Nonlocal initial value problems for implicit differential equations with Hilfer – Hadamard fractional derivative

Fractional differential equations (FDEs) have been applied in many fields such as physics, mechanics, chemistry, engineering etc. There has been a significant development in ordinary differential equations involving fractional-order derivatives, one can see the monographs of Hilfer [19], Kilbas [16] and Podlubny [18] and the references therein. Moreover, Hilfer [19] studied applications of a generalized fractional operator having the Riemann– Liouville and the Caputo derivatives as specific cases. Hilfer fractional derivative has been receiving more and more attention in recent times; see, for example, [8–11, 14, 21, 24]. Benchohra et al. [4, 5] studied implicit differential equations(IDEs) of fractional order in various aspects. Recently, some mathematicians have considered FDEs depending on the Hadamard fractional derivative [2, 6, 7].


Introduction
Fractional differential equations (FDEs) have been applied in many fields such as physics, mechanics, chemistry, engineering etc.There has been a significant development in ordinary differential equations involving fractional-order derivatives, one can see the monographs of Hilfer [19], Kilbas [16] and Podlubny [18] and the references therein.Moreover, Hilfer [19] studied applications of a generalized fractional operator having the Riemann-Liouville and the Caputo derivatives as specific cases.Hilfer fractional derivative has been receiving more and more attention in recent times; see, for example, [8-11, 14, 21, 24].Benchohra et al. [4,5] studied implicit differential equations(IDEs) of fractional order in various aspects.Recently, some mathematicians have considered FDEs depending on the Hadamard fractional derivative [2,6,7].

D. Vivek et al.
In this paper, we consider the Hilfer-Hadamard-type IDE with nonlocal condition of the form where H D α,β 1 + is the Hilfer-Hadamard fractional derivative of order α and type β.Let X be a Banach space, f : J × X × X → X is a given continuous function and H I 1−γ 1 + is the left-sided mixed Hadamard integral of order 1 − γ.
In passing, we remark that the application of nonlocal condition H I 1−γ 1 + x(1) = m i=1 c i x(τ i ) in physical problems yields better effect than the initial condition H I 1−γ 1 + x(1) = x 0 .For sake of brevity, let us take A new and important equivalent mixed-type integral equation for our system (1) can be established.We adopt some ideas in [24] to establish an equivalent mixed-type integral equation where In the theory of functional equations, there is some special kind of data dependence [3,13,17,20].For the advanced contribution on Ulam stability for FDEs, we refer the reader to [12,23,25,26].In this paper, we study different types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for the IDEs with Hilfer-Hadamard fractional derivative.Moreover, the Ulam-Hyers stability for FDEs with Hilfer fractional derivative was investigated in [1,22].
The paper is organized as follows.A brief review of the fractional calculus theory is given in Section 2. In Section 3, we will prove the existence and uniqueness of solutions for problem (1).In Section 4, we discuss the Ulam-Hyers stability results.Finally, an example is given in Section 5 to illustrate the usefulness of our main results.

Fundamental concepts
In this section, we introduce some definitions and preliminary facts, which are used in this paper.
Let C[J, X] be the Banach space of all continuous functions from J into X with the norm where C γ,log [J, X] is the weighted space of the continuous functions f on the finite interval [1, b].
Obviously, C γ,log [J, X] is the Banach space with the norm , n ∈ N.
Remark 1.Clearly, (i) The operator H D α,β 1 + also can be rewritten as and In particular, if β = 1 and α 0, then the Hadamard fractional derivative of a constant is not equal to zero: T ] there exist the following properties:

Existence results
In this section, we introduce spaces that helps us to solve and reduce system (1) to an equivalent integral equation ( 2): It is obvious that ] is a solution of Hilfer-type fractional IDE: if and only if x satisfies the following Volterra integral equation: Further details can be found in [15].From Lemma 7 we have the following result.
] is a solution of system (1) if and only if x satisfies the mixed-type integral (2).
Proof.According to Lemma 7, a solution of system (1) can be expressed by Next, we substitute t = τ i into the above equation: multiplying both sides of (4) by c i , we can write Thus, we have which implies Submitting ( 5) to (3), we derive that (2).It is probative that x is also a solution of the integral equation ( 2) when x is a solution of (1).The necessity has been already proved.Next, we are ready to prove its sufficiency.Applying H I 1−γ 1 + to both sides of (2), we have using Lemmas 1 and 2, , Lemma 4 can be used when taking the limit as t → 1: https://www.mii.vu.lt/NASubstituting t = τ i into (2), we have Then we derive It follows ( 6) and ( 7) that Now by applying H D γ 1 + to both sides of (2), it follows from Lemmas 1 and 5 that Since x ∈ C γ 1−γ,log [J, X] and by the definition of C γ 1−γ,log [J, X], we have that f satisfy the conditions of Lemma 3.
Next, by applying H I to both sides of (8) and using Lemma 3, we can obtain where (I Hence, it reduces to ).The results are proved completely.
First, we list the following hypotheses to study the existence and uniqueness results: (H3) There exist positive constants K, L > 0 such that The existence result for problem (1) will be proved by using the Schaefer's fixed-point theorem.
Theorem 1. Assume that (H1) and (H2) are satisfied.Then system (1) has at least one solution in For sake of clarity, we split the proof into a sequence of steps.
Consider the operator N : It is obvious that the operator N is well defined.
Step 1. N is continuous.Let x n be a sequence such that x n → x in C 1−γ,log [J, X].Then for each t ∈ J, Since K x is continuous (i.e., f is continuous), then we have Step 2. N maps bounded sets into bounded sets in Indeed, it is enough to show that η > 0, there exists a positive constant l such that For computational work, we set and by (H2), We estimate A 1 , A 2 terms separately.By (10) we have Bringing inequalities (11) and ( 12) into ( 9), we have Step 3. N maps bounded sets into equicontinuous set of Using the fact f is bounded on the compact set J × B η (thus sup (t,x)∈J×Bη K x (t) := C 0 < ∞), we will get As t 1 → t 2 , the right-hand side of the above inequality tends to zero.As a consequence of Steps 1-3 together with Arzela-Ascoli theorem, we can conclude that N : is continuous and completely continuous.
Step 4. A priori bounds.Now it remains to show that the set ) for some 0 < δ < 1.Thus, for each t ∈ J, we have This implies by (H2) that for each t ∈ J, we have This shows that the set ω is bounded.As a consequence of Schaefer's fixed-point theorem, we deduce that N has a fixed point, which is a solution of system (1).The proof is completed.
Our second theorem is based on the Banach contraction principle.
Proof.Let the operator N : By Lemma 8 it is clear that the fixed points of N are solutions of system (1).Let x 1 , x 2 ∈ C 1−γ,log [J, X] and t ∈ J, then we have and By replacing (15) in inequality (14) we get Hence, From ( 13) it follows that N has a unique fixed point, which is solution of system (1).The proof of Theorem 2 is completed.
Definition 4. Equation ( 1) is Ulam-Hyers stable if there exists a real number C f > 0 such that for each > 0 and for each solution z ∈ C γ 1−γ,log [J, X] of the inequality there exists a solution Definition 5. Equation ( 1) is generalized Ulam-Hyers stable if there exists ψ https://www.mii.vu.lt/NADefinition 6. Equation ( 1) is Ulam-Hyers-Rassias stable with respect to ϕ ∈ C 1−γ,log [J, X] if there exists a real number C f > 0 such that for each > 0 and for each solution z ∈ C γ 1−γ,log [J, X] of the inequality there exists a solution Definition 7. Equation ( 1) is generalized Ulam-Hyers-Rassias stable with respect to ϕ ∈ there exists a solution  One can have similar remarks for inequalities (17) and (18).Lemma 9. Let 0 < α < 1, 0 β 1.If a function z ∈ C γ 1−γ,log [J, X] is a solution of inequality (16), then x is a solution of the following integral inequality: Thus, system (1) is generalized Ulam-Hyers-Rassias stable.The proof is completed.

An example
As an application of our results, we consider the following problem of Hilfer-Hadamard IDE: e −(log t) (9 + e log t ) Notice that this problem is a particular case of (1), where α = 2/3, β = 1/2, and choose γ = 5/6.Set f (t, u, v) = e −(log t) (9 + e log t ) for any u, v ∈ X.
Clearly, the function f satisfies the condition of Theorem 1.
For any u, v, u, v ∈ X and t ∈ J, f (t, u, v) − f (t, u, v)