Existence results of solution for fractional Sturm–Liouville inclusion involving composition with multi-maps

ABSTRACT This paper is introduced as complementary studies based on fractional Sturm–Liouville problems in a Banach space. We explore the existence results for new considered problems which can be considered as mixture of equations and inclusions. For the sake of that, we use jointly continuous composed functions with multi-valued maps and denote this form by eq-inclusion problems. The form of the solutions is calculated by the rules of Caputo derivative and the corresponding integral. The concept “continuous image of multi-valued maps” is useful to show that the strong results will be under inclusion hypothesis. The argument and fit technicals used here consider both Lipschitz and non-Lipschitz cases with using nonlinear alternative Leray Schauder type and Covitiz and Nadler theorems.


Introduction
Most researchers in applied mathematics are using differential operators to describe a lot of kinds of modelling that have strong effects in various applied sciences. Many scholar teams are attracted to study partial differential operators having useful extents to present the physical equations. They use different methods to calculate their solutions, for instance, the obtained solutions in the shape of hyperbolic, trigonometric, elliptic functions including dark, bright, singular, combined, kink wave solitons, travelling wave, solitary wave and periodic wave. These kinds of solutions play vital role in mathematical physics, optical fiber, plasma physics and other various branches of applied sciences. There are some noteworthy results investigated by Seadawy et al. in [1][2][3][4][5][6][7][8][9][10][11]. They studied by different mathematical methods the general form of solutions for modified and the nonlinear damped modified Kortewege-de Vries, (2+1)-dimensional nonlinear Nizhnik-Novikov-Vesselov, Kadomtsev-Petviashvili modified equal width, modify unstable nonlinear Schrodinger, Zakharov-Kuznetsov-modified equal width, and Kudryashov-Sinelshchikov dynamical equations, Camassa-Holm and the nonlinear longitudinal wave equations. They also [12][13][14] studied some systems of equations like dynamical system of nonlinear wave propagation, three coupled system of nonlinear partial differential equations, and the system of dynamical equations.
As a mathematical analysis, fractional calculus studies some different possibilities to define real number powers or complex number powers of the differential operator D = d/dt and then some strong generalization by fractional number powers, see [15]. For instance, Sturm-Liouville, Langevin, Evolution, Duffing, Navier-Stokes, and Hybrid operators are all of the most important and famous fractional differential operators [16][17][18][19][20].
The importance of fractional powers attracted mathematical scientific teams to study different considerable results of fractional differential equations and inclusions that have gained substantial noteworthiness derived from their applications in various sciences. Not long ago, Zhou [21] has investigated some results of the solvability for a Cauchy problem of Riemann-Liouville type fractional differential equations given by: where α ∈ [0, 1), R D α and I 1−α are respectively Riemann-Liouville derivative and integral of orders α and 1 − α. After that, he in [22] has improved the previous problem up to more general by the next Riemann-Liouville fractional differential evolution equations: R D α w(ι) = Aw(ι) + f (ι, w(ι)), ι ∈ (0, ∞), where α ∈ [0, 1), R D α and I 1−α are respectively Riemann-Liouville derivative and integral of orders α and 1 − α.
In [23], he has initiated the attractively question of solutions for fractional evolution equations with almost sectorial operators. In fact, these equations have been studied first before more than 30 years ago.
The most interesting and the finest of our knowledge is recalling general forms of problems that are studying more cases at the same time (For example: equation and inclusion, ordinary and fractional operators). Here, we pick out one kind of nonlinear fractional problems that can be studied as equations and inclusions at one time. It is considered with a jointly continuous composed functions (ι, ω, K) with multi-valued maps K. This kind is not be presented before and new in inclusion field.
Consider the following problem: where 0 < α ≤ 1, n − 1 < β ≤ n, n > 1, c D r is the symbol of the Caputo fractional derivative with respect to the order r, p(ι) is a positive function such that p(ι) ∈ C[0, ϒ], is a multi-valued map, and We say that if and only if ∃ k(ι, ω(ι)) ∈ K(ι, ω(ι)), in which that: Up to now, fractional Sturm-Liouville problems are the main problems of applied science. They become more advantageous than the classical models and has drawn interest so much. Furthermore, Sturm-Liouville operators and its properties have attractive huge applications in physics, applied mathematics, engineering filed and science, applications of wide in quantum, classical mechanics and wave phenomena. Gerald in (2009) [28], has explored some mathematical methods in quantum mechanics under the vision of Sturm-Liouville operator. In the previous scientific studies and contributions we can see some related results in [29] given in (2019) for the following problems: For the sake of finding new influential results and applications, we pick out the fractional inclusion given in (1) associated with composite functions with multivalued maps.
In fact, a continuous maps composite with a multivalued maps maybe take a single-valued for every points in their domains as follows: (4) And absolutely on other times, they give multi-valued maps like: Basically, we are going to study the existence results under the inclusion arguments clarified as well as in [30][31][32][33][34]. The results as in equation case will be easy and clearly. It should be mentioned the existence results for the higher ordinary case in sense that: under same conditions in (2) can be included in the provided results. This paper is organized to start with some needed preliminaries in the next section. After that, we will illustrate the main results by section three. It should be add a section to gived some related examples for the adopted results. That will be in section four. Finally, in section five we try to conclude all studied points and mention to new open problem.

Prefatory Facts
As necessary for the contemporary paper, we will introduce some definitions, basic facts, and some useful rules.
A multi-valued K : → P cl ( ) is known as convex (closed) if for every ν ∈ , K(ν) is convex (closed) [24,36]. It is completely continuous if K( ) is relatively compact for every ∈ P b ( ).
The map K assimilates to be upper semi-continuous if ∀ W ∈ P cl ( ); K −1 (W) ∈ P cl ( ). By the other word, If we adopt K as a completely continuous function with non-empty compact values, then it is upper semicontinuous if and only if its graph is closed [37] (i.e.), if ν n → ν * , y n → y * , then y n ∈ K(ν n ) implies that y * ∈ K(ν * ).
For any E ⊂ R, the characteristic function of E is defined as follows [38]:

Definition 2.1 (Jointly Continuity [39]): Suppose
. . ., A n , C are all topological spaces and the map Then, the sentences come after are all equivalent:

Definition 2.3 (Lipschitz Condition [24]):
Take ( , . ) as a normed space, and d be the metric map conformed from the norm. Then, a multi-valued map : → P cl ( ) is adopted as: (2) a contraction if the first statement is hold with γ < 1.

Definition 2.4 (Riemann-Liouville Fractional Integral [24]):
The fractional integral of order α > 0 of Riemann-Liouville vision is given by the relation: in case that the integral exists.

Definition 2.5 (Caputo Derivative [24]):
Caputo formula for fractional derivative of order α for n-times absolutely continuous map g is defined as:
Then, has non-empty values if Now, Lemma 2.2 allows us to present the next Lemma.

Lemma 2.3:
Let α and β be positive reals and p(ι) ∈ C[0, ϒ] such that p(ι) > 0. Then, the differential equation Again by using the third item of Lemma 2.2, we get the desired result.

Discussion and Results
It is so interesting for some scientific teams to describe so many models by using the fractional (ordinary) differential operators. See for examples related details given in [49,50]. In this field, the general form of exact solutions created under the vision of derivatives and corresponding integrals with their rules and arguments. For example, Riemann-Liouville, Caputo, Caputo-Fabrizio, Hadmard, and the ordinary derivatives.
In the previous literature, the researchers used different fixed point theorems to study separately the attractivity of types of solutions for differential equations and inclusions and systems of equations and inclusions in different spaces. See [16,19,20,23,26,27,29] and the references there in.
By the actual work, we suggest to use composite functions with multi-valued maps to explore the solvability of some equation and inclusion problems at one time. We will start by the results associated with Sturm-Liouville operator and hope other researchers to work with different operators. Our results essentially reveal the characteristics of solutions with Caputo derivative. So, we are going to create some portability results of solving the problem (1)-(2).
Backing to Definition 2.6, we define Then, in view of Lemma 2.4 consider : → P( ) as: with: (17) At this time, we are ready to survey the main results.

Convex Case
The result here is followed by assuming that both maps and K are convex and overviewed by applying (Leray-Schauder Theorem 2.1).

Proof:
The fit technical to prove this argument is going through five steps. These steps together will show that defined by (16) and (17), satisfies all arguments of (Leray-Schauder Theorem 2.1).
Step 1: By this step, (ω) will be convex valued. According to convexity of , the set S ,ω so does. Let Thus, It is easy to see by using convexity of S ,ω that: which leads to (ω) is convex.
Step 2: Through this step, we prove that (ω) is bounded on a bounded set. In order to see that, consider the open ball B R = {ω : ω < R} and v ∈ (ω), then we get: Through few simple calculations and by the inequality (15), we can see that: Hence, by taking μ = inf ι∈J p(ι) we have: In fact, (19) will be true if and only if (H 3 ) be hold for the value R.
Step 3: By this step, (B R ) will be equicontinuous. Take 0 < ι 1 (17) and (18), we find that: By (15) we get Independently of being ω ∈ B R , v ∈ (ω) we get: Step 4: By this step, the graph of will be closed and then is upper semi-continuous. Let Existence of ψ n (ι) is depending on the existence of a suitable sequence of the functions k n (t) ∈ S K,ω n where: R) is so as: Hence, Under the vision of Lemma 2.5, the operator • S has a closed graph which implies the existence of k * and then ψ * with ψ * (ι) = (ι, ω * , k * ). That follows v * (ι) = (ψ * )(ι), ψ * (ι) ∈ S ,ω * . Then, we conclude that v * ∈ (ω * ) which completes the proof. The previous steps can explain that there exists fixed point of the map . It remains to make sure about the priori bounds on the solutions and that will be in the next step.
Step 5: Let ω ∈ ∂B R , v ∈ λ (ω), λ ∈ (0, 1). Hence, there exists In view of step 2, we have: and set Since B R is an open subset of and due to first four steps (step 1-step 4) with Arzela Ascoli theorem, we conclude that: : B R → P cp,cv ( ).
Backing to choice of B R (20-21), we have no v ∈ ∂B R v ∈ λ (ω) for some λ ∈ (0, 1). Hence, it can be deduced by nonlinear alternative Theorem 2.1 that there is a fixed point ω ∈ B R of the operator . Therefore, there is a solution at least for the problem (1)-(2).

Lower Semi-Continuous Case
While this case is similar to the convex case in some conditions and arguments, but we are here going through non-convex hypothesis.

Theorem 3.2:
Consider in addition of (H 1 ) and (H 2 ) that the following condition is hold: Then, the problem (1)-(2) is able to has solutions at least one.
Proof: Define the operator The operator (ω) is called Nemytskii's operator associated with (see Definitions 1.2 and 1.3 in [24]) where is lower semi-continuous if is lower semicontinuous and has closed and decomposable values, see Lemma 4.4 of [51]. Note that This drives us to see clearly that there exists a continuous selection [52] ψ :

Now, consider the problem
Define the operator : We need to show that is continuous and equicontinuous. The other steps to apply alternative theorem is will be similarly in convex case.
First: Claim that is continuous. Since ω, k, are all continuous, then for > 0 be given there is δ, σ > 0 such that for ω 1 Thus, Moreover, we can see that .

Lipschitz Case
In the way to dispute the result of existence under Lipschitz condition, we will follow Covitz and Nadler Theorem 2.2.
Proof: Let be espoused as in convex case. Since ι → (ι, ω, w) is measurable and has closed values, then there exists at least one measurable selection and then the set S ,ω = ∅.

Remark 3.1:
Note that the sufficient conditions of Continuity in these results can be applicable to explore finite time stability conditions for the differential equations (inclusions). For instance, compare to the active stability conditions in the papers [53][54][55] and more stability results related to Continuity, Lipschitz continuity, non-lipschitz continuity and Holder continuity of the settling-time functions.

Related Examples
Over the current section, we will work to present examples related to the stated upshots.

Example 4.1:
Let us have the problem below: where For any k(ι) ∈ K(ι, ω), we have: which implies from (H 2 ) that h 2 (ι) = 4 and Backing to (H 1 ), we have for any ψ(ι) ∈ (ι) that |ψ(ι)| ≤ 1 + b 2 ≤ 1 + η 2 . It follows that we can take: Thus,  It is known that n-variable function G over convex set is a concave function if and only if −G is convex in the same set. That is why K(ω) is concave. And because the sin ν is concave on the interval [0, π] and 0 < |K| < π, that makes (K(ω)) is concave function.
Applying the lower semi-continuous case (Theorem 3.2), we have: In view of (H 4 ) since ω, K, are all continuous, then the composite (K(ω)) is also continuous. It means that the points in (H 4 ) are all hold. The property 0 < |K| < π helps us to see that: h 2 (ι) = π, θ 3 (ι) = 1, which drives the ability of satisfying (H 2 ).
There for, (H 1 ) is hold.

Conclusion
Based on composed functions with multi-valued maps, we discuss one of the strong suggestions to create new kind (1)-(2) of nonlinear fractional differential boundary value problems. We prove by some examples (4)-(5) that this form of problems is able to be a generalization of equations and inclusions problems related to this form. That is why we call this form by eq-inclusion problems. In inclusion field, we particularized per chances to solve this problem involving Sturm-Liouville operators on a bounded domain. The obtained solutions for this problem are subjected to Caputo derivative. The chosen argument surveyed the advanced results into convexity and non-convexity cases. And the suitable theorems used here are (Leray-Schauder nonlinear alternative type ) and (Covitz and Nadler). As necessary, we applied all provided results in some related examples. What is more, we mentioned how to connect these results with some applications in stability field. It is worth to invoke the new concept in the next work that will be about the positive solutions at resonance of nonlinear fractional differential inclusions in the half real line. Besides, we expect that our results would make improvements for the previous studies into new extents.