Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses

Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, finance, hydrology, electromagnetics, and viscoelasticity. There are numerous books and articles focused on linear and nonlinear initial and boundary value problems for fractional differential equations involving different kinds of fractional derivatives, see, for example, [1–6]. Impulsive fractional differential equations have been considered by many authors (see, for instance, [7–12]). Recent results involving different fractional derivatives can be found in [13–21] and the references therein. Ulam was the first who raise the concept of stability of functional equations [22]. In 1941, Hyers [23] provided the first answer to Ulam’s question. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [24] was able to make a remarkable generalization of UlamHyers stability of mappings by considering variables. Considerable attention has been given to the study of the Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of functional equations; one can see the monograph of Abbas et al. [3] and the paper by Rus [25] who discussed the Ulam-Hyers stability for operational equations (see also [26–29]). Recently, in [30], Harikrishnan et al. investigated existence theory and different kinds of stability in the sense of Ulam, for the following initial value problem with nonlinear generalized Hilfer-type fractional differential equation and impulses:

Ulam was the first who raise the concept of stability of functional equations [22]. In 1941, Hyers [23] provided the first answer to Ulam's question. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [24] was able to make a remarkable generalization of Ulam-Hyers stability of mappings by considering variables.
Considerable attention has been given to the study of the Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of functional equations; one can see the monograph of Abbas et al. [3] and the paper by Rus [25] who discussed the Ulam-Hyers stability for operational equations (see also [26][27][28][29]).
The present paper is organized as follows. In Section 2, some notations are introduced and we recall some preliminaries about generalized Hilfer fractional derivative and auxiliary results. In Section 3, three results for problems (2)-(4) are presented which are based on the Banach contraction principle and Krasnoselskii's and Schaefer's fixed-point theorems. In Section 4, we discuss the Ulam-Hyers-Rassias stability for problems (2)-(4). Finally, we give examples to illustrate the applicability of our main results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let 0 < a < b, J = ½a, b. By C, we denote the Banach space of all continuous functions from J into ℝ with the norm We consider the weighted spaces of continuous functions with the norms Consider the Banach space PCðJÞ = fu : ða, b → ℝ : u ðtÞ ∈ CðJkÞ, k = 0, ⋯, m, and there exist uðt − k Þ and ð ρ J with the norm Consider the space X p c ða, bÞðc ∈ ℝ, 1 ≤ p≤∞Þ of those complex-valued Lebesgue measurable functions f on ½a, b for which k f k X p c < ∞, where the norm is defined by In particular, when c = 1/p, the space X p c ða, bÞ coincides with the L p ða, bÞ space: X p 1/p ða, bÞ = L p ða, bÞ.
Abstract and Applied Analysis PC γ,ρ ðJÞ. Then, u ∈ PC γ γ,ρ ðJÞ is a solution of the differential equation: if and only if u satisfies the following Volterra integral equation: where γ = α + β − αβ.
Lemma 20. implies that If t ∈ J 1 , then Lemma 13 implies If t ∈ J 2 , then Lemma 13 implies Repeating the process in this way, the solution uðtÞ for t ∈ J k , k = 1, ⋯, m, can be written as Applying ρ J 1−γ t + m on both sides of (39), using Lemma 6 and taking t = b, we obtain Abstract and Applied Analysis Multiplying both sides of (40) by c 2 and using condition (31), we obtain which implies that Substituting (42) into (39) and (36), we obtain (34). Reciprocally, applying ρ J 1−γ t + k on both sides of (34) and using Lemma 6 and Theorem 3, we get Next, taking the limit t → a + of (43) and using Lemma 5, Now, taking t = b in (43), we get From (44) and (45), we find that which shows that the boundary condition c 1 ð ρ J 1−γ a + uÞða + Þ + c 2 ð ρ J 1−γ t + m uÞðbÞ = c 3 is satisfied. Next, apply operator ρ D γ t + k on both sides of (34), where k = 0, ⋯, m. Then, from Lemma 6 and Lemma 12, we obtain Since u ∈ PC γ γ,ρ ðJÞ and by definition of PC γ γ,ρ ðJÞ, we have As ψð·Þ ∈ C γ,ρ ðJÞ and from Lemma 4, it follows that From (48) and (49) and by the definition of the space PC n γ,ρ ðJÞ:, we obtain Applying operator ρ J βð1−αÞ t + k on both sides of (47) and using Lemma 8, Lemma 5, and Property 10, we have Abstract and Applied Analysis that is, (29) holds. Also, we can easily show that This completes the proof. As a consequence of Theorem 19, we have the following result.
Proof. The proof will be given in two steps.
Step 1. We show that the operator Ψ defined in (53) has a unique fixed point u * in PC γ,ρ ðJÞ. Let u, w ∈ PC γ,ρ ðJÞ and t ∈ ða, b; then, we have where h, g ∈ C γ,ρ ðJÞ such that By (H2), we have 7 Abstract and Applied Analysis Then, Therefore, for each t ∈ ða, b, Thus, By Lemma 6, we have Hence, which implies that By (61), the operator Ψ is a contraction. Hence, by Theorem 14, Ψ has a unique fixed point u * ∈ PC γ,ρ ðJÞ.
Step 2. We show that such a fixed point u * ∈ PC γ,ρ ðJÞ is actually in PC γ γ,ρ ðJÞ. Since u * is the unique fixed point of operator Ψ in PC γ,ρ ðJÞ, then for each t ∈ J k , with k = 0, ⋯, m, we have where h ∈ C γ,ρ ðJÞ such that Applying ρ D γ t + k to both sides and by Lemma 6 and Lemma 12, we get Since γ ≥ α, by (H1), the right-hand side is in PC γ,ρ ðJÞ and thus, ρ D γ t + k u * ∈ PC γ,ρ ðJÞ which implies that u * ∈ PC γ γ,ρ ðJÞ.

Abstract and Applied Analysis
As a consequence of Steps 1 and 2 together with Theorem 22, we can conclude that problems ( (2))-( (4)) have a unique solution in PC γ γ,ρ ðJÞ. Our second result is based on Schaefer's fixed-point theorem.
Proof. We shall use Schaefer's fixed-point theorem to prove in several steps that the operator Ψ defined in (53) has a fixed point.
Step 1. Ψ is continuous. Let fu n g be a sequence such that u n → u in PC γ,ρ ðJÞ. Then, for each t ∈ ða, b, we have where h n , h ∈ C γ,ρ ðJÞ such that Since u n → u, then we get h n ðtÞ → hðtÞ as n → ∞, for each t ∈ ða, b, and since f and L k are continuous, then we have Step 2. We show that Ψ maps bounded set into bounded set of PC γ,ρ ðJÞ. For η > 0, there exists a positive constant r such that for u ∈ B η = fu ∈ PC γ,ρ ðJÞ: kuk PC γ,ρ ≤ ηg, we have kΨðuÞk PC γ,ρ ≤ r.
By (H4) and from (53), we have for each t ∈ J k , k = 0, ⋯, m, which implies that Then, Thus, (53) implies Abstract and Applied Analysis By Lemma 6, we have Step 3. Ψ maps bounded sets into equicontinuous sets of PC γ,ρ .
Step 4. A priori bound. Now it remains to show that the set is bounded. Let u ∈ G, then u = λ * ΨðuÞ for some 0 < λ * < 1: By (H4), we have for each t ∈ J which implies that This implies, by (53) and (H5) and by letting the estimation of Step 2, that for each t ∈ J, we have where k = 0, ⋯, m, and h : ða, b → ℝ is a function satisfying the functional equation By Lemma 6, we have This means that Q 2 is uniformly bounded on B η . Next, we show that Q 2 B η is equicontinuous. Let any w ∈ B η and a < ϵ 1 < ϵ 2 ≤ b. Then, Note that This shows that Q 2 B η is equicontinuous. Therefore, Q 2 B η is relatively compact. By PC γ -type Arzela-Ascoli theorem, Q 2 is compact. As a consequence of Theorem 16, we deduce that Ψ has at least a fixed point u * ∈ PC γ,ρ ðJÞ and by the same way of the proof of Theorem 22, we can easily show that u * ∈ P C γ,ρ ðJÞ. Using Lemma 21, we conclude that problems ((2))-((4)) have at least one solution in the space PC γ γ,ρ ðJÞ.