Terminal Value Problem for Differential Equations with Hilfer–Katugampola Fractional Derivative

: We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical ﬁxed point theory such as the Banach contraction principle and Krasnoselskii’s ﬁxed point theorem. We illustrate our main ﬁndings, with a particular case example included to show the applicability of our outcomes.


Introduction
Recently, by means of different tools from nonlinear analysis, many classes of differential equations with Caputo fractional derivative have extensively been studied in books [1][2][3][4][5] and in some papers, for example, [6][7][8][9][10][11]. In order to solve fractional differential equations, we mention the works [12,13] where the authors propose and prove the equivalence between an initial value problem and the Volterra integral equation.
It is well known [22] that the comparison principle for initial value problems of ordinary differential equations is a very useful tool in the study of qualitative and quantitative theory. Recently, attempts have been made to study the corresponding comparison principle for terminal value problems (TVP) [23].
Motivated by the works above, we establish in this paper existence and uniqueness results to the terminal value problem of the following Hilfer-Katugampola type fractional differential equation: ρ D α,β a + y (t) = f t, y(t), ρ D α,β a + y (t) , for each , t ∈ (a, T], a > 0 (1) y(T) = c ∈ R (2) where ρ D α,β a + is the Hilfer-Katugampola fractional derivative (to be defined below) of order α ∈ (0, 1) and type β ∈ [0, 1] and f : (a, T] × R × R → R is a given function. To our knowledge, no papers on terminal value problem for implicit fractional differential equations exist in the literature, in particular for those involving the Hilfer-Katugampola fractional derivative. This paper is organized as follows. In Section 2, some notations are introduced and we recall some concepts of preliminaries about Hilfer-Katugampola fractional derivative. In Section 3, two results for Equations (1) and (2) are presented: The first one is based on the Banach contraction principle, the second one on Krasnoselskii's fixed point theorem. Finally, in Section 4, we give an example to show the applicability of our main results.

Preliminaries
In this part, we present notations and definitions that we will use throughout this paper. Let 0 < a < T, J = [a, T]. By C(J, R) we denote the Banach space of all continuous functions from J into R with the norm: We consider the weighted spaces of continuous functions: and: with the norms: and: In particular, when

Theorem 3 ([27]
). (Krasnoselskii's fixed point theorem). Let M be a closed, convex and nonempty subset of a Banach space X, and A, B be the operators such that: A is compact and continuous 3.
B is a contraction mapping Then there exists z ∈ M such that z = Az + Bz.
As a consequence of Theorem 4, we have Theorem 5.

Now, we state and prove our existence result for Equations
then the Equations (1) and (2) has unique solution in C Proof. The proof will be given in two steps: Step 1: We show that the operator N defined in Equation (13) has a unique fixed point y * in C 1−γ,ρ (J). Let y, u ∈ C 1−γ,ρ (J) and t ∈ (a, T], then, we have: where g, h ∈ C 1−γ,ρ (J) such that: By (H2), we have: Hence, for each t ∈ (a, T]: By Lemma 3, we have: which implies that: By Equation (14), the operator N is a contraction. Hence, by Banach's contraction principle, N has a unique fixed point y * ∈ C 1−γ,ρ (J).
We present now the second result, which is based on Krasnoselskii fixed point theorem.
then Equations (1) and (2) have at least one solution.
Proof. Consider the set: We define the operators P and Q on B η * by: Then the fractional integral Equation (13) can be written as the operator equation: The proof will be given in several steps: Step 1: We prove that Py + Qu ∈ B η * for any y, u ∈ B η * . For operator P, multiplying both sides of By (H3), we have for each t ∈ (a, T]: Multiplying both sides of the above inequality by t ρ − a ρ ρ 1−γ , we get: Then, for each t ∈ (a, T], we have: Thus, Equation (18) and Lemma 3, imply: This gives: Using Equation (19) and Lemma 3, we have: Thus: Linking Equations (20) and (21), for every y, u ∈ B η * we obtain: Step 2: P is a contraction. Let y, u ∈ C 1−γ,ρ (J) and t ∈ (a, T]; then, we have: where g, h ∈ C 1−γ,ρ (J) such that: By (H2), we have: Then, Therefore, for each t ∈ (a, T]: By Lemma 3, we have: which implies that: By Equation (15) the operator P is a contraction.
Step 3: Q is compact and continuous. The continuity of Q follows from the continuity of f . Next we prove that Q is uniformly bounded on B η * .
Let any u ∈ B η * . Then by Equation (21) we have: This means that Q is uniformly bounded on B η * . Next, we show that QB η * is equicontinuous. Let any u ∈ B η * and 0 < a < τ 1 < τ 2 ≤ T. Then: Note that: This shows that Q is equicontinuous on J. Therefore, Q is relatively compact on B η * . By C 1−γ , type Arzela-Ascoli Theorem Q is compact on B η * .
As a consequence of Krasnoselskii's fixed point theorem, we conclude that N has at least a fixed point y * ∈ C 1−γ,ρ (J) and by the same way of the proof of Theorem 6, we can easily show that y * ∈ C γ 1−γ,ρ (J). Using Lemma 5, we conclude that Equations (1) and (2) have at least one solution in the space C γ 1−γ,ρ (J).
The condition: is satisfied with with T = 2 and a = 1. It follows from Theorem 7 that Equations (22) and (23)

Conclusions
We have provided sufficient conditions ensuring the existence and uniqueness of solutions to a class of terminal value problem for differential equations with the Hilfer-Katugampola type fractional derivative. The arguments are based on the classical Banach contraction principle, and the Krasnoselskii's fixed point theorem. An example is included to show the applicability of our results.