# Some Qualitative Analyses of Neutral Functional Delay Differential Equation with Generalized Caputo Operator

Authors: Abdellatif Boutiara; Mohammed M. Matar; Mohammed K. A. Kaabar; Francisco Martínez; Sina Etemad; Shahram Rezapour
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, a new class of a neutral functional delay differential equation involving the generalizedψ-Caputo derivative is investigated on a partially ordered Banach space. The existence and uniqueness results to the given boundary value problem are established with the help of the Dhage’s technique and Banach contraction principle. Also, we prove other existence criteria by means of the topological degree method. Finally, Ulam-Hyers type stability and its generalized version are studied. Two illustrative examples are presented to demonstrate the validity of our obtained results.


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## Body

## 1. Introduction

Fractional calculus has demonstrated high visibility and capability in the applications of various topics linked to physics, signal processing, mechanics, electromagnetics, economics, biology, and many more [1–3]. Even recently, fractional differential equations (FDEqs) have acquired particular attention because of their numerous applications in the fractional modeling [4–12]. FDEqs involving hybrid nonlinearity have been gained much attention during the past few years. This class of equations arises from various mathematical and physical phenomena such as three-layer beam, electromagnetic waves, curved beam’s deflection with a constant or varying cross-section, and gravity-driven [13–19].Almeida [20] introduced a new fractional derivative, named the ψ-fractional order derivative (FOD), with respect to another function, which extended the classical fractional derivative. Therefore, the generalizations of existing results in fractional calculus and FBVPs have been established by several mathematicians [21–25].The qualitative analysis of FDEqs such as the solution’s existence and uniqueness is the most popular problems that many researchers focus on. Various fixed point theorems are considered as the most effective tools for dealing with such problems. In this work, we follow some results presented by Ragusa et al. [26, 27] concerning the qualitative properties of some suitable FDEqs.In the last decade, a new technique was developed by Dhage [28], named Dhage iteration principle, for investigating the numerical solutions’ existence and approximation of integral and FDEqs by constructing a sequence of successive approximations with initial lower or upper solution. Dhage [29–32] provided a generalized form of hybrid fixed point theorem in the context of a metric space having the partial order without applying any geometric condition. In Dhage’s research study, with the help of the measure of noncompactnessan, an algorithm for studying the solutions’ existence of a certain nonlinear functional integral equation was investigated under weaker conditions. The advantages of the applied method were studied by Dhage to compare with the standard approaches that involve Banach, Schauder’s, and Krasnoselskii’s fixed point theorems. As a result, the iteration method due to Dhage has recently became an important tool for investigating the solution’s existence and approximate results of nonlinear hybrid FDEqs that have various scientific applications such as air motion, electricity, fluid dynamics, process control with nonlinear structures, and electromagnetism. In addition, this method can be extended to other functional differential equations (FuDEqs) classes. On the other side, in recent years, the topological degree method has been considered as one of the main tools for studying the existence results to different fractional differential equations and inclusions. This method will be used in our research study to derive desired results in relation to the solutions of the proposed problem. For more details, see [33–38].The FuDEqs’ stability was first proposed by Ulam [39] and then by Hyers [40]. Later on, this type of stability and its generalization were called of the Ulam-Hyers (UH) and generalized Ulam-Hyers (GUH) type, respectively. Investigating the UH and GUH stability has been given a special attention in studying all FuDEqs kinds and FDEqs in particular [41–44].Motivated by the novel developments inψ-fractional calculus, the solution’s existence, uniqueness, and UH stability of the proposed neutral functional differential equation (NFuDEq) is investigated in this research work. The NFuDEq is expressed as:
(1)Dca+v;ψϖτ−Fτ,ϖτ=ℍτ,ϖτ,τ∈J≔a,b,ϖτ=ϕτ,τ∈a−δ,a,where the ψ-Caputo FOD, denoted by cDa+ν;ψ, of order ν∈0,1, given F,ℍ:J×ℝ⟶ℝ are continuous functions such that Fa,ϖa=0, and ϕ:a−δ,a⟶ℝ is a continuous function with ϕa=0. For any function u defined on a−δ,a and any τ∈J, it is given by
(2)uτρ=uτ+ρ,ρ∈−δ,0.The main aim of this research work is to apply an iteration principle due to Dhage to ensure the solutions’ existence along with approximation of (1) under weaker partial continuity and partial compactness type conditions.This article is constructed as follows: some important definitions and lemmas which are needed for our results are provided in Section2. The solutions’ existence and approximation of (1) are proven in Section 3 via the Dhage iteration principle. In Section 4, a theorem, based on the coincidence degree theory for condensing maps, is established on the solutions’ existence of the proposed NFuDEq (1). In Section 5, the solution’s uniqueness for the NFuDEq (1) is proven by the Banach contraction principle of solutions. Moreover, we investigate the UH and GUH stability for the NFuDEq (1). Some illustrative examples for supposed problem are provided at the end to validate our theoretical results.


## 2. Fundamental Preliminaries

Some important definitions, theorems, and lemmas concerning advanced fractional calculus and nonlinear analysis are stated in this section which are needed for our approach in the next parts.Consider the space of all continuous real-valued functionsC=CJ,ℝ endowed with the norm
(3)ϖC=supτ∈Jϖτ.Also,Cδ=C−δ,0,ℝ is endowed with norm
(4)ϕCδ=supτ∈−δ,0ϕτ,andϖτCδ=supρ∈−δ,0ϖτ+ρ.Consider the Banach spaceCb=Ca−δ,b,ℝ defined on a−δ,b with the norm
(5)ϖCb=ϖCδ+ϖC=supτ∈a−δ,bϖτ.The order relation≼ is defined as follows:
(6)ϖ≼ω⇔ϖt≤ωt∀t∈a−δ,b,which gives a partial ordering in Cb.From the research study in [29], let us now state some necessary definitions and preliminary results for our research work. Assume that X¯=X,≼,. displays a real partial order on X. If for ϖ,ω in X, either ϖ≼ω or ω≼ϖ, then ϖ and ω are termed as comparable elements, and also when all members of ∅≠C⊂X are comparable, then C is named either totally ordered or a chain. If there exists a nondecreasing (resp., nonincreasing) sequence ϖnn∈N and ϖ∗ in X such that ϖn⟶ϖ∗ as n⟶∞, then X is regular (ϖn≼ϖ∗ (resp. ϖn±ϖ∗)) for all n∈ℕ. By assuming this fact that there are lower and upper bounds in X for every both members of X, in that case, the partially ordered Banach space X is named regular and lattice.Definition 1 (see [29]).
An operator:Q:X⟶X is termed as nondecreasing or isotone if Q maintains the order relation ≼, i.e., when ϖ≼ω, it means that Qϖ≼Qω for all ϖ,ω∈X.Definition 2 (see [29]).
A mappingQ:X⟶X has the compactness specification if QX is a set in X with the relative compactness. In addition, Q is totally bounded if QS has the relative compactness property in X, where S⊆X is an arbitrary bounded set.Every operator having the continuity and total boundedness properties will be completely continuous.Definition 3 (see [29]).
Q:X⟶X has the partial continuity property at a∈X, if for each ε>0, δ>0 exists so that Qϖ−Qa<ε whenever ϖ−a<δ and ϖ and a are comparable. Assuming Q as an operator with the partial continuity on X, it is well-known that Q is continuous on each chain C⊂X. Furthermore, if QC is bounded for every C⊆X, then Q is partially bounded. In addition, Q is uniformly partially bounded if all existing chains QC⊆X involve the boundedness by a bound uniquely.Definition 4 (see [29]).
Q:X⟶X has the partial compactness if QC⊂X has the relative compactness with respect to all chains C⊆X. It has the partial total boundedness property if for each bounded and totally ordered set C contained in X, QC⊂X possesses the relative compactness.Every operator with the partial continuity and the partial total boundedness is named as partially completely continuous on the underlying space.Remark 5.
Assume thatQ is a nondecreasing selfmap on X and C is an arbitrary chain in it. In this case, Q possesses the partial compactness or the partial boundedness specifications whenever QC is relatively compact or bounded in X.Definition 6 (see [28]).
Regardd and ≼ as a metric and an order relation on X. We say that d and ≼ are compatible if ϖnn∈ℕ⊂X is monotone, and if a subsequence ϖnkk∈N of ϖnn∈ℕ tends to ϖ∗, then ϖnn∈ℕ tends to ϖ∗. Similar definition can be applied on a partially order norm space. A subset S of X is named Janhavi if the order relation ≼ and the metric d (or the norm ·) are compatible in it. Particularly, if S=X, then we say that X is Janhavi metric (or Janhavi Banach space).Definition 7 (see [29]).
An operatorQ:X⟶X is D-Lipschitz if there exists an upper semicontinuous nondecreasing function: Ψ:ℝ+⟶ℝ+ with Ψ0=0 such that
(7)Qϖ−Qω≼Ψϖ−ω,for all ϖ,ω∈X.Definition 8 (see [29]).
The same above operatorQ is termed as partially nonlinear D-Lipschitz whenever a D-function Ψ:ℝ+⟶ℝ+ exists provide
(8)Qϖ−Qω≼Ψϖ−ω,∀ϖ,ω∈X. In addition, when Q is nonlinear D-Lipschitz subject to Ψτ<τ for τ>0, in that case, Q is nonlinear D-contraction.Let us at present introduce a novel procedure, named Dhage iterative method, which is very useful for obtaining a scheme for the approximation of solutions to problems with nonlinearity.Theorem 9 (see [29]).
LetX,≼,· be a complete regular normed linear algebra via the partial order so that ≼ and · are compatible. Consider two nondecreasing operators K,H:X⟶X such that
(a)
K is partially nonlinear D-Lipschtiz and partially bounded with D-function ψK(b)
H has the partial continuity and the compactness(c)
∃ an element ϖ0∈X such that ϖ0≼Kϖ0+Hϖ0 or ϖ0±Kϖ0+Hϖ0
Then,Kϖ+Hϖ=ϖ possesses a solution ϖ∗ in X, and the sequence of the successive iterations ϖnn=0∞, expressed as ϖn+1=Kϖn+Hϖn, approaches to ϖ∗ monotonically.Theorem 10 (see [30]).
LetH:X⟶X be a nondecreasing and partially nonlinear D-contraction. Assume that ϖ0∈X exists with ϖ0≼Hϖ0 or ϖ0±Hϖ0. If X is regular or H is continuous, then a fixed point ϖ∗ is found, and the sequence of successive iterations Hnϖ0 tends to ϖ∗ monotonically. In addition, ϖ∗ is unique if each of both members of X possesses a lower and an upper bound.Remark 11 (see [31]).
Let every set contained inX with the partial compactness includes the compatibility specification with respect to ≼ and ·. Then, every compact chain of X is Janhavi. This implication can be simply applied to establish the existence property of solutions in our research work.Remark 12.
The regularity property ofX in Theorem 9 can be replaced with another strong continuity condition of the operators K and H on X where Dhage in [28] proved this result.Remark 13 (see [30]).
(1)
In a partially normed linear space, every compact operator has the partial compactness, and all partially compact operators has the partial total boundedness, while the converse is not valid(2)
Each completely continuous operator has the partial complete continuity, and each partially completely continuous operator has the continuity and the partial total boundedness, while the converse is not valid
In such a situation, the hypotheses regarding to the partial continuity and the partial compactness of an operator in Theorem9 can be replaced by the continuity and compactness of that operator.We state here the results below given by [45–47].Definition 14.
The mappingκ:MC⟶0,∞ is named Kuratowski measure of non-compactness (KMNC) if
(9)κB=infε>0:BcanbecoveredbyfinitelymanysetswithDIAMB≤ε,where MC represents a class of all bounded mappings in C.Proposition 15.
The following are fulfilled for KMNC:(1)
A⊂E⇒κA≤κE(2)
κA=κA¯=κconvA, where A¯ and convA represent the closure and the convex hull of A, respectively(3)
κA+E≤κA+κE and κcA=∣c∣κA,c∈ℝDefinition 16.
Assume thatK:A⟶C be a continuous bounded mapping and A⊂C. The operator K is said to be κ-Lipschitz if we can find a constant ℓ≥0 satisfying the following condition:
(10)κKB≤ℓκB,foreveryB⊂A.Moreover,K is called strict κ-contraction subject to ℓ<1.Definition 17.
K is called κ-condensing when
(11)κKB<κB,for every bounded and nonprecompact subset B of A. So,
(12)κKB≥κB,whichimpliesκB=0.
Further, we haveK:A⟶C is Lipschitz if we can find ℓ>0 such that
(13)Ku−Kv≤ℓu−v,forallu,v∈A,if ℓ<1, K is said to be strict contraction.The following three interesting results are based on [48]:Proposition 18.
LetK,H:A⟶C be κ-Lipschitz with constants ℓ1 and ℓ2. Then, K+H:A⟶C is κ-Lipschitz with ℓ1+ℓ2.Proposition 19.
Every compact mappingK:A⟶C is κ-Lipschitz with ℓ=0.Proposition 20.
Every Lipschitz mappingK:A⟶C with ℓ is κ-Lipschitz with ℓ.Isaia [48] used the topological degree theory to introduce the following interesting results:Theorem 21.
LetF:A⟶C be κ-condensing and
(14)Θ=u∈C:∃ξ∈0,1s.t.x=ξFu.
IfΘ⊂C is bounded, i.e., r>0 exists subject to Θ⊂Br0; then, the degree
(15)degI−ξF,Br0,0=1,forallξ∈0,1.As a result, it is found a fixed-point forF and all possible fixed-points of F are contained in Br0.Letψ∈C1=C1J,ℝ be an increasing differentiable function such that ψ′τ≠0, ∀τ∈J. Now, we start by defining ψ-FODs as follows:Definition 22 (see [2]).
Theψ-Riemann-Liouville fractional integral of order α>0 for an integrable function ϖ:J⟶ℝ is given by
(16)Ia+α;ψϖτ=1Γα∫aτψ′sψτ−ψsα−1ϖsds,where the Gamma function is denoted by Γ.Definition 23 (see [2]).
Letn−1<α<nn∈ℕ, ϖ:J⟶ℝ be an integrable function, and ψ∈CnJ,ℝ. Then, the ψ-Riemann–Liouville FOD of a function ϖ of order α is expressed as:
(17)Da+α;ψϖτ=Dtψ′τnIa+n−α;ψϖτ,where n=α+1 and Dt=d/dt.Definition 24 (see [20]).
Forn−1<α<nn∈ℕ and ϖ,ψ∈CnJ,ℝ, the ψ-Caputo FOD of a function ϖ of order α is given by
(18)cDa+α;ψϖτ=Ia+n−α;ψϖψnτ,where ϖψnτ=Dt/ψ′τnϖτ.From the above definition, we can expressψ-Caputo FOD by the following formula:
(19)cDa+α;ψϖτ=∫aτψ′sψτ−ψsn−α−1Γn−αϖψnsds,ifα∉ℕ,ϖψnτ,ifα∈ℕ.Also, theψ–Caputo FOD of order α of ϖ is defined as
(20)cDa+α;ψϖτ=Da+α;ψϖτ−∑k=0n−1ϖψkak!ψτ−ψak.For more details, see ([20], Theorem 3).Lemma 25 (see [2]).
Forα,β>0, and ϖ∈CJ,ℝ, we have
(21)Ia+α;ψIa+β;ψϖτ=Ia+α+β;ψϖτ,a.e.τ∈J.Lemma 26 (see [22]).
Assume thatα>0. If ϖ∈CJ,ℝ, then
(22)cDa+α;ψIa+α;ψϖτ=ϖτ,τ∈J,and if ϖ∈Cn−1J,ℝ, then
(23)Ia+α;ψcDa+α;ψϖτ=ϖτ−∑k=0n−1ϖψkak!ψτ−ψak,τ∈J.It is easily to deduce that(24)cDa+k;ψIa+α,ψ=Ia+α−k,ψ.Lemma 27 (see [2, 20]).
Letτ>a,α≥0, ς>0 and let χτ=ψτ−ψa. Then:
(i)
Ia+α;ψχτς−1=Γς/Γς+αχτς+α−1(ii)
cDa+α;ψχτβ−1=Γς/Γς−αχτς−α−1(iii)
cDa+α;ψχτk=0,forallk∈0,⋯,n−1,n∈ℕ


## 3. Existence and Approximation Results via Dhage’s Technique

The solutions’ existence and approximation of problem (1) are studied in this section.Lemma 28.
Assume thatCb,≼,. is a partially ordered Banach space with the norm ., and the order relation ≼ defined by (5) and (6), respectively. Then, every partially compact subset of Cb is Janhavi.Proof (see [31]).
Let us now discuss exactly the problem (1).☐Definition 29.
A functionϖ∈Cb is a lower solution for the NFuDEq (1) if:
(1)
ϖτ∈Cδ, ∀τ∈J(2)
the functionτ↦ϖτ−Fτ,ϖτ is continuously differentiable on J and settles
(25)Dca+ν;ψϖτ−Fτ,ϖτ≼ℍτ,ϖτ,τ∈J,ϖτ≼ϕτ.Similarly, a differentiable functionω∈Cb is named an upper solution of the NFuDEq (1) if the above inequality is satisfied with reverse sign.To demonstrate the solutions’ existence to (1), we state this lemma:Lemma 30.
Assume that0<ν<1,ϕa=0, and g,h:J⟶ℝ are continuous with ha=0. The linear problem
(26)cDa+ν;ψϖτ−hτ=gτ,τ∈J;ϖτ=ϕτ,τ∈a−δ,a,has a unique solution ϖτ defined by:
(27)hτ+Ia+ν,ψgτ,ifτ∈J,ϕτ,ifτ∈a−δ,a.For the proof of Lemma30, it is useful to refer to [2, 23, 41, 49].With the help of the following hypothesis, we can investigate our results:(H1) The functionsFτ,ϖ and ℍτ,ϖ are monotone nondecreasing with respect to ϖ for any τ∈J.(H2)∃ a D-function Ψ that satisfies ΨR<R for R>0 such that
(28)0≤Fτ,ϖ−Fτ,ω≤Ψϖ−ω∀τ∈Jandϖ,ω∈ℝwithϖ≥ω.(H3)∃M>0 such that ℍτ,ϖ≤M,∀τ∈J, and ϖ∈ℝ.(H4)∃L>0 such that Fτ,ϖ≤L, ∀τ∈J, and ϖ∈ℝ.(H5) The FBVP (1) possesses a lower solution x∈Cb.(H6)∃ a positive constant Lℍ such that
(29)ℍτ,ϖ−ℍτ,ω≤Lℍϖ−ω,∀τ∈Jandϖ,ω∈ℝwithϖ≥ω.Theorem 31.
If the hypotheses (H1)-(H5) are fulfilled, then the NFuDEq (1) includes a solution ϖ∗ formulated on a−δ,b, and ϖn containing the successive approximations expressed as:
(30)ϖ0=x,ϖn+1τ≔Ia+ν,ψℍτ,ϖn,τ+Fτ,ϖn,τ,ifτ∈J,ϕτ,ifτ∈a−δ,a,where ϖn,τρ=ϖnτ+ρ,ρ∈−δ,0, tends to ϖ∗ monotonically.Proof.
TakeX=Cb=Ca−δ,b,ℝ. Then, using Lemma 28, each compact chain C⊂Cb admits the compatibility property in · and ≤ such that C is Janhavi in Cb. On the other side, K and H can be defined on Cb as follows:
(31)Kϖτ≔Fτ,ϖτ,ifτ∈J,0,ifτ∈a−δ,a,(32)Hϖτ≔Ia+ν,ψℍτ,ϖτ,ifτ∈J,ϕτ,ifτ∈a−δ,a.
According to the structure of integral, it is obvious thatK,H:Cb⟶Cb are well-defined. In addition, the studied problem (1) can be reformulated by:
(33)Kϖτ+Hϖτ=ϖτ,τ∈a−δ,b.
To investigate the solutions’ existence to this operator equation, we can sufficiently show that the operatorsK and H satisfy all items of Theorem 9. We follow our argument split into five steps.☐Step I.
K and H are nondecreasing on Cb.
Forϖ,ω∈Cb with ϖτ≥ωτ, using (H1), we get
(34)Kϖτ=Fτ,ϖτ,ifτ∈J,0,ifτ∈a−δ,a≥Fτ,ωτ,ifτ∈J,0,ifτ∈a−δ,a=Kωτ,for all τ∈a−δ,b. It means that K:Cb⟶Cb is a nondecreasing operator.Similarly, we obtain(35)Hϖτ=Ia+ν,ψℍτ,ϖτ,ifτ∈J,ϕτ,ifτ∈a−δ,a≥Ia+ν,ψℍτ,ωτ,ifτ∈J,ϕτ,ifτ∈a−δ,a=Hωτ,for all τ∈a−δ,b. Thus, it is concluded that H:Cb⟶Cb is a nondecreasing operator.Step II.
K is a nonlinear D-contraction on Cb.
Forϖ,ω∈Cb with ϖ≥ω and by (H2), we get that
(36)Kϖτ−Kωτ≤Fτ,ϖτ−Fτ,ωτ≤Ψϖτ−ωτCδ≤Ψϖ−ω,
∀τ∈a−δ,b. By taking the supremum over τ, we get
(37)Kϖ−Kω≤Ψϖ−ωCb,
∀ϖ,ω∈Cb,ϖ≥ω, where r>Ψr for r>0. Therefore, according to Definition 8, our result is derived.Step III.
H is partially continuous on Cb.
Regardϖnn∈ℕ in a chain C∋ϖn⟶ϖ as n⟶∞. Then, ϖn,s⟶ϖs for any s∈J letting n⟶∞. The continuity of ℍ yields
(38)limn⟶∞Hϖnτ=1Γνlimn⟶∞∫aτψ′sψτ−ψsν−1ℍs,ϖn,sds,ifτ∈J,ϕτ,ifτ∈a−δ,a=1Γν∫abψ′sψτ−ψsν−1limn⟶∞ℍs,ϖn,sds,ifτ∈J,ϕτ,ifτ∈a−δ,a=Ia+ν,ψℍτ,ϖτ,ifτ∈J,ϕτ,ifτ∈a−δ,a=Hϖτ,
∀τ∈a−δ,b. Hence, Hϖn converges to Hϖ pointwise on a−δ,b.In the following two cases, we prove thatHϖnn∈ℕ is an equicontinuous sequence of functions in Cb.Case A.
Takeτ1, τ2∈J, with τ1<τ2. Then,
(39)Hϖnτ2−Hϖnτ1≤Ia+ν,ψℍτ2,ϖn,τ2−Ia+ν,ψℍτ1,ϖn,τ1≤1Γν∫aτ1ψ′sψτ1−ψsν−1−ψτ2−ψsν−1ℍs,ϖn,sds+1Γν∫τ1τ2ψ′sψτ2−ψsν−1ℍs,ϖn,sds≤LΓν+1ψτ2−ψaν−ψτ1−ψaν+2ψτ2−ψτ1ν,which tends to zero as τ1⟶τ2.Case B.
Forτ1,τ2∈a−δ,a. Then,
(40)Hϖnτ2−Hϖnτ1=ϕτ2−ϕτ1⟶0,asτ1⟶τ2.
Clearly, ifτ1∈a−δ,a and τ2∈J such that τ1⟶τ2 has only one possibility that they are close to a at which Hϖn is close to zero.Thus,(41)Hϖnτ2−Hϖnτ1⟶0,asτ1⟶τ2,uniformly ∀n≥1. This proves that Hϖn is equi–continuous on a−δ,b. Thus, the pointwise convergence of Hϖn on a−δ,b implies the uniform convergence, so Hϖn converges to Hϖ uniformly on a−δ,b. Consequently, the selfmap H possesses the partial continuity on Cb.Step IV.
H has the partial compactness property on Cb.
Regard the chainC in Cb and ω∈HC. Then ∃ϖ∈C∋ω=Hϖ. Using hypothesis (H3), if τ∈a−δ,a, we have
(42)ωτ=Hϖτ≤ϕτ≤ϕCδ≤ϕCb.Otherwise, ifτ∈J, then
(43)Hϖτ≤Ia+ν,ψℍτ,ϖτ≤MIa+ν,ψ1τ≤ψτ−ψaνΓν+1M.Hence,(44)HϖCb≤ϕCb+ψτ−ψaνΓν+1M≔R,∀τ∈a−δ,b. Thus, we obtain ω≤Hϖ≤R for any ω∈HC. Thus, HC is a uniformly bounded subset of Cb.Let us now prove thatHC is an equi–continuous set in Cb. Let τ1,τ2∈J, with τ1<τ2. Then, according to Step III arguments, it is concluded that
(45)ωτ2−ωτ1=Hϖτ2−Hϖτ1⟶0asτ1⟶τ2,uniformly for any ω∈HC which illustrates the equi–continuity of HC in Cb. So, HC is compact in reference to Arzelà-Ascoli criterion. As a result, the selfmap H:Cb⟶Cb admits the partial compactness property on Cb.Step V.
ϖ satisfies ϖ≤Kϖ+Hϖ.
By (H5),W is a lower solution of the NFuDEq (1) defined on a−δ,b. Then, according to the lower solution definition, we get
(46)Dca+v;ψWτ−Fτ,Wτ≤ℍτ,Wτ,τ∈J=a,b,Wτ≤ϕτ,τ∈a−δ,a.Let us integrate the above inequality froma to τ, we obtain
(47)Wτ≤Ia+ν,ψℍτ,Wτ+Fτ,Wτ,ifτ∈J,ϕτ,ifτ∈a−δ,a.=KWτ+HWτ,∀τ∈a−δ,b. Thus, W≤KW+HW. Obviously, both operators K and H satisfy all of the items of Theorem 9; therefore, the operator equation Kϖ+Hϖ=ϖ has a solution ϖ∗ defined on a−δ,b. Furthermore, the sequence ϖnn=0∞ of successive approximations defined by (30) tends to ϖ∗ monotonically. So, our proof is ended.Remark 32.
Above theorem’s conclusion also remains true if the hypothesis (H5) is replaced with (H7) such that the NFuDEq (1) has an upper solution: y∈Cb.Similarly, its proof under this replaced condition can be shown by the observation of the same arguments with some modifications.Theorem 33.
Let (H1), (H5), and (H6) be valid. Then, the problem (1) has a unique solution ϖ∗ defined on a−δ,b provided that ΩR<R,R>0, where
(48)ΩR=Lℍψb−ψaνRΓν+1+ΨR.Moreover, the sequenceϖn of successive approximations defined by (30) converges monotonically to ϖ∗.Proof.
First, the operator:G:Cb⟶Cb is defined by
(49)Gϖτ≔Ia+ν,ψℍτ,ϖτ+Fτ,ϖτ,ifτ∈J,ϕτ,ifτ∈a−δ,a,for τ∈a−δ,b,. To prove this theorem, we establish the satisfaction of all items of Theorem 10 for G in Cb. We know that G is nondecreasing and continuous.
The details are similar as in the proof of Theorem31, so we omit them. Therefore, it is needed to be verified that G is a partially D-contraction on Cb. To arrive at such an aim, by taking ϖ,ω∈Cb such that ϖ≥ω, if τ∈a−δ,a, then it is obvious that
(50)Gϖτ−Gϖτ=0.
Otherwise, letτ∈J, it follows from (H1) and (H6), that
(51)Gϖτ−Gϖ¯τ≤Ia+ν,ψℍτ,ϖτ−ℍτ,ϖ¯τ+Fτ,ϖτ−Fτ,ϖ¯τ≤Lℍϖτ−ϖ¯τCδIa+ν,ψ1τ+Ψϖτ−ϖ¯τCδ≤Lℍψb−ψaνΓν+1ϖτ−ϖ¯τCδ+Ψϖτ−ϖ¯τCδ≤Ωϖτ−ϖ¯τCδ,for all τ∈J, where ΩR<R,R>0. Let us now take the supremum over τ, we get
(52)Gϖ−Gϖ¯≤Ωϖ−ϖ¯Cb,for all ϖ,ϖ¯∈Cb, with ϖ≥ϖ¯. As a result, G is a partially nonlinear D-contraction on Cb. In addition, by using Theorem 31, it is proven that the given function x in (H5) satisfies the operator inequality x≤Gx on a−δ,b. Therefore, from Theorem 10, it is found a solution ϖ∗ uniquely for the NFuDEq (1), and ϖn defined by (30) tends to ϖ∗ monotonically.☐


## 4. Existence Result via Topological Degree Theory

The existence problem of the NFuDEq (1) is investigated in this section based on the Topological Degree Theory due to Isaia [48]. Let us first introduce the following hypothesis for convenience:(M1) The functionsF and ℍ satisfy the following growth conditions for constants Mi,Ni>0, i=1,2, p∈0,1:
(53)Ft,ϖ≤M1ϖp+N1,ℍt,ϖ≤M2ϖp+N2,for each t∈J and each ϖ∈ℝ.(M2) For eachτ∈J, and for each, ϖ,ω∈ℝ, ∃ constants LF,Lℍ>0, provided
(54)Fτ,ϖ−Fτ,ϖ¯≤LFϖ−ϖ¯,ℍτ,ϖ−ℍτ,ϖ¯≤Lℍϖ−ϖ¯.In view of Lemma30, we consider two operators K,H:Cb⟶Cb given by (31) and (32), respectively. Then, we write the integral equation (27) as an operator equation:
(55)Fϖt=Kϖt+Hϖt,t∈a−δ,b.The continuity ofF and ℍ shows that the operator F is well-defined, and its fixed points are the same solutions of the existing equation (27) in Lemma 30.Lemma 34.
If (M1) and (M2) hold, then the operatorK is Lipschitz with constant LF and satisfies
(56)Kϖ≤M1ϖp+N1,foreveryϖ∈Cb.Proof.
Letϖ,ϖ¯∈Cb, then we get
(57)Kϖτ−Kϖ¯τ=Fτ,ϖ−Fτ,ϖ¯≤LFϖ−ϖ¯,
∀τ∈J. Let us take the supremum over τ, so we get
(58)Kϖ−Kϖ¯≤LFϖ−ϖ¯.
Hence,K:Cb⟶Cb is a Lipschitzian on Cb with Lipschitz constant LF. From Proposition 20, K is κ–Lipschitz with constant LF. In addition, we get
(59)Kϖt≤M1ϖp+N1,for every ϖ∈Cb. This finishes the proof.☐Lemma 35.
If (M1) holds, thenH is continuous and satisfies the growth condition
(60)Hϖ≤M¯ϖp+N¯,foreveryϖ∈Cb.Proof.
Choose a bounded subsetDr=ϖ∈Cb:ϖ≤r⊂Cb and consider a sequence ϖn∈Dr via ϖn⟶ϖ by letting n⟶∞ in Dr. We shall prove that Hϖn−Hϖ⟶0, letting n⟶∞. From the continuity of ℍ, it follows that ℍs,ϖn⟶ℍs,ϖ, as n⟶∞. In view of (M1), we get Hϖnτ≤λτ, where
(61)λτ=ϕτ,ifτ∈a−δ,a,ψτ−ψaνΓν+1M2rp+N2,ifτ∈J,which is Lebesgue’s integrable bounded function. The Labesgue dominated convergence theorem ensures that ∥Hϖn−Hϖ∥⟶0, letting n⟶+∞, which confirms the continuity of H.
Next, it is easy as above to deduce that(62)Hϖτ≤ϕCb,ifτ∈a−δ,a,ψb−ψaνΓν+1M2ϖp+N2,ifτ∈J.
Therefore,(63)Hϖ≤ψb−ψaνΓν+1M2ϖp+N2+ϕCb=M¯ϖp+N¯,where M¯=ψb−ψaν/Γν+1M2 and N¯=ψb−ψaν/Γν+1N2+ϕCb. This completes the proof.☐Lemma 36.
If (M1) holds, then the operatorH:Cb⟶Cb is compact. As a result, H is κ-Lipschitz with zero constant.Proof.
Take a bounded setΩ¯⊂Dr. We need to establish the relative compactness of HΩ¯ in Cb. For ϖ∈Ω¯, with the help of the estimate (63), we can obtain
(64)Hϖ≤M¯rp+N¯,which shows that HΩ is uniformly bounded.
Now, we prove the equi–continuity ofH. For τ∈J, we can estimate the derivative operator using (24) as follows:
(65)Hϖ′τ≤Ia+ν−1,ψℍτ,ϖτ≤M2ϖp+N2Ia+ν−1,ψ1τ≤ψb−ψaν−1ΓνMϖp+N≔ℓ.
Hence, for eachτ1,τ2∈J with a<τ1<τ2<b, we get
(66)Hϖτ2−H2ϖτ1≤∫τ1τ2Hϖ′sds≤ℓτ2−τ1,which tends to zero independently of ϖ as τ2−τ1⟶0. So, H is equi–continuous. The equi–continuity for the case τ1,τ2∈a−δ,a is obvious. From the foregoing arguments along with Arzela-Ascoli theorem, we deduce that H is compact on Dr. Thus, from Proposition 19, H is κ–Lipschitz with zero constant. This completes our proof.☐Theorem 37.
If (M1) and (M2) hold, then the NFuDEq (1) has at least one solution ϖ∈Cb provided that LF<1, and the set of the solutions is bounded in Cb.Proof.
Assume thatK,H,F are the operators defined by (31), (32) and (55), respectively, which all of them are bounded and continuous, and also, by Lemma 34, K is κ–Lipschitz with LF and by Lemma 36, H is κ–Lipschitz via constant 0. Thus, by Proposition 18, F is κ–Lipschitz with LF. Hence, F is strict κ–contraction with LF>0. Since LF<1, F is κ-condensing.
Now, let us consider the following set:(67)Θ=ϖ∈Cb:thereexistsς∈0,1suchthatx=ςFϖ.
We will show that the setΘ is bounded. For ϖ∈Θ, we have ϖ=ςFϖ=ςKϖ+Hϖ, which implies that
(68)ϖ≤ςKu+Hϖ≤ςM1ϖp+N1+M¯ϖp+N¯=ςM1+M¯ϖp+N1+N¯=Mϖp+N,where M=M+M¯ and N=N+N¯. If Θ is unbounded in Cb, in that case, we divide the obtained inequality by a≔ϖ and supposing a⟶∞, we get
(69)1≤lima⟶∞Map+Na=0,which is impossible, and Θ is bounded. Accordingly, it is found a fixed point for F which is interpreted as the solution of the NFuDEq (1). This finishes the proof.☐Remark 38.
If (M1) is represented forp=1, then Theorem 37 is true so that M<1.


## 5. Uniqueness Result and UH Stability

The uniqueness of the solution for the NFuDEq (1) will be investigated below by using the standard Banach fixed point theorem. Moreover, The UH stability of the NFuDEq (1) will be also checked.Theorem 39.
Suppose that assumption (M2) holds. Assume that(70)Δ≔ψb−ψaνΓν+1Lℍ+LF<1.Then,∃ a unique solution for (1) on the interval a−δ,b.Proof.
Define the set(71)U≔ϖ∈Cb:ϖa−δ,a∈Cδ,ϖJ∈C;cDa+νϖ∈C,and the operator G:U⟶U:
(72)Gϖτ≔Ia+ν,ψℍτ,ϖτ+Fτ,ϖτ,ifτ∈J,ϕτ,ifτ∈a−δ,a.
Notice thatG is well defined. Indeed, for ϖ∈U,τ↦Gϖτ is continuous, for any τ∈a−δ,b. In addition, ∀τ∈J,cDa+a,ψGϖτ−Fτ,ϖτ=ℍτ,ϖτ exists, and it is continuous too due to the continuity of ℍ and Lemma 26.
Now, we need to show thatG is a contraction. If ϖ,ϖ¯∈U and τ∈a−δ,a, then, ∣Gϖτ−Gϖ¯τ∣ equals to zero. On the contrary, for τ∈J, by (M2), it is derived that
(73)Gϖτ−Gϖ¯τ≤Ia+ν,ψℍτ,ϖτ−ℍτ,ϖ¯τ+Fτ,ϖτ−Fτ,ϖ¯τ≤Lℍϖτ−ϖ¯τCδIa+ν,ψ1τ+LFϖτ−ϖ¯τCδ≤ψb−ψaνΓν+1Lℍ+LFϖτ−ϖ¯τCδ≤ψb−ψaνΓν+1Lℍ+LFϖ−ϖ¯Cb,which implies
(74)Gϖ−Gϖ¯Cb≤Δϖ−ϖ¯Cb.
SinceΔ<1, the operator G is a contraction. Hence, Banach fixed point theorem shows that G admits a unique fixed point. This finishes the proof.☐Here, we discuss the UH and GUH stability types of (1).Definition 40.
The NFuDEq (1) is UH stable when ∃c∈ℝ+ so that ∀ε∈ℝ+ and ∀ϖ¯∈Cb satisfying
(75)cDa+ν;ψϖ¯τ−Fτ,ϖ¯τ−ℍτ,ϖ¯τ≤ε,τ∈J,ϖ¯τ−ϕτ≤ε,τ∈a−δ,a,exactly one solution ϖ∈Cb of (1) exists with
(76)ϖ¯−ϖ≤cε.Definition 41.
The NFuDEq (1) is GUH stable if ∃σ∈Cℝ+,ℝ+, σ0=0 so that ∀ε∈ℝ+ and ∀ϖ¯∈Cb satisfying (75), one and exactly one solution ϖ∈Cb of (1) exists with
(77)ϖ−ϖ¯≤σε.Remark 42.
A functionϖ¯∈Cb is a solution of the inequality (75) iff∃ a function η∈C such that
(1)
ητ≤ε,τ∈J(2)
cDa+ν;ψϖ¯τ−Fτ,ϖ¯τ=ℍτ,ϖ¯τ+ητ,τ∈JTheorem 43.
Suppose that (M2) and (70) hold. In this case, the solution of (1) is UH and GUH stable.Proof.
Assume that each of these two membersε∈ℝ+ and ϖ¯∈Cb satisfy (75). Then, ∃η∈C such that ητ≤ε,τ∈J, and
(78)cDa+ν;ψϖ¯τ−Fτ,ϖ¯τ=ℍτ,ϖ¯τ+ητ,τ∈J,ϖ¯τ=ϕτ,τ∈a−δ,a.
Using Lemma30, the NFuDEq (78) has a solution given as
(79)ϖ¯t≔Ia+ν,ψℍτ,ϖ¯τ+ηt+Fτ,ϖ¯τ,ifτ∈J,ϕτ,ifτ∈a−δ,a.☐Theorem 44.
Ensures the existence of a unique solutionϖ∈Cb of the NFuDEq (1) which satisfies
(80)ϖt≔Ia+ν,ψℍτ,ϖτ+Fτ,ϖτ,ifτ∈J,ϕτ,ifτ∈a−δ,a.
Therefore, for anyτ∈J, we get:
(81)ϖ¯τ−ϖτ≤Ia+ν,ψℍτ,ϖ¯τ−ℍτ,ϖτ+Ia+ν,ψητ+Fτ,ϖ¯τ−Fτ,ϖτ≤ψτ−ψaνΓν+1Lℍϖ¯τ−ϖτCδ+ψτ−ψaνΓν+1ε+LFϖ¯τ−ϖτCδ≤ψb−ψaνΓν+1Lℍ+LFϖ¯−ϖCb+κε,where κ=ψb−ψaν/Γν+1. Therefore, we have proved that
(82)ϖ¯−ϖCb≤Δϖ¯−ϖCb+κε.
By the condition in Theorem39, one can deduce that
(83)ϖ¯−ϖCb≤κ1−Δε.
Forc=κ/1−Δ>0, we infer that the solution of (1) is UH stable. In a similar manner, it is shown the existence of σ∈Cℝ>0,ℝ>0 by σε=κ/1−Δε with σ0=0. Hence, the solution of (1) is GUH stable.


## 6. Examples

Two illustrative examples are provided in this section to apply and validate our obtained results.Example 45.
Let us consider the NFuDEq according to (1) such that
(84)Dca+3/4;lnτϖτ−Fτ,ϖτ=ℍτ,ϖτ,τ∈J≔1,e,ϖτ=0,τ∈1−δ,1,where
ψτ=lnτ, a=1, b=e, ν=3/4,
andFτ,uτ, ℍτ,uτ are given as
(85)Fτ,u=lnτcosu100+lnτ,ℍτ,u=1t+12u+1+u2.
To explain Theorem39, let us take ℍτ,u and Fτ,u given by (85) and F1,u1=0. Clearly, the condition (M2) holds with LF=1/10 and Lℍ=1/4. In addition, Δ≈0.3720<1. Hence, all hypotheses of Theorem 39 are satisfied. Therefore, it is found exactly one solution for the NFuDEq (84) on 1,e.Example 46.
Consider the NFuDEq as follows:(86)Dca+1/2;τzτ−Fτ,zτ=ℍτ,zτ,τ∈J≔0,1,z0=0.
Notice that(87)ψτ=τ,ϕτ=0,a=0,b=1,ν=12.
To illustrate Theorem37, let us take
(88)ℍτ,uτ=1e−πτ+9uτ1+uτ+τ,Fτ,vτ=3+τ2∣v∣10+1+τ2.
It is obvious that(89)ℍτ,u−ℍτ,v≤110u−vCb,Fτ,u−Fτ,v≤15u−vCb.
Hence, (M2) also holds withLℍ=1/10, LF=1/5. Further, from the above-given data, we can easily calculate
(90)ψb−ψaνΓν+1Lℍ=0.1128.
On the contrary,∀τ∈J,u∈ℝ, we get
(91)ℍτ,u≤1+110u,Fτ,u≤2+15u.
Hence, (M1) holds withM1=1/10, M2=1/5, p=N1=1, and N2=2. In view of Theorem 37,
(92)Θ=u∈Cb:thereexistsς∈0,1suchthatu=ςKuis the solution set, then
(93)u≤ξKuCb+HuCb≤MuCb+N.
Using the Matlab program, we have(94)uCb≤N1−M=4.8298.
By Theorem37, the NFuDEq (86) with the data (87) and (88) has at least a solution.


## 7. Conclusion

In this paper, we considered and studied a fractional neutral functional delay differential equation involving aψ-Caputo fractional derivative on a partially ordered Banach space. To do this, we proved the existence results with the help of the Dhage approximation technique, and then by topological degree method for condensing maps. We established the uniqueness result by the well-known Banach contraction principle. The different kinds of Hyers-Ulam stability were checked in the sequel. Finally, we supported the validity of our findings by providing two examples. This study can be extended to more general structures by using generalized fractional operators with singular or nonsingular kernels due to their high accuracy.



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