Mild Solutions for w -Weighted, Φ -Hilfer, Non-Instantaneous, Impulsive, w -Weighted, Fractional, Semilinear Differential Inclusions of Order µ ∈ ( 1, 2 ) in Banach Spaces

: The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w -weighted, Φ -Hilfer, fractional derivative of order µ ∈ ( 1,2 ) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w -weighted Φ -Laplace transform, w -weighted ψ -convolution and the measure of non-compactness. Since the operator, the w -weighted Φ -Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.


Introduction
Fractional differential inclusions and equations have many applications in our life [1][2][3][4].Impulsive differential equations and impulsive differential inclusions are suitable models for studying the dynamics of actions in which a sudden change in state occurs.If this change occurs instantaneously, it is called an instantaneous impulse [5,6], but if this change continues for a period of time, it is called a non-instantaneous impulse [7][8][9].
In this article, we will prove the existence of a mild solution to a semi-linear differential inclusion involving the w-weighted Φ-Hilfer fractional differential operator.Because the fractional differential operators introduced by Caputo, Riemann-Liouville, Caputo-Hadamard, Hilfer and Hilfer-Katugampola are special cases of w-weighted Φ-Hilfer fractional differential operators, our work generalizes many of the abovementioned results by replacing the fractional differential operator considered in these papers with the w-weighted Φ-Hilfer fractional differential operator.
In order to formulate the problem, we mention some symbols that will be used during this paper. - -E is a Banach space.
-A is the infinitesimal generator of a strongly continuous cosine family{C(σ)} σ∈R , where C(σ) maps E into itself.
-F : ℑ × E → P ck (E) (the family of non-empty, convex and compact subsets of E ) -g i , g * i : [σ i , ϑ i ] × E → E; i ∈ N 1 are continuous functions, and x 0 , x 1 ∈ E are fixed points.
-AC(ℑ, E) is the Banach space of absolutely continuous functions from ℑ to E.
In this paper, and by using the properties of w-weighted Φ-Laplace transform, we derive at first the formula of a mild solution to the following differential inclusion containing the w-weighted Φ-Hilfer fractional derivative order µ and of type υ with the existence of non-instantaneous impulses in Banach spaces with infinite dimensions: x(σ) ∈ Ax(σ) + F(σ, x(σ)), a.e., σ ∈ (ϑ i , σ i+1 ], i ∈ N 0 , Then, without assuming the compactness of C(σ); σ > 0, we find the sufficient conditions that ensure that the mild solution set of Problem (1) is not empty or compact in the Banach space PC 2−γ,Φ,w (ℑ, E), which will be defined latter.
To further explain our arguments that clarify our motivation for studying Problem (1) addressed in this manuscript, as well as the importance of our purpose, we state the following: In a very recently published paper, Alsheekhhussain et al. [34] considered Problem (1) when the operator A is the zero operator.Zhou et al. [10] and He et al. [11] investigated the existence of mild solutions to Problem (1) when Thongsalee et al. [13] proved that solutions for Problem (1) exist; when A is the zero operator, F is a single-valued function, v = 0, and Φ(σ) = σ ∈ ℑ, Shu et al. [14] studied Problem (1) in the particular cases v = 0, w(σ) = 1 and Φ(σ) = σ, ∀σ ∈ ℑ, Gu et al. [17] was the first to consider Problem (1) when µ ∈ (0, 1), Working with the Laplace transform and density function, Gu et al. [17] was the first to define the mild solution for the semilinear differential equation: where D δ,η 0,σ is the Hilfer fractional derivative of order δ, η ∈ (0, 1), ρ = δ + η − δη, Ξ is the infinitesimal generator of C 0 -semigroup of linear bounded operators, f : [0, b] × E → E and x 0 ∈ E is a fixed point.Jaiwal et al. [18] presented the definition of a mild solution for (2) when Ξ is an almost sectorial operator, and then they found the sufficient conditions that guarantee that the solution exists.

Remark 1. Our work is novel and interesting because:
1-To date, none of the researchers in the field have considered studying semilinear differential equations or semilinear differential inclusions containing the w-weighted Φ-Hilfer fractional derivative of order µ ∈ (1, 2) and of type v ∈ [0, 1].
2-Our studied problem is considered with the existence of non-instantaneous impulses and in infinite-dimensional Banach spaces.
3-Our problem contains the w-weighted Φ-Hilfer fractional derivative, which interpolates many fractional differential operators, and hence, it includes the majority of problems cited above.
The following summarizes the focal contributions of our work.

•
A new class of differential inclusions is formulated, involving the w-weighted Φ-Hilfer , of order µ ∈ (1, 2) and of type υ in Branch spaces with finite dimension, when the linear term is the infinitesimal generator of a strongly continuous cosine family, and the nonlinear term is a multi-valued function

•
The conditions that ensure that the mild solution set for Problem (1) is not empty or compact are obtained (Theorem 1).

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An example is given to show the possibility of applying our results (Example 1).

•
Our method helps interested researchers to generalize the majority of the aforementioned works to the case where the non-linear term is a multifunction and the space is infinite-dimensional.

•
Since a large class of fractional differential operators can be obtained from D µ,v,Φ,w 0,σ , the works in many results mentioned above can be generalized by replacing the considered fractional differential operator with D µ,v,Φ,w 0,σ and making the dimension of the space infinite, and this is considered as a suggestion for future research work as a result of our work.

•
One can obtain a broad class of fractional differential equations and inclusions as a particular case of Problem (1) (see Remark 1).
We organize our work as follows: in Section 2, we present definitions and results from previous work that we will need to obtain our results.In the third section, we obtain the relation between Problem (1) and the correlating fractional integral equation and the representation of mild solutions.Moreover, we prove that the mild solution set for Problem (1) is not empty or compact.Finally, an example is presented to clarify the possibility of the application of our results.

Preliminaries and Notations
We commence this section by recalling some symbols that will be used later.For any p ≥ 1, denote by L p,Φ w ((0, b), E) the Banach space of all Lebesgue measurable functions f such that f w(Φ ′ )

Definition 2 ([50]
).Let n ∈ N and α ∈]n − 1, n[.The w-weighted Riemann-Liouville fractional derivative whose order α where the lower limit at a of a function f : [a, b] → E in regard to Φ is given by: assuming that the right-hand side is well defined.
f exits almost everywhere and Definition 3 ([50]).Let n ∈ N and α ∈]n − 1, n[.The w-weighted Φ−Caputo fractional derivative of order α where the lower limit at a of a function f ; [a, b] → E in regard to to Φ is given by: assuming that the right-hand side is well defined.
As a result of definitions ( 3) and ( 1), we give in the following definition the concept of the w-weighted Φ-Hilfer derivative operator.Definition 4. The w-weighted Φ-Hilfer derivative of order µ and of type v where the lower limit at a for a function f : where γ = µ + 2vs.− µvs., assuming that the right-hand side is well defined.
f (σ) exists a.e., and consequently D . Therefore, by Lemma 1, ( 5) and ( 6), we obtain for σ ∈ (a, b], Definition 5 ([51]).We call a one-parameter family of bounded linear operators {C(σ)} σ∈R which maps the Banach space E into itself a strongly cosine family if and only if
Lemma 5 ([51]).Let {C(σ)} σ∈R be a strongly cosine family on E.Then, the following are true.Then, for λ with Rel λ > ς, λ 2 belongs to the resolvent set of A, λ(λ where ς is defined in the sixth item 6 of Lemma 5.
4-If the w−weighted Φ−Laplace transform of f and h exist for λ > 0, then Lemma 8 ([52], Corollary 3.3.1.).Assume that U is a not empty, closed and convex subset of E and Υ : U → P ck (E) is χ−condensing and with a closed graph, where χ is a non-singular measure of noncompactness defined on subsets of U.Then, Υ has a fixed point.
Lemma 9 ([52], Propostion 3.5.1.).Assume that U is as in Lemma 8 and Υ : U → P ck (E) is χ−condensing on all bounded subsets of U, where χ is a monotone measure of noncompactness defined on E. If Υ has a closed graph, and the fixed-points set for Υ, Fix(Υ), is a bounded subset of E, then it is compact.
As a consequence of Lemma 10, we obtain the next definitions: Definition 11.A function x ∈ C 2−γ,Φ,w (ℑ, E) is called a mild solution for Problem (14) when it satisfies the next fractional integral equation: Definition 12.A function x ∈ PC 2−γ,Φ,w (ℑ, E) is called a mild solution for Problem (1) if it satisfies the following fractional integral equation: where and this coincides with Definition (8) in [33].
The next lemma illustrates some properties of K δ (., .).
Lemma 11.Suppose that the operator A satisfies the next condition: (A) A is the infinitesimal generator of a strongly continuous cosine family {C(σ)} σ∈R , and there is M > 0 such that sup σ≥0 ||C(σ)|| ≤ M.
In the next theorem, we demonstrate that the mild solution set for Problem (4) is not empty or compact.
(F 2 ) There is a function φ ∈ L 1 Φ,w (ℑ, [0, ∞)) such that for every u ∈ E and any k ∈ N 0 , where χis the Hausdorff measure of non-compactness on E.
(H 1 ) For every i ∈ N 1 , g i : [σ i , ϑ i ] × E → E such that for every σ ∈ [σ i , ϑ i ], g i (σ, .)map, every bounded set to a relatively compact subset and for every bounded set D, ⊆ E, defined such that it maps bounded sets to relatively compact sets, and there is h * i > 0 with Hence, Problem (1) has a mild solution assuming that the next inequalities are satisfied.
where h = max i∈N 1 {h i , h * i }.Moreover, the set of mild solutions is compact in Banach space C 2−γ,Φ,w (ℑ, E).
Step 1.For any u ∈ C 2−γ,Φ,w (ℑ, E), ℜ(u) is convex.This is, clearly, achieved since the set of values of F is convex.
Step 2. The graph of ℜ is closed on (45) is fulfilled.From ( 49), ( f n ) is uniformly bounded, and hence, it has a weakly convergent subsequence.We denote it, again, by ( f n ) to a function f in L 1 ((0, b), E).From Mazur's lemma, there exists a subsequence of ( f n ), (z n ), which converges almost everywhere to f .For any n ∈ N, let Obviously, (v * n ) is a subsequence of (v n ) and converges to the function Then, v * = v; moreover, the upper semi-continuity of F(σ, .)implies f (σ) ∈ F(σ, u(σ)), a.e., and so, v ∈ ℜ(u).
Step 3. Let £ = ℜ(℧ n 0 ).For every k ∈ N 0 , and every i ∈ N 1 , let According to the definition of ℜ, there exists u ∈ ℧ n 0 and f (σ Due to the continuity of Φ and (41), lim σ 1 →σ 1 ∆ 11 = 0.Moreover, using (35), we obtain Then, by Definition 3, Therefore, by (38), Next, from (47), we obtain Relation (35) implies that lim Again, by (35), one has lim Note that assumption (F 2 ) leads to Then, relations (35) and (50) .a.e.for ϑ ∈ (0, τ 2 ]. From (51), we have lim Next, using (51) and the continuity of Φ, it follows that lim For ∆ 33 , we have lim Finally, due to (36) and (51), it yields lim Case 2. Let τ 1 , τ 2 ∈ T i and τ 1 < τ 2 .Since ℧ n 0 , then due to (H 1 ), Following the same arguments used in case 1, one can show that £ |ℑ i ; i ∈ N 1 is equicontinuous.As a result of the above discussion, the proof of the results in this step is complete. Step Then, (℧ n ) is a decreasing sequence of not empty, bounded, convex subsets.In this step, our aim is to show that ℧ is not empty or compact.Using the Cantor intersection property, it remains to be shown that lim where χ C 2−γ,Φ,w (ℑ,E) is the measure of noncompactness on C 2−γ,Φ,w (ℑ, E), which is defined in the introduction section.
Assume n ∈ N is fixed and ϵ > 0 is arbitrarily small.By Lemma 5 in [54], one can find a sequence (v r ) r≥1 in ℧ n with where D = {v r : r ≥ 1}.From Step 3, it yields where χ E is the measure of non-compactnes in E, Since, for any v ∈ D, and any i ∈ N 1 , v(σ) = g i (σ, v(σ − i )), ∀ σ ∈ T i , and since g i (σ, ) maps bounded sets into relatively compact sets, it follows that max i∈N 1 max σ∈T i {v * (σ) : v * ∈ D | T i } = 0, and hence inequality (55) becomes where f r ∈ L 1 ((0, b), E) with f r (σ) ∈ F(σ, u r (σ)), a.e.In view of (F 3 ), it holds for a.e.ϑ ∈ (0, σ] Set Therefore, from (35), ( 56) and ( 57) and the properties of the measure of noncompactness, we obtain Because both g i (σ, .)and g * i (σ, .)map bounded sets to relatively compact sets, it yields where As in ( 58) From ( 56), ( 58) and ( 59), one has Then, Step 5.By applying the Cantor intersection property, the set ℧ is not empty or compact.Then, the multi-valued function ℜ : ℧ → P ck (℧) satisfies the assumptions in Lemma 8, and hence, the fixed-points set of the function is not empty.Moreover, Using Lemma 9, the set of fixed points of ℜ is compact in C 2−γ,Φ,w (ℑ, E).

Discussion and Conclusions
There are many definitions for the fractional differential operator, and some of them include others.Therefore, it is useful to consider fractional differential equations and fractional differential inclusions that contain a fractional differential operator which includes a large number of other fractional differential operators.Since the w-weighted Φ-Hilfer fractional derivative, D µ,v,Φ,w 0,σ , interpolates the fractional derivative differential operators that were presented by Riemann-Liouville, Caputo, Hadamard, Φ-Riemann-Liouville, Φ-Caputo, Katugampola, Hilfer-Hadamard, Hilfer, Hilfer-Katugampola and Φ-Hilfer derivatives, it contains a large number of fractional differential operators.In this work, the representation for a mild solution to a semilinear differential inclusion involving the w-weighted Φ-Hilfer fractional derivative of order µ ∈ (1, 2) and of type v ∈ (0, 1) is derived in the presence of non-instantaneous impulses, and then the non-emptiness and compactness of the set of mild solution for the considered problem is proved in infinite dimensional Banach spaces.The nonlinear part of the considered problem is the infinitesimal generator of the strongly continuous cosine family, and the nonlinear part is a multi-valued function.Our results are novel and interesting because no researchers have previously studied such semilinear differential inclusion.Moreover, since the fractional differential operator D µ,v,Φ,w 0,σ interpolates many other known fractional differential operators, our objective problem includes many problems which are considered in many cited papers in the introduction section.In addition, our technique can be used to generalize many cited papers in the introduction to the case when the considered fractional differential operator is replaced by D µ,v,Φ,w 0,σ and the dimension of the space is infinite, and this can be considered as a suggestion for future research work as a result of this paper.   2 ,Φ,w ϑ i ,σ