1Γγ∫0tt−sγ−1max−τ≤s1≤Tv¯s1−w¯s1ds+LΓγ∫0tt−sγ−1max−τ≤s1≤Tv¯s1−w¯s1ds≤λψψt+c2v¯t1−w¯t1+c1+LΓγ+1v¯t1−w¯t1Tγ≤λψψt+c2+TγΓγ+1c1+Lv¯t−w¯t.
Therefore,(59)v¯t−w¯t≤c1ψψt,wherec1ψ=λψΓγ+11−c2Γγ+1−c1+LTγ.
Hence, equation (36) is local generalized Ulam–Hyers–Rassias stable.


## 5. Example

In this section, we present an example to explain the applicability of main results.Example 1.
Consider the problem(60)D01/2xt−xt−τ=txt−xt−τ,t∈0,1,xt=0,t∈−τ,0.
One can easily prove that this problem has a unique fixed point and that it has a positive solution, i.e.,(61)xt−xt−τ=1Γ1/2∫0tt−s1/2−1txtds−1Γ1/2∫0tt−s1/2−1xt−sds,xt=xt−τ−1Γ1/2∫0tt−s−1/2txtds−1Γ1/2∫0tt−s−1/2xt−sds.
Define the operatorP as(62)Pxt=xt−τ−1Γ1/2∫0tt−s−1/2txtds−1Γ1/2∫0tt−s−1/2xt−sds,t∈0,1,0,t∈−τ,0,Px1t−Px2t≤x1t−x2t+1Γ1/2x1t−x2t∫0tt−s−1/2ds+1Γ1/2x1t−x2t∫0ttt−s−1/2ds≤1+t3/2Γ3/2−t1/2Γ1/2x1−x2,with1+t3/2Γ3/2−t1/2Γ1/2≤1,ift∈0,1.
For stability, we take (60) as(63)D01/2xt−xt−τ=txt−xt−τ,t∈0,1,xt=zt,t∈−τ,0,x0=0,and the inequalities(64)D01/2zt−zt−τ−tzt−zt−τ≤ϵ1,t∈0,1,(65)D01/2zt−zt−τ−tzt−zt−τ≤ψt,t∈0,1.
To prove that equation (63) is local generalized Ulam–Hyers stable, we take the conditions as in Theorem 7, i.e., a function z∈C0,1,ℝ is a solution of inequality (63) if and only if there exists h∈C0,1,ℝ such that(66)ht≤ϵ1,t∈0,1,D01/2zt−zt−τ=tzt−zt−τ+ht,t∈0,1.
ApplyingI0+1/2 on both sides, we obtain(67)zt−zt−τ−1Γ3/2t3/2zt+1Γ1/2∫0tt−s−1/2zt−sds≤ϵΓ3/2.
Here, we take unique solutionxt of equation (63) as(68)xt=xt−τ+1Γ1/2∫0tt−s−1/2txtds−1Γ1/2∫0tt−s−1/2xt−sds,xt=zt−τ+1Γ1/2∫0tt−s−1/2txtds−1Γ1/2∫0tt−s−1/2xt−sds.
By further simplification, we obtain(69)zt−xt≤c1ϵ;c1=1Γ3/2−t1/2t−1.Hence, (63) is local generalized Ulam–Hyers stable.Remark 4.
If we replace equation (66) by the inequality,(70)h˜t≤ψt,t∈0,1,D01/2zt−zt−τ=tzt−zt−τ+h˜t,t∈0,1.
By repeating the same process as in the above example, one can easily verify the main results of Theorem8.


## 6. Graphical Presentation

For graphical representation of the solution of the problem given in equation (60), we adopted Adams–Bashforth–Moulton scheme [38] to obtain the numerical solution for this fractional differential equation. To analyze the effect and contribution of time delayed factor, the modified predictor-corrector scheme [39] is incorporated for simulation. To check and demonstrate the stability of the consider model, graphical representation of the solution with different variations of the time delay factor along with other parameters is made. From the numerical results, we could note the UH-stability of the system with varying orders and delays. With the higher orders, the system will achieve the Ulam–Hyers stability more rapidly. It also holds true with increasing delays. One can see the stability of system (60) in Figures (1)–(4) for various values of time delays and fractional derivatives.Figure 1
Trajectories for different values ofτ=0.1,0.2,0.3, and 0.4 with α=0.5.Figure 2
Trajectories for different values ofτ=0.6,0.7,0.8, and 0.9 with α=0.5.Figure 3
Trajectories for different values ofα=0.1,0.2,0.3, and 0.4 with time delay τ=0.1.Figure 4
Trajectories for different values ofτ=0.1,0.2,0.3, and 0.4 with α=0.5.


## 7. Conclusions

We present some new results about the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We also establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results. The current concepts have significant applications since it means that if we are studying local generalized Hyers–Ulam–Rassias stable (or Hyers–Ulam stable) system then one does not have to reach the exact solution. We just need to get a function which satisfies a suitable approximation inequality. In other words, local generalized Hyers–Ulam–Rassias stability (or Hyers–Ulam stability) guarantees that there exists a close exact solution. This is altogether useful in many applications where finding the exact solution is quite difficult such as optimization, numerical analysis, biology, and economics. It also helps, if the stochastic effects are small, to use deterministic model to approximate a stochastic one.



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