On a Caputo conformable inclusion problem with mixed Riemann–Liouville conformable integro-derivative conditions

We discuss some existence criteria for a new category of the Caputo conformable differential inclusion furnished with four-point mixed Riemann–Liouville conformable integro-derivative boundary conditions. In this way, we employ some analytical techniques on α-ψ-contractive mappings and operators having the approximate endpoint property to reach desired theoretical results. Finally, we provide an example to illustrate our last main result.

paper published in 2017, Jarad et al. [17] wanted to answer the question if one can extend the standard Riemann-Liouville integral of fractional order so that we obtain a unification to other fractional operators including Riemann-Liouville, Caputo, Hadamard, Caputo-Hadamard, and other derivatives [18]. To reach this goal, the authors introduced novel integration and differentiation operators of fractional order based on conformable operators. Indeed, they defined new functional spaces and established some basic properties of these new combined operators. After that manuscript, a limited number of papers, which rely on these novel operators, have been published so far. For instance, in the following paper for the first time the authors applied new Caputo and Riemann-Liouville conformable operators in their BVP. In fact, Aphithana et al. [19] sketched a new problem as the conformable differential equation of Caputo type with four-point integral conditions a is the conformable derivative of Caputo type of order ∈ (1, 2] with υ ∈ (0, 1]. Also, RC I υ,β a is a conformable integral of Riemann-Liouville type of order β > 0. They employed some functional analysis techniques to obtain their desired results. Further, different kinds of Ulam stability of the solutions are investigated by them [19]. Also, one can find different applied type works, where researchers use fractional models [20][21][22][23][24][25][26][27][28][29][30][31][32].
By utilizing these new operators introduced in [17] and motivated by the abovementioned work, we designed the following Caputo fractional conformable differential inclusion: so that CC D υ, a is the Caputo conformable derivative of fractional order ∈ (1, 2] with υ ∈ (0, 1], RC D υ,p * a is the Riemann-Liouville conformable derivative of fractional order p * ∈ (0, 1], and RC I υ,q * a is the Riemann-Liouville conformable integral of fractional order q * > 0. Also ξ , σ ∈ (a, T), μ 1 , μ 2 ∈ R andȒ : [a, T] × R → P(R) is a set-valued map endowed with some properties which are stated in the sequel. Our main goal in the present manuscript is to obtain some existence criteria for the mentioned Caputo conformable differential inclusion. In this way, we employ some analytical techniques on the α-ψcontractive mappings and operators having the approximate endpoint property to reach the desired theoretical results. Note that unlike other published papers in the field of the existence theory, this inclusion problem supplemented with newly defined Caputo and Riemann-Liouville conformable operators is unique, and this type of mixed inclusion problems has not been investigated in any literature. We arrange the contents of the paper as follows. Some auxiliary definitions and notions are assembled in Sect. 2. Then in Sect. 3, we utilize two concepts of fixed point and endpoint to obtain the existence criteria corresponding to the given BVP by (1)- (2). In the last section, we propose an illustrative example to support our findings from a numerical point of view.

Preliminaries
In the current section, we state some fundamental and auxiliary concepts. As we know, the notion of the Riemann-Liouville integral of order > 0 of a function w : [0, +∞) → R is given by R I 0 w(s) = s 0 (s-r) -1 Γ ( ) w(r) dr provided that the value of the integral is finite [33,34]. In this position, let us assume that ∈ (n -1, n) so that n = [ ] + 1. For a function w ∈ AC (n) R ([0, +∞)), the fractional derivative of Caputo type is given by provided that the integral is finite-valued [33,34]. The left conformable derivative at the initial point s 0 = a for a function w : [a, ∞) → R with υ ∈ (0, 1] is defined as follows: so that the limit exists [16]. [16]. In [17], Jarad et al. extended the conformable operators to arbitrary orders in the Caputo and Riemann-Liouville setting. Assume that ∈ C with Re( ) ≥ 0. Then the Riemann-Liouville fractional conformable integral of a function w of order with υ ∈ (0, 1] is defined by 1-if the value of integral exists [17]. One can easily see that if a = 0 and υ = 1, then RC I υ, a w(s) is reduced to the usual Riemann-Liouville integral R I 0 w(s). On the other hand, the Riemann-Liouville conformable derivative of a function w of order with υ ∈ (0, 1] is given by 1 where the notation I υ a ϕ(s) = s a ϕ(r) dυ(r, a) = s a ϕ(r) dr (r-a) 1-υ stands for the left conformable integral of ϕ by putting dυ(r, a) = dr (r-a) 1-υ [15]. Also, if a = 0, then we write dυ(r) =

Lemma 2 ([19])
Let n -1 < Re( ) ≤ n and w ∈ C n a,υ ([a, b]). Then, for υ ∈ (0, 1], we have In view of the above lemma, it is verified that the general solution of the homogeneous equation ( CC D υ, a w)(s) = 0 is given by In the following, we review some notions about the set-valued maps theory. For this purpose, consider the normed space (W, · W ). Also, for convenience, we use the notations P(W), P cls (W), P bnd (W), P cmp (W), and P cvx (W) for the representation of the collection of all subsets, all closed subsets, all bounded subsets, all compact subsets, and all convex subsets of W, respectively. The element w * ∈ W is a fixed point for the given setvalued mapȒ : W → P(W) whenever w * ∈Ȓ(w * ) [35]. We represent the family of all fixed points ofȒ by notation FIX (Ȓ) [35]. In the following, the Pompeiu-Hausdorff [35]. A setvalued mapȒ : W → P cls (W) is Lipschitzian with positive constantλ if the inequality PH d W (Ȓ(w),Ȓ(w )) ≤λd W (w, w ) holds for all w, w ∈ W. A Lipschitz mapȒ is supposed to be defined contraction ifλ ∈ (0, 1) [35]. In the sequel,Ȓ is said to be completely continuous ifȒ(K) is relatively compact for each K ∈ P bnd (W), whereasȒ : [35,36]. Also,Ȓ is upper semi-continuous whenever for every w * ∈ W the setȒ(w * ) belongs to P cls (W) and for each open We construct a graph of the set-valued mapȒ : W → P cls (Z) by The Graph(Ȓ) is closed whenever, for two arbitrary convergent sequences {w n } n≥1 in W and {z n } n≥1 in Z with w n → w 0 , z n → z 0 and z n ∈Ȓ(w n ), we have the inclusion z 0 ∈Ȓ(w 0 ) [35,36]. In view of [35], it is deduced that if the set-valued mapȒ : W → P cls (Z) has an upper semi-continuity property, then Graph(Ȓ) is a closed subset of W × Z. On the contrary, ifȒ has the complete continuity and closed graph property, thenȒ is upper semi-continuous [35]. In addition,Ȓ has convex values ifȒ(k) ∈ P cvx (W) for each w ∈ W. Furthermore, a collection of selections ofȒ for (a.e.) all s ∈ [0, 1] [35,36]. Note that if we assume thatȒ is an arbitrary set-valued map, 1] {|q| : q ∈Ȓ(s, w)} ≤ ϕ μ (s) for all |w| ≤ μ and for almost any s ∈ [0, 1] [35,36].
Samet et al. [37] introduced a new collection of nondecreasing and nonnegative func- By considering the properties of these functions, it is obvious that ψ(s) < s for all s > 0 [37]. Later, Mohammadi et al. constructed a new structure for set-valued maps with the following definition [38]. A set-valued mapȒ : for each w, w ∈ W [38]. In addition, we say that W has the property (C α ) if for every [39]. Besides, we say that S has an approximate endpoint property if we have inf w∈W sup z∈Ȓw d W (w, z) = 0 [39]. The following theorems are our required tool for establishing the desired results in this research.
be a complete metric space. Assume that α is a nonnegative map on W × W, ψ ∈ Ψ is a strictly increasing map, andȒ : W → P cls,bnd (W) is an αadmissible and α-ψ-contractive set-valued map so that α(w, w ) ≥ 1 for some w ∈ W and w ∈Ȓ(w). ThenȒ has a fixed point whenever the space W has the property (C α ).
ThenȒ has a unique endpoint iffȒ has the approximate endpoint property.

Main results
Then (W, · W ) is a Banach space. Further, consider the following constants for convenience: In the following lemma, we introduce an equivalent integral structure for the solution of the four-point Caputo conformable inclusion BVP (1)-(2).
Proof In the beginning, w 0 is supposed to satisfy the Caputo conformable equation (4). Clearly, we have CC D υ, a w 0 (s) =ĥ(s). Now, by taking th order Riemann-Liouville conformable integral on the last equality, we obtain the following equation: where we wish to seek two constants b 0 , b 1 ∈ R. On the other hand, by taking the Riemann-Liouville conformable derivative and integral with respect to s on both sides of equation (8), we get By applying the four-point mixed boundary conditions, we get and where k 0 , k 1 , k 2 , k 3 are given in (3). By some direct computations on equations (9) and (10), we obtain If we insert the values b 0 and b 1 into equation (8), then we have which shows that the function w 0 satisfies the integral equation (6). In the opposite direction, one can easily check that w 0 is a solution for the four-point Caputo conformable BVP (4)-(5) whenever w 0 satisfies the integral equation (6).
For the sake of convenience in writing, we set In this position, for each w ∈ W, we represent the family of selections ofȒ as follows:

Theorem 8 Let us suppose thatȒ : [a, T] × W → P cmp (W) is a compact set-valued map.
Furthermore, assume that all six assumptions are valid: (Hp1)Ȓ is a bounded and integrable operator, and also the set (Hp5) There are two elements w 0 ∈ W and z ∈ K(w 0 ) such thatζ (w 0 (s), z(s)) ≥ 0 for all s ∈ [a, T], where K : W → P(W) is the same operator defined by (12);

Then the four-point Caputo conformable inclusion BVP (1)-(2) has a solution on [a, T].
Proof It is a well-known fact that the solution for the four-point Caputo conformable inclusion BVP (1)-(2) is as a fixed point of the operator K : W → P(W) given by (12).
From assumption (Hp1), the measurability of the set-valued map s →Ȓ(s, w(s)) is evident, and thus it is closed-valued for each w ∈ W. Hence,Ȓ has measurable selection and (SEL)Ȓ ,w = ∅. Here, it is suitable to prove that K(w) is a closed subset of W for each w ∈ W. To do this, we consider a sequence {w n } n≥1 of K(w) having the property w n → w.
For each n, chooseθ n ∈ (SEL)Ȓ ,w provided that for almost all s ∈ [a, T]. For the sake of compactness of the set-valued mapȒ, we may pass into a subsequence (if necessary) to obtain a convergent subsequence {θ n } n≥1 which converges to someθ ∈ L 1 ([a, T]). Therefore, we haveθ ∈ (SEL)Ȓ ,w , and so for each s ∈ [a, T]. From the above argument, we realize that w ∈ K(w) and thus K has closed values. By assumption of theorem, we know thatȒ is a compact set-valued map, thus one can easily confirm that K(w) is a bounded set for each w ∈ W. In this position, we are going to prove that the operator K is α-ψ-contractive. To observe that, formulate the nonnegative function α : W × W → [0, ∞) as α(w, w ) = 1 ifζ (w(s), w (s)) ≥ 0 and α(w, w ) = 0 otherwise. We also assume that w, w ∈ W and z 1 ∈ K(w ) are arbitrary. Select ϑ 1 ∈ (SEL)Ȓ ,w so that for any s ∈ [a, T]. Condition (13) verifies that for all w, w ∈ W having the propertyζ (w(s), w (s)) ≥ 0 for any s ∈ [a, T]. Hence, there exists h ∈Ȓ(s, w(s)) such that In what follows, we build a new set-valued map B * : [a, T] → P(W) as follows: for each s ∈ [a, T]. Becauseθ 1 and =ȃψ(|ww |) 1 M ȃ are measurable, so the intersection of two set-valued maps B * (·) ∩Ȓ(·, w(·)) is measurable. For continuing the deduction, we chooseθ 2 belonging toȒ(s, w(s)) provided that for any s ∈ [a, T]. Then, one can compute the following estimate: for all s ∈ [a, T]. Hence, we find that and so we get α(w, w )PH d W (K(w), K(w )) ≤ ψ( ww ) for each w, w ∈ W showing that the set-valued map K is α-ψ-contractive. Now, consider two elements w ∈ W and w ∈ K(w) with α(w, w ) ≥ 1. In the light of the definition ofζ , we have an inequalitỹ ζ (w(s), w (s)) ≥ 0, and so there is a function h ∈ K(w ) such thatζ (w (s), h(s)) ≥ 0. Hence, α(w , h) ≥ 1 and this states that K is α-admissible.
To finish the rest of the proof, we suppose that w 0 ∈ W and w ∈ K(w 0 ) are such that ζ (w 0 (s), w (s)) ≥ 0 for each s. Thus, we get α(w 0 , w ) ≥ 1. On the other hand, consider the sequence {w n } n≥1 of W with w n → w and α(w n , w n+1 ) ≥ 1 for each n. Then we havẽ ζ (w n (s), w n+1 (s)) ≥ 0. Now, with the help of (Hp4), we find that there is a subsequence {w n l } l≥1 of {w n } such thatζ (w n l (s), w(s)) ≥ 0 for each s ∈ [a, T]. Consequently, α(w n l , w) ≥ 1 for all l ≥ 1, and so it is verified that W has the property (C α ). Ultimately, in the light of Theorem 4, we realize that the set-valued map K has a fixed point which is as a solution for the four-point Caputo conformable inclusion BVP (1)- (2).
By continuing the current process, we obtain another existence criterion for the fourpoint Caputo conformable inclusion BVP (1)-(2) under new analytical conditions. In other words, we shall prove our desired existence result under a new concept due to Amini-Harandi [39]. In this way, we use the approximate endpoint property for K which is defined by (12).
for all s ∈ [a, T] and w, w ∈ W, where sup s∈[a,T] |δ(s)| = δ and M is defined by (11); (Hp10) The operator K defined by (12) has the approximate endpoint property.

Then the four-point Caputo conformable inclusion BVP (1)-(2) has a solution on [a, T].
Proof As a general goal, we shall show the existence of an endpoint for the set-valued map K : W → P(W). In this direction, we need to prove that the set K(w) is closed for each w ∈ W. If we pay attention to hypothesis (Hp8), then for the sake of the measurability of the map s →Ȓ(s, w(s)) and closeness of it for all w ∈ W, we realize thatȒ has a measurable selection, and so (SEL)Ȓ ,w = ∅ for each w ∈ W. Therefore, similar to the proof of last Theorem, it is easy to check that K(w) is a closed subset of W, and hence we omit it. Moreover, we know that the set K(w) is bounded for each w ∈ W with due attention to the compactness ofȒ. We conclude the proof by proving the inequality PH d W (K(w), K(w )) ≤ ψ( ww ) for every two arbitrary members of W. To see this, assume that w, w ∈ W and z 1 ∈ K(w ). Chooseθ 1 ∈ (SEL)Ȓ ,w so that for almost all s ∈ [a, T]. Since, by considering inequality (14) presented in hypothesis (Hp9), we have for any s ∈ [a, T], thus there is h * ∈Ȓ(s, w(s)), for which we get We know thatθ 1 and = δψ(|ww |) 1 M δ are measurable, so we can easily deduce that the intersection set-valued map Q(·) ∩Ȓ(·, w(·)) is measurable. Now, we choose the functionθ 2 (s) ∈Ȓ(s, w(s)) so that for any s ∈ [a, T]. We also select z 2 ∈ K(w) provided that for all s ∈ [a, T]. Therefore, by repeating a similar process in the proof of Theorem 8, we obtain the following relations: This yields the inequality PH d W (K(w), K(w )) ≤ ψ( ww ) for each w, w ∈ W. Furthermore, hypothesis (Hp10) confirms that K has the approximate endpoint property, so with due attention to Theorem 5, we arrive at the desired conclusion which involves this property that the operator K has a unique endpoint, i.e., there is w * ∈ W so that K(w * ) = {w * }. Thus, w * is a solution for the four-point Caputo conformable inclusion problem (1)-(2). Now, we provide an example to illustrate Theorem 9. Riemann-Liouville conformable integro-derivative boundary conditions. Note that unlike other published papers in the field of existence theory, this inclusion problem supplemented with newly defined Caputo and Riemann-Liouville conformable operators is unique and this type of mixed inclusion problems has not been investigated in any literature. In this way, we utilize two concepts of fixed point and endpoint to obtain the existence criteria corresponding to the given BVP by (1)- (2). Indeed, some analytical techniques on the α-ψ-contractive mappings and operators having the approximate endpoint property are employed to reach the desired theoretical results. Also, we provide an example to illustrate our last main result.