Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type

This article deals with some existence and Ulam-Hyers-Rassias stability results for a class of functional differential equations involving the Hilfer-Hadamard fractional derivative. An application is made of a Schauder fixed point theorem for the existence of solutions. Next we prove that our problem is generalized Ulam-Hyers-Rassias stable.


Introduction
Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering, and other applied sciences. For some fundamental results in the theory of fractional calculus and fractional ordinary and partial differential equations, we refer the reader to the monographs of Abbas  Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative; see [-]. Motivated by the Hilfer fractional derivative (which interpolates the Riemann-Liouville derivative and the Caputo derivative), Qassim et al. [, ] considered a new type of fractional derivative (which interpolates the Hadamard derivative and its Caputo counterpart). Motivated by the above papers, in this article we discuss the existence and the Ulam stability of solutions for the following problem of Hilfer-Hadamard fractional differential equations of the form The present paper initiates the Ulam stability for differential equations involving the Hilfer-Hadamard fractional derivative.

Preliminaries
Let C be the Banach space of all continuous functions v from I into R with the supremum (uniform) norm As usual, AC(J) denotes the space of absolutely continuous functions from J into R. We denote by AC  (J) the space defined by where [q] is the integer part of q. Define the space Let γ ∈ (, ], by C γ ,ln (J), C γ (J) and C  γ (J), we denote the weighted spaces of continuous functions defined by In the following, we denote w C γ ,ln by w C . Now, we give some results and properties of fractional calculus.
where (·) is the (Euler's) gamma function defined by Notice that for all r, r  , r  >  and each w ∈ C, we have I r  w ∈ C, and Definition . ([-]; Riemann-Liouville fractional derivative) The Riemann-Liouville fractional derivative of order r >  of a function w ∈ L  (J) is defined by where n = [r] +  and [r] is the integer part of r.
In particular, if r ∈ (, ], then Let r ∈ (, ], γ ∈ [, ) and w ∈ C -γ (J). Then the following expression leads to the left inverse operator as follows: , then the following composition is proved in []: In particular, if r ∈ (, ], then Let us recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [, ] for a more detailed analysis. Definition . ([, ]; Hadamard fractional integral) The Hadamard fractional integral of order q >  for a function g ∈ L  (I, E) is defined as provided the integral exists. Set Analogous to the Riemann-Liouville fractional calculus, the Hadamard fractional derivative is defined in terms of the Hadamard fractional integral in the following way.
Definition . ([, ]; Hadamard fractional derivative) The Hadamard fractional derivative of order q >  applied to the function w ∈ AC n δ is defined as In particular, if q ∈ (, ], then It has been proved (see, e.g., Kilbas [], Theorem .) that in the space L  (J) the Hadamard fractional derivative is the left-inverse operator to the Hadamard fractional integral, i.e., Analogous to the Hadamard fractional calculus, the Caputo-Hadamard fractional derivative is defined in the following way.
Definition . (Caputo-Hadamard fractional derivative) The Caputo-Hadamard fractional derivative of order q >  applied to the function w ∈ AC n δ is defined as In particular, if q ∈ (, ], then In [], Hilfer studied applications of a generalized fractional operator having the Riemann-Liouville and the Caputo derivatives as specific cases (see also [-]).
Moreover, the parameter γ satisfies . The generalization () for β =  coincides with the Riemann-Liouville derivative and for β =  with the Caputo derivative.
. If D γ  w exists and in L  (J), then From the Hadamard fractional integral, the Hilfer-Hadamard fractional derivative (introduced for the first time in []) is defined in the following way.
This new fractional derivative () may be viewed as interpolating the Hadamard fractional derivative and the Caputo-Hadamard fractional derivative. Indeed, for β = , this derivative reduces to the Hadamard fractional derivative, and when β = , we recover the Caputo-Hadamard fractional derivative.
From Theorem  in [], we concluded the following lemma.

Lemma . Let f : I × E → E be such that f (·, u(·)) ∈ C γ ,ln (J) for any u ∈ C γ ,ln (J). Then problem () is equivalent to the problem of the solutions of the Volterra integral equation
Now, we consider the Ulam stability for problem (). Let >  and : I → [, ∞) be a continuous function. We consider the following inequalities:

Definition . ([, ]) Problem () is Ulam-Hyers stable if there exists a real number
c f >  such that for each >  and for each solution u ∈ C γ ,ln of inequality () there exists a solution v ∈ C γ ,ln of () with

Definition . ([, ]) Problem () is generalized Ulam-Hyers stable if there exists c f :
) with c f () =  such that for each >  and for each solution u ∈ C γ ,ln of inequality () there exists a solution v ∈ C γ ,ln of () with

Definition . ([, ]) Problem () is Ulam-Hyers-Rassias stable with respect to if
there exists a real number c f , >  such that for each >  and for each solution u ∈ C γ ,ln of inequality () there exists a solution v ∈ C γ ,ln of () with Definition . ([, ]) Problem () is generalized Ulam-Hyers-Rassias stable with respect to if there exists a real number c f , >  such that for each solution u ∈ C γ ,ln of inequality () there exists a solution v ∈ C γ ,ln of () with Remark . It is clear that (i) Definition . ⇒ Definition ., (ii) Definition . ⇒ Definition ., (iii) Definition . for (·) =  ⇒ Definition ..

One can have similar remarks for inequalities () and ().
In the sequel we will make use of the following fixed point theorem.

Existence of solutions
Let us start by defining what we mean by a solution of problem ().
Definition . By a solution of problem () we mean a measurable function u ∈ C γ ,ln that satisfies the condition ( H I The following hypotheses will be used in the sequel. (H  ) The function t → f (t, u) is measurable on I for each u ∈ C γ ,ln , and the function u → f (t, u) is continuous on C γ ,ln for a.e. t ∈ J, (H  ) There exists a continuous function p : |u| for a.e. t ∈ J and each u ∈ R. Set Now, we shall prove the following theorem concerning the existence of solutions of problem ().

Theorem . Assume that hypotheses (H  ) and (H  ) hold. Then problem () has at least one solution defined on J.
Proof Consider the operator N : C γ ,ln → C γ ,ln defined by Clearly, the fixed points of the operator N are solution of problem (). For any u ∈ C γ ,ln and each t ∈ J, we have Thus This proves that N transforms the ball B R := B(, R) = {w ∈ C γ ,ln : w C ≤ R} into itself. We shall show that the operator N : B R → B R satisfies all the assumptions of Theorem .. The proof will be given in several steps.
Step . N : B R → B R is continuous. Let {u n } n∈N be a sequence such that u n → u in B R . Then, for each t ∈ J, we have Since u n → u as n → ∞ and f is continuous, by the Lebesgue dominated convergence theorem, equation () implies Step . N(B R ) is uniformly bounded. This is clear since N(B R ) ⊂ B R and B R is bounded. Step Theorem . Assume that hypotheses (H  ), (H  ) and the following hypotheses hold.
(H  ) There exists λ >  such that for each t ∈ J, we have (H  ) There exists q ∈ C(J, [, ∞)) such that for each t ∈ J, we have Then problem () is generalized Ulam-Hyers-Rassias stable.
Proof Consider the operator N : C γ ,ln → C γ ,ln defined in (). Let u be a solution of inequality (), and let us assume that v is a solution of problem (). Thus, we have From inequality (), for each t ∈ J, we have Set q * = sup t∈J q(t).
In the sequel, we will use the following theorem.
Theorem . Let ( , d) be a generalized complete metric space and : → be a strictly contractive operator with a Lipschitz constant L < . If there exists a nonnegative integer k such that d( k+ x, k x) < ∞ for some x ∈ , then the following propositions hold true: (A) The sequence ( k x) n∈N converges to a fixed point x * of ; (B) x * is the unique fixed point of in * = {y ∈ | d( k x, y) < ∞}; (C) If y ∈ * , then d(y, x * ) ≤  -L d(y, x).
Let X = X(I, R) be the metric space, with the metric Theorem . Assume that (H  ) and the following hypothesis hold.
Hence, we get d N(u), N(v) = sup t∈J (Nu)(t) -(Nv)(t) C (t) ≤ L uv C , from which we conclude the theorem.