On Fractional Cauchy-type Problems Containing Hilfer's Derivative

In the paper we study fractional systems with generalized Riemann–Liouville derivatives. A theorem on the existence and uniqueness of a solution to a fractional nonlinear ordinary Cauchy problem is proved. Next a formula for the solution to a linear problem of such a type is presented.


Introduction
In our paper we study the following fractional differential equation (D α,β a+ y)(t) = g(t, y(t)), t ∈ [a, b] a.e. (1.1) with the initial condition where 0 a+ denotes the generalized Riemann-Liouville derivative operator introduced by Hilfer in [8].It is easy to see that the D α,β a+ derivative is considered as an interpolator between the Riemann-Liouville and Caputo derivative (cf.[7]).
In paper [7] the existence and uniqueness of a solution to such problem in a some weighted space of continuous functions has been investigated.The main idea of the proof relies on the change of such problem over to the equivalent integral equation and next, using the constructive method based on the Banach fixed point theorem, solving this equation.
We also investigate the question of the existence and uniqueness of a solution to problem (1.1)-(1.2) but in a different space of solutions, namely in the space so called "γ-absolutely continuous functions" denoted by AC γ a+ ([a, b], R n ) (generally this is the space of non-continuous Corresponding author.Email: rafkam@math.uni.lodz.plfunctions).In our opinion such space of solutions is more useful in applications than the space of continuous functions (for example in the fields of control theory or calculus of variations).Similarly as in paper [7] we use the Banach contraction principle and additionally a notion of the Bielecki norm in the space of solutions.Such approach makes the proofs of our results not complicated and rather short.
Detailed description of our method is the following.First we consider a homogeneous problem (with zero initial condition).We prove that such problem is equivalent to integral equation (3.2).Next, in order to prove the existence of a solution to this integral equation, we use mentioned notion of the Bielecki norm in the space and the Banach fixed point theorem.The point of existence of a solution to nonhomogeneous problem reduces to point of existence of a solution to homogeneous problem.
In the second part of this work we consider the linear problem given by where . Using a constructive method, provided by the Banach fixed point theorem, we obtain existence of a solution to such problem under different (less complicated) assumptions than in the case of the nonlinear problem.Moreover we give a formula for this solution.For the linear problem involving the Riemann-Liouville derivative such formula was derived in paper [10].
The paper is organized as follows.Section 2 contains some notions and facts concerning the fractional integrals and derivatives.In section 3, we prove theorems on the existence and uniqueness of a solution to problem (1.1) with zero and nonzero initial conditions (1.2). Results of a such type, for the linear problem (1.3), were obtained in Section 4.Moreover, a formula for the solution to such problem is given.

Preliminaries
In this section we recall some basic definitions and results concerning the fractional calculus, that we will use in the next sections (cf.[7,14,17]).
Let α > 0 and it is natural to put Similarly, we put We have the following semigroup properties (cf.[17, formula 2.21]) The following rule of fractional integration by parts holds (cf.[17, formula 2.20]).
Theorem 2.3.Let α > 0, p ≥ 1, q ≥ 1 and We say that the function f possesses the left-sided Riemann-Liouville derivative D α a+ f of order α, if the function I 1−α a+ f is absolutely continuous on [a, b] and In view of Remark 2.1, we put By I α a+ (L 1 ) we denote the set (cf. [14]) In [17,Theorem 2.3] the following characterization of the space From the above proposition it follows that if f ∈ I α a+ (L 1 ) then f possesses the left-sided Riemann-Liouville derivative D α a+ f = g, where g is the function from the definition of I α a+ (L 1 ).Let us introduce in the space I α a+ (L 1 ) the norm given by We have the following theorem.
Proof.Let (u k ) k∈N ⊂ I α a+ (L 1 ) be a Cauchy sequence.So, From the definition of the norm in the space I α a+ (L 1 ) and a linearity of the operator D α a+ it follows that for m, n > N we have This means that the sequence (D α a+ u k ) k∈N is a Cauchy sequence in the space Let us put u = I α a+ x.
We shall prove the following lemma.
The proof is completed.
We have the following composition properties.
We say that the function f possesses the left-sided generalized Riemann-Liouville derivative (so called Hilfer derivative) D α,β a+ f of order α and type β, if the function The operator D α,β a+ f , given by (2.2), was introduced by Hilfer in [8].We have the following comments (cf.[7,Remark 19]).

The Hilfer derivative D
α,β a+ f can be written as

The D α,β
a+ f derivative is considered as an interpolator between the Riemann-Liouville and Caputo derivative since (cf.Remark 2.1) 3. The parameter γ satisfies Now, we shall prove the following composition properties for the Hilfer derivative.

Cauchy problem
In this section we investigate the problem (1.1)-(1.2).First, we consider it with zero initial condition.We shall prove a theorem on the existence and uniqueness of a solution to such problem.Next, using obtained result, we shall prove the result of a such type for problem (1.1) with nonzero initial condition (1.2).

Homogenous Cauchy problem
Let us consider the following Cauchy problem where 0 By a solution to this problem we shall mean a function x ∈ I γ a+ (L 1 ) satisfying the above equation almost everywhere on [a, b] (from proposition 2.4 it follows that each function belonging to I γ a+ (L 1 ) satisfies the initial condition).We have the following theorem.
Applying the operator D α a+ to both sides of the last equality and using Proposition 2.7 (a) we obtain Hence and from equality (3.3) we conclude that x satisfies equation (3.1).Since x ∈ I γ a+ (L 1 ), therefore the inital condition is satisfied.
The proof is completed.

Remark 3.2. It is easy to verify that the condition
satisfies the Lipschitz condition with respect to the second variable and the function [a, b] Now, we prove the following theorem.
then problem (3.1) possesses a unique solution x ∈ I γ a+ (L 1 ).Proof.Let us consider the operator S : It is easy to check that S is well defined.Now, let us consider in I α a+ (L 1 ) the Bielecki norm given by where k > 0 is a fixed constant.We shall show that S is contractive.Using Proposition 2.7 (b), assumption (2 h ) and Theorem 2.
Let us note that (cf.[10, proof of Since Nk −α ∈ (0, 1) for sufficiently large k, therefore the operator S has a unique fixed point.It means that integral equation (3.2) possesses a unique solution x * ∈ I α a+ (L 1 ).From assumption (1 h ) it follows that there exists a function ).The proof is completed.
By a solution to such problem we shall mean a function y ∈ AC γ a+ ([a, b], R n ), where (cf.[9]) ) is a solution to problem (3.1) with the function h of the form then the function 2) with the function g of the form So, using Theorem 3.3, we can prove the following Proof.In order to prove the existence part of the above theorem it suffices to show that if g satisfies assumptions (1 g ), (2 g ), then the function h given by (3.4) satisfies conditions (1 h ), (2 h ) from Theorem 3.3.Indeed, the fact that h satisfies the Lipschitz condition with respect to the second variable is obvious.Moreover, for any x ∈ I α a+ (L 1 ) we have A uniqueness of the solution to problem (1.1)-(1.2) follows from the uniqueness of the solution to homogeneous problem.
The proof is completed.

Linear Cauchy problem
In the previous section we obtained the existence of a unique solution to nonlinear Cauchy problem (1.1)-(1.2).Similarly as in paper [7] our method relies on the change of such problem over to the equivalent integral equation and next, using the Banach fixed point theorem, solving this equation.The obtained solution belongs to the space AC γ a+ ([a, b], R n ) (generally, in contrast to the paper [7], this is the space of non-continuous functions).An advantage of our paper is the fact that proofs of main results are not complicated and rather short.Unfortunately, the existence results were proved under the key assumption (1 h ) ((1 g )), which generally is difficult to check (except the case β = 0 -cf.Remark 3.2).
In this section we shall consider the linear Cauchy problem of a type (1.1)-(1.2).We shall show that in this case the mentioned assumption reduces to a condition, which is easier to verify.Moreover, we give the formula for a solution to such problem.
In our opinion the obtained results concerning the linear problem are useful in applications -for example in linear control systems involving the Hilfer derivative.

Homogenous problem
Let us consider the following linear Cauchy problem where 0 ), then Lemma 2.6 guarantees satisfying assumption (1 h ) from Theorem 3.3.Consequently, there exists a unique solution x ∈ I γ a+ (L 1 ) to such problem.Now, we shall show that the existence result can be obtained for any 0 < α < 1 and 0 β 1.Indeed, from the proof of Theorem 3.3 it follows that the operator S : has a unique fixed point x * ∈ I α a+ (L 1 ).So there exists a function ϕ From [17, Theorem 2.6] it follows that ) for all m ∈ N.Moreover, there exists m ∈ N such that (m + 1)α γ and δ := (m + 1)α − γ ∈ (0, 1).Consequently Using once again Theorem 2.6 from [17] we assert that A m I (m+1)α a+ ϕ * ∈ I γ a+ (L 1 ).So we showed that all terms of the equality (4.2) belong to the space I γ a+ (L 1 ).Thus and from Theorem 3.1 we conclude that there exists a unique solution x * to problem (4.1) belonging to I γ a+ (L 1 ).
almost everywhere on[a, b],where ϕ ∈ L 1 ([a, b], R n ) is a function such that x = I γ a+ ϕ.Consequently, there exists the derivative D then x is a solution to problem (3.1) if and only if x satisfies Now, let assume that x ∈ I γ a+ (L 1 ) satisfies (3.2).Then there exists the derivative D γ a+ x = ϕ