Analysis of Hilfer fractional integro-differential equations with almost sectorial operators

We investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove existence results by applying Schauder's fixed point technique. Moreover, we show fundamental properties of the solution representation by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.

During the last decades, mathematical modeling has been supported by the field of fractional calculus, with several successful results and fractional operators showing to be an excellent tool to describe the hereditary properties of various materials and processes. Recently, this combination has gained a lot of importance, mainly because fractional differential equations have become powerful tools in modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering, see, for instance, the basic text books [1][2][3][4] and recent researches [5][6][7]. In fact, abrupt changes, such as shocks, harvesting, or natural disasters, may happen in the dynamics of evolving processes. These short-term perturbations are often treated in the form of impulses. Recently, in so many published works, Hilfer fractional differential equations have received attention [8][9][10][11][12][13][14][15][16][17].
In [15], Jaiswal and Bahuguna study equations of Hilfer fractional derivatives with almost sectorial operators in the abstract sense: We also refer to the work in [8], where Ahmed et al. study the question of existence for nonlinear Hilfer fractional differential equations with controls. Sufficient conditions are also established, where the time fractional derivative is the Hilfer derivative. In [18], Zhang and Zhou study fractional Cauchy problems with almost sectorial operators of the form is the Riemann-Liouville integral of order 1 − q, 0 < q < 1, A is an almost sectorial operator on a complex Banach space, and f is a given function. Motivated by these results, here we extend the previous available results of the literature to a class of Hilfer fractional integro-differential equations in which the closed operator is almost sectorial. Moreover, we also consider both compactness and noncompactness cases of the semigroup operator.
The paper is structured as follows. In Section 2, we present necessary information about the Hilfer derivative, almost sectorial operators, measure of non-compactness, mild solutions of equations (1)-(2), along with some useful definitions, results, and lemmas. We discuss fundamental results for mild solutions for the equations (1)- (2) in Section 3. In Section 4, we prove the solvability question in two cases, when associated semigroup is compact and noncompact, respectively. An example is then given in Section 5, to illustrate our main results. We end with Section 6 of conclusions.

Preliminaries
In this section we recall some necessary theory that will be used throughout the work in order to get the new results.

Fractional derivatives
We start by a short introduction of the main definitions in fractional calculus [13,19]. Definition 1. The left-sided Riemann-Liouville fractional integral of order α > 0 with lower limit a for a function h : [a, +∞) → R is defined as provided the right hand side is defined a.e. on [a, +∞).
provided the right hand side is defined a.e. on [a, +∞).

Measure of non-compactness
The motivation to consider our problem can be found in [18,19]. Here we generalize the results in [18,19]. Let L ⊂ Y be bounded. The Hausdorff measure of non-compactness is considered as while the Kurtawoski measure of noncompactness Φ on a bounded set B ⊂ Y is given by with the following properties: for n ∈ N and m ∈ L 1 (I, R + ). Then,

as the family of all closed and linear operators
where R(z, A) = (zI − A) −1 is the resolvent operator and A ∈ Θ β ω is said to be an almost sectorial operator on Y.

Q(t) is analytic and d n dt n Q(t)
We use the following Wright-type function [19]: For −1 < σ < ∞, r > 0, the following properties hold: Theorem 1 (See Theorem 4.6.1 of [19]). For each fixed t ∈ S 0 π 2 −ω , S α (t) and T α (t) are bounded linear operators on Y. Moreover, where C 1 and C 2 are constants dependent on α and β.
Theorem 2 (See [19]). The operators S α (t) and T α (t) are continuous in the uniform operator topology, for t > 0. For s > 0, the continuity is uniform on [s, ∞]. Define Our main results are proved under the following hypotheses: and almost all t on J and (H3) Function h : C(J , Y ) → Y is completely continuous and there exists a positive constant k such that h(u) ≤ k.
(H4) We assume that For the next two lemmas, we refer to [9,12].

Definition 6. By a mild solution of the Cauchy problem
We define operator P : Lemma 3 (See [15]). The operators R α (t) and S α,γ (t) are bounded linear operators on Y for every fixed t ∈ S 0 π 2 −ω . Also, for t > 0, we have Proposition 5 (See [15]). The operators R α (t) and S α,γ (t) are strongly continuous for t > 0.

Auxiliary results
Now, we are in position to start our original contributions.
Applying Theorem 1, we have which tends to 0 as n → ∞, i.e., P y n → P y pointwise on J . Moreover, Theorem 3 implies that P y n → P y uniformly on J as n → ∞, that is, P is continuous.

Main results
We prove existence of a mild solution to problem (1)-(2) when the associated semigroup is compact (Theorem 5) and noncompact (Theorem 6).

Compactness of the semigroup
Here we assume Q(t) to be compact.

Conclusion
In this paper, we applied Schauder's fixed point theorem to investigate the solvability of a class of Hilfer fractional integro-differential equations involving almost sectorial operators. We discussed both cases of compactness and noncompactness, related to associated semigroup operators. The obtained existence results were subject to an appropriate set of sufficient conditions. As a future direction of research, it is desirable to consider the study of ψ-Hilfer fractional nonlocal nonlinear stochastic systems involving almost sectorial operators and impulsive effects, generalizing the current work. Another open line of research consists to develop numerical methods to approximate the mild solutions predicted by our Theorems 5 and 6.