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

Laboratory of Mathematics and Applied Sciences, University of Ghardaia, 47000, Algeria Department of Mathematics, Al-Azhar University-Gaza, Gaza Strip, State of Palestine Jabalia Camp, United Nations Relief andWorks Agency (UNRWA) Palestinian Refugee Camp, Gaza Strip Jabalya, State of Palestine Department of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

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][22][23][24][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][30][31][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][34][35][36][37][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][42][43][44].
Motivated by the novel developments in ψ-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 NFu-DEq is expressed as: where the ψ-Caputo FOD, denoted by c D ν;ψ a + , of order ν ∈ ð0, 1Þ, given F, ℍ : J × ℝ ⟶ ℝ are continuous functions such that Fða, ϖ a Þ = 0, and ϕ : ½a − δ, a ⟶ ℝ is a continuous function with ϕðaÞ = 0. For any function u defined on ½a − δ, a and any τ ∈ J, it is given by 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 Section 2. 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.

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 functions C = CðJ, ℝÞ endowed with the norm Also, C δ = Cð½−δ, 0, ℝÞ is endowed with norm The order relation ≼ is defined as follows: which gives a partial ordering in C b . From the research study in [29], let us now state some necessary definitions and preliminary results for our research work. Assume that X = ðX, ≼, k:kÞ 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 ðϖ n Þ n∈N and ϖ * in X such that ϖ n ⟶ ϖ * as n ⟶ ∞, then X is regular (ϖ n ≼ϖ * (resp. ϖ n ± 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 2 (see [29]). A mapping Q : X ⟶ X has the compactness specification if QðXÞ is a set in X with the relative compactness. In addition, Q is totally bounded if QðSÞ 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 2 Journal of Function Spaces kQϖ − Qak < ε whenever kϖ − ak < 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 QðCÞ is bounded for every C ⊆ X, then Q is partially bounded. In addition, Q is uniformly partially bounded if all existing chains QðCÞ ⊆ X involve the boundedness by a bound uniquely.
Definition 4 (see [29]). Q : X ⟶ X has the partial compactness if QðCÞ ⊂ 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, QðCÞ ⊂ 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 that Q 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 QðCÞ is relatively compact or bounded in X.
Definition 6 (see [28]). Regard d and ≼ as a metric and an order relation on X. We say that d and ≼ are compatible if fϖ n g n∈ℕ ⊂ X is monotone, and if a subsequence fϖ n k g k∈N of fϖ n g n∈ℕ tends to ϖ * , then fϖ n g n∈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 k·k) are compatible in it. Particularly, if S = X, then we say that X is Janhavi metric (or Janhavi Banach space).
Definition 8 (see [29]). The same above operator Q is termed as partially nonlinear D-Lipschitz whenever a D-function Ψ : ∀ϖ, ω ∈ X. In addition, when Q is nonlinear D-Lipschitz subject to ΨðτÞ < 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]). Let ðX, ≼, k·kÞ be a complete regular normed linear algebra via the partial order so that ≼ and k·k 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 ψ K (b) H has the partial continuity and the compactness Then, Kϖ + H ϖ = ϖ possesses a solution ϖ * in X, and the sequence of the successive iterations fϖ n g ∞ n=0 , expressed as ϖ n+1 = Kϖ n + H ϖ n , approaches to ϖ * monotonically.
Theorem 10 (see [30]). Let H : X ⟶ X be a nondecreasing and partially nonlinear D -contraction. Assume that ϖ 0 ∈ X exists with ϖ 0 ≼H ϖ 0 or ϖ 0 ± H ϖ 0 . If X is regular or H is continuous, then a fixed point ϖ * is found, and the sequence of successive iterations fH n ϖ 0 g 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 in X with the partial compactness includes the compatibility specification with respect to ≼ and k·k. 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 of X 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.
(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 Theorem 9 can be replaced by the continuity and compactness of that operator.
where M C represents a class of all bounded mappings in C.

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Proposition 15. The following are fulfilled for KMNC: (2) κðAÞ = κð AÞ = κðconvðAÞÞ, where A and convðAÞ represent the closure and the convex hull of A, respectively (3) κðA + EÞ ≤ κðAÞ + κðEÞ and κðcAÞ = |c | κðAÞ, c ∈ ℝ Definition 16. Assume that K : 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: Moreover, K is called strict κ-contraction subject to ℓ < 1 .
for every bounded and nonprecompact subset B of A. So, if ℓ < 1, K is said to be strict contraction.
The following three interesting results are based on [48]: Isaia [48] used the topological degree theory to introduce the following interesting results: If Θ ⊂ C is bounded, i.e., r > 0 exists subject to Θ ⊂ B r ð0Þ; then, the degree As a result, it is found a fixed-point for F and all possible fixed-points of F are contained in B r ð0Þ.

Existence and Approximation Results via Dhage's Technique
The solutions' existence and approximation of problem (1) are studied in this section.

Lemma 28.
Assume that ðC b , ≼, k:kÞ is a partially ordered Banach space with the norm k:k, and the order relation ≼ defined by (5) and (6), respectively. Then, every partially compact subset of C b is Janhavi.
Proof (see [31]). Let us now discuss exactly the problem (1). ☐ Definition 29. A function ϖ ∈ C b is a lower solution for the NFuDEq (1) if: Similarly, a differentiable function ω ∈ C b is named an upper solution of the NFuDEq (1) if the above inequality is satisfied with reverse sign.
With the help of the following hypothesis, we can investigate our results: (H1) The functions Fðτ, ϖÞ and ℍðτ, ϖÞ are monotone nondecreasing with respect to ϖ for any τ ∈ J.
Proof. Take X = C b = Cð½a − δ, b, ℝÞ. Then, using Lemma 28, each compact chain C ⊂ C b admits the compatibility property in k·k and ≤ such that C is Janhavi in C b . On the other side, K and H can be defined on C b as follows:

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According to the structure of integral, it is obvious that K, H : C b ⟶ C b are well-defined. In addition, the studied problem (1) can be reformulated by: To investigate the solutions' existence to this operator equation, we can sufficiently show that the operators K and H satisfy all items of Theorem 9. We follow our argument split into five steps. ☐ Step I. K and H are nondecreasing on C b . For ϖ, ω ∈ C b with ϖ τ ≥ ω τ , using (H1), we get for all τ ∈ ½a − δ, b. It means that K : Similarly, we obtain for all τ ∈ ½a − δ, b. Thus, it is concluded that H : C b ⟶ C b is a nondecreasing operator.
Step III. H is partially continuous on C b .
In the following two cases, we prove that fH ϖ n g n∈ℕ is an equicontinuous sequence of functions in C b .
Case A. Take τ 1 , τ 2 ∈ J, with τ 1 < τ 2 . Then, which tends to zero as τ 1 ⟶ τ 2 : Clearly, if τ 1 ∈ ½a − δ, a and τ 2 ∈ J such that τ 1 ⟶ τ 2 has only one possibility that they are close to a at which H ðϖ n Þ is close to zero. Thus, uniformly ∀n ≥ 1. This proves that fH ϖ n g is equi-continuous on ½a − δ, b. Thus, the pointwise convergence of fH ϖ n g on ½a − δ, b implies the uniform convergence, so H ϖ n converges to H ϖ uniformly on ½a − δ, b. Consequently, the selfmap H possesses the partial continuity on C b .
Let us now prove that H ðCÞ is an equi-continuous set in C b . Let τ 1 , τ 2 ∈ J, with τ 1 < τ 2 . Then, according to Step III arguments, it is concluded that uniformly for any ω ∈ H ðCÞ which illustrates the equi-continuity of H ðCÞ in C b . So, H ðCÞ is compact in reference to Arzelà-Ascoli criterion. As a result, the selfmap H : C b ⟶ C b admits the partial compactness property on C b .
Step V. ϖ satisfies ϖ ≤ Kϖ + H ϖ. By (H5), W is a lower solution of the NFuDEq (1) defined on ½a − δ, b. Then, according to the lower solution definition, we get Let us integrate the above inequality from a to τ, we obtain ∀τ ∈ ½a − δ, b. Thus, W ≤ KW + H W. Obviously, both operators K and H satisfy all of the items of Theorem 9; therefore, the operator equation Kϖ + H ϖ = ϖ has a solution ϖ * defined on ½a − δ, b. Furthermore, the sequence fϖ n g ∞ n=0 of successive approximations defined by (30) tends to ϖ * monotonically. So, our proof is ended.
Moreover, the sequence fϖ n g of successive approximations defined by (30) converges monotonically to ϖ * .
Proof. First, the operator: G : for τ ∈ ½a − δ, b,. To prove this theorem, we establish the satisfaction of all items of Theorem 10 for G in C b . We know that G is nondecreasing and continuous. The details are similar as in the proof of Theorem 31, so we omit them. Therefore, it is needed to be verified that G is a partially D-contraction on C b . To arrive at such an aim, by taking ϖ, ω ∈ C b such that ϖ ≥ ω, if τ ∈ ½a − δ, a, then it is obvious that Otherwise, let τ ∈ J, it follows from (H1) and (H6), that for all τ ∈ J, where ΩðRÞ < R, R > 0. Let us now take the supremum over τ, we get for all ϖ, ϖ ∈ C b , with ϖ ≥ ϖ. As a result, G is a partially nonlinear D-contraction on C b . In addition, by using Theorem 31, it is proven that the given function x in (H5) satisfies the operator inequality x ≤ Gx on ½a − δ, b. Therefore, from Theorem 10, it is found a solution ϖ * uniquely for the NFuDEq (1), and fϖ n g defined by (30) tends to ϖ * monotonically. ☐

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 functions F and ℍ satisfy the following growth conditions for constants M i , N i > 0, i = 1, 2, p ∈ ð0, 1Þ: for each t ∈ J and each ϖ ∈ ℝ.
(M2) For each τ ∈ J, and for each, ϖ, ω ∈ ℝ, ∃ constants In view of Lemma 30, we consider two operators K, H : C b ⟶ C b given by (31) and (32), respectively. Then, we write the integral equation (27) as an operator equation: The continuity of F 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.
Proof. Let ϖ, ϖ ∈ C b , then we get ∀τ ∈ J. Let us take the supremum over τ, so we get Hence, K : for every ϖ ∈ C b . This finishes the proof. ☐ Now, let us consider the following set: We will show that the set Θ is bounded. For ϖ ∈ Θ, we have ϖ = ςFϖ = ςðKðϖÞ + H ðϖÞÞ, which implies that where M = ðM + MÞ and N = ðN + NÞ. If Θ is unbounded in C b , in that case, we divide the obtained inequality by a ≔ kϖk and supposing a ⟶ ∞, we get 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 for p = 1, then Theorem 37 is true so that M < 1:

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.
Proof. Define the set and the operator G : U ⟶ U: Notice that G is well defined. Indeed, for ϖ ∈ U, τ ↦ GðϖÞðτÞ is continuous, for any τ ∈ a − δ, b: In addition, ∀τ ∈ J, c D a,ψ a + ½GðϖÞðτÞ − Fðτ, ϖ τ Þ = ℍðτ, ϖ τ Þ exists, and it is continuous too due to the continuity of ℍ and Lemma 26. Now, we need to show that G is a contraction. If ϖ, ϖ ∈ U and τ ∈ a − δ, a, then, |GðϖÞðτÞ − Gð ϖÞðτÞ | equals to zero. On the contrary, for τ ∈ J, by (M2), it is derived that which implies Since Δ < 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).

Examples
Two illustrative examples are provided in this section to apply and validate our obtained results.

Conclusion
In this paper, we considered and studied a fractional neutral functional delay differential equation involving a ψ-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.