Qualitative Study on Solutions of Piecewise Nonlocal Implicit Fractional Differential Equations

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Introduction
It merits noticing that fractional calculus (FC) has gotten significant thought from scientists and researchers. It is a result of its wide scope of uses in different fields and disciplines. The crucial concepts and definitions of FC have been presented in [1,2]. In [3,4], the authors introduced some fundamental history of fractional calculus and its applications to engineering and different areas of science.
Many classes of fractional differential equations (FDEs) have been intensively investigated in the last decades, for instance, theories involving the existence of unique solutions have been notarized [5][6][7]. Numerical and analytical methods have been evolving with the target to solve such equations [8][9][10]. These equations have been tracked as useful in modeling some real-world problems with incredible achievement.
The qualitative properties of solutions represent a very important aspect of the theory of FDEs. The formerly aforesaid region has been studied well for classical differential equations. However, for FDEs, there are many aspects that require further studying and reconnoitering. The attention on the existence and uniqueness has been especially focused by applying Riemann-Liouville (R-L), Caputo, Hilfer, and other FDs, see [11][12][13][14][15].
In this regard, Agarwal et al. [16] investigated the existence of solutions of the following Caputo type FDE: The basic theory of implicit FDEs with Caputo FD has been investigated by Kucche et al. [17]. Wahash et al. [18] considered the following nonlocal implicit FDEs with ψ-Caputo FD Problem (2) with ψðϰÞ = ϰ has been studied by Benchohra and Bouriah [19].
Motivated by the above works and inspired by [20], we consider the piecewise Caputo implicit FDE (PC-IFDE) of the type: , ℝÞ, and PC D ϑ 0 + represent the piecewise Caputo FD of order ϑ defined by where D f ðϰÞ ≔ ðd/dϰÞf ðϰÞ is a classical derivative on 0 ≤ ϰ ≤ ϰ 1 and C D ϑ ϰ 1 is standard Caputo FD on ϰ 1 ≤ ϰ ≤ b. It is essential to note that the utilization of nonlinear condition υð0Þ + gðυÞ = υ 0 in physical issues yields better impact than the initial condition υð0Þ = υ 0 (see [21]).
We pay attention to the topic of the novel piecewise operators. As far as we could possibly know, no outcomes in the literature are addressing the qualitative aspects of the aforesaid problems by using the piecewise FC. Consequently, by conquering this gap, we will examine the existence, uniqueness, and Ulam-Hyers stability results of piecewise Caputo problems (3) and (4) based on the standard fixed point theorems due to Banach-type and Schauder-type. Furthermore, we present similar results containing piecewise Caputo-Fabrizio (PCF) type and piecewise Atangana-Baleanu (PAB) type. An open problem with respect to another function is suggested.
The substance of this paper is coordinated as follows: Section 2 presents a few required outcomes and fundamentals about piecewise FC. Our key outcomes for problem (4) are proved in Section 3. Two examples to make sense of the gained outcomes are built in Section 4. Toward the end, we encapsulate our study in the end section.

Primitive Results
In this section, we present some concepts of a piecewise FC. Let Obviously C is a Banach space under kηk.

Main Results
In this section, we give some qualitative analyses of the PC-IFDE and PC-NIFDE.
Next, we prove the uniqueness theorem for (4) based on Banach's theorem.

An Analogous Results.
In this part, we show some analogous results according to our preceding outcomes.
Based on PAB-NIFDE (45), the results in Theorems 7 and 8 can be presented by [26]): Remark 9. Following the strategy of proof utilized in the previous part, we can get the existence results for nonlinear problems (42) and (45).

UH Stability Analysis.
In this portion, we give the UH Stability of problem (4). (4) is UH stable if there exists a K f > 0, such that for all ε > 0 and each solution ω ∈ C of the inequality.
Journal of Function Spaces By part (ii) of Remark 11, we have Then, the solution of problem (52) is Again by (i) of Remark 11, we obtain Theorem 13. Under the assumptions of Theorem 8. Then, the solution of PC-NIFDE (4) is HU and GHU stable. Proof.
Let ω ∈ C be a solution of inequality (48), and υ ∈ C be a unique solution of the following PC-NIFDE.

Conclusions
Somewhat recently, numerous methodologies have been proposed to portray behaviors of some complex world problems emerging in numerous scholarly fields. One of these problems is the multistep behavior shown by certain problems. In this regard, Atangana and Araz [20] introduced the concept of piecewise derivative. As an extra contribution to this subject, existence, uniqueness, and UH stability results for PC-NIFDE (4) involving a piecewise Caputo FD have been obtained. Our approach to this work has been based on Banach's and Schaefer's fixed-point theorem and Gronwall's Lemma. In light of our current results, the solution form for analogous problems containing piecewise Caputo-Fabrizio and Atangana-Baleanu operators have been presented. Finally, we have created two examples to validate the results obtained.
As an open problem, it will be very interesting to study the present problems on piecewise fractional operators with another function that is more general; precisely, one has to consider in problem (2) with PC D ϑ;ψ where D ψ ≔ ðð1/ðψ ′ ðϰÞÞÞðd/dϰÞÞ and C D ϑ;ψ 0 + are ψ-Caputo FD of order ϑ introduced by Almeida [28]:

Data Availability
No real data were used to support this study. The data used in this study are hypothetical.

Conflicts of Interest
No conflicts of interest are related to this work.