Qualitative Analysis of Implicit Dirichlet Boundary Value Problem for Caputo-Fabrizio Fractional Differential Equations

Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 06790, Turkey


Introduction and Preliminaries
The concerned area of fractional order differential equations (FODEs) have many concentrations in real-world problems and have paid close attention to numerous researchers in the past few decades [1][2][3][4][5]. The mentioned area has been studied from several aspects, such as the existence and uniqueness of solutions via using the classical fixed-point theory, the numerical analysis, the optimization theory, and also the theory of stability corresponding to various fractional differential operators like Caputo, Hamdard, and Riemann-Liouville (we refer few as [6][7][8][9]). In the aforementioned operators, there exists a singular kernel. Therefore, recently some authors introduced some new types of fractional derivative operators in which they have replaced a singular kernel by a nonsingular kernel. The nonsingular kernel derivative has been proved as a good tool to model real-world problems in different fields of science and engineering [10,11]. In fractional, it is called nonsingular exponential type or Caputo-Fabrizio fractional differential (CFFD) operator. The CFFD operator introduced two researchers, Caputo and Fabrizio for the first time in 2015 [12]. They replaced the singular kernel in the usual Caputo and Riemann-Liouville derivative by an exponential nonsingular kernel. The new operator of this type was found to be more practical than the usual Caputo and Riemann-Liouville fractional differential operators in some problems (see some detailed references such as [13][14][15]). Recently, many researchers have studied the existence and uniqueness of the solutions at the initial value problems for FODEs under the said operator. But the investigation has been limited to initial value problems only. On the other hand, boundary value problems have significant applications in engineering and other physical sciences during modeling numerous phenomena (we refer to see [16][17][18][19]). Furthermore, during optimization and numerical analysis of the mentioned problems, researchers need stable results from theoretical as well as practical sides. A stable result may lead us to a stable process. Therefore, the stability theory has also got proper attention during the last many decades. It is well known fact that stability analysis plays an important role. Various stability concepts such as exponential stability, Mittag-Lefler stability and Hayers-Ulam's stability have been adopted in literature to study the stability of different systems of FODEs. The analysis of Hyers-Ulam's stability has been recognized as a simple form of investigation. For historical background on the stability of Hyers-Ulam, we refer to see previous articles [20][21][22][23]. But recently, that type of problem has not been adequately studied for a new type of CFFD operator. Therefore, in this work, we will investigate an implicit class of FODEs involving the CFFD operator under DBCs where CF D is used for CFFD and I = ½c, d, f : I × R × R → R is a continuous function. In this article, we investigate uniqueness and existence of solutions to the proposed problem (1) by classical fixed-point theorems due to Banach's and Krasnoselskii. Further, we investigate some pertinent analysis about the stability theory due to Ulam, and Hyers is investigated for the mentioned problem (1). For the authenticity of the presented work, two concrete examples are also studied. Throughout the paper, C½I, R is a Banach space with norm kzk = max w∈I jzðwÞj.

Results and Discussion
In this part, we investigate the solution of the proposed problem (1) and also study the uniqueness and existence of the solutions.
is given by Proof. Let zðwÞ be a solution to problem (6). Applying Caputo-Fabrizio integral on both sides and then using Lemma 4 and Definition 3, we have which implies that Using boundary conditions zðcÞ = zðdÞ = 0, we have Putting c 0 , c 1 in (9), we get

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For simplification, use some notations; we use G μ = ð2 − μÞ/Dðμ − 1Þ, G * μ = ðμ − 1Þ/Dðμ − 1Þ and give the solution of (1) as bellow. Corollary 6. In view of 6, the solution of the considered problem (1) is given by Further, for the existence and uniqueness of the solution of problem (1), we use some fixed point theorems. For this, we need to define an operator as N : To proceed further, using Corollary (6) to convert the proposed problem (1) is to a fixed point problem as NzðwÞ = zðwÞ, where the operator N is given by (13). Therefore, Problem (1) has a solution if and only if the operator N has a fixed point, where λðwÞ = f ðw, zðwÞ, λðwÞÞ and λðwÞ= CF c D μ w zðwÞ. We assume that (H 1 ) There exist certain constant D f > 0 and 0 for all z, z, λ, λ ∈ R: Theorem 7. Under the hypothesis (H 1 ), the mentioned problem (1) has a unique solution if Proof. Suppose zðwÞ, zðwÞ ∈ C½I, R, we have where λðwÞ, λðwÞ ∈ C½I, R are given by λðwÞ = f ðw, zðwÞ, λ ðwÞÞ and λðwÞ = f ðw, zðwÞ, λðwÞÞ by using hypothesis (H 1 ), we have Repeating the above process, we get Using (18) in (16), we have Applying maximum on both sides, we have

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Thus, operator N is a contraction; therefore, the operator N has a unique fixed point. Hence, the corresponding problem (1) has a unique solution.
Our next result is to show the existence of the solution to the proposed problem (1) which is based on Krasnoselskii's fixed-point theorem. Therefore, the given hypothesis hold.
(H 2 ) There exist constant p f , q f , r f > 0 with 0 < r f < 1 such that Theorem 8 (see [26]). Let H ⊂ C½I, R be a closed, convex nonempty subset of C½I, R; then, there exist N 1 , N 2 operators such that (2) N 1 is a contraction, and N 2 is compact and continuous Then, there exist at least one solution z ∈ H such that N 1 z + N 2 z = z: Theorem 9. If the hypothesis (H 2 ) is satisfied, then (1) has at least one solution if Proof. Suppose we define two operators from (13) as Let us define a set F = fz ∈ C½I, R: ∥z∥≤rg, since f is continuous, so we show that the operator N 1 is contraction. For this z, z ∈ C½I, R, we have using (18), and then taking the maximum on both sides, we have Hence, N 1 is contraction. Next, to prove that the operator N 2 is compact and continuous, for this zðwÞ ∈ C½I, R, we have where λðwÞ ∈ R, λðwÞ = f ðw, zðwÞ, λðwÞÞ; now, using hypothesis (H 2 ), we have repeating the above process, so we get Now, using (28) in (26) and then taking the maximum on both sides, we have Journal of Function Spaces Which implies that Therefore, N 2 is bounded. Next, let w 1 < w 2 in I, we have Now, using (28) in (31), we have Applying maximum on right-hand side of the above inequality, we take Obviously, from (33), we see that w 1 → w 2 ; then, the right-hand side of (33) goes to zero, so |N 2 zðw 2 Þ − N 2 zðw 1 Þ | → 0 as w 1 → w 2 . Hence, the operator N 2 is continuous. Also, NðHÞ ⊂ H; therefore, the operator N 2 is compact, and by the Arzela-Ascoli theorem, the operator N has at least one fixed point. Therefore, the mentioned problem (1) has at least one solution.

Stability Theory
In this portion, we develop several consequences concerning the stability of Hyers-Ulam and generalize Hyers-Ulam type. Before progressing further, we provide various notions and definitions: Definition 10. The proposed problem (1) is Hyers-Ulam stable if at any ε > 0 for the given inequality there exist a unique solution zðwÞ with a constant K f such that Further, the considered problem (1) will generalize Hyers-Ulam stable if there exists nondecreasing function ϕ : ðc, dÞ → ð0,∞Þ such that with ϕðcÞ = 0 and ϕðdÞ = 0.
Also, we state an important remark as: Remark 11. Let there exist a function ψðwÞ which depends on z ∈ C½I, R with ψðcÞ = 0 and ψðdÞ = 0 such that where G μ = ð2 − μÞ/Dðμ − 1Þ, G * μ = ðμ − 1Þ/Dðμ − 1Þ, and λðwÞ = f ðw, zðwÞ, λðwÞÞ: Moreover, the solution of the given inequality, we have 5 Journal of Function Spaces Proof. The solution of (39) can be acquired straightforward by using Lemma 5. Although from the solution, it is clear to become result (40) by using Remark 11.