Fractional hybrid inclusion version of the Sturm–Liouville equation

The Sturm–Liouville equation is one of classical famous differential equations which has been studied from different of views in the literature. In this work, we are going to review its fractional hybrid inclusion version. In this way, we investigate two fractional hybrid Sturm–Liouville differential inclusions with multipoint and integral hybrid boundary conditions. Also, we provide two examples to illustrate our main results.


Preliminaries
We consider the norm u = sup t∈[0,1] |u(t)| on the space C R ([0, 1]) and u L 1 = 1 0 |u(s)| ds on L 1 [0, 1]. The Riemann-Liouville fractional integral of order α for a function f is defined by I α f (t) = 1 (α) t 0 (ts) α-1 f (s) ds (α > 0) and the Caputo derivative of order α for a function f is defined by c D α f (t) = I n-α d n dt n f (t) = 1 [38,39]). Suppose that (X , · X ) is a normed space. Denote by P(X ), P cl (X ), P b (X ), P cp (X ), and P cv (X ) the set of all subsets of X , the set of all closed subsets of X , the set of all bounded subsets of X , the set of all compact subsets of X and the set of all convex subsets of X , respectively. We say that a set-valued map has convex values whenever the set (z) is convex for each element z ∈ X . A set-valued map is called upper semicontinuous (u.s.c.) whenever for each z * ∈ X and open setV containing (z * ) there exists an open neighborhoodÛ 0 of z * such that (Û 0 ) ⊆V [40]. An element z * ∈ X is called a fixed point for the multivalued map : X → P(X ) whenever z * ∈ (z * ). The set of all fixed points of the multifunction is denoted by Fix( ) [40].
A multifunction is said to be a completely continuous operator whenever the set (W) is relatively compact for all W ∈ P b (X ). If the multifunction : X → P cl ( )) is upper semicontinuous, then Graph( ) is a subset of the product space X × with the closedness property. Conversely, if the set-valued mapping is completely continuous and has a closed graph, then is upper semicontinuous (see [40], Proposition 2.1). A setvalued map : [0, 1] × R → P(R) is said to be a Caratheodory multifunction whenever t → (t, z) is a measurable mapping for all z ∈ R and z → (t, z) is an upper semicontinuous mapping for almost all t ∈ [0, 1] (see [40,41]). Also, a Caratheodory multifunction for all |z| ≤ μ and for almost all t ∈ [0, 1] (see [40,41]). The set of selections of a multifunction at a point z ∈ C R ([0, 1]) is defined by (SEL) ,z := {y ∈ L 1 ([0, 1], R) : y(t) ∈ (t, z)} for almost all t ∈ [0, 1] (see [40,41]). Let be a set-valued map. It is known that (SEL) ,z = ∅ for all z ∈ C R ([0, 1]) whenever dim X < ∞ [40]. We need the following results. (ii)B 2 is u.s.c. and compact;

Lemma 4 Let y ∈ L 1 ([0, 1], R). A function z is a solution for the fractional hybrid Sturm-
with multipoint hybrid boundary conditions if and only if z is a solution for the integral equation where Proof First assume that z is a solution for the hybrid fractional equation (5). Note that equation (5) can be written as Then, Since ( z(t) g(t,z(t)) ) t=0 = (p (t)f (z(t)) p(t) ) t=0 , one has p(t)( z(t) g(t,z(t)) ) =p(t)f (z(t)) + I α y(t), and so By integrating from 0 to t, we get and Now by subtracting (10) from (11) and . Now by substituting the value of in (9), we get Conversely, to complete the equivalence between integral equation (7) and problem (5)-(6), by using (8), we obtain and ( z(t) g(t,z(t)) ) t=0 = (p (t) p(t)f (z(t))) t=0 . Also by using simple computations and (7), we obtain ). This completes the proof.

Definition 5
We say that an absolutely continuous function z : [0, 1] → R is a solution for the fractional hybrid Sturm-Liouville differential inclusion (1)-(2) whenever there is an and for all t ∈ [0, 1].

Theorem 6 Assume that
for all x ∈ R and almost all t ∈ [0, 1]; and It is easy to check that fixed point of the set-valued mapK is solution for the fractional hybrid Sturm-Liouville inclusion problem (1)- (2). Define the mapsB 1 ,B 3 :V ζ (0) → X and the set-valued-mapB 2 : Note thatK(z) =B 1 zB 2 z +B 3 z for all z ∈V ζ (0). We show that the operatorsB 1 ,B 2 , andB 3 satisfy the conditions of Theorem 2. First, we prove that the set-valued mapB 2 is convex- for k = 1, 2. Let λ ∈ (0, 1). Then, we have for almost all t ∈ [0, 1]. Since is convex-valued, (SEL) ,z is convex, that is, Hence, λϕ 1 (t) + (1λ)ϕ 2 (t) ∈B 2 z, and soB 2 z is a convex set for all z ∈ X . Now, we show that the operatorB 2 is completely continuous and upper semicontinuous on X . To establish the complete continuity of the operatorB 2 , we should prove thatB 2 (X ) is an equicontinuous and uniformly bounded set. To do this, first we prove thatB 2 maps all bounded sets into bounded subsets of X . LetV be a bounded subset ofV ζ (0). Choose 0 < κ * ≤ ζ such that z ≤ κ * for all z ∈V. For each z ∈V and ϕ ∈B 2 (V), there exists y ∈ (SEL) ,z such that Hence, , and so ϕ ≤ σ (α+2)¯ . Thus,B 2 (V) is a uniformly bounded. Now, we prove that the op-eratorB 2 maps bounded sets onto equicontinuous sets. Let z ∈V and ϕ ∈B 2 z. Choose y ∈ (SEL) ,z such that for all t ∈ [0, 1]. For each t 1 , t 2 with t 1 < t 2 , we have Since the right-hand side of the above inequality tends to zero as t 1 → t 2 , by using the Arzela-Ascoli theorem, the operatorB 2 :V ζ (0) → P(X ) is completely continuous. Here, we show thatB 2 has a closed graph, and this implies thatB 2 is upper semicontinuous. For this aim, suppose that z n ∈V and ϕ n ∈B 2 z n with z n → z * and ϕ n → ϕ * . We show that ϕ * ∈B 2 z * . For each ϕ n ∈B 2 z n , choose y n ∈ (SEL) ,z n such that It is sufficient to be prove that there exists a function y * ∈ (SEL) ,z * such that for all t ∈ [0, 1]. Consider the continuous linear operatorΥ : for all t ∈ [0, 1]. Then, we get dτ ds → 0 (as n → ∞).
Choose κ * > 0 such that z < κ * for all z ∈ V. For every z ∈ V and ϕ ∈B 2 (V), there exists y ∈ (SEL) ,z such that Hence, Since |ν||R| Hence, ϕ ≤ * . This meansB 2 (V) is a uniformly bounded set. Here, we show that the operatorB 2 maps bounded sets onto equicontinuous sets. Let z ∈ V and ϕ ∈B 2 z. Choose y ∈ (SEL) ,z such that for all t ∈ [0, 1]. For each t 1 , t 2 with t 1 < t 2 , we can write Note that the right-hand side of the inequality tends to zero as t 1 → t 2 . Now by using the Arzela-Ascoli theorem, the operatorB 2 : X → P(X ) is completely continuous. We show thatB 2 has a closed graph, and this implies thatB 2 is upper semicontinuous. For this, suppose that z n ∈ V and ϕ n ∈B 2 z n with z n → z * and ϕ n → ϕ * . We show that ϕ * ∈B 2 z * .