Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

In the past years, fractional differential equations have attracted a lot of attention from many research studies as they have played a key role in many basic sciences such as chemistry, control theory, biology, and other arenas [1–3]. In addition, boundary conditions of differential models are the strongest tools to extend applications of those equations [4–6]. In fact, fractional differential equations can be extended by creating different types of boundary conditions. Newly, many authors have studied various types of boundary conditions to obtain new results of differential models. *e following hybrid differential equation was studied by Dhage and Lakshmikantham [7]:


Introduction
In the past years, fractional differential equations have attracted a lot of attention from many research studies as they have played a key role in many basic sciences such as chemistry, control theory, biology, and other arenas [1][2][3]. In addition, boundary conditions of differential models are the strongest tools to extend applications of those equations [4][5][6]. In fact, fractional differential equations can be extended by creating different types of boundary conditions. Newly, many authors have studied various types of boundary conditions to obtain new results of differential models. e following hybrid differential equation was studied by Dhage and Lakshmikantham [7]: d dt
In the present paper, we combine mixed fractional derivatives and hybrid fractional differential equations. More precisely, we investigate the existence of solutions for the following hybrid boundary value problem which contains both left Caputo and right Riemann-Liouville fractional derivatives and integrals and nonlocal hybrid conditions of the form: where c D , and δ, θ ∈ R. An existence result is obtained via a new extension of Darbo's theorem associated with measures of noncompactness. e structure of the paper has been organized as follows. Section 2 presents some basic definitions and lemmas which will be applied in the future. In Section 3, we prove an existence result for problem (4). Finally, we present an example to illustrate the obtained result.

Preliminaries
Now, some basic notations are recalled from [2].

Definition 1.
For an integrable function ϕ: (0, ∞) ⟶ R, we define the left and right Riemann-Liouville fractional integrals of order β > 0, respectively, by Definition 2. For the function ϕ: (0, ∞) ⟶ R in which ϕ ∈ C n (0, ∞), we define the left Riemann-Liouville fractional derivative and the right Caputo fractional derivative of order β ∈ (n − 1, n], respectively, by Lemma 1. If p > 0 and q > 0, then the following relations hold almost everywhere on [a, b]: As the technique of measure of noncompactness will be applied to obtain our main result, we recall some basic facts about the notion of measure of noncompactness. Assume that Z is the real Banach space with the norm ‖ · ‖ and zero element θ. For a nonempty subset X of Z, the closure and the closed convex hull of X will be denoted by X and Conv(X), respectively. Also, M Z and N Z denote the family of all nonempty and bounded subsets of Z and its subfamily consisting of all relatively compact sets, respectively.
Definition 3 (see [32]). We say that a mapping h: M Z ⟶ [0, ∞) is a measure of noncompactness, if the following conditions hold true: In [33], some generalizations of Darbo's theorem have been proved by Samadi and Ghaemi. Also, in [34], Darbo's theorem was extended, and the following result was presented which is basis for our main result.

Theorem 1. Let T be a continuous self-mapping operator on the set D,
where D denotes a nonempty, bounded, closed, and convex subset of a Banach space Z. Assume that, for all nonempty subset X of D, we have where h is an arbitrary measure of noncompactness defined in Z and (θ 1 , θ 2 ) ∈ U. en, T has a fixed point in D.
In eorem 1, let U indicate the set of all pairs (θ 1 , θ 2 ) where the following conditions hold true: for each strictly increasing sequence t n (U 2 ) θ 2 is strictly increasing function (U 3 ) If α n be a sequence of positive numbers, then lim n⟶∞ α n � 0 ⇔ lim n⟶∞ θ 2 (α n ) � − ∞ (U 4 ) Let l n be a decreasing sequence in which l n ⟶ 0 and θ 1 (l n ) < θ 2 (l n ) − θ 2 (l n+1 ), then we have ∞ n�1 l n < ∞ Next, the definition of a measure of noncompactness in the space C([0, 1]) is recalled which will be applied later. Fix 1] , and for ε > 0 and y ∈ Y, we define Banas and Goebel [32] Lemma 2 (see [32]). e measure of noncompactness φ 0 on C(I) satisfies the following condition: for all X, Y⊆C(I).

Main Existence Result
In this section, an existence result of problem (4) is investigated. In view of [13], Lemma 2, we present the following lemma which is an essential tool in our consideration.
, and Λ ≠ 0. en, the solution of the problem has the form: Journal of Mathematics where Now, the hypotheses which will be applied to prove the main result of this section are presented.
{ } is a continuous function, and there exists a positive real number d > 0 provided that where t ∈ I and x 1 , where has a positive solution r 0 . Also, assume that where Proof. Due to Lemma 3, assume that the operator T has been defined on C(I), I ≔ [0, 1] as follows: where G 1 u(t) � g(t, u(t)), First, we show that T 1 u ∈ C(I) in which u ∈ C(I). In view of assumption (H 1 ), we conclude that G 1 u ∈ C(I), u ∈ C(I). Consequently, by proving F 1 u, F 2 u ∈ C(I), the claim is obtained. Let l n be a sequence in [0, 1] such that l n ⟶ l. en, due to our assumptions, we get 4 Journal of Mathematics Hence, F 1 u ∈ C(I). To obtain that F 1 u ∈ C(I), by the definitions of a 1 and a 2 , we have |a 1 (l n ) − a 1 (l)| ⟶ 0 and Now, we prove that the ball D r 0 � u ∈ C(I): ‖u‖ ≤ r 0 is mapped into itself by the operator T. Let us fix u ∈ C(I).
Hence, due to existence assumptions, for t ∈ I, we have Consequently, according to assumption (H 3 ) we conclude that T maps the ball D r 0 into itself. Now, the continuity property of the operator T is considered on the ball D r 0 . To do this, fix ε > 0 and take u, v ∈ D r 0 such that ‖u − v‖ ≤ ε. en, for t ∈ I, we have Journal of Mathematics en, we have Consequently, the continuity property of T is obtained on the ball D r 0 .

Conclusion
We have studied a nonlocal hybrid boundary value problem which contains both left Caputo and right Riemann-Liouville fractional derivatives and integrals and nonlocal hybrid conditions. An existence result is proved by applying a new generalization of Darbo's fixed point theorem associated with measures of noncompactness. e result obtained in this paper is new and significantly contributes to the existing literature on the topic.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.