Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fr´echet Spaces

Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leﬄer–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coeﬃcient in Banach and Fr´echet Spaces. The techniques used are a variety of tools for functional analysis. More speciﬁcally, we apply Weissinger’s ﬁxed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new ﬁxed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorﬀ measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Gr¨onwall’s inequality, the Mittag–Leﬄer–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s ﬁxed point theorem in Fr´echet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.


Introduction
Fractional differential equations gained much attention due to their applications in various fields of science and engineering (see, for instance, [1][2][3][4][5][6][7] and the references therein). For more information about the basic theory of fractional differential equations, we can refer to the monographs [8][9][10][11] and references cited therein. Besides the classical and fractional-order differential and integral operators, there is another kind of fractional derivatives that appears in the literature called ψ-Caputo fractional derivative, which was introduced by Almeida in [12], where the kernel operator contains a special function of an arbitrary exponent. According to this idea, a wide class of well-known fractional derivatives are obtained like Caputo and Caputo-Hadamard for particular choices of ψ(t). Additionally, some interesting details about the ψ-fractional derivatives and integrals can be found in [13][14][15][16][17][18][19][20][21][22][23]. Moreover, fixed point theory is a very useful tool in the theory of the existence of solutions to functional and differential equations; the reader is advised to see references [24][25][26][27][28][29] in which many scholars turned to the existence and uniqueness of solutions for differential equations involving different kinds of fractional derivatives under various boundary conditions. On the contrary, the notion of measure of noncompactness was first introduced by Kuratowski [30] in 1930 which was further extended to general Banach spaces by Banás and Goebel [31]. Later Darbo formulated his celebrated fixed point theorem in 1955 for the case of the Kuratowski measure of noncompactness (cf. [32]) which generalizes both the classical Schauder fixed point principle and (a special variant of ) Banach's contraction mapping principle. After that, the Darbo fixed point theorem has been generalized in many different directions; we suggest some works for reference [33][34][35][36]. e reader may also consult [37][38][39][40][41][42][43] and references therein where several applications of the measure of noncompactness can be found.
Very recently Dudek [44] proved a new fixed point theorem using the concept of measures of noncompactness in Fréchet spaces which generalize the famous Darbo's fixed point theorem. To see more applications about the usefulness of this new fixed point theorem to prove the existence of solutions for certain classes of functional integral equations in Fréchet spaces, the reader can refer to [45][46][47][48][49][50].
Inspired by the above works, our goal is to extend the studies in [29,37,45,72]. More precisely, we consider first the problem of the existence, uniqueness, and Mittag-Leffler-Ulam-Hyers stability for the following initial value problem of the fractional differential equation with constant coefficient λ > 0 in Banach spaces of the form: where c D α;ψ a + is the ψ-Caputo fractional derivatives such that 0 < α ≤ 1, f: J × X ⟶ X is a given function satisfying some assumptions that will be specified later, X is a Banach space with norm ‖·‖, and ϕ 0 ∈ X. Moreover, we also extend the above problem to give a uniqueness results on unbounded domains in a Banach space via Banach contraction principle coupled with Bielecki-type norm.
Next, we turn our attention to the existence of solutions for the same problem (1) in the Fréchet spaces. In precise terms, we investigate the existence of solutions for the following problem: e structure of the present work is organized as follows: in Section 2, we collect some basic concepts on the fractional integrals and derivatives, auxiliary results, lemmas and notions of measures of noncompactness, and fixed point theorems that are used throughout this paper. In Section 3, based on Weissinger's fixed point theorem combined with the Chebyshev norm, we give a uniqueness result for problem (1) on a compact interval in a Banach space. In Section 4, using the ideas of Hausdorff measure of noncompactness and Meir-Keeler condensing operator, we present the existence of solutions of IVP (1) in Banach spaces. In Section 5, we discuss the Mittag-Leffler-Ulam stability results for the problem at hand. In Section 6, we apply the Banach fixed point theorem coupled with a Bielecki-type norm to derive the uniqueness of solution on unbounded domains in a Banach space. In Section 7, we look into the existence of solutions for the IVP (2) in the Fréchet spaces via Darbo's fixed point theorem. e last section provides a couple of examples to illustrate the applicability of the results developed.

Preliminaries and Background Materials
In this section, we present some basic notations, definitions, and preliminary results, which will be used throughout this paper.
Let J ≔ [a, b](0 < a < b < ∞) be a finite interval and ψ: J ⟶ R be an increasing function with ψ ′ (t) ≠ 0, for all t ∈ J, and let C(J, X) be the Banach space of all continuous functions x from J into X with the supremum (uniform) norm: A measurable function x: J ⟶ X is Bochner integrable if and only if ‖x‖ is Lebesgue integrable.
By L 1 (J, X), we denote the space of the Bochner integrable functions x: J ⟶ X, with the norm Now, we define the Hausdorff measure of noncompactness and give some of its important properties.
Definition 1 (see [31]). Let X be a Banach space and B a bounded subset of X.
en the Hausdorff measure of noncompactness of B is defined by χ(B) � inf ε > 0: B can be covered by finitely many balls with radius < ε .

(5)
To discuss the problem in this paper, we need the following lemmas. Lemma 1. Let A, B ⊂ X be bounded. en the Hausdorff measure of noncompactness has the following properties. For more details and the proof of these properties see [31]: , where A and convA represent the closure and the convex hull of A, respectively (5) International Journal of Differential Equations Now, we recall some fixed point theorems that will be used later Theorem 1 (Weissinger's fixed point theorem [74]). Assume (E, d) to be a nonempty complete metric space and let β j ≥ 0 for every j ∈ N such that ∞ j�0 β j converges. Furthermore, let the mapping T : E ⟶ E satisfy the following inequality: for every j ∈ N and every u, v ∈ E. en, T has a unique fixed point u * . Moreover, for converges to this fixed point u * .
On the contrary, in 1969, the concepts of the Meir-Keeler contraction mapping were introduced by Meir and Keeler.
Definition 2 (see [75]). Let (E, d) be a metric space. en a mapping T on E is said to be a Meir-Keeler contraction (MKC, for short); if for any ε > 0, there exists δ > 0 such that In [34], the authors defined the notion of the Meir-Keeler condensing operator on a Banach space and gave some fixed point results.
Definition 3 (see [34]). Let C be a nonempty subset of a Banach space X and μ arbitrary measure of noncompactness on X. We say that an operator T: C ⟶ C is a Meir-Keeler condensing operator if for any ε > 0, there exists δ > 0 such that for any bounded subset Ω of C. e following fixed point theorem with respect to the Meir-Keeler condensing operator which is introduced by Aghajani et al. [34] plays a key role in the proof of our main results.
Theorem 2 (see [34]). Let Ω be a nonempty, bounded, closed, and convex subset of a Banach space X. Also, let μ be an arbitrary measure of noncompactness on X. If T: Ω ⟶ Ω is a continuous and Meir-Keeler condensing operator, then T has at least one fixed point and the set of all fixed points of T in Ω is compact.
The following lemmas are needed in our argument.
, and χ C is the Hausdorff measure of noncompactness defined on the bounded sets of C(J, X). Furthermore if B is equicontinuous, then t ⟶ χ(B(t)) is continuous on J, and Lemma 3 (see [77]). Let X be a Banach space and let B ⊂ X be bounded. en for each ε, there is a sequence x n ∞ n�1 ⊂ B, such that We call B ⊂ L 1 (J, X) uniformly integrable if there exists η ∈ L 1 (J, R + ) such that ‖x(t)‖ ≤ η(t), for all x ∈ B and a.e. t ∈ J. (11) Lemma 4 (see [78]).
Before introducing the basic facts on fractional operators, we recall three types of functions that are important in fractional calculus: the gamma, beta, and Mittag-Leffler functions Definition 4 (see [79]). e gamma function, or the secondorder Euler integral, denoted Γ(·) is defined as Definition 5 (see [79]). e beta function, or the first-order Euler function, can be defined as We use the following formula which expresses the beta function in terms of the gamma function: e next function is a direct generalization of the exponential series.
International Journal of Differential Equations Now, we give some results and properties from the theory of fractional calculus. We begin by defining ψ-Riemann-Liouville fractional integrals and derivatives, in what follows.
Definition 8 (see [2,12]). For α > 0, the left-sided ψ-Riemann-Liouville fractional integral of order α for an integrable function x: J ⟶ R with respect to another function ψ: J ⟶ R that is an increasing differentiable function such that ψ ′ (t) ≠ 0, for all t ∈ J, is defined as follows: (19) where Γ is the gamma function.
Note that equation (19) is reduced to the Riemann-Liouville and Hadamard fractional integrals when ψ(t) � t and ψ(t) � ln t, respectively. e integer order of the differential operator x [1] ψ with respect to another function ψ: J ⟶ R that is an increasing differentiable function such that ψ ′ (t) ≠ 0, for all t ∈ J is defined by Furthermore, for n ∈ N, we use the symbol x [n] ψ to indicate the n-th composition of x [1] ψ with itself; that is, we put Definition 9 (see [12]). Let n ∈ N and let ψ, x ∈ C n (J, R), be two functions such that ψ is increasing and ψ ′ (t) ≠ 0, for all t ∈ J. e left-sided ψ-Riemann-Liouville fractional derivative of a function x of order α is defined by where n � [α] + 1.
Definition 10 (see [12]). Let n ∈ N and let ψ, x ∈ C n (J, R), be two functions such that ψ is increasing and ψ ′ (t) ≠ 0, for all t ∈ J. e left-sided ψ-Caputo fractional derivative of x of order α is defined by where n � [α] + 1 for α ∉ N and n � α for α ∈ N. From the definition, it is clear that is generalization (24) yields the Caputo fractional derivative operator when ψ(t) � t. Moreover, for ψ(t) � ln t, it gives the Caputo-Hadamard fractional derivative.
Some basic properties are listed in the following lemma.
Lemma 5 (see [2,12]). Let α, β > 0, and x ∈ C(J, R). en for each t ∈ J, we have Note that for an abstract function x: J ⟶ X, the integrals which appear in the previous definitions are taken in Bochner's sense (see, for instance, [80]).
In the sequel, we will make use of the following generalizations of Grönwall's lemmas Theorem 3 (see [23]). Let u, v be two integrable functions and w continuous, with domain J. Let ψ ∈ C 1 (J, R + ) be an increasing function such that ψ ′ (t) ≠ 0, ∀t ∈ J. Assume that (1) u and v are nonnegative (2) w is nonnegative and nondecreasing. If Corollary 1 (see [23]). Under the hypotheses of eorem 3, let v be a nondecreasing function on J. en, we have where E α (·) is a Mittag-Leffler function with one parameter.

International Journal of Differential Equations
Lemma 6 (see [14]). Let α, β > 0. en for all t ∈ J, we have Remark 2. Observe that from Lemma 6 if β � 1, we can get the following inequality: For the existence of solutions for the problem (1), we need the following lemma.
if and only if it is the solution of the integral equation: Proof. Let x(t) be a solution of the problem (30). Define taking the ψ-Riemann-Liouville fractional integral of order α to the above equation, we get Since we get Using the definition of h(t), we obtain equation (31). Conversely, suppose that x(t) is the solution of the equation (31). en, it can be written as where h(t) � λx(t) + f(t, x(t)). Since h(t) is continuous and ϕ 0 is constant, operating the ψ-Caputo fractional differential operator c D α;ψ a + on both sides of equation (37), we obtain Using Lemma 5, the following is obtained: From equation (37), we get x(a) � ϕ 0 . is proves that x(t) is the solution of the Cauchy problem (30) which completes the proof. Now, we are ready to present our main results.

Uniqueness Result with respect to the Chebyshev Norm and Weissinger's Fixed Point Theorem
First of all, we define what we mean by a solution of equation (1).
x(t)) on J and the condition x(a) � ϕ 0 .
for any x, y ∈ X and t ∈ J.
Then there exists a unique solution of equation (1) on J.
Proof. In view of Lemma 7, we introduce an operator T: C(J, X) ⟶ C(J, X) associated with equation (1) as follows: Clearly, the fixed points of the operator T are solutions of equation (1). Weissinger's fixed point theorem will be used to prove that T has a fixed point. For this reason, we shall show that T is a contraction. Let x, y ∈ C(J, X). en, for every n ∈ N and t ∈ J, using (H2), we have International Journal of Differential Equations erefore, we conclude for each n ∈ N and all x, y ∈ C(J, X). Now let By Definition (6), we have erefore, the existence of the unique fixed point of T follows from Weissinger's fixed point theorem. at is, (1) fd1 has a unique solution. is completes the proof.

Existence Result via Meir-Keeler Condensing Operators
In this section, we can weaken the condition (H2) to a linear growth condition. But now eorem 2 that we apply will only guarantee the existence not also the uniqueness of the solution.
Theorem 5. Assume that the hypothesis (H1) holds. Furthermore, we impose the following: (H3) ere exist continuous functions μ, ]: J ⟶ R + such that for any x ∈ X and t ∈ J.
(H4) For each bounded set B ⊂ X, and each t ∈ J, the following inequality holds: Then the problem (1) has at least one solution defined on J provided that 4 ] * + λ ℓ α,ψ < 1, where Proof. Consider the operator T defined by equation (41) and define a bounded closed convex set with We shall show that the operator T satisfies all the assumptions of eorem 2. We split the proof into four steps: Step 1. e operator T maps the set Ω r into itself. By the assumption (H3), we have 6 International Journal of Differential Equations us is proves that T transforms the ball Ω r into itself.
Step 2. e operator T is continuous. Suppose that x n is a sequence such that x n ⟶ x in Ω r as n ⟶ ∞. It is easy to see that f(s, x n (s)) ⟶ f(s, x(s)), as n ⟶ + ∞ due to the continuity of f. On the contrary, taking (H3) into consideration, we get the following inequality: We notice that since the function s ⟼ 2(μ(s)+ is fact together with the Lebesgue dominated convergence theorem implies that It follows that ‖Tx n − Tx‖ ∞ ⟶ 0 as n ⟶ + ∞, which implies the continuity of the operator T.
Step 4. Now, we prove that T: Ω r ⟶ Ω r is a Meir-Keeler condensing operator. To do this, suppose ε > 0 is given. We will prove that there exists δ > 0 such that For every bounded and equicontinuous subset B ⊂ Ω r and ε ′ > 0 using Lemma 3 and the properties of χ, there exist sequences x n ∞ n�1 ⊂ B such that Next, by Lemma 4 and (H4), we have As the last inequality is true, for every ε ′ > 0, we infer Since T(B) ⊂ Ω r is bounded and equicontinuous, we know from Lemma 3 that erefore, we have Observe that from the last estimates Let us now take so we get which means that T: Ω r ⟶ Ω r is a Meir-Keeler condensing operator. It follows from eorem 2 that the operator T defined by (41) has at least one fixed point x ∈ Ω r , which is just the solution of the initial value problem (1). is completes the proof of eorem 5.

Mittag-Leffler-Ulam-Hyers Stability Analysis
In this section, we discuss the Mittag-Leffler-Ulam-Hyers stability analysis of the solutions to the proposed problem (1). Now, we consider the Mittag-Leffler-Ulam-Hyers stability for problem (1). 8 International Journal of Differential Equations Let ε, λ > 0 and Φ: J ⟶ R + be a continuous function. We consider the following inequalities: Definition 12 (see [71,73]). Equation (1) is Mittag-Leffler-Ulam-Hyers stable, with respect to E α ((ψ(t) − ψ(a)) α ) if there exists a real number c > 0 such that for each ε > 0 and for each solution y ∈ C(J, X) of the inequality (67), there exists a solution x ∈ C(J, X) of equation (1) with Definition 13 (see [71,73]). Equation (1) is generalized Mittag-Leffler-Ulam-Hyers stable, with respect to with ω(0) � 0 such that for each ε > 0 and for each solution y ∈ C(J, X) of the inequality (67), there exists a solution x ∈ C(J, X) of equation (1) with Remark 3 (see [71,73]). It is clear that Definition 12 ⟹ Definition 13, Remark 4 (see [71,73]). A function y ∈ C(J, X) is a solution of the inequality (67) if and only if there exists a function z ∈ C(J, X) (which depends on solution y) such that , t ∈ J Now we are ready to state our Mittag-Leffler-Ulam-Hyers stability of solution to the problem (1). e arguments are based on the Grönwall inequality equation (27).

Theorem 6 Assume that (H1) and (H2) hold. en problem (1) is Mittag-Leffler-Ulam-Hyers stable on J and consequently generalized Mittag-Leffler-Ulam-Hyers stable.
Proof. Let ε, λ > 0 and let y ∈ C(J, X) be a function which satisfies the inequality (67) and let x ∈ C(J, X) be the unique solution of the following problem: By Lemma 7, we have Since we have assumed that y is a solution of the inequality (67), we have the following by Remark 4: Again by Lemma 7, we have On the contrary, we have, for each t ∈ J, Hence, using Remark 2 and part (i) of Remark 4 and (H2), we can get Applying Corollary 1 (the Grönwall inequality equation (27)) to the above inequality with u(t) � ‖y(t)− x(t)‖, v(t) � εE α ((ψ(t) − ψ(a)) α ), and w(t) � L + λ/Γ(α).
Since v(t) is a nondecreasing function on J, we conclude that which yields Taking for simplicity International Journal of Differential Equations then (78) becomes In consequence, it follows that us, the problem (1) is Mittag-Leffler-Ulam-Hyers stable. Furthermore, if we set ω(ε) � cε; ω(0) � 0, then the problem (1) is generalized Mittag-Leffler-Ulam-Hyers stable. is completes the proof.
for all t ∈ J ′ and each x, y ∈ X.
Proof. Consider the Banach space C(J ′ , X) equipped with a Bielecki norm type ‖·‖ B defined as below: where β > 0 will be chosen later and E α (·) is the Mittag-Leffler function which is given in Definition 6. (for more properties on the Bielecki-type norm, see [25,28]). Consider the operator A: where Now, we prove that the operator A is a contraction mapping on C(J ′ , X) with respect to the Bielecki norm. To this end, we apply the Banach fixed point theorem to prove that A has a fixed point. Given x, y ∈ C(J ′ , X) and t ∈ J ′ , using the inequality (82), and Lemma 6, we can get Hence, we have Note that E α (·) is a monotone increasing function on J ′ , then we get Since we can choose β > 0 sufficiently large such that it follows that the mapping A is a contraction with respect to the Bielecki norm. Hence, by the Banach fixed point theorem, A has a unique fixed point which is a unique solution of the initial value problem (1) in the space C(J ′ , X). is completes the proof.

An Existence Result in Fréchet Spaces via Darbo's Fixed Point Theorem
By using Darbo's fixed point theorem, we give in this section our last existence theorem concerning the IVP (2) in the Fréchet spaces. Firstly, we need to fix the notation. Let J ′ � [a, +∞) and let J n ≔ [a, n], n ∈ N * .
In this section, we let E ≔ C(J ′ , X) to be the Fréchet space of all continuous functions x from J ′ into X, equipped with the family of seminorms: Next we present some facts concerning the notion of a sequence of measures of noncompactness in the Fréchet spaces [44,50].
In what follows, F will be a real Fréchet space. If B is a nonempty subset of F, then B and ConvB denote the closure and the closed convex closure of B, respectively. Also, we denote by m F the family of all nonempty and bounded subsets of F and n F the family of all relatively compact subsets of F. (1) e family Kerμ n � B ∈ m F ; μ n (B) � 0, for n ∈ N is nonempty and Kerμ n ∈ n F (2) , and lim i⟶∞ μ n (B i ) � 0 for each n ∈ N, then B ∞ ≔ ∩ ∞ i�1 B i ≠ ∅ e following lemmas are needed in our argument.
for each t ∈ J n . Definition 16. Let Ω be a nonempty subset of a Fréchet space F, and let A: Ω ⟶ F be a continuous operator which transforms bounded subsets of Ω onto bounded ones. One says that A satisfies the Darbo condition with constants k n n∈N * with respect to a family of measures of noncompactness μ n n∈N * , if for each bounded set B ⊂ Ω and n ∈ N * . If k n < 1, n ∈ N * , then A is called a contraction with respect to μ n n∈N * . e following generalization of the classical Darbo fixed point theorem for the Fréchet spaces plays a key role in the proof of our main results. Theorem 8. (see [44]). Let Ω be a nonempty, bounded, closed, and convex subset of a Fréchet space F and let A: Ω ⟶ Ω be a continuous mapping. Suppose that A is a contraction with respect to a family of measures of noncompactness μ n n∈N * . en A has at least one fixed point in the set Ω.

Definition 17.
A function x ∈ C(J ′ , X) is said to be a solution of equation (2) , x(t)) on J ′ , and the condition x(a) � ϕ 0 . Now, we shall prove the following theorem concerning the existence of solutions of problem (2).
Theorem 9. Let f: J ′ × X ⟶ X be a continuous function such that the following assumptions hold: (H5) ere exists a continuous functions p: J ′ ⟶ R + such that ‖f(t, x)‖ ≤ p(t)(1 +‖x‖), for all t ∈ J ′ , and each x ∈ X.
(95) (H6) For each bounded set D ⊂ X and for each t ∈ J ′ , we have where μ is a measure of noncompactness on the Banach space X.
For n ∈ N * , let Define on E the family of measures of noncompactness by where D is a bounded and equicontinuous set of E, and for each n ∈ N * , then the problem (2) has at least one solution.
Proof. Consider the operator A defined by (84), but in the Fréchet space E ≔ C(J ′ , X). Note that, the fixed points of the operator A are solutions of the problems (2). For any n ∈ N * , let R n be a positive real number with and we consider the ball International Journal of Differential Equations Notice that B R n is closed, convex, and bounded subset of the Fréchet space E. We shall show that the operator A satisfies all the assumptions of eorem 8. We split the proof into three steps.
Step 1. e operator A maps the set B R n into itself. For any n ∈ N * , and each x ∈ B R n and t ∈ J n , by (H5), we have is proves that A transforms the ball B R n into itself.
Step 2. e operator A is continuous. Suppose that x k k∈N is a sequence such that x k ⟶ x in B R n as k ⟶ ∞. en for each t ∈ J n , we have Since x k ⟶ x as k ⟶ + ∞, the Lebesgue dominated convergence theorem implies that which implies the continuity of the operator A.
Step 3. Our aim in this step is to show that A is μ n contraction on B R n . For every bounded equicontinuous subset D ⊂ B R n and ε > 0 using Lemma 8, there exist sequences x k ∞ k�1 ⊂ D such that for all t ∈ J n , we have Next, by Lemma 9 and (H6), we have As the last inequality is true, for every ε > 0, we infer us μ n (A(D)) ≤ 4 p * n + λ ℓ n,ψ μ n (D).
Using the condition (99), we claim that A is a k n contraction on B R n . It follows from eorem 8 that the operator A defined by (84) has at least one fixed point x ∈ B R n , which is just the solution of initial value problem (2). is completes the proof of eorem 9.

Examples
In this section, we give a couple of examples to illustrate the usefulness of our main result. Let be the Banach space of real sequences converging to zero, endowed its usual norm 12 International Journal of Differential Equations Example 1. Consider the following initial value problem of a fractional differential posed in c 0 : u(1) � 0.5, 0.25, . . . , 0.5 n , . . . .
and E ≔ C[0, +∞), X be the Fréchet space of all continuous functions x from [0, +∞) into X, equipped with the family of seminorms: In order to illustrate eorem 9, we take ψ(t) � σ(t) where σ(t) is the sigmoid function [19] which can be expressed as in the following form: and a convenience of the sigmoid function is its derivative: Taking also f: [0, +∞) × c 0 ⟶ c 0 given by for t ∈ [0, +∞), x � x k k≥1 ∈ c 0 .

Data Availability
ere are no data used in this work.

Conflicts of Interest
e authors declare they have no conflict of interest.