A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A ψ -HILFER FRACTIONAL OPERATOR

. Boundary value problems driven by fractional operators has drawn the attention of several researchers in the last decades due to its applicability in several areas of Science and Technology. The suitable deﬁnition of the fractional derivative and its associated spaces is a natural problem that arise on the study of this kind of problem. A manner to avoid of such problem is to consider a general deﬁnition of fractional derivative. The purpose of this manuscript is to contribute, in the mentioned sense, by presenting the ψ − fractional spaces H α,β ; ψ p ([0 , T ] , R ). As an application we study a problem, by using the Mountain Pass Theorem, which includes an wide class of equations.


Introduction
In the last decades the Fractional Calculus has drawn the attention of several researchers due to some advantages with respect to the usual one which occurs for example in problems involving memory, see for instance [10,15,30,32,35]. An important fact is its applicability, see for example Sousa et. al. [38,39], where it is considered the fractional version of a mathematical model that describes, under certain conditions, the blood concetration of nutrients and its relation with the erythrocyte sedimentation. We also quote the references [5,7,11,14,16,17,22,25,26,31,34,43,46] Let Ω ⊂ R N (N ≥ 2) be a bounded domain with smooth boundary. Motivated from the usual Calculus, we have the development of the Sobolev spaces W k,p (Ω) with k ∈ N and p ≥ 1 and its applications. An important one is the Variational approach for the Dirichlet problem where f : Ω × R → R is a Caratheodory function, whose main idea consists in to associate the solutions of the problem above with critical points of C 1 energy functional of the form where H(x, t) = t 0 f (x, ξ)dξ and W 1,2 0 (Ω) := H 1 0 (Ω) denotes the functions of W 1,2 (Ω) that are null on the boundary in the sense of the trace operator. There is a vast literature regarding such subject, thus we only mention some classical ones, see for instance [3,4].
With the wide number of definitions of integrals and fractional derivatives, it is interesting to consider a general notion of fractional derivative of a function f with respect to another function. Such question was recently considered in Sousa & Oliveira [37], where the authors introduced the ψ-Hilfer fractional derivative and exhibited an wide class of examples . Thus from [37] it is natural to construct a suitable space and study its properties to consider, by using a variational approach, the problem where H D α,β;ψ T − (·), H D α,β;ψ 0+ (·) are the right and left ψ-Hilfer fractional derivatives respectively of order α ∈ (1/2, 1] and type 0 ≤ β ≤ 1 and f : [0, T ] × R → R is a function satisfying the conditions : ( In what follows we describe in details the contributions of this work. (i) It is presented a suitable space (denoted by H α,β;ψ p ([0, T ], R)) to study the problem (P ). (ii) Several important results are proved for the space H α,β;ψ p ([0, T ], R) such as completeness, reflexivity and some embeddings. Such properties will be needed to consider a variational approach for (P ). (iii) A notion of weak solution for (P ) is introduced and it is obtained the existence of a weak solution by using the classical Mountain Pass Theorem. To the best of our knowledge it is the first time that a Dirichlet problem with an operator which involves the ψ−Hilfer fractional derivative is studied in the literature. Moreover, the results of [41] are obtained for a larger class of equations. The rest of the paper is organized as follows: Section 2 is devoted to present the fractional Riemann-Lioville integral with respect to another function, the ψ−Hilfer fractional derivative and some results that will be often used. In Section 3 the spaces H α,β;ψ p ([0, T ], R) and examples are presented and several properties of such spaces are proved in Section 4. As an application of the mentioned results, it is proved in Section 5 the existence of solution for (P ) by using the Mountain Pass Theorem.

Preliminaries
Let [a, b] be a finite interval and C[a, b], AC n [a, b], C n [a, b] be the spaces of continuous functions, n−times absolutely continuous functions, n−times continuously differentiable functions on [a, b], respectively.
The space of the continuous functions f on [a, b] with the norm defined by On the order hand, we have n−times absolutely continuous given by The weighted space C γ;ψ [a, b] is defined by with the norm    .2. [36, 37] Consider that ψ (x) = 0 (−∞ ≤ a < x < b ≤ ∞) and α > 0, n ∈ N. The Riemann-Liouville derivatives of a function u with respect to ψ of order α correspondent to the Riemann-Liouville, are defined by , R) two functions such that ψ is increasing and ψ (x) = 0, for all x ∈ I. The left ψ-Caputo fractional derivative of u of order α is given by , R) two functions such that ψ is increasing and ψ (x) = 0, for all x ∈ I. The ψ-Hilfer fractional derivative left-sided and right-sided H D α,β;ψ a+ (·) and H D α,β;ψ b− (·) of function of order α and type 0 ≤ β ≤ 1, are defined by The ψ-Hilfer fractional derivative as above defined, can be written in the following form and In what follows we consider the integration by parts rule for ψ-Riemann-Liouville fractional integral and for the ψ-Hilfer fractional derivative.
By Almeida [1], we know that the relation eq.218 eq.218 is valid. Now we present the integration by parts rule for the ψ-Hilfer fractional derivative, which plays a key role in the variational formulation of problem (P ).
Moreover, c can be characterized as In this section we present the abstract spaces that will be used to study (P ) in the variational framework. Consider then, (3.1) can be rewritten as Motivated by this equality we introduce the following ψ−fractional spaces (·) is the ψ-Hilfer fractional derivative with 0 < α ≤ 1 and 0 ≤ β ≤ 1.
Proof. In fact, since L p ([0, T ] , R) is reflexive and separable, the cartesian product space )) 2 is also reflexive and separable. Consider the space Ω = u, H D α,β;ψ 0+ u : u ∈ H α,β;ψ p which is a closed subset of (L p ([0, T ] , R)) 2 as H α,β;ψ p is closed. Therefore, Ω is also reflexive and separable Banach space with respect to the norm (4.1) for We form the for t ∈ [0, T ].
In particular the embedding H α,β;ψ Below we point out some examples regarding the previous definition.

Fractional nonlinear Dirichlet problem
The goal of this section is to prove the existence of solution for the fractional nonlinear Dirichlet problem (P ). The notion of solution that will be considered is given below.
Consider the energy functional given by where H is the primitive of f , that is, for u, v ∈ H α,β;ψ 2 . Thus, the solutions of (P ) are given by the critical points of A. The result of this section is provided below. t4.1 Theorem 5.1. Let 1/2 < α ≤ 1, 0 ≤ β ≤ 1 and suppose that f satisfy (f 1 ) and (f 2 ). Then problem (P ) has a nontrivial weak solution u ∈ H α,β;ψ 2 .  f (t, u k (t)) u k (t) dt.
Then by (5.4) we get Then by (5.5) and (5.6) we obtain that Since µ > 2 it follows that {u k } is bounded in H α,β;ψ 2 . From Proposition 4.1 we have that H α,β;ψ 2 is reflexive space. Thus going to a subsequence if necessary, we may assume that u k u in H α,β;ψ 2 . Therefore , u k − u . 4.9 (5.7) Taking limit with k → ∞ on both sides of the inequality (5.7) we get From Propositions 4.5 and 4.6, we get that u k is bounded in C ([0, T ]) and we can also assume that lim k→∞ u k − u ∞ = 0.