CUBO, A Mathematical Journal Vol. 24, no. 01, pp. 83–94, April 2022 DOI: 10.4067/S0719-06462022000100083 Existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative fractional differential equation Abderrahim Guerfi 1 Abdelouaheb Ardjouni 1,2 1Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, University of Annaba, P.O. Box 12, Annaba 23000, Algeria. abderrahimg21@gmail.com 2Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria. abd ardjouni@yahoo.fr ABSTRACT In this work, we study the existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an itera- tive Caputo fractional differential equation by first inverting it as an integral equation. Then we construct an appropri- ate mapping and employ the Schauder fixed point theorem to prove our new results. At the end we give an example to illustrate our obtained results. RESUMEN En este trabajo, estudiamos la existencia, unicidad, depen- dencia continua y estabilidad de Ulam de soluciones mild para una ecuación diferencial fraccionaria de Caputo itera- tiva, invirtiéndola primero como ecuación integral. Luego construimos una aplicación apropiada y empleamos el teo- rema del punto fijo de Schauder para demostrar nuestros nuevos resultados. Finalmente damos un ejemplo para ilus- trar los resultados obtenidos. Keywords and Phrases: Iterative fractional differential equations, fixed point theorem, existence, uniqueness, continuous dependence, Ulam stability. 2020 AMS Mathematics Subject Classification: 34K40, 34K14, 45G05, 47H09, 47H10. Accepted: 20 December, 2021 Received: 02 April, 2021 c©2022 A. Guerfi et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462022000100083 https://orcid.org/0000-0002-8221-9545 https://orcid.org/0000-0003-0216-1265 mailto:abderrahimg21@gmail.com mailto:abd_ardjouni@yahoo.fr 84 A. Guerfi & A. Ardjouni CUBO 24, 1 (2022) 1 Introduction Fractional differential equations have gained considerable importance due to their applications in various sciences, such as physics, mechanics, chemistry, engineering, etc. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives, see the monographs of Kilbas et al. [10], Miller and Ross [12], Podlubny [14]. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1]–[4], [6]–[16], [18] and the references therein. Recently, iterative functional differential equations of the form x′ (t) = H ( x[0] (t) ,x[1] (t) ,x[2] (t) , . . . ,x[n] (t) ) , have appeared in several papers, where x[0] (t) = t, x[1] (t) = x(t) , x[2] (t) = x(x(t)) , . . . , x[n] (t) = x[n−1] (x(t)) are the iterates of the state x(t). Iterative differential equations often arise in the modeling of a wide range of natural phenomena such as disease transmission models in epidemiology, two-body problem of classical electrodynam- ics, population models, physical models, mechanical models and other numerous models. This kind of equations which relates an unknown function, its derivatives and its iterates, is a special type of the so-called differential equations with state-dependent delays, see [5, 9, 19] and the references therein. In this paper, inspired and motivated by the references [1]–[16], [18, 19], we concentrate on the existence, uniqueness, continuous dependence and Ulam stability of mild solutions for the nonlinear iterative fractional differential equation    CDα 0+ x(t) = f ( x[0] (t) ,x[1] (t) ,x[2] (t) , . . . ,x[n] (t) ) , t ∈ J, x(0) = x′ (0) = 0, (1.1) where J = [0,T ], CDα 0+ is the standard Caputo fractional derivative of order α ∈ (1,2) and f is a positive continuous function with respect to its arguments and satisfies some other conditions that will be specified later. To reach our desired end we have to transform (1.1) into an integral equation and then use the Schauder fixed point theorem to show the existence and uniqueness of mild solutions. The organization of this paper is as follows. In Section 2, we introduce some definitions and lemmas, and state some preliminary results needed in later sections. Also, we present the inversion of (1.1) and state the Schauder fixed point theorem. For details on the Schauder theorem we refer the reader to [17]. In Section 3, we present our main results on the existence, uniqueness, continuous CUBO 24, 1 (2022) Existence, uniqueness, continuous dependence and Ulam stability... 85 dependence and Ulam stability of mild solutions for the problem (1.1) and provide an example to illustrate our results. 2 Preliminaries Let C (J,R) be the Banach space of all real-valued continuous functions defined on the compact interval J, endowed with the norm ‖x‖ = sup t∈J |x(t)| . For 0 < L ≤ T and M > 0, define the sets C (J,L) = {x ∈ C (J,R) : 0 ≤ x(t) ≤ L, ∀t ∈ J} , and CM (J,L) = {x ∈ C (J,L) : |x(t2) − x(t1)| ≤ M |t2 − t1| , ∀t1, t2 ∈ J}. Then, CM (J,L) is a closed convex and bounded subset of C (J,R). Furthermore, we suppose that the positive function f is globally Lipschitz in xi, that is, there exist positive constants c1, c2, . . . , cn such that |f (t,x1,x2, . . . ,xn) − f (t,y1,y2, . . . ,yn)| ≤ n ∑ i=1 ci |xi − yi| . (2.1) We introduce the constants ρ = sup t∈J {f (t,0,0, . . . ,0)} , ζ = ρ + L n ∑ i=1 ci i−1 ∑ j=0 Mj, where Mj = M × Mj−1. Definition 2.1 ([10]). The fractional integral of order α > 0 of a function x : R+ −→ R is given by Iα0+x(t) = 1 Γ (α) ∫ t 0 (t − s) α−1 x(s)ds, provided the right side is pointwise defined on R+, where Γ is the gamma function. For instance, Iα 0+ x exists for all α > 0, when x ∈ C(R+) then Iα 0+ x ∈ C (R+) and moreover Iα 0+ x(0) = 0. Definition 2.2 ([10]). The Caputo fractional derivative of order α > 0 of a function x : R+ −→ R is given by CDα0+x(t) = I n−α 0+ x(n) (t) = 1 Γ (n − α) ∫ t 0 (t − s) n−α−1 x(n) (s)ds, where n = [α] + 1, provided the right side is pointwise defined on R+. 86 A. Guerfi & A. Ardjouni CUBO 24, 1 (2022) Lemma 2.3 ([10]). Suppose that x ∈ Cn−1 ([0,+∞)) and x(n) exists almost everywhere on any bounded interval of R+. Then ( Iα C0+ D α 0+x ) (t) = x(t) − n−1 ∑ k=0 x(k) (0) k! tk. In particular, when α ∈ (1,2) , ( Iα C 0+ Dα 0+ x ) (t) = x(t) − x(0) − x′ (0)t. Definition 2.4. A function x ∈ CM (J,L) is a mild solution of the problem (1.1) if x satisfies the corresponding integral equation of (1.1). From Lemma 2.3, we deduce the following lemma. Lemma 2.5. Let x ∈ CM (J,L) is a mild solution of (1.1) if x satisfies x(t) = 1 Γ (α) ∫ t 0 (t − s) α−1 f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ds, t ∈ J. (2.2) Lemma 2.6 ([19]). If ϕ,ψ ∈ CM (J,L), then ∥ ∥ ∥ ϕ[m] − ψ[m] ∥ ∥ ∥ ≤ m−1 ∑ j=0 Mj ‖ϕ − ψ‖ , m = 1,2, . . . Theorem 2.7 (Schauder fixed point theorem [17]). Let M be a nonempty compact convex subset of a Banach space (B,‖·‖) and A : M → M is a continuous mapping. Then A has a fixed point. 3 Main results In this section, we use Theorem 2.7 to prove the existence of mild solutions for (1.1). Moreover, we will introduce the sufficient conditions of the uniqueness of mild solutions of (1.1). To transform (2.2) to be applicable to the Schauder fixed point, we define an operator A : CM (J,L) → C (J,R) by (Aϕ)(t) = 1 Γ (α) ∫ t 0 (t − s) α−1 f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ds, t ∈ J. (3.1) Since CM (J,L) is a compact set as a uniformly bounded, equicontinuous and closed subset of the space C (J,R). To prove that operator A has at least one fixed point, we will prove that A is well defined, continuous and A(CM (J,L)) ⊂ CM (J,L), i. e. Aϕ ∈ CM (J,L) for all ϕ ∈ CM (J,L) . Lemma 3.1. Suppose that (2.1) holds. Then the operator A : CM (J,L) → C (J,R) given by (3.1) is well defined and continuous. CUBO 24, 1 (2022) Existence, uniqueness, continuous dependence and Ulam stability... 87 Proof. Let A be defined by (3.1). Clearly, A is well defined. To show the continuity of A. Let ϕ,ψ ∈ CM (J,L), we have |(Aϕ)(t) − (Aψ)(t)| ≤ 1 Γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) −f ( ψ[0] (s) ,ψ[1] (s) ,ψ[2] (s) , . . . ,ψ[n] (s) ) ∣ ∣ ∣ ds. By (2.1), we obtain |(Aϕ)(t) − (Aψ)(t)| ≤ 1 Γ (α) ∫ t 0 (t − s) α−1 n ∑ i=1 ci ∥ ∥ ∥ ϕ[i] − ψ[i] ∥ ∥ ∥ ds. It follows from Lemma 2.6 that |(Aϕ)(t) − (Aψ)(t)| ≤ 1 Γ (α) ∫ t 0 (t − s) α−1 n ∑ i=1 ci i−1 ∑ j=0 Mj ‖ϕ − ψ‖ds ≤ T α Γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 Mj ‖ϕ − ψ‖ , which proves that the operator A is continuous. Lemma 3.2. Suppose that (2.1) holds. If ζT α Γ (α + 1) ≤ L, (3.2) and ζT α−1 Γ (α) ≤ M, (3.3) then A(CM (J,L)) ⊂ CM (J,L). Proof. For ϕ ∈ CM (J,L), we get |(Aϕ)(t)| ≤ 1 Γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ ds. But ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ = ∣ ∣ ∣ f ( s,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) − f (s,0,0, . . . ,0) + f (s,0,0, . . . ,0) ∣ ∣ ∣ ≤ ∣ ∣ ∣ f ( s,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) − f (s,0,0, . . . ,0) ∣ ∣ ∣ + |f (s,0,0, . . . ,0)| ≤ ρ + n ∑ i=1 ci i−1 ∑ j=0 Mj ‖ϕ‖ ≤ ρ + L n ∑ i=1 ci i−1 ∑ j=0 Mj = ζ, 88 A. Guerfi & A. Ardjouni CUBO 24, 1 (2022) then |(Aϕ)(t)| ≤ ζ Γ (α) ∫ t 0 (t − s) α−1 ds ≤ ζT α Γ (α + 1) ≤ L. From (3.2), we have 0 ≤ (Aϕ)(t) ≤ |(Aϕ)(t)| ≤ L. Let t1, t2 ∈ J with t1 < t2, we have |(Aϕ) (t2) − (Aϕ) (t1)| ≤ 1 Γ (α) ∫ t1 0 ∣ ∣ ∣ (t2 − s) α−1 − (t1 − s) α−1 ∣ ∣ ∣ ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ ds + 1 Γ (α) ∫ t2 t1 (t2 − s) α−1 ∣ ∣ ∣ f ( ϕ[0] (s) ,ϕ[1] (s) ,ϕ[2] (s) , . . . ,ϕ[n] (s) ) ∣ ∣ ∣ ds ≤ ζ Γ (α) ( ∫ t1 0 ( (t2 − s) α−1 − (t1 − s) α−1 ) ds + ∫ t2 t1 (t2 − s) α−1 ds ) ≤ ζ Γ (α + 1) (tα2 − t α 1 ) ≤ ζT α−1 Γ (α) |t2 − t1| . Using (3.3), we obtain |(Aϕ) (t2) − (Aϕ) (t1)| ≤ M |t2 − t1| . Therefore, Aϕ ∈ CM (J,L) for all ϕ ∈ CM (J,L). So, we conclude that A(CM (J,L)) ⊂ CM (J,L). Theorem 3.3. Suppose that conditions (2.1), (3.2) and (3.3) hold. Then (1.1) has at least one mild solution x in CM (J,L). Proof. From Lemma 2.5, the problem (1.1) has a mild solution x on CM (J,L) if and only if the operator A defined by (3.1) has a fixed point. From Lemmas 3.1 and 3.2, all conditions of the Schauder fixed point theorem are satisfied. Consequently, A has at least one fixed point on CM (J,L) which is a mild solution of (1.1). Theorem 3.4. In addition to the assumptions of Theorem 3.3, if we suppose that T α Γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 Mj < 1, (3.4) then (1.1) has a unique mild solution in CM (J,L). Proof. Let ϕ and ψ be two distinct fixed points of the operator A. Similarly as in the proof of Lemma 3.1 we have |ϕ(t) − ψ (t)| = |(Aϕ) (t) − (Aψ) (t)| ≤ T α Γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 Mj ‖ϕ − ψ‖ . CUBO 24, 1 (2022) Existence, uniqueness, continuous dependence and Ulam stability... 89 It follows from (3.4) that ‖ϕ − ψ‖ < ‖ϕ − ψ‖ . Therefore, we arrive at a contradiction. We conclude that A has a unique fixed point which is the unique mild solution of (1.1). Theorem 3.5. Suppose that the conditions of Theorem 3.4 hold. The unique mild solution of (1.1) depends continuously on the function f. Proof. Let f1,f2 : J × R n → [0,+∞) two continuous functions with respect to their arguments. From Theorem 3.4, it follows that there exist two unique corresponding functions x1 and x2 in CM (J,L) such that x1 (t) = 1 Γ (α) ∫ t 0 (t − s) α−1 f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ds, and x2 (t) = 1 Γ (α) ∫ t 0 (t − s) α−1 f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) ds. We get |x2 (t) − x1 (t)| ≤ 1 Γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ ds. But ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ = ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) − f2 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) + f2 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ . Using (2.1) and Lemma 2.6, we arrive at ∣ ∣ ∣ f2 ( x [0] 2 (s) ,x [1] 2 (s) ,x [2] 2 (s) , . . . ,x [n] 2 (s) ) −f1 ( x [0] 1 (s) ,x [1] 1 (s) ,x [2] 1 (s) , . . . ,x [n] 1 (s) ) ∣ ∣ ∣ ≤ ‖f2 − f1‖ + n ∑ i=1 ci i−1 ∑ j=0 Mj ‖x2 − x1‖ . Hence ‖x2 − x1‖ ≤ T α Γ (α + 1) ‖f2 − f1‖ + T α Γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 Mj ‖x2 − x1‖ . Therefore ‖x2 − x1‖ ≤ T α Γ(α+1) 1 − T α Γ(α+1) n ∑ i=1 ci i−1 ∑ j=0 Mj ‖f2 − f1‖ . This completes the proof. 90 A. Guerfi & A. Ardjouni CUBO 24, 1 (2022) Now, we investigate the Ulam-Hyers stability and generalized Ulam-Hyers stability for the problem (1.1). Definition 3.6 ([18]). The problem (1.1) is said to be Ulam-Hyers stable if there exists a real number Kf > 0 such that for each ǫ > 0 and for each mild solution y ∈ CM (J,L) of the inequality ∣ ∣ ∣ CDα0+y (t) − f ( y[0] (t) ,y[1] (t) ,y[2] (t) , . . . ,y[n] (t) ) ∣ ∣ ∣ ≤ ǫ, t ∈ J, (3.5) with y (0) = y′ (0) = 0, there exists a mild solution x ∈ CM (J,L) of the problem (1.1) with |y (t) − x(t)| ≤ Kfǫ, t ∈ J. Definition 3.7 ([18]). The problem (1.1) is generalized Ulam-Hyers stable if there exists ψ ∈ C (J,R+) with ψ (0) = 0 such that for each ǫ > 0 and for each mild solution y ∈ CM (J,L) of the inequality (3.5) with y (0) = y′ (0) = 0, there exists a mild solution x ∈ CM (J,L) of the problem (1.1) with |y (t) − x(t)| ≤ ψ (ǫ) , t ∈ J. Theorem 3.8. Assume that the assumptions of Theorem 3.4 hold. Then the problem (1.1) is Ulam-Hyers stable. Proof. Let y ∈ CM (J,L) be a mild solution of the inequality (3.5) with y (0) = y ′ (0) = 0, i.e.    ∣ ∣ CDα 0+ y (t) − f ( y[0] (t) ,y[1] (t) ,y[2] (t) , . . . ,y[n] (t) ) ∣ ∣ ≤ ǫ, t ∈ J, y (0) = y′ (0) = 0. (3.6) Let us denote by x ∈ CM (J,L) the unique mild solution of the problem (1.1). By using Lemma 2.5, we get x(t) = 1 Γ (α) ∫ t 0 (t − s) α−1 f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ds, t ∈ J. By integration of (3.6), we have ∣ ∣ ∣ ∣ y (t) − 1 Γ (α) ∫ t 0 (t − s) α−1 f ( y[0] (s) ,y[1] (s) ,y[2] (s) , . . . ,y[n] (s) ) ds ∣ ∣ ∣ ∣ ≤ tα Γ (α + 1) ǫ ≤ T α Γ (α + 1) ǫ. On the other hand, we obtain, for each t ∈ J |y (t) − x(t)| = ∣ ∣ ∣ ∣ y (t) − 1 Γ (α) ∫ t 0 (t − s) α−1 f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ds ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ y (t) − 1 Γ (α) ∫ t 0 (t − s) α−1 f ( y[0] (s) ,y[1] (s) ,y[2] (s) , . . . ,y[n] (s) ) ds ∣ ∣ ∣ ∣ + 1 Γ (α) ∫ t 0 (t − s) α−1 ∣ ∣ ∣ f ( y[0] (s) ,y[1] (s) ,y[2] (s) , . . . ,y[n] (s) ) −f ( x[0] (s) ,x[1] (s) ,x[2] (s) , . . . ,x[n] (s) ) ∣ ∣ ∣ ds ≤ T α Γ (α + 1) ǫ + T α Γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 Mj ‖y − x‖ . CUBO 24, 1 (2022) Existence, uniqueness, continuous dependence and Ulam stability... 91 Thus, in view of (3.4) ‖y − x‖ ≤ T α Γ(α+1) 1 − T α Γ(α+1) n ∑ i=1 ci i−1 ∑ j=0 Mj ǫ. Then, there exists a real number Kf = T α/ ( Γ (α + 1) − T α n ∑ i=1 ci i−1 ∑ j=0 Mj ) > 0 such that |y (t) − x(t)| ≤ Kfǫ, t ∈ J. (3.7) Thus, the problem (1.1) is Ulam-Hyers stable, which completes the proof. Corollary 3.9. Suppose that all the assumptions of Theorem 3.8 are satisfied. Then the problem (1.1) is generalized Ulam-Hyers stable. Proof. Let ψ (ǫ) = Kfǫ in (3.7) then ψ (0) = 0 and the problem (1.1) is generalized Ulam-Hyers stable. Example 3.10. Let us consider the following nonlinear fractional initial value problem    CD 3 2 0+ x(t) = 1 4 + 1 4 cost + 1 18 cos2 (t)x[1] (t) + 1 19 sin2 (t) x[2] (t) , t ∈ [0,1] , x(0) = x′ (0) = 0, (3.8) where T = 1, J = [0,1] and f (t,x,y) = 1 4 + 1 4 cost + 1 18 xcos2 (t) + 1 19 y sin2 (t) . We have |f (t,x1,x2) − f (t,y1,y2)| ≤ 1 18 |x1 − y1| + 1 19 |x2 − y2| , then |f (t,x1,x2) − f (t,y1,y2)| ≤ 2 ∑ i=1 ci ‖xi − yi‖ . with c1 = 1 18 , c2 = 1 19 . Furthermore, if L = 1 and M = 4 in the definition of CM (J,L), then f is positive, ρ = sup t∈J {f (t,0,0)} = 1 2 and ζ = 0.5 + ( 1 18 + 4 19 ) ≃ 0.766. For α = 3 2 , we get ζT α Γ (α + 1) = 0.766 Γ ( 5 2 ) ≃ 0.576 ≤ L = 1, and ζT α−1 Γ (α) = 0.766 Γ ( 3 2 ) ≃ 0.864 ≤ M = 4. So, T α Γ (α + 1) n ∑ i=1 ci i−1 ∑ j=0 Mj = 1 Γ ( 5 2 ) ( 1 18 + 4 19 ) ≃ 0.2 < 1. Then, by Theorems 3.4 and 3.5, (3.8) has a unique mild solution which depends continuously on the function f. Also, from Theorem 3.8, (3.8) is Ulam-Hyers stable, and from Corollary 3.9, (3.8) is generalized Ulam-Hyers stable. 92 A. Guerfi & A. Ardjouni CUBO 24, 1 (2022) 4 Conclusion In the current paper, under some sufficient conditions on the nonlinearity, we established the existence, uniqueness, continuous dependence and Ulam stability of a mild solution for an iterative Caputo fractional differential equation. The main tool of this work is the Schauder fixed point theorem. The obtained results have a contribution to the related literature. Acknowledgement The authors would like to thank the anonymous referee for his/her valuable comments and good advice. CUBO 24, 1 (2022) Existence, uniqueness, continuous dependence and Ulam stability... 93 References [1] S. Abbas, “Existence of solutions to fractional order ordinary and delay differential equations and applications”, Electron. J. Differential Equations, no. 9, 11 pages, 2011. [2] A. A. Amer and M. Darus, “An application of univalent solutions to fractional Volterra equa- tion in complex plane”, Transylv. J. Math. Mech., vol. 4, no. 1, pp. 9–14, 2012. [3] M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay”, J. Math. Anal. Appl., vol. 338, no. 2, pp. 1340–1350, 2008. [4] H. Boulares, A. Ardjouni and Y. Laskri, “Existence and uniqueness of solutions to fractional order nonlinear neutral differential equations”, Appl. Math. E-Notes, vol. 18, pp. 25–33, 2018. [5] S. Cheraiet, A. Bouakkaz and R. Khemis, “Bounded positive solutions of an iterative three- point boundary-value problem with integral boundary conditions”, J. Appl. Math. Comput., vol. 65, no. 1-2, pp. 597–610, 2021. [6] K. Diethelm, “Fractional differential equations, theory and numerical treatment”, TU Braun- schweig, Braunschweig, 2003. [7] A. M. A. El-Sayed, “Fractional order evolution equations”, J. Fract. Calc., vol. 7, pp. 89–100, 1995. [8] C. Giannantoni, “The problem of the initial conditions and their physical meaning in linear differential equations of fractional order”, Appl. Math. Comput., vol. 141, no. 1, pp. 87–102, 2003. [9] A. A. Hamoud, “Uniqueness and stability results for Caputo fractional Volterra-Fredholm integro-differential equations”, Zh. Sib. Fed. Univ. Mat. Fiz., vol. 14, no. 3, pp. 313–325, 2021. [10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Amsterdam: Elsevier Science B. V., 2006. [11] J. T. Machado, V. Kiryakova and F. Mainardi, “Recent history of fractional calculus”, Com- mun. Nonlinear Sci. Numer. Simul., vol. 16, no. 3, pp. 1140–1153, 2011. [12] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: John Wiley & Sons, Inc., 1993. [13] K. B. Oldham and J. Spanier, The fractional calculus, Mathematics in Science and Engineer- ing, vol. 111, New York-London: Academic Press, 1974. 94 A. Guerfi & A. Ardjouni CUBO 24, 1 (2022) [14] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, San Diego, CA: Academic Press, Inc., 1999. [15] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu and R. Magin, “Generalized fractional order Bloch equation with extended delay”, Internat. J. Bifur. Chaos, vol. 22, no. 4, 1250071, 15 pages, 2012. [16] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, Yverdon: Gordon and Breach Science Publishers, 1993. [17] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, London-New York: Cambridge University Press, 1974. [18] J. Wang, L. Lv and Y. Zhou, “Ulam stability and data dependence for fractional differential equations with Caputo derivative”, Electron. J. Qual. Theory Differ. Equ., no. 63, 10 pages, 2011. [19] H. Y. Zhao and J. Liu, “Periodic solutions of an iterative functional differential equation with variable coefficients”, Math. Methods Appl. Sci., vol. 40, no. 1, pp. 286–292, 2017. Introduction Preliminaries Main results Conclusion