Existence results to a ψ - Hilfer neutral fractional evolution equation with infinite delay

: In this paper, we prove the existence and uniqueness of a mild solution to the system of ψ - Hilfer neutral fractional evolution equations with infinite delay H D α , β ; ψ [0, b ], b > 0 and x ( t ) = ϕ ( t ), t ∈ (−∞, 0]. We first obtain the Volterra integral equivalent equation and propose the mild solution of the system. Then, we prove the existence and uniqueness of solution by using the Banach contraction mapping principle and the Leray-Schauder alternative theorem.


Introduction
In this paper, we consider the following ψ-Hilfer neutral fractional di erential equations with in nite delays: where H D α,β;ψ + (.) is the ψ-Hilfer fractional derivative of order < α ≤ , with respect to function ψ ∈ L([ , b], X) and type ≤ β ≤ . Also, x(t) ∈ X is the state vector, h(t) is a continuous function where h : [ , b] × P −→ X, t ∈ [ , b] and ϕ(t) ∈ P, where P is the admissible phase space. f : [ , b] × X × P −→ X and f ∈ C is a nonlinear function and x t stands for the history of the state function up to the time t, i.e. x t (θ) = x(t + θ) ∀ θ ∈ (−∞, ]. Let A be an in nitesimal generator of an analytic semigroup {T(t)} t≥ of uniformly bounded linear operators de ned on in nite Banach space X. We denote C = C b ((−∞, b], X) the Banach space of all continuous and bounded functions from (−∞, b] into X endowed with the topology of uniform convergence. and let B(X), · B(X) be the Banach space of all linear and bounded operators from X to X.

Remark 1. Since T(t) t≥ is an analytic semigroup on X, there exists K ≥ such that T(t) ≤ K.
(For more information on semigroups please see [4]).
In recent years, fractional calculus has been used and known as an excellent instrument for the description of hereditary properties of various processes and applications in many elds such as physics, biology, nance, etc. [6,20,24]. In the recent decades, many scienti c researchers have become interested to study and investigate the existence and uniqueness of solution to the system of fractional di erential equations. As one of the special and important problems in fractional calculus, there were some studies on the neutral type di erential equations. Some of the authors proved the existence and uniqueness of the solution to semilinear neutral type fractional di erential equations with in nite delays by using the Banach Contraction mapping theorem such as [19,26]. In [2,14], researchers investigated the existence of solution by applying the Kuratowski's measure of noncompactness. Also, in [18], the authors proved the existence and uniqueness of solution using Banach xed point theorem and the Leray-Schauder Alternative Theorem.
In nite dimensional case, the existence and uniqueness of solutions to problems of type (1) are widely studied in [26] by means of Schauder's xed point theorem. Moreover, Zhou and Jiao [27] investigated the existence of nonlocal Cauchy problem by applying the Krasnoselskii's xed point theorem and Banach xed point theorem.
Recently, general fractional operators, especially such as the ψ-Hilfer fractional operator and the Riemann-Liouville fractional operator with respect to another function have been proposed. Then in 2018, Vanterler and Capelas de Oliveira [23] presented a fractional di erential operator of a function with respect to another function, the so-called ψ-Hilfer derivative. Obviously, the class of fractional derivatives derived from the ψ-Hilfer operator is in fact larger, making the fractional operator a generalization of the fractional operators. There are some authors who have worked on the existence and uniqueness of ψ-Hilfer and ψ-Caputo fractional di erential equations as in [22]. In addition, Liu, et al. [15] presented some results on the existence, uniqueness, and Ulam-Hyers-Mittag-Le er stability of solutions to a class of ψ-Hilfer fractional-order delay di erential equations. They used the Picard operator method and a generalized Gronwall inequality involved in a Riemann-Liouville fractional integral. Also, Abdo, et al. [1] considered the fractional integro-di erential equation with nonlocal condition involving a general form of Hilfer fractional derivative. They proved that Cauchy-type problem is equivalent to a Volterra fractional integral equation and used the Banach xed point theorem and Krasnoselskii's xed point theorem to obtain existence and uniqueness of solutions. Moreover, Luo, etc. [16] proved the existence and uniqueness of solution by Banach contraction mapping principle and investigated the Ulam-Hyers stability of solutions.
Motivated by all the above mentioned, we are interested to consider the system (1) of ψ-Hilfer neutral fractional evolution equations with in nite delay which have not been used to be investigated. In our work, we aim to investigate and prove the existence and uniqueness of solution by means of Banach contraction mapping principle and the Leray-Schauder alternative theorem. In this case, rst we have obtain the Volterra integral equivalent equation and propose the mild solution of system (1) in nite dimensional space.
The rest of our paper is organized as follows: In section 2 we present some preliminary results that are useful in this paper to prove the results. In Section 3, we prove the existence and uniqueness of solution by using Banach contraction mapping principle which states that there exists a unique mild solution to the problem (1). Also, in Section 3, we apply the Leray-Schauder alternative theorem to prove that problem (1) has at least one mild solution.

Preliminaries
In this section, we present some de nitions and properties of fractional calculus which will be used throughout this paper.
De nition 2.1. [11] The Laplace transform of a function f is denoted and de ned by: In addition, if F(s) = L f (t) (s) and G(s) = L g(t) (s), then De nition 2.2. [5] Let f : [ , ∞) −→ R be a real-valued function and ψ be a non-negative increasing function such that ψ( ) = . Then the Laplace transform of f with respect to ψ is de ned by

Remark 2. The Laplace transform of the one-sided stable probability density
is given by Furthermore, for any ≤ δ ≤ , we have (see [21]): For more details on the above preliminaries, we refer to [4,5,12,17,23].

Existence and uniqueness of the mild solution to the Cauchy problem (1).
We rst recall the Banach xed point principle [8] which we are going to use in this section. Then, we obtain the Volterra integral equivalent equation and mild solution to (1). At the end, we prove the existence and uniqueness of the mild solution based on Banach xed point principle and the Leray-Schauder alternative theorem. Then, F has a uniquely determined xed point z * . Furthermore, for any z ∈ Z, the sequence F j z ∞ j= converges to this xed point z * .
Proof. The proof of this lemma can be found in [8].
Applying the operator I α;ψ + to both sides of the rst equation of (1), we have: By properties of ψ-Hilfer fractional derivative and using (10), we obtain the left hand side of (16) as: In addition, on the right-hand side of (16) we have: and Substituting (17), (18) and (19) into (16), we obtain the Volterra integral equation (15).
Then, we prove that the Cauchy problem (1) holds.
The rst equation of (15) can also be written as Applying the operator H D α,β;ψ + on both sides of the equality (20) while taking into account (11), we have: Therefore, we have the following estimate: Since, H D α,β;ψ Hence, combining (22) and , we conclude that if x ∈ C satis es the Volterra integral equation (15), then x is solution to the Cauchy problem (1). This ends the proof of Lemma 3.2. (15) holds, then we have the following integral equation:

Lemma 3.3. If the rst equation of
where h and the operators Q α ψ (t) and where is the probability density function de ned on ( , ∞), that is, ζα(η) ≥ for η ∈ ( , ∞) and . Applying the generalized Laplace transform on both sides of the rst equation of (15), we have: By using (3), (4) and (12) and considering X( x t ) (p) as follows: We obtain: Now we consider the following change of variable: τ =t α and dτ = αt α− dt. Equation (30) becomes: Taket = ψ(t) − ψ( ). Then, we get: Now, equation (32) becomes: Taking into account (13) which is the Laplace transform of the one-sided probability density, we obtain: Now we consider the change of variable

Equation (37) becomes
Thus, we have: Therefore, where h and the operators Q α ψ (t) are de ned by (23), (24) and (25), respectively and the proof of Lemma 3.3 is complete.
Motivated by Lemma 3.3, we give the following de nition of the mild solution of the Cauchy problem (1).
In particular, for δ = , we deduce the following: For any x ∈ X, by remark 1 we have: In addition, we have for any x ∈ X the following: From inequalities (46) and (47), we deduce that the operators Q α ψ (t) and R α ψ t∈ [ ,b] are linear and bounded.
We assume that (H ) : h : [ , b] × P −→ X is a continuous function and there exists N > , such that for any t ∈ I and x t , x * t ∈ P : (H ) : f : [ , b] × X × P −→ X is a continuous function and there exists C , C > , such that for any t ∈ I and x, x * ∈ X and y, y * ∈ P ||f (t, x, y) − f (t, x * , y * )|| ≤ C ||x − x * || + C ||y − y * || P .

Theorem 3.1. Under the assumptions (H ) − (H ), the Neutral Cauchy problem (1) has a unique mild solution provided that A to be a bounded linear operator and there exists a constant < Lα < such that:
Proof. Consider the operator N : C −→ C. We have: Let y(.) : (−∞, b] −→ X be the function as: which we have y = ϕ( ). Then for each z ∈ C ([ , b], X) with z( ) = , we de ne the function Z by the following: If x(.) veri es the mild solution (42), then by and, . Therefore, (W , . W ) is a Banach space.
We de ne the operator G : W −→ W by: By assumptions (H ) and (H ), we deduce that the operator G is well-de ned and the operator N has a unique xed point if and only if G has a unique xed point. So we need to prove further that G has a unique xed point. Consider z, z * ∈ W . So, for any t ∈ [ , b] we obtain: By De nition 2.7 related to phase space and assumption (H ), let µ * (t) = sup µ (t) . Then, since z = we have: Also, by De nition 2.7 related to phase space and assumption (H ), let µ * (t) = sup µ (t) t∈ [ ,b] . Then, since z = we have: By (54) and (55), we obtain: We need to state and prove the following lemmas and the Leray-Schauder alternative theorem in order to prove the next theorem. ) f (t, y(t) + z(t), y t + z t ) ≤ C * H + C * (µ * ϕ P + µ * sup z(s) Proof.
On the other hand, Therefore, we obtain: and, Lemma 3.6. (Gronwall type Inequality) Suppose b ≥ , α > and a(t) is a nonnegative function locally integrable on ≤ t ≤ T (for some T ≤ +∞), and suppose u(t) is nonnegative and locally integrable on ≤ t ≤ T with On this interval, then Proof. The proof of this lemma can be found in [10,25]. (1) f has a xed-point, The proof of this lemma can be found in [3,13]. Proof. To prove this result, we aim to use the Leray-Schauder Alternative Theorem based on the following ve steps: Let G : W −→ W be de ned as in Theorem 3.1 and let A be a bounded linear operator on X.
Step (1): To show that G is continuous, let (z n ) be a sequence such that z n −→ z in W as n −→ ∞.
Now, for all t ∈ [ , b] by assumptions and Lemma (3.5), we obtain: ys + z n s ) − f (s, y(s) + z(s), ys + zs))ψ Now, there exists a positive number λ such that Step (3): To show that G maps bounded sets of W into equi-continuous sets of W , let q be de ned as in Step (2). Let t , t ∈ [ , b], t > t and z ∈ Pq and assume that for each bounded set B, the set {t → h(t, z t ) : z ∈ B} is equicontinuous. Then, By using the fact that z ∈ Pq, we obtain: Hence, where, and, which is the Mittage-Le er function. Thus, we obtain: which proves that (z)(t) W is bounded. Therefore, by proving the steps (1)-(5) and using Leray-Schauder Alternative Theorem, we result that G has a xed point which is a mild solution of problem (1) on (−∞, b].