#### Abstract

In this paper, a new class of a neutral functional delay differential equation involving the generalized -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.

#### 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 -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 in -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: where the -Caputo FOD, denoted by , of order , given are continuous functions such that , and is a continuous function with . For any function defined on and any , 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.

#### 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 functions endowed with the norm

Also, is endowed with norm

Consider the Banach space defined on with the norm

The order relation is defined as follows: which gives a partial ordering in .

From the research study in [29], let us now state some necessary definitions and preliminary results for our research work. Assume that displays a real partial order on . If for in , either or , then and are termed as comparable elements, and also when all members of are comparable, then is named either totally ordered or a chain. If there exists a nondecreasing (resp., nonincreasing) sequence and in such that as , then is regular ( (resp. )) for all . By assuming this fact that there are lower and upper bounds in for every both members of , in that case, the partially ordered Banach space is named regular and lattice.

*Definition 1 (see [29]). *An operator: is termed as nondecreasing or isotone if maintains the order relation , i.e., when , it means that for all .

*Definition 2 (see [29]). *A mapping has the compactness specification if is a set in with the relative compactness. In addition, is totally bounded if has the relative compactness property in , where is an arbitrary bounded set.

Every operator having the continuity and total boundedness properties will be completely continuous.

*Definition 3 (see [29]). * has the partial continuity property at , if for each , exists so that whenever and and are comparable. Assuming as an operator with the partial continuity on , it is well-known that is continuous on each chain . Furthermore, if is bounded for every , then is partially bounded. In addition, is uniformly partially bounded if all existing chains involve the boundedness by a bound uniquely.

*Definition 4 (see [29]). * has the partial compactness if has the relative compactness with respect to all chains . It has the partial total boundedness property if for each bounded and totally ordered set contained in , 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 is a nondecreasing selfmap on and is an arbitrary chain in it. In this case, possesses the partial compactness or the partial boundedness specifications whenever is relatively compact or bounded in .

*Definition 6 (see [28]). *Regard and as a metric and an order relation on . We say that and are compatible if is monotone, and if a subsequence of tends to , then tends to . Similar definition can be applied on a partially order norm space. A subset of is named Janhavi if the order relation and the metric (or the norm ) are compatible in it. Particularly, if , then we say that is Janhavi metric (or Janhavi Banach space).

*Definition 7 (see [29]). *An operator is -Lipschitz if there exists an upper semicontinuous nondecreasing function: with such that
for all .

*Definition 8 (see [29]). *The same above operator is termed as partially nonlinear -Lipschitz whenever a -function exists provide
. In addition, when is nonlinear -Lipschitz subject to for , in that case, is nonlinear -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 be a complete regular normed linear algebra via the partial order so that and are compatible. Consider two nondecreasing operators such that
*(a)* is partially nonlinear -Lipschtiz and partially bounded with -function *(b)* has the partial continuity and the compactness*(c)* an element such that or **Then, possesses a solution in , and the sequence of the successive iterations , expressed as approaches to monotonically.*

Theorem 10 (see [30]). *Let be a nondecreasing and partially nonlinear -contraction. Assume that exists with or . If is regular or is continuous, then a fixed point is found, and the sequence of successive iterations tends to monotonically. In addition, is unique if each of both members of possesses a lower and an upper bound.*

*Remark 11 (see [31]). *Let every set contained in with the partial compactness includes the compatibility specification with respect to and . Then, every compact chain of 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 in Theorem 9 can be replaced with another strong continuity condition of the operators and on 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 validIn 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.

We state here the results below given by [45–47].

*Definition 14. *The mapping is named Kuratowski measure of non-compactness (KMNC) if
where represents a class of all bounded mappings in .

Proposition 15. *The following are fulfilled for KMNC:
*(1)*(2)**, where and represent the closure and the convex hull of , respectively*(3)* and *

*Definition 16. *Assume that be a continuous bounded mapping and . The operator is said to be -Lipschitz if we can find a constant satisfying the following condition:

Moreover, is called strict -contraction subject to .

*Definition 17. * is called -condensing when
for every bounded and nonprecompact subset of . So,
Further, we have is Lipschitz if we can find such that
if , is said to be strict contraction.

The following three interesting results are based on [48]:

Proposition 18. *Let be -Lipschitz with constants and . Then, is -Lipschitz with .*

Proposition 19. *Every compact mapping is -Lipschitz with .*

Proposition 20. *Every Lipschitz mapping with is -Lipschitz with .*

Isaia [48] used the topological degree theory to introduce the following interesting results:

Theorem 21. *Let be -condensing and
**If is bounded, i.e., exists subject to ; then, the degree
*

As a result, it is found a fixed-point for and all possible fixed-points of are contained in .

Let be an increasing differentiable function such that , . Now, we start by defining -FODs as follows:

*Definition 22 (see [2]). *The -Riemann-Liouville fractional integral of order for an integrable function is given by
where the Gamma function is denoted by .

*Definition 23 (see [2]). *Let , be an integrable function, and . Then, the -Riemann–Liouville FOD of a function of order is expressed as:
where and .

*Definition 24 (see [20]). *For and , the -Caputo FOD of a function of order is given by
where

From the above definition, we can express -Caputo FOD by the following formula:

Also, the –Caputo FOD of order of is defined as

For more details, see ([20], Theorem 3).

Lemma 25 (see [2]). *For and , we have
*

Lemma 26 (see [22]). *Assume that . If then
and if then
*

It is easily to deduce that

Lemma 27 (see [2, 20]). *Let , and let . Then:
*(i)*(ii)**(iii)*

#### 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 that 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 is Janhavi.*

*Proof (see [31]). *Let us now discuss exactly the problem (1).☐

*Definition 29. *A function is a lower solution for the NFuDEq (1) if:
(1), (2)the function is continuously differentiable on and settles

Similarly, a differentiable function 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 that and are continuous with . The linear problem
has a unique solution defined by:
*

For the proof of Lemma 30, it is useful to refer to [2, 23, 41, 49].

With the help of the following hypothesis, we can investigate our results:

(H1) The functions and are monotone nondecreasing with respect to for any .

(H2) a -function that satisfies for such that

(H3) such that , and .

(H4) such that , , and .

(H5) The FBVP (1) possesses a lower solution .

(H6) a positive constant such that

Theorem 31. *If the hypotheses (H1)-(H5) are fulfilled, then the NFuDEq (1) includes a solution formulated on , and containing the successive approximations expressed as:
where , tends to monotonically.*

*Proof. *Take . Then, using Lemma 28, each compact chain admits the compatibility property in and such that is Janhavi in . On the other side, and can be defined on as follows:
According to the structure of integral, it is obvious that 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 and satisfy all items of Theorem 9. We follow our argument split into five steps.☐

*Step I. * and are nondecreasing on .

For with , using (H1), we get
for all . It means that is a nondecreasing operator.

Similarly, we obtain for all . Thus, it is concluded that is a nondecreasing operator.

*Step II. * is a nonlinear -contraction on .

For with and by (H2), we get that
By taking the supremum over we get
where for Therefore, according to Definition 8, our result is derived.

*Step III. * is partially continuous on .

Regard in a chain as . Then, for any letting . The continuity of yields
. Hence, converges to pointwise on .

In the following two cases, we prove that is an equicontinuous sequence of functions in .

*Case A. *Take , , with . Then,
which tends to zero as

*Case B. *For . Then,
Clearly, if and such that has only one possibility that they are close to at which is close to zero.

Thus, uniformly . This proves that is equi–continuous on . Thus, the pointwise convergence of on implies the uniform convergence, so converges to uniformly on . Consequently, the selfmap possesses the partial continuity on .

*Step IV. * has the partial compactness property on .

Regard the chain in and . Then Using hypothesis (H3), if , we have

Otherwise, if , then

Hence,

. Thus, we obtain for any . Thus, is a uniformly bounded subset of .

Let us now prove that is an equi–continuous set in . Let with . Then, according to Step III arguments, it is concluded that uniformly for any which illustrates the equi–continuity of in . So, is compact in reference to Arzelà-Ascoli criterion. As a result, the selfmap admits the partial compactness property on .

*Step V. * satisfies .

By (H5), is a lower solution of the NFuDEq (1) defined on . Then, according to the lower solution definition, we get

Let us integrate the above inequality from to , we obtain

. Thus, . Obviously, both operators and satisfy all of the items of Theorem 9; therefore, the operator equation has a solution defined on . Furthermore, the sequence of successive approximations defined by (30) tends to 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: .

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 defined on provided that , where
*

Moreover, the sequence of successive approximations defined by (30) converges monotonically to .

*Proof. *First, the operator: is defined by
for . To prove this theorem, we establish the satisfaction of all items of Theorem 10 for in . We know that 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 is a partially -contraction on . To arrive at such an aim, by taking such that , if then it is obvious that
Otherwise, let , it follows from (H1) and (H6), that
for all , where . Let us now take the supremum over we get
for all with . As a result, is a partially nonlinear -contraction on . In addition, by using Theorem 31, it is proven that the given function in (H5) satisfies the operator inequality on . Therefore, from Theorem 10, it is found a solution uniquely for the NFuDEq (1), and defined by (30) tends to 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 functions and satisfy the following growth conditions for constants , , : for each and each .

(M2) For each , and for each, , constants , provided

In view of Lemma 30, we consider two operators given by (31) and (32), respectively. Then, we write the integral equation (27) as an operator equation:

The continuity of and shows that the operator 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 operator is Lipschitz with constant and satisfies
*

*Proof. *Let , then we get
. Let us take the supremum over , so we get
Hence, is a Lipschitzian on with Lipschitz constant . From Proposition 20, is –Lipschitz with constant . In addition, we get
for every . This finishes the proof.☐

Lemma 35. *If (M1) holds, then is continuous and satisfies the growth condition
*

*Proof. *Choose a bounded subset and consider a sequence via by letting in . We shall prove that , letting . From the continuity of , it follows that as . In view of (M1), we get , where
which is Lebesgue’s integrable bounded function. The Labesgue dominated convergence theorem ensures that , letting , which confirms the continuity of .

Next, it is easy as above to deduce that
Therefore,
where and . This completes the proof.☐

Lemma 36. *If (M1) holds, then the operator is compact. As a result, is -Lipschitz with zero constant.*

*Proof. *Take a bounded set . We need to establish the relative compactness of in . For , with the help of the estimate (63), we can obtain
which shows that is uniformly bounded.

Now, we prove the equi–continuity of . For , we can estimate the derivative operator using (24) as follows:
Hence, for each with , we get
which tends to zero independently of as . So, is equi–continuous. The equi–continuity for the case is obvious. From the foregoing arguments along with Arzela-Ascoli theorem, we deduce that is compact on . Thus, from Proposition 19, is –Lipschitz with zero constant. This completes our proof.☐

Theorem 37. *If (M1) and (M2) hold, then the NFuDEq (1) has at least one solution provided that , and the set of the solutions is bounded in .*

*Proof. *Assume that are the operators defined by (31), (32) and (55), respectively, which all of them are bounded and continuous, and also, by Lemma 34, is –Lipschitz with and by Lemma 36, is –Lipschitz via constant . Thus, by Proposition 18, is –Lipschitz with . Hence, is strict –contraction with . Since , is -condensing.

Now, let us consider the following set:
We will show that the set is bounded. For , we have , which implies that
where and . If is unbounded in , in that case, we divide the obtained inequality by and supposing , we get
which is impossible, and is bounded. Accordingly, it is found a fixed point for which is interpreted as the solution of the NFuDEq (1). This finishes the proof.☐

*Remark 38. *If (M1) is represented for , then Theorem 37 is true so that

#### 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
*

Then, a unique solution for (1) on the interval .

*Proof. *Define the set
and the operator :
Notice that is well defined. Indeed, for is continuous, for any In addition, exists, and it is continuous too due to the continuity of and Lemma 26.

Now, we need to show that is a contraction. If and , then, equals to zero. On the contrary, for by (M2), it is derived that