Solvability of fractional differential inclusions with nonlocal initial conditions via resolvent family of operators


 In this paper, we consider mild solutions to fractional differential inclusions with nonlocal initial conditions. The main results are proved under conditions that (i) the multivalued term takes convex values with compactness of resolvent family of operators; (ii) the multivalued term takes nonconvex values with compactness of resolvent family of operators and (iii) the multivalued term takes nonconvex values without compactness of resolvent family of operators, respectively.


Introduction
A differential inclusion is a generalization of the notion of an ordinary differential equation, which is often used to deal with differential equations with a discontinuous righthand side or an inaccurately known right-hand side [1,2]. Differential inclusions are also from the control problem, for instance, for a control problem x′ = f(x, u), u ∈ U, where u denotes a control parameter. It is founded that the aforementioned control system has the same trajectories as the differential inclusion x′ ∈f (x, U) ⋃ u∈U f (x, u). If the set of controls is dependent upon the state x, i.e. U = U(x), then the differential inclusion x′ ∈ F(x, U(x)) is also achieved. This equivalence between control systems and differential inclusions plays a key role in proving existence theorems in optimal control theory. Differential inclusion has wide applications to models in economics, sociology and bioecology etc., and thus, it has been considerably investigated in last decades, see, for instance, [2][3][4][5][6][7][8] and references therein.
The concept of nonlocal initial condition has been introduced to extend the study of classical initial-valued problems. As indicated in [9], this notion can be more natural and more precise in describing nature phenomena than the classical notion since some additional information is taken into account. For nonlocal initial conditions of abstract differential inclusions, we can refer to [4,6,10,11] and references therein.
In a recent paper, some new properties on the compactness of resolvent family of operators related to fractional differential equations have been established [35]. This new characterization of compactness of resolvent family of operators provides a new way to consider mild solutions of abstract fractional differential equations.

Preliminaries
Let (X, ∥ ⋅ ∥) be a Banach space. We denote P cl (X) {Y∈ P(X) : Y closed}, P b (X) {Y ∈ P(X) : Y bounded}, P cp (X) {Y ∈ P(X) : Y compact} and P cv (X) {Y ∈ P(X) : Y convex}. We also denote by L(X) the space of bounded linear operators from X into X.
A multivalued map G : The multivalued map G : X → P(X) is called upper semicontinuous (u.s.c.) on X if for each x 0 ∈ X, the set G(x 0 ) is a nonempty, closed subset of X, and if for each open set N of X containing G(x 0 ), there exists an open neighbourhood 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 * , y n ∈ G(x n ) imply y * ∈ G(x * ).
The u.s.c. multivalued map G is said to be condensing if for any B ∈ P b (X) with ν(B) ≠ 0, we have ν(G(B)) < ν(B), where ν denotes the Kuratowski measure of noncompactness.
for all ‖x‖ ≤ r and for a.e. t ∈ J.
Lemma 2.1. Let X be a Banach space. Let G : J × X→ P cp, cv (X) be an L 1 -Carathéodory multivalued map with and let Γ be a linear continuous mapping from L 1 (J, X) to C(J, X), then the operator for all u, v ∈ A and all measurable subsets N of J, the function uχ N + vχ J−N ∈ A, where χ denotes the characteristic function.
Let F : J × X → P cp (X). Assign to F the multivalued operator The operator F is called the Niemytzki operator associated to F.
Let Y be a separable metric space and let N : Y → P(L 1 (J, X)) be a multivalued operator. We say that N has property (BC) if (1) N is l.s.c.; (2) N has nonempty closed and decomposable values.
Let (X, d) be a metric space induced by the normed space (X, ∥ ⋅ ∥). Let H d : P(X) × P(X) → R + ⋃ {∞} be defined as for each x, y ∈ X; (ii) a contraction if it is γ-Lipschitz with γ < 1.
For more detailed results on multivalued maps and differential inclusions, we refer to [1,2,4,7,8]. We now give some important properties of resolvent family of operators.
Definition 2.6. [35] Let A be a closed and linear operator with domain D(A) defined on a Banach space X and α > 0. We call A the generator of an (α, 1)-resolvent family if there exists ω ≥ 0 and a strongly continuous function S α : R + → L(X) such that {λ α : Reλ > ω} ⊆ ρ(A) and In this case, the family {S α (t)} t≥0 is called an (α, 1)-resolvent family generated by A.
Definition 2.7. [35] Let A be a closed and linear operator with domain D(A) defined on a Banach space X and 1 ≤ α ≤ 2. We say that A is the generator of an (α, α)-resolvent family if there exist ω ≥ 0 and a strongly continuous function R α : R + → L(X) such that {λ α : Reλ > ω} ⊆ ρ(A) and In this case, the family {R α (t)} t≥0 is called an (α, α)-resolvent family generated by A.
Recall that a strongly continuous family {T(t)} t≥0 ⊆ L(X) is said to be of type (M, ω) if there exist constants M > 0 and ω ∈ R such that ||T(t)|| ≤ Me ωt for all t ≥ 0.
Suppose that S α (t) is continuous in the uniform operator topology for all t > 0, then the following assertions are equivalent: , ω and the following assertions are equivalent: Next, we list some well-known fixed point theorems.
Let Ξ be a bounded, convex and closed subsets of a Banach space X and let ϒ : Ξ → Ξ be a condensing map. Then, ϒ has a fixed point in Ξ.
Lemma 2.6. [1] Let Ξ be a bounded and convex set in Banach space X. ϒ : Ξ → P(Ξ) is an u.s.c., condensing multivalued map. If for every x ∈ Ξ, ϒ(x) is a closed and convex set in Ξ, then ϒ has a fixed point in Ξ.
where Fix(G) denotes the fixed point set of G.

Existence results
In this section, we shall investigate some existence results for mild solutions to Eqs.  For the problem (1.1)-(1.2), according to [35], we have the following definition.
Definition 3.1. Let A be the generator of an (α, 1)-resolvent family S α (t); the mild solutions of the problem (1.1)-(1.2) are defined as follows: We list the following assumptions: Remark 3.1. (i) Of concern, for useful criteria for the continuity of S α (t) in the uniform operator topology, one can refer to the work [37]. For instance, this property holds true for the class of analytic resolvent.
(ii) According to Lemma 2.3, the condition (A1) implies S α (t) is compact for all t > 0.
Proof. Consider the operator N : C(J, X) → P(C(J, X)) defined by where v ∈ S F,x . Clearly, the fixed points of N are mild solutions to (1.1)-(1.2). We shall show that N satisfies all the hypothesis of Lemma 2.6. The proof will be given in several steps.
Step 1. There exists a positive number r such that N( If it is not true, then for each positive number r, there exists a function x r such that h r ∈ N(x r ) but ‖h r (t)‖ > r for some t ∈ J, where v r ∈ S F, x r . However, on the other hand, we have Dividing both sides by r and taking the lower limit as r → ∞, we obtain the following equation: which contradicts the relation (3.1). Step Let θ ∈ (0, 1). Then for each t ∈ J, we have Due to the fact that F has compact values, we may pass to a subsequence if necessary to get that v n converges to v in L 1 (J, X) and hence v ∈ S F,x . Then for each t ∈ J, Step 4. N is u.s.c. and condensing. Now, we decompose N as N 1 + N 2 as We only need to prove that N 1 is a contraction and N 2 is completely continuous.
To show that N 1 is a contraction, for arbitrary x 1 , x 2 ∈ B r and each t ∈ J, we have from (A3) From the relation (3.1), we conclude that N 1 is a contraction.
Next, we show that N 2 is u.s.c. and condensing.
Then there exists a selection v ∈ S F,x such that Then, For the term I 1 , as t 1 → t 2 , we have Next for the term I 2 , we have

Now take into account that
and S α (t 1 − s) − S α (t 2 − s) → 0 in L(X), as t 1 → t 2 (see (A1)). By the Lebesgue's dominated convergence theorem, For t = 0, the conclusion obviously holds. Let 0 < t ≤ b and ε be a real number satisfying 0 < ε < t. For x ∈ B r and v ∈ S F,x such that In view of (A1) and Lemma 2.3, we have S α (t) which is compact for t > 0. Therefore, the set Therefore, let ε → 0, we see that there are relatively compact sets arbitrarily close to the set V(t) = {m(t):m(t) ∈ N 2 (B r )}. Hence, the set V(t) = {m(t):m(t) ∈ N 2 (B r )} is relatively compact in X.
As a consequence of the above steps and the Arzela-Ascoli theorem, we can deduce that N 2 is completely continuous. (iv) N 2 has a closed graph. Let x n → x * , m n ∈ N 2 (x n ) and m n → m * . We shall show that m * ∈ N 2 (x * ). Now m n ∈ N 2 (x n ) implies that there exists v n ∈ S F, x n such that We must prove that there exists v * ∈ S F, x * such that Consider the linear continuous operator defined by From Lemma 2.1, it follows that Γ∘S F is a closed graph operator. Moreover, we have m n (t) ∈ Γ(S F, x n ).
Since x n → x * and m n → m * , it follows again from Lemma 2.1 that m * (t) ∈ Γ(S F, x * ). That is, there must exists v * ∈ S F, x * such that Therefore, N 2 is u.s.c. On the other hand, N 1 is a contraction, hence N = N 1 + N 2 is u.s.c. and condensing. By the fixed point theorem Lemma 2.6, there exists a fixed point x(⋅) for N on B r . Thus, the problem (1.1)-(1.2) admits a mild solution.
▫ Replace the condition (A2)(b) by (b′). There exists a constant τ ∈ (0, 1) and a function ϕ ∈ L 1 (J, R + ) such that From the above proof of Theorem 3.1, we can obtain the following result.
where 1 < α < 2, v ∈ L 1 (J, X). By Laplace transform, we have that is Thus, we have Now, we can give the following definition.  4) can be given as follows: Let us list the following basic assumptions: (A4) Let 1 < α < 2 and A generates an (α, 1)-resolvent family {S α (t)} t≥0 of type (M, ω). (λ α − A) −1 is compact for all λ > ω. (A5) q : C(J, X) → C(J, X) is continuous and there exists L q > 0 such that where v ∈ S F,x . Clearly, the fixed points of N are mild solutions to (1.1)-(1.2). We shall show that N satisfies all the hypothesis of Lemma 2.6. The proof will be given in several steps.
Step 1. There exists a positive number r such that N(B r ) ⊆ B r , where B r {x ∈ C(J, X) : ‖x‖ ∞ ≤ r}. If it is not true, then for each positive number r, there exists a function x r such that h r ∈ N(x r ) but ‖h r (t)‖ > r for some t ∈ J, where v r ∈ S F, x r . However, on the other hand, we have Dividing both sides by r and taking the lower limit as r → ∞, we obtain 1 ≤M L p + bL q + ϕ L 1 , which contradicts the relation (3.3).
Step 2. N(x) is convex for each x ∈ C(J, X).
Let δ ∈ (0, 1). Then for each t ∈ J, we have Because S F,x is convex (since F has convex values), Step 3. N(x) is closed for each x ∈ C(J, X). Let{h n } n≥0 ∈ N(x) such that h n → h in C(J, X). Then h ∈ C(J, X) and there exist {v n } ∈ S F,x such that for Due to the fact that F has compact values, we may pass to a subsequence if necessary to get that v n converges to v in L 1 (J, X) and hence v ∈ S F,x . Then for each t ∈ J, Step 4. N is u.s.c. and condensing. Now, we decompose N as N 1 + N 2 as We only need to prove that N 1 is a contraction and N 2 is completely continuous.
To show that N 1 is a contraction, for arbitrary x 1 , x 2 ∈ B r and each t ∈ J, we have from (A3) and (A5) From the relation (3.3), we conclude that N 1 is a contraction.
Next, we show that N 2 is u.s.c. and condensing. (i) N 2 (B r ) is obviously bounded. (ii) N 2 (B r ) is equicontinuous.
Indeed, let x ∈ B r , m ∈ N 2 (x) and take t 1 , t 2 ∈ J with t 2 < t 1 . Then, there exists a selection v ∈ S F,x such that Then For the term I 1 , as t 1 → t 2 , we have Next for the term I 2 , we have

Now take into account that
, as t 1 → t 2 (see (A4)). By the Lebesgue's dominated convergence theorem, For t = 0, the conclusion obviously holds. Let 0 < t ≤ b and ε be a real number satisfying 0 < ε < t. For x ∈ B r and v ∈ S F,x such that In view of (A4) and Lemma 2.4, we have R α (t) which is compact for t > 0. Therefore, the set K {R α (t − s)v(s), 0 ≤ s ≤ t − ε} is relatively compact. Then, conv K is compact. Considering m ε (t) ∈ tconv K for all t ∈ J, the set V ε (t) {m ε (t) : m ε (t) ∈ N 2 (B r )} is relatively compact in X for every ε, 0 < ε < t. Moreover, for m ∈ N(B r ), Therefore, let ε → 0, we see that there are relatively compact sets arbitrarily close to the set V(t) = {m(t):m(t) ∈ N 2 (B r )}. Hence, the set V(t) = {m(t):m(t) ∈ N 2 (B r )} is relatively compact in X.
As a consequence of the above steps and the Arzela-Ascoli theorem, we can deduce that N 2 is completely continuous. (iv) N 2 has a closed graph. Let x n → x * , m n ∈ N 2 (x n ) and m n → m * . We shall show that m * ∈ N 2 (x * ). Now m n ∈ N 2 (x n ) implies that there exists v n ∈ S F, xn such that We must prove that there exists v * ∈ S F, x * such that Consider the linear continuous operator defined by From Lemma 2.1, it follows that Γ∘S F is a closed graph operator. Moreover, we have m n (t) ∈ Γ(S F, x n ).
Since x n → x * and m n → m * , it follows again from Lemma 2.1 that m * (t) ∈ Γ(S F, x * ). That is, there must exist v * ∈ S F, x * such that Therefore, N 2 is u.s.c. On the other hand, N 1 is a contraction, hence N = N 1 + N 2 is u.s.c. and condensing. By the fixed point theorem Lemma 2.6, there exists a fixed point x(⋅) for N on B r . Thus, the problem (1.1)-(1.2) admits a mild solution.
▫ According to the above proof of Theorem 3.2, we can also have the following result. Let X be a separable Banach space X. We list the following condition: Proof. Hypotheses (A2)(b) and (C1) imply that F is of l.s.c. type. In view of Lemma 2.2, there exists a continuous function f : C(J, X) → L 1 (J, X) such that f (x) ∈ F (x) for all x ∈ C(J, X). Now consider the following equation:

6)
Notice that if x ∈ C(J, X) is a solution of the problem (3.5)-(3.6), then x is also a solution of the problem (1.1)-(1.2). Next, we transform the problem (3.5)-(3.6) into a fixed point problem by defining N : C(J, X)→ C(J, X) as We shall show that N satisfies all the hypothesis of Lemma 2.5. The proof will be given in several steps.
Step 1. There exists a positive number r such that This can be conducted similarly as Step 1. in the proof of Theorem 3.1.
We decompose N as N 1 + N 2 as Step 2. N 2 is continuous on B r . Let {x n } be a sequence such that x n → x in B r . Then Note that ϕ ∈ L 1 (J, R + ), ∫ t 0 ||f (x n )(s) − f (x)(s)|| ds → 0, n → ∞ by the Lebesgue's dominated convergence theorem. Hence, N 2 is continuous.
Step 3. N is condensing. Similarly conducted as the proof of Theorem 3.1, we can prove that N 1 is a contraction and N 2 is completely continuous.
From the above three steps, we can complete the proof via Lemma 2.5.
Similarly conducted as the proof of Theorems 3.2 and 3.3, we can prove that N 1 is a contraction and N 2 is completely continuous. Thus, Lemma 2.5 can be applied to complete the proof.
Proof. Transform the problem (1.1)-(1.2) into a fixed point problem. Let the multivalued operator N : C(J, X)→ P(C(J, X)) be defined as in Theorem 3.1. We shall prove that N admits at leas one fixed point. We divide the proof into two steps.
This can be proved just as Step 3 in the proof of Theorem 3.1.
Step 2. For each x,x ∈ C(J, X), there exists a constant 0 < γ < 1 such that H d (N(x), N(x)) ≤ γ x −x ∞ . Let x,x ∈ C(J, X) and h ∈ N(x). Then there exists v ∈ S F,x such that for each t ∈ J Consider U : J → P(X) defined as Because U(t) W(t)⋂ F(t,x) is measurable (see [38,Proposition III.4]), there exists a functionṽ(t), which is a measurable selection for U. Hence,ṽ(t) ∈ F(t,x(t)) and For each t ∈ J, we now definẽ Then for each t ∈ J, we have By an analogous relation, obtained by interchanging the roles ofx and x, we can obtain Owing to relation (3.7), we conclude that N is a contraction. Thus, by Lemma 2.7, N admits a fixed point, which just is one mild solution to the problem (1.1)-(1.2). ▫ Theorem 3.6. Let 1 < α < 2 and A generates an (α, 1)resolvent family {S α (t)} t≥0 of type (M, ω). Suppose that conditions (A3), (A5) and (A6) are satisfied, then the problem (1.3)-(1.4) has at least one mild solution on J provided thatM Proof. Transform the problem (1.3)-(1.4) into a fixed point problem. Let the multivalued operator N : C(J, X)→ P(C(J, X)) be defined as in Theorem 3.2. We shall prove that N admits at least one fixed point. We divide the proof into two steps.
Step 1. For each x ∈ C(J, X), N(x) ∈ P cl (C(J, X)). This can be proved just as Step 3 in the proof of Theorem 3.2.
Step 2. N is a contraction.
By an analogous relation, obtained by interchanging the roles ofx and x, we can obtain H d (N(x), N(x)) ≤M L p + bL q + ||l|| L 1 x −x ∞ .
Owing to relation (3.8), we conclude that N is a contraction. Thus, by Lemma 2.7, N admits a fixed point, wh1ich just is one mild solution to the problem (1.3)-(1.4). ▫ Example 3.1. As a simple application, we consider the following equations: It is well known that A generates a compact and analytic (and hence norm continuous for all t > 0) C 0 -semigroup {T(t)} t≥0 on X such that ‖T(t)‖ ≤ 1. Now, we can extract an (α, α)-resolvent family {R α (t)} t≥0 of type (1, 1) (see [39]). Meanwhile, the compactness of T(t) implies that (λ α − A) −1 is compact.
According to Theorem 3.2, the problem (3.9)-(3.11) has at least one mild solution on J.

Conclusions
In this paper, we establish some sufficient conditions to guarantee the existence of mild solutions to abstract fractional differential inclusions with nonlocal initial conditions under conditions that (i) the multivalued term takes convex values with compactness of resolvent family of operators; (ii) the multivalued term takes nonconvex values with compactness of resolvent family of operators and (iii) the multivalued term takes nonconvex values without compactness of resolvent family of operators.
The main results are based upon theories of resolvent family of operators, multivalued analysis and fixed point approach. It is noted that several partial differential equations arising in physics and applied sciences can be described by fractional differential equations of degenerate type (cf. [40,41]); we propose to investigate the existence of solutions to fractional stochastic equations of degenerate type via the resolvent family in future works.