On a new structure of the pantograph inclusion problem in the Caputo conformable setting

In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.


Introduction
Over the years, human beings have needed to be acquainted with various natural phenomena more and more. One possible way to achieve this aim is to utilize the techniques and tools available in mathematics and particularly the mathematical operators in modeling of different processes. Numerous fractional operators have been introduced during years and their applicability is becoming increasingly apparent to researchers every day that passes. In this direction, it is better that we formulate and investigate various complicated modelings of processes from all aspects by applying the fractional operators in boundary problems.
More recently, Abdeljawad [42] extended some notions presented in [43] and studied some applied specifications of the well-behaved conformable derivatives of arbitrary order. Next, Jarad et al. [44] proceeded to answer this key problem if we can generalize the usual fractional Riemann-Liouville integral provided that we obtain a unification to remaining useful operators such as Caputo, Riemann-Liouville, Hadamard, and Caputo-Hadamard derivatives [45]. To achieve this purpose, they tried to derive two corresponding integration and differentiation operators of arbitrary order based on the existing conformable operators. In this way, the authors first designed functional spaces and then verified some fundamental applied specifications of both newly-defined combined operators.
Until now, there have been published a limited number of papers based on these novel operators. For example, the authors employed new Riemann-Liouville and Caputo conformable operators in the following BVP for the first time. Indeed, Aphithana, Ntouyas and Tariboon [46] regraded a modern BVP including the Caputo conformable differential equation along with integral conditions: indicates the conformable derivative in the Caputo frame of order ν * ∈ (1, 2) along with ζ ∈ (0, 1]. Also, RC I ζ ,p * c stands for the conformable integral in the Riemann-Liouville frame of order p * > 0. The authors utilized several techniques to establish desired theorems. Furthermore, different types of Ulam stability of the proposed problem were studied by authors [46]. Recently, Baleanu, Etemad and Rezapour [47] turned to the differential inclusion in the Caputo fractional conformable frame illustrated by stand for the Caputo-and the Riemann-Liouville conformable derivatives and the Riemann-Liouville conformable integral of order q * > 0, respectively. The main aim of the authors in that manuscript is to discuss the existence aspects for mentioned BVP by employing several methods based on the α-ψ-contractives and operators involving approximate endpoint specification [47]. One of the most famous categories of differential equations is related to the pantograph equation. This kind of equation is considered as proportional delay functional differential equations and they have many applications in applied and pure mathematics. In other words, pantograph equations arise in rather various contexts: control systems, quantum mechanics, electrodynamics, probability, etc. For the first time, Balachandran et al. [48] formulated a pantograph equation of fractional order and derived existence and also uniqueness criteria for the proposed problem. After that, different researchers studied fractional pantograph equations with the help of various numerical methods such as the operational method, the spectral-collocation method, and the Hermite wavelet method [49][50][51]. Recently, other researchers investigated various versions of fractional pantograph equations relying on analytical methods (see [52][53][54]). By taking into account the aforementioned new operators introduced by Jarad et al. [44] and inspired by some existing ideas in the above articles, in the current manuscript, for the first time, we formulate an inclusion version of the pantograph boundary problem in the fractional Caputo conformable settings subject to three-point Riemann-Liouville conformable integral conditions as follows: indicates the derivative in the Caputo conformable settings of order ν * ∈ (1, 2) along with ζ ∈ (0, 1] and RC I ζ ,θ * c stands for the integral in the Riemann-Liouville conformable frame of order θ * > 0. Furthermore, σ ∈ (c, M), μ * 1 , μ * 2 , ξ * ∈ R, λ * ∈ (0, 1) and is a multifunction furnished with several necessary specifications which are indicated in the rest of the manuscript. It is important that the reader pays attention to the fact that this structure of a pantograph inclusion problem in the Caputo conformable operators is novel and such a kind of construction has not been discussed in any literature so far. In fact, we reformulate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann-Liouville settings simultaneously for the first time. We demonstrate the contents of the current research manuscript as follows. In Sect. 2, we briefly review fundamental and auxiliary concepts and notions. In Sect. 3, we employ some well-known analytical techniques to establish existence criteria corresponding to the given pantograph inclusion BVP (1). In this way, we deduce key results in three cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. In fact, we derive desired existence results for three different structures considered on the set-valued maps and this cover a vast range of multifunctions satisfying our given conditions. the last part of the present research is devoted to proposing two numerical simulative examples to demonstrate the consistency of the analytical findings.

Auxiliary notions
Now, we review some fundamental and auxiliary notions and some specifications of the fractional Riemann-Liouville and Caputo conformable operators. As we see in much of the literature, the concept of the Riemann-Liouville integral of order ν * > 0 for a real dq such that the RHS integral possesses finite values [55,56]. In the current position, we assume that ν * ∈ (k -1, k) so that k = [ν * ] + 1. For a given function φ ∈ AC (k) R ([0, +∞)), the fractional derivative in the Caputo settings is defined by so that the existing R.H.S integral involves the finite values [55,56]. Subsequently, the left conformable derivative at s 0 = c for φ : [c, ∞) → R along with ζ ∈ (0, 1] was introduced as provided that the limit exists [43]. Notice that, if D ζ c φ(s) exists on (c, d), in this case we have D ζ c φ(c) = lim s→c + D ζ c φ(s). Also, if we assume that the given function φ is differentiable, then it is clear that 1-ζ whenever the RHS integral is finite-valued [43]. Jarad et al. [44] presented a new formulation of integro-derivative operators which generalize conformable operators to fractional orders in both Riemann-Liouville and Caputo settings. To see this, let ν * ∈ C with Re(ν * ) ≥ 0. In this phase, the Riemann-Liouville conformable integral for φ of order ν * along with ζ ∈ (0, 1] is introduced as follows: 1-ζ so that the RHS integral is finite [44]. One can simply deduce that, if c = 0 and ζ = 1, then Moreover, the Riemann-Liouville conformable derivative for φ of order ν * along with ζ ∈ (0, 1] is formulated as when we have c = 0 and ζ = 1.

Lemma 1 ([44]
) Take Re(ν * ) > 0, Re( * ) > 0 and Re(σ ) > 0. Then, for ζ ∈ (0, 1] and for each s > c, the following hold: . Then, for ζ ∈ (0, 1], the following identity is valid: In the light of the above lemma, one can deduce that the general solution of the homogeneous equation such that k -1 < Re(ν * ) < k andr * 0 ,r * 1 , . . . ,r * k-1 ∈ R. In the sequel, we intend to devote the rest of this section to reviewing some primary definitions and key properties on the set-valued maps. To achieve this goal, we regard the normed space (Y, · Y ). In addition to this, we introduce the notations P(Y), P cls (Y), P bnd (Y), P cmp (Y) and P cvx (Y) for the illustration of the collection of all nonempty subsets, all closed subsets, all bounded subsets, all compact subsets and all convex subsets of Y, respectively. An element φ * ∈ Y is defined to be a fixed point forÕ : Y → P(Y) when we have φ * ∈Õ(φ * ) [57]. In this case, we illustrate the set of all fixed points ofÕ by symbol FIX(Õ) [57]. In the subsequent text, the Pompeiu-Hausdorff metric PH d Y : Notice that a Lipschitz mapÕ is defined to be a contraction ifĉ ∈ (0, 1) [57]. The multifunctionÕ is called completely continuous ifÕ(K) is relatively compact for any K ∈ P bnd (Y) and alsoÕ : [57,58]. In addition to the above notions, we say thatÕ possesses an upper semi-continuity specification if for each φ * ∈ Y, the setÕ(φ * ) belongs to P cls (Y) and, for every open set V which containsÕ(φ * ), there exists a neighborhood U * 0 of φ * so thatÕ(U * 0 ) ⊆ V [57]. The graph ofÕ : [57,58]. With due attention to [57], it is concluded that, ifÕ : Y → P cls (X) is a set-valued map having the upper semi-continuity property, then GR(Õ) is a closed subset of Y × X. In the opposite direction, ifÕ possesses the complete continuity and closed graph specifications, in this case,Õ has an upper semi-continuity property [57]. Moreover, it is clear thatÕ is convex-valued ifÕ(φ) ∈ P cvx (Y) for any φ ∈ Y. We illustrate the family of all existing selections ofÕ at for each s ∈ [0, 1] (a.e.) [57,58]. It is necessary to pay attention to the fact that by assum-ingÕ to be an arbitrary multi-valued function, then, for [57]. The multi-valued mapÕ : [57,58]. In addition, a Carathéodory mapÕ : Then N is said to be a Niemytzki operator associated withÕ [58]. Moreover,Õ : is said to be of lower semi-continuous type (l.s.c. type) whenever its relevant Niemytzki operator N is lower semi-continuous and involves nonempty closed decomposable values [58]. The next theorems are regarded as our required tools for verifying desired results in the current research.
Theorem 3 (Bohnenblust-Karlin theorem, [59]) Regard Y as a Banach space and E = ∅ as a subset contained in Y which is convex, bounded and closed. Assume thatÕ : O possesses a fixed point.

Main results
After reviewing and introducing some auxiliary concepts in previous sections, we proceed to deduce desired existence theorems. To arrive at this goal, we regard the norm φ Y = In the next result, we derive an integral construction for the solution of the proposed three-point Caputo conformable pantograph BVP (1).

Lemma 9
Regard˘ ∈ Y. In this phase, φ 0 is regarded as a solution for the fractional linear differential equation in the Caputo conformable settings subject to three-point Riemann-Liouville conformable integral boundary conditions iff φ 0 satisfies integral equation where a nonzero constant˜ is defined by (2).
Proof First, we regard φ 0 as a function which satisfies the Caputo conformable equation (4). Then we see that CC D ζ ,ν * c φ 0 (s) =˘ (s). Now, we integrate both sides of the latter equation in the ν * th order Riemann-Liouville conformable settings. We have so that we wish to find constant coefficientsr * 0 ,r * 1 ∈ R. Prior to seeking these constants, by taking the integral of the Riemann-Liouville conformable type with respect to s on both sides of (7), we obtain The first boundary condition causesr * 0 to be zero. Now, according to the second integral boundary condition, we get By inserting the obtained valuesr * 0 andr * 1 into (7), we obtain indicating that φ 0 satisfies (6). In the reverse direction, we can simply verify that φ 0 satisfies the given three-point Caputo conformable problem (4)-(5) whenever φ 0 satisfies the integral equation (6).
In this position, we deal with several existence criteria for the proposed pantograph fractional BVP (1)

The upper semi-continuity case
Here, we assume that values of the set-valued mapÕ belong to P cvx (Y). The first existence criterion is derived due to both Bohnenblust-Karlin's theorem, Theorem 3, and the closed graph theorem, Theorem 4.

Theorem 10
Let the following be valid: where lim inf μ→∞ Proof To transform the given Caputo conformable pantograph BVP (1) into a well-known fixed point problem, we regard a multifunction : Y → P(Y) formulated by We claim that satisfies all existing hypotheses of Theorem 3 and so possesses a fixed point which is regarded as a solution function for the proposed Caputo conformable pantograph BVP (1). In the first stage, we are going to check the convexity of (φ) for each φ ∈ Y. For this purpose, let ψ 1 , ψ 2 ∈ (φ). Then there are two functionsg 1 ,g 2 ∈ SELÕ ,φ so that, for any s ∈ [c, M], we get Take 0 ≤ κ ≤ 1. In this phase, for any s ∈ [c, M], one may write As SELÕ ,φ is convex (Õ is convex-valued), so it is deduced that [κψ 1 + (1κ)ψ 2 ] ∈ (φ). Next, we verify that is a bounded operator on B μ , where B μ = {φ ∈ Y : φ Y ≤ μ} for every constant μ > 0. Obviously, B μ is a convex bounded and closed set belonging to Y. We claim that μ ∈ R + exists so that (B μ ) ⊆ B μ . To confirm this claim, we assume that, for any μ ∈ R + , there is a function φ μ ∈ B μ and ψ μ ∈ (φ μ ) with (φ μ ) Y > μ and In view of hypothesis (HP 2 ) and taking the supremum, we obtain In the following, we multiply both sides of (11) by 1/μ and take the lower limit of it when μ goes to infinity. Then we find that and this is a contradiction by considering the condition (9). Therefore there is μ ∈ R + provided that (B μ ) ⊆ B μ . This means that is a set-valued map from B μ to B μ .
In the sequel, we check that (φ) is equi-continuous. Let φ be arbitrary member belonging to B μ and ψ ∈ (φ). In this case, there existsg ∈ SELÕ ,φ so that, for each s ∈ [c, M], we have Therefore for any s , s ∈ [c, M] with s < s , we get As s → s , we realize that the RHS of the latter inequality approaches 0 without any dependence to φ ∈ B μ . This points to the fact that is equi-continuous. By virtue of the well-known Ascoli-Arzelá theorem, we deduce that the set-valued map possesses a complete continuity specification.
Eventually, we verify that possesses a closed graph. To reach this goal, let φ n → φ * , ψ n ∈ (φ n ) and ψ n → ψ * . Also, chooseg n ∈ SELÕ ,φ n . Our aim is to prove ψ * ∈ (φ * ). Hence, for each s ∈ [c, M], we have In this case, we want to prove that a functiong * ∈ SELÕ ,φ * exists so that, for each s ∈ [c, M], To achieve this purpose, we define a new continuous linear operator ϒ * : It is evident that ψ nψ * Y → 0 as n → ∞. So in the light of Theorem 4, we realize that ϒ * • SELÕ ,φ is a closed graph operator. Furthermore, ψ n (s) ∈ ϒ * (SELÕ ,w n ). As φ n → φ * , Theorem 4 yields 1-ζ for someg * ∈ SELÕ ,φ * . Consequently, we realize that is a compact and upper semicontinuous multifunction furnished with closed and convex values. Hence, by considering Theorem 3, we realize that possesses a fixed point, which is the same solution as for the proposed three-point Caputo conformable pantograph inclusion problem (1). This completes the proof.
Our second criterion is derived with the help of Martelli's fixed point result given by Theorem 5.
Theorem 11 Let the following be valid: Proof Let us regard as given in Theorem 10. Then, in a similar manner, we can simply confirm the convexity and the complete continuity of the operator . Thus, it just remains to check the boundedness of the set = {φ ∈ Y : ηφ ∈ (φ), η > 1)}. To investigate this, let φ ∈ . Hence ηφ ∈ (φ) for some η > 1 and a functiong ∈ SELÕ ,φ exists provided that With due attention to condition (HP 6 ), there is a number M so that M = φ Y . Let us assume By proceeding similar to the proof of Theorem 10, it is easily verified that : U → P(Y) is a compact and upper semi-continuous multifunction having closed and convex values. So we observe that there exists no φ ∈ ∂U so that φ ∈ η (φ) for some η ∈ (0, 1) in view of the choice of U. Hence, by Theorem 6 one concludes that is a multifunction including a fixed point φ ∈ U and eventually we find that the proposed three-point Caputo conformable pantograph inclusion BVP (1) involves a solution on [c, M].

The lower semi-continuity case
In the current position, we derive other existence criterion in the lower semi-continuous phase. Here, the set-valued mapÕ has not necessarily convex values. We discuss the next result by applying nonlinear alternative of Leray-Schauder along with the selection result due to Colombo and Bressan (Theorem 7) for all lower semi-continuous mappings having decomposable values.

Theorem 13
Let the hypotheses (HP 5 ) and (HP 6 ) along with the following condition be valid: In this moment, we regard the following reformulated BVP: Notice that, if φ ∈ AC 2 R ([c, M]) is regarded as a solution of problem (12), then φ will be as a solution of main inclusion problem (1). Define an operator as follows: In this way, the aforementioned Caputo conformable problem (12) is reduced to a standard fixed point problem. Finally, one can simply prove that the newly-defined operator is completely continuous and continuous. The remaining proof is implemented as one in Theorem 12 and thus we omit it again. This finishes the proof process and yields the required existence result.

The Lipschitzian case
Here, we discuss the existence criterion whenÕ has non-convex values. To reach the desired purpose, we utilize a fixed point result attributed to Covitz and Nadler (Theorem 8) on set-valued maps.
Proof We again regard : Y → P(Y) similar to the one defined in the proof of Theorem 10. In this case, the three-point Caputo conformable pantograph inclusion problem (1) is transformed into a standard fixed point problem. At first, we verify that (φ) = ∅ for any φ ∈ Y and also is closed set for everyg ∈ SELÕ ,φ . To see this, it is clear thatÕ(·, φ(·),φ(·)) is measurable in view of the measurable selection theorem ( [65], Theorem III.6) and so a measurable selectiong ∈ L 1 R ([c, M]) exists and thusÕ is integrable bounded. This means that SELÕ ,φ = ∅. In addition, (φ) ∈ P cls (Y) for each φ ∈ Y as is verified in Theorem 10. Thus (φ) is a closed set for each φ ∈ Y. In the sequel, we show that there is a constant c < 1 so that for any φ 1 , φ 2 ,φ 1 ,φ 2 ∈ Y. To confirm this, let φ 1 , φ 2 ,φ 1 ,φ 2 ∈ Y and ψ 1 (s) ∈ (φ). Hence, for each s ∈ [c, M], there existsg 1 (s) ∈Õ(s, φ 1 (s),φ 1 (s)) so that In a similar way, by interchanging the roles of φ 1 and φ 2 , the following holds: Then, in the light of the condition (13), we find that is a contraction. In conclusion, with the help of a fixed point result attributed to Nadler and Covitz (Theorem 8), we deduce that involves a fixed point which is regarded as a solution for the proposed three-point Caputo conformable pantograph inclusion problem (1). This completes the proof.
As one can see, all hypotheses of Theorem 14 are valid. Then at least one solution exists for the proposed three-point Caputo conformable pantograph inclusion problem (14) along withÕ(s, φ,φ) defined in (16) on the interval s ∈ [0, 1].

Conclusion
Over the years, the human beings have needed to be acquainted with various natural phenomena more and more. One possible way to achieve this aim is to utilize the techniques and tools available in mathematics and particularly the mathematical operators in modeling of different processes. In the current manuscript, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann-Liouville settings simultaneously for the first time. In fact, we derive required existence criteria of solutions corresponded to inclusion version of three-point Caputo conformable pantograph BVP subject to Riemann-Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to demonstrate the consistency of our findings.