On More General Fractional Differential Equations Involving Mixed Generalized Derivatives with Nonlocal Multipoint and Generalized Fractional Integral Boundary Conditions

This paper deals with the existence of solutions for a new boundary value problem involving mixed generalized fractional derivatives of Riemann-Liouville and Caputo supplemented with nonlocal multipoint boundary conditions. The existence results for inclusion case are also discussed. The nonlinear term belongs to a general abstract space, and our results rely on modern theorems of fixed point. Ulam stability is also presented. We provide some examples that well-illustrate our main results.


Introduction and Preliminaries
In this paper, we study a new generalized boundary value problem (GBVP) that involves both generalized Riemann-Liouville and Caputo derivatives via nonlocal multipoint and generalized fractional integral boundary conditions given by where the derivatives RL D α,ρ 0 + and C D β,ρ 0 + denote the generalized Riemann-Liouville fractional derivative of order 0 < α ≤ 1 and generalized Caputo fractional derivative of order 0 < β ≤ 1, respectively. I q,ρ 0 + is the generalized Riemann-Liouville fractional integral of order 0 < q < 1, and the nonlinear term g is given in the Orlicz space L G ½0, 1.
We also investigate the corresponding inclusion problem which is given by where a i , b i ∈ ℝ ði = 1, 2Þ; δ i , r i ∈ ℝ + ði = 1, ⋯, pÞ; σ i , μ i ∈ ℝ + ði = 1, ⋯, mÞ; γ = t 1−ρ ðd/dtÞ; and ρ > 0, and G : ½0, 1 × ℝ ⟶ P ðℝÞ is a multivalued map. P ðℝÞ is the family of all subsets of ℝ. The concept of generalized fractional derivatives and integrals was introduced by Katugampola in [1,2]. This new form of fractional integral operator generalizes Riemann-Liouville and Hadamard fractional derivative into a single form and satisfy a semigroup property. Jarad et al. [3] proposed a new generalized fractional calculus based on a special case of proportional derivatives where the kernel of the fractional operator involves an exponential function.
This generalization contains Riemann-Liouville and Caputo fractional derivatives and integrals as special cases. Also, there were works on general fractional derivative and integral containing a Mittag-Leffler function. For more new definitions of generalized fractional calculus, one can see [4][5][6][7]. These generalizations are helpful to reach a better form of the real models that appeared in various fields such as physics, chemistry, aerodynamics, and electrorheology employing fractional differential equations (see [8][9][10][11][12][13][14][15][16][17]). The existence and uniqueness results of fractional boundary value problems involving fractional derivatives and derivative of a generalized type were established by many authors (for example, see [18][19][20][21]).
On the other hand, Birnbaum and Orlicz [22] generalized the classical Lebesgue space L p , 0 < p < ∞, into a general abstract space, which was called Orlicz space. This generalization is based on replacing the function x p with more general function f .
In what follows, we recall some basic concepts of fractional calculus [1,2,4] and Orlicz space [22] that are needed throughout the paper.
Definition 1. For −∞ < a < t < b < ∞ , the left-sided generalized fractional integral of f ∈ X p c ða, bÞ of order α > 0 and ρ > 0 is defined by Definition 2. For 0 ≤ a < x < b < ∞ ,the generalized Riemann-Liouville fractional derivative associated with generalized fractional integrals (2) is defined by Definition 3. The generalized Caputo-type fractional derivative of f ∈ AC n δ ½a, b of order α ≥ 0 is defined by where n = ½α + 1.
The following shows some important properties which are used in the sequel.
Alternatively, the function G is an N-function iff G is continuous, even, and convex with lim x⟶0 ðGðxÞ/xÞ = 0, lim x⟶∞ GðxÞ/x = ∞ and GðxÞ > 0 if x > 0: Note that a function G : ½0,∞Þ ⟶ ½0,∞Þ is called a Young function if it is convex and satisfies the conditions For an N-function G, we define where φ −1 is the right inverse of the right derivative of φ of G and is called the complementary of G and it satisfies the condition Notice that the function G * is also an N-function and the complementary pairs G and G * satisfy the following Young inequality: Definition 9. For an N -function G , the Orlicz space L G ð½0, 1Þ is the space of measurable functions x : ½0, 1 ⟶ ℝ such that Ð 1 0 Gð|xðtÞ | Þdt < ∞ . This space is endowed with the Luxemburg norm, i.e., ∥x∥ G = inf λ > 0 : and the pair ðL G ð½0, 1Þ, k:k G Þ is a Banach space.
For an Orlicz space, the Hölder inequality holds; that is, where u ∈ L G ð½0, 1Þ and v ∈ L G * ð½0, 1Þ.
The existence and uniqueness of solutions of GBVP (1) rely on the following fixed point theorems [23].
Theorem 10. Let P be a closed, convex, bounded, and nonempty subset of a Banach space X .Let T 1 , T 2 be operators such that (i) T 1 ðu 1 Þ + T 2 ðu 2 Þ belong to P whenever u 1 , u 2 ∈ P (ii) T 1 is a compact and T 2 is a contraction mapping Then, there exists u 0 ∈ P such that u 0 = T 1 ðu 0 Þ + T 2 ðu 0 Þ: Theorem 11. Let X be a Banach space. Assume that T : X ⟶ X is a completely continuous operator and the set V = fu ∈ X : u = εTu, 0 < ε < 1g is bounded. Then, T has a fixed point in X.
For computational convenience, 3 Journal of Function Spaces Lemma 12. Let h ∈ L G ð½0, 1Þ . Then, the solution of the following GBVP: is given by where ϕ and ψ 2 are given in (23).
Proof. To obtain the integral equation modeled by the GBVP(24), we apply the generalization Riemann-Liouville fractional integral of order α to both sides of (24), and we get Next, applying the generalization Caputo fractional integral of order β to both sides of (26), we have where c 1 and c 2 ∈ ℝ. By using Lemma 7, we get So, the quantities γyðtÞ and I q,ρ γyðtÞ are given, respectively, by In view of the boundary conditions given in (24), the con-stants c 1 and c 2 are defined by where ψ 1 , ψ 2 , ψ 3 , and ϕðtÞ are given in (23), and by substituting of values of c 1 and c 2 , we obtain (25).

Main Results
In this section, we are looking to prove the existence and uniqueness of solutions of GBVP (1). We will utilize in our proofs the fixed point theorems. We shall assume that g is in the Orlicz space L G ½0, 1. Let Y = fy : y ∈ Cð½0, 1, ℝÞg denote the Banach space of all continuous functions on ½0, 1 into ℝ endowed with the norm kyk = sup fjyðtÞj: t ∈ ½0, 1g. Now, we define an operator F : Y ⟶ Y associated with problem (1) by Therefore, the GBVP (1) has a solution if and only if the operator F has a fixed point.
The following lemma plays a curial role in the sequel.
Proof. Let α, β ∈ ð0, 1, q ∈ ð0, 1Þ, and y ∈ Y. Define a function Journal of Function Spaces We show that Λ 1 ∈ L G * ½0, 1. By using appropriate substitution and properties of the Young functions, one obtains By the assumption of the theorem, we get Then, we can get Λ 2 , Λ 3 , and Λ 4 ∈ L G * ½0, 1. Next, we show that F is well defined, i.e., FyðtÞ ∈ Cð½0, 1, where

<
: The functions Journal of Function Spaces increasing function with Jð0Þ = 0. Using the Hölder inequality, we get Hence, for 0 < |t ρ − τ ρ | <ε and by continuity of J, we see that F is continuous, which completes the proof.
For convenience, we define Now, we are in a position to state and prove the first existence result. The following result is based on Banach fixed point theorem.
Proof. We define B r = fy ∈ Y : ∥y∥≤rg, with r ≥ g * ψ/ð1 − Ω ψÞ, and ψ is defined by (39). Set We shall show that FB r ⊂ B r : For any y ∈ B r , by assumption (S1), we obtain that Therefore, Now, for y,ỹ ∈ Cð½0, 1, ℝÞ, and for each t ∈ ½0, 1, we get Journal of Function Spaces So, As Ωψ < 1, then F is a contraction. Hence, by the Banach fixed point theorem, F has a unique fixed point which is a unique solution of boundary value problem (1) on ½0, 1.
In the next result, we prove the existence of problem (1) by applying Schaefer's fixed point theorem.
Proof. The proof is divided into three steps.
Step 1. Let K be a bounded subset of Y. By (S2), for all t ∈ ½ 0, 1 and y ∈ K, we have Then, ∥Fy∥≤ν * , and so F maps bounded sets into bounded sets in Y.
Step 2. Let t 1 , t 2 ∈ ½0, 1, t 1 < t 2 , and K be a bounded set of Y. Thus, for y ∈ K and by (S2), we have It follows that the right-hand side tends to zero as ðt 2 − t 1 Þ ⟶ 0 independent of y; by Arzela-Ascoli theorem, we conclude that F is completely continuous.
Step 3. It remains to show that the set M = fy ∈ Y : y = aF ðyÞ for some 0 < a < 1g is a bounded set. Let y ∈ M, and then, yðtÞ = aðFyÞðtÞ, 0 < a < 1, t ∈ ½0, 1. Hence, by (S2), we have This shows that M is bounded. Schaefer's fixed point theorem guarantees the existence of a fixed point of F. Hence, the integral equation (31) has a solution on [0,1]. Now, we establish the existence of solutions of problem (1) via Krasnoselskii's fixed point theorem.
Proof. We define and choose a suitable constant r * such that r * ≥ ∥ζ∥ψ, where ψ is given by (39). Furthermore, we split F into two 8 Journal of Function Spaces operators F 1 and F 2 on B r * = fy ∈ Y : ∥y∥≤r * g by Þ, for all t ∈ 0, 1 ½ , For y ∈ B r * , we get which shows that F 1 y + F 2 y ∈ B r * . Now, we need to prove that F 2 is a contraction. By assumption (S1), we have Therefore, By using (S1) and (48), F 2 is a contraction. Since g is continuous, we have F 1 which is continuous. Now,

Ulam Stability
In this section, we establish the criteria of Ulam stability of problem (1) by means of its equivalent integral equation where w ∈ Y and g : ½0, 1 × ℝ ⟶ ℝ is a continuous function. We define a continuous nonlinear operator Now, we recall some basic concepts of the Ulam stability. For more details, one can see [24][25][26][27].
Definition 17. GBVP is said to be Ulam-Hyers stable if there exists a real number k > 0 such that for each ε > 0 and for each solution w ∈ Y of ((1)) satisfying the inequality there exists a solution y ∈ Y of GBVP satisfying the inequality where ε * is a positive real number depending on ε.
Definition 18. GBVP is generalized Ulam-Hyers stable if there exists τ ∈ Cðℝ + , ℝ + Þ such that for each solution w ∈ Y of GBVP, there exists a solution y ∈ Y of GBVP with Definition 19. GBVP is Ulam-Hyers Rassias stable with respect to ς ∈ Cð½0, 1, ℝ + Þ if there exists a real number k > 0 such that for each ε > 0 and for each solution w ∈ Y of GBVP we can find a solution y ∈ Y of GBVP satisfying the inequality where ε * is a positive real number depending on ε.
Proof. We know that y ∈ Y is a unique solution of GBVP as in Theorem 14. Let w ∈ Y be another solution of GBVP satisfying (64). Observe that the operator F and F − I are equivalent for every solution w ∈ Y (given by (62)) of GBVP. Therefore, by the fixed point property of operator F, we have where ε > 0 and Ωψ < 1. Taking the norm of (69) for t ∈ ½0, 1 and solving for ∥w − y∥, we obtain If we let ε * = ε/ð1 − ΩψÞ and take k = 1, then the Ulam-Hyers stability condition is satisfied. More generally, defining τðεÞ = ε/ð1 − mψÞ, the generalized Ulam-Hyers stability condition is also satisfied.
Proof. Following the idea of the proof of Theorem 19, we can obtain that with ε * = ε/ð1 − ΩψÞ.

Existence Results for Multivalued Maps
In this section, we extend the results to cover the multivalued case. We recall some basic definitions on multivalued maps. For more details, one can see [28]. For a normed space ðX, k:kÞ, let P cl ðXÞ = fy ∈ P ðxÞ ; y is closedg, P b ðXÞ = fy ∈ P ðxÞ ; y is boundedg, P cp ðXÞ = fy ∈ P ðxÞ ; y is compactg, and P cp,c ðXÞ = fy ∈ P ðxÞ ; y is compact and convexg: A multivalued map G : X ⟶ P ðXÞ is convex (closed) valued if GðxÞ is convex (closed) for all x ∈ X. The map G is bounded on bounded sets if GðBÞ = ∪ x∈B GðxÞ is bounded in X for all B ∈ P b ðXÞ, i.e., G is called upper semicontinuous (u.s.c.) on X if for each x 0 ∈ X, the set Gðx 0 Þ is a nonempty closed subset of X, and if for each open set N of X containing Gðx 0 Þ, there exists an open neighbourhood N 0 of x 0 such that GðN 0 Þ ⊆ N. G is said to be completely continuous if GðBÞ is relatively 11 Journal of Function Spaces compact for every B ∈ P b ðXÞ. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., x n ⟶ x * , y n ⟶ y * , G has a fixed point if there is x ∈ X such that x ∈ GðxÞ. The fixed point set of multivalued operator G will be denoted by FixG. A multivalued map G : ½0, 1 ⟶ P cl ðℝÞ is said to be measurable if for every y ∈ ℝ, the function is measurable.
(H3) there exists a number L > 0 such that where ψ is given in (39).
Then, the B.V.P has at least one solution on [0,1].
Proof. Let the operatorF : Cð½0, 1, ℝÞ ⟶ P ðCð½0, 1, ℝÞ be defined bỹ > : or g ∈ S G,y . It is clear the fixed points ofF are the solutions of problem (2). So, we need to verify that the operator satisfies all the conditions of the Leray-Schauder nonlinear alternative. This will be done in five steps.
Step 3.FðyÞ maps bounded sets into equicontinuous sets in Cð½0, 1, ℝÞ. Let y be any element in B ε and t 1 , t 2 ∈ ½0, 1, t 1 < t 2 ; then, by using (H2), we get that for each t ∈ ½0, 1, we find that It follows that the right-hand side tends to zero independently of y ∈ B ε as ðt 2 − t 1 Þ ⟶ 0. Combining the outcomes of Steps 1-3 with the Arzela-Ascoli theorem, we conclude thatF is completely continuous.