Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations


 In this paper, we prove some existence results of solutions for a class of nonlocal initial value problem for nonlinear fractional hybrid implicit differential equations under generalized Hilfer fractional derivative. The result is based on a fixed point theorem on Banach algebras. Further, examples are provided to illustrate our results.


Introduction
Fractional calculus is a branch of classical mathematics, which is concerned with the generalization of the integer order di erentiation and integration of a function to non-integer order, its is a solid and growing eld both in theory and in its applications [2-4, 15, 32]. In the last few decades, fractional di erentiation and fractional integration have found many applications in various elds of science and engineering. There are numerous kinds of fractional derivatives, such as, Riemann-Liouville fractional derivative, Caputo fractional derivative, Hilfer fractional derivative, Hadamard fractional derivative, Erdélyi-Kober, Katugampola and others; see [1, 5-8, 10, 11, 14, 21, 26-29] and the references therein. For some recent applications, see [22,23,25].
Another interesting class of problems involves hybrid fractional di erential equations and has received attention of several researchers [9,12,19,31].
In [13], the authors discussed the following terminal value problem for fractional di erential equations with generalized Hilfer fractional derivative : Their reasoning is mainly based upon di erent types of classical xed point theory such as the Banach contraction principle and the Krasnoselskii xed point theorem.
Using Krasnoselskii, Schaefer and Schauder xed point theorems, Wang and Zhang [30] proved some existence results for the following nonlocal initial value problem for di erential equations involving Hilfer fractional derivative : a + u(t) = f (t, u(t)), t ∈ (a, b], Derbazi et al. [16] studied the existence and uniqueness of solutions of the following three-point boundary value problem for fractional hybrid di erential equations with Caputo fractional derivative : The proved results rely on a hybrid xed point theorem for a sum of three operators due to Dhage. Motivated by the works of the papers mentioned above, we establish in this paper, existence results to the nonlocal initial value problem (IVP for short) with nonlinear implicit hybrid generalized Hilfer type fractional di erential equation : where α D ϑ,r a + , α J −ξ a + are the generalized Hilfer fractional derivative of order ϑ ∈ ( , ) and type r ∈ [ , ] and generalized fractional integral of order − ξ , (ξ = ϑ + r − ϑr) respectively, c i , i = , . . . , m, are real numbers, ϵ i , i = , . . . , m, are pre-xed points satisfying a < ϵ ≤ . . .
. Further details and de nitions are given in Section 2.
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, an existence result for the problem (1)-(2) is presented which is based on a xed point theorem in Banach algebras [17,18]. Finally, in the last section, we give an example to illustrate the applicability of our results.
with the norm Consider the space X p c (a, b), (c ∈ R, ≤ p ≤ ∞) of those complex-valued Lebesgue measurable functions f on [a, b] for which f X p c < ∞, where the norm is de ned by In particular, when c = p , the space X p c (a, b) coincides with the L p (a, b) space: De nition 2.1. [20] Let ϑ ∈ R+, c ∈ R and h ∈ X p c (a, b). The generalized fractional integral of order ϑ is de ned by De nition 2.2. [20] Let ϑ ∈ R+ \ N and α > . The generalized fractional derivative α D ϑ a + of order ϑ is de ned by where n = [ϑ] + and δ n α = t −α d dt n . Lemma 2.3. [20,24] Let ϑ > , ≤ ξ < . Then, α J ϑ a + is bounded from C ξ ,α (J) into C ξ ,α (J).
Consider the following parameters ϑ, r, ξ satisfying We de ne the spaces a + x.
Lemma 2.9. [17] Let B be a closed, convex, bounded and nonempty subset of a Banach algebra (X, · ), and let P, R : X → X and Q : B → X be three operators such that 1) P and R are Lipschitzian with Lipschitz constants η and η , respectively, 2) Q is compact and continuous, Then the operator equation PxQx + Rx = x has a solution in B.

Existence of Solutions
Let , and the function χ ∈ C(J × R, R). We consider the following simpler fractional di erential equation related to (1)- where < ϑ < , ≤ r ≤ , α > , with the nonlocal condition ≠ . The following theorem shows that the problem (3)-(4) have a solution given by Theorem 3.1. The function x satis es equations (3) and (4) if and only if it satis es (5).
Proof. Assume x satis es the equations (3) and (4) and such that the function σ : ∈ C ξ ξ ,α (J). We prove that x is a solution to the equation (5). From the de nition of the space C ξ ξ ,α (J) and by using Lemma 2.3 and De nition 2.2, we have Using Lemma 2.8 we have Then, which implies that Next, we substitute t = ϵ i into (6), then we multiply c i to both sides, we obtain Then by using condition (4), we have which implies Substituting (7) into (6), we obtain (5).
Reciprocally, assume x satis es the equation (5) such that the function σ : ∈ C ξ ξ ,α (J). Applying operator α D ξ a + on both sides of ( ), and since f (t, x(t)) ≠ for all t ∈ J, then, from Lemma 2.8 we obtain Since σ ∈ C ξ ξ ,α (J) we have α D ξ a + σ ∈ C ξ ,α (J), then (8) implies that As v(·) ∈ C ξ ,α (J) and from Lemma 2.3, follows From (9), (10) and by the de nition of the space C n ξ ,α (J), we obtain Applying operator α J r( −ϑ) a + on both sides of (9) and using Lemma 2.5 and Property 2.7, we have that is, (3) holds. Now, applying α J −ξ a + on both sides of ( ) we get Taking the limit t → a + of ( ) we obtain Substituting t = ϵ i into ( ), we have Then, we have From ( ) and ( ), we nd that which shows that the initial condition ( ) is satis ed.
∈ C ξ ξ ,α (J), then x satis es the problem ( ) − ( ) if and only if x is the xed point of the operator : C ξ ,α (J) → C ξ ,α (J) de ned by where K = −
Proof. We de ne a subset Ω of C ξ ,α (J) by We consider the operator de ned in (14), and de ne three operators S, N : and Then we get x = SxTx + Nx.
Step 1: The operators S and N are Lipschitzian on C ξ ,α (J). Let x, y ∈ C ξ ,α (J) and t ∈ (a, b]. Then by (Ax2) we have then for each t ∈ (a, b] we obtain Also, for each t ∈ (a, b] we have Step 2: The operator T is completely continuous on Ω.
We rstly show that the operator T is continuous on Ω. Let {xn} be sequence in Ω such that xn → x in Ω. Let x, y ∈ C ξ ,α (J). Then for each t ∈ (a, b], we have where vn , v ∈ C ξ ,α (J) such that

vn(t) = φ(t, xn(t), vn(t)), v(t) = φ(t, x(t), v(t)).
Since xn → x and φ is a continuous function on J then we get vn(t) → v(t) as n → ∞ for each t ∈ (a, b], so by Lebesgue dominated convergence theorem, we have Then T is continuous. Next we prove that T(Ω) is uniformly bounded on C ξ ,α (J). Let any x ∈ Ω. By (Ax3), we have for each t ∈ (a, b] For t ∈ (a, b], by ( ) we have Then we obtain This prove that the operator T is uniformly bounded on Ω. Next we prove that the operator TΩ is equicontinuous. We take x ∈ Ω and a < ε < ε ≤ b. Then, Then we have for each t ∈ (a, b] This proves that TΩ is equicontinuous on J. Therefore by the Arzelà-Ascoli theorem, T is completely continuous on Ω. Step 3: The third hypothesis of Lemma 2.9 is satis ed. Let x ∈ C ξ ,α (J) and y ∈ Ω be arbitrary such that Then, for t ∈ (a, b] we have then, Then x ∈ Ω, thus the third hypothesis of Lemma 2.9 is satis ed. Step 4: The fourth hypothesis of Lemma 2.9 is satis ed. We show that p * Ψ ξ (b, a)L + q * < , where L = T(Ω) C ξ ,α = sup{ Ty C ξ ,α : y ∈ Ω}.
That is, the last hypothesis of Lemma 2.9 is satis ed. Thus, the operator equation x = SxTx + Nx = x has at least one solution x * ∈ C ξ ,α , which is a xed point for the operator .

Example 4.1. Consider the problem
where I = [ , ], a = , b = and and , t ∈ I, x, y ∈ R.
Same as the last example, it follows by (15) and (16) that the constant satis es the inequality . ≤ < e ( e − ln( ))( Then the problem (23)− (24) has at least one solution in C , (I).