MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS∗

Abstract The purpose of this paper is concerned with the existence of mild L-quasi-solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces E. By employing mixed monotone iterative technique, measure of noncompactness and Sadovskii’s fixed point theorem, we obtain the existence of mild L-quasi-solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provide to illustrate the feasibility of our main results.


Introduction
Fractional differential equations provide an excellent instrument for the description of memory and hereditary properties of various materials and processes and there has been a significant development in fractional differential equations theory. Hilfer [14] proposed a generalized Riemann-Liouville fractional derivative, for short, Hilfer fractional derivative, which includes Riemann-Liouville fractional derivative and Caputo fractional derivative. This operator appeared in the theoretical simulation of dielectric relaxation in glass forming materials.
In recent years, many authors began to consider Hilfer fractional differential equations, we refer the reader to [1,2,9,11,12,14,15,30]. Hilfer fractional evolution equations has also been widely concerned by many scholars. In [11], Gu and Trujillo investigated a class of Hilfer fractional evolution equations, and established the existence results of mild solutions by using fixed point theorem.
Later, the nonlocal problems have better effects in applications than the initial problem, many contributions have been made in applications of fractional evolution equations with nonlocal conditions, see [20,23,24] and the reference therein. In [20], Liang and Yang investigated the exact controllability of the nonlocal Cauchy problem for the fractional integro differential evolution equations in Banach spaces      D q x(t) + Ax(t) = f (t, x(t), Gx(t)) + Bu(t), t ∈ J, where D q denotes the Caputo fractional derivative of order q ∈ (0, 1), −A : D(A) ⊂ E → E is the infinitesimal generator of a C 0 -semigroup T (t)(t ≥ 0) of uniformly bounded linear operator, B is a linear bounded operator; f is a given function and the operator is given by In [1], Hamdy M. Ahmed  where (t, v(t)) = (t, u(t), u(b 1 (t))), . . . , u(b m (t))) and (t, η(t)) = (t, u(t), u(a 1 (t))), . . . , u(a n (t))), D ν,µ 0+ denotes the Hilfer fractional derivative 0 ≤ ν ≤ 1, 0 < µ < 1, −A is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t), t ≥ 0 on a separable Hilbert space.
On the other hand, by employing the method of lower and upper to study the existence of extremal mild solution for fractional evolution equation is an interesting issue, which has been attention in [6,21,23,24,26]. In [6], Chen and Li used monotone iterative technique in the presence of coupled lower and upper L-quasi-solutions to discuss the existence of mild solutions to the initial value problem of impulsive evolution equations in an ordered Banach space E: ∆u| t=t k = I k (u(t k ), u(t k )), k = 1, 2, . . . , m, is an impulsive function, k = 1, 2, . . . , p; u 0 ∈ E.
In [27], Vikram Singh et al. investigated the existence and uniqueness of mild solutions for Sobolev type fractional impulsive differential systems with nonlocal conditions By applying monotone iterative technique combined with the method of lower and upper solutions.
However, there are few papers that study Hilfer fractional evolution equations with nonlocal problems by applying the mixed monotone iterative technique and coupled L-quasi-upper and lower solutions. Motivated above discussion, in this paper, we use the fixed point theorem combined with mixed monotone iterative technique to discuss the existence of mild L-quasi-solutions for Hilfer fractional evolution equations with nonlocal conditions where D ν,µ 0+ denotes the Hilfer fractional derivative of order µ and type ν, which will be given in the next section, is given functions satisfying some assumptions, u 0 ∈ E and τ i (i = 1, 2, . . . , m) are prefixed points satisfying 0 < τ 1 ≤ · · · ≤ τ m < b and λ i are real numbers. Here the nonlocal can be applied in physical problem better effect than the initial conditions I 1−γ 0+ u(0) = u 0 . The rest of this paper is organized as follows: In Section 2, we review some Lemmas and notations. In Section 3, we prove the existence of mild L-quasi-solutions for Hilfer fractional differential system (1.1). In Section 4, an example is given to illustrate the effectiveness of the our results.

Preliminaries
Throughout this paper, by C(J, E) and C(J ′ , E), we denote the spaces of all continuous functions from J to E and J ′ to E, respectively. Let E be an ordered Banach space with the norm ∥ · ∥ and partial order ≤, whose positive cone is a Banach space with the norm ∥u∥ γ = sup t∈J ′ |t 1−γ u(t)|. And C 1−γ (J, E) is also an ordered Banach space with the partial order ≤ induced by the positive cone which is also normal with the same normal constant N . First, we recall some definitions and basic results on fractional calculus, for more details see [9,11,15,19,30].
provided the right side is point-wise defined on [0, ∞). Definition 2.2. The Riemann-Liouville derivative of order γ with the lower limit zero for a function f : [0, ∞) → R can be written as Definition 2.3. The Caputo fractional derivative of order γ for a function f : [0, ∞) → R can be written as where n = [γ] + 1 and [γ] denotes the integer part of γ.
Definition 2.4 (Hilfer fractional derivative see [14]). The generazlied Riemann-Liouville fractional derivative of order 0 ≤ ν ≤ 1 and 0 < µ < 1 with lower limit a is defined as for functions such that the expression on the right hand side exists.

Lemma 2.5 ( [11]). Assume that −A is the infinitesimal generator of a
if and only if u satisfies the following integral equation: the function ξ µ is the function of Wright type: (ii) The operators S ν,µ (t) and K µ (t) are strongly continuous for all t ≥ 0.
is an equicontinuous semigroup, then S ν,µ (t) and K µ (t) are equicontinuous in E for t > 0. Definition 2.5. A function u ∈ C 1−γ (J, E) is said to be a mild solution of (2.2) if u 0 ∈ E the integral equation Next, we present useful lemma which plays an important role in our main results. (2.4)

Proof.
Let λ > 0, we consider the one sided stable probability density as follows whose Laplace transform is given by Then, using (2.5), we have where ξ µ is a probability density function defined on (0, ∞) such that Since the Laplace inverse transform of λ ν(µ−1) is where δ(t) is the Delta function. It follows from (2.6), (2.7) and Laplace transform, it is obvious to see that where * denotes the convolution of functions. By Remark 2.2, we obtain Combing (2.8) and (2.9) yields Similarly, we have Thus, it follows from (2.10) and (2.11) that For the convenience of discussion, we assume that In view of [6] and [20], we present the following lemma.

Lemma 2.8. Assume that (H0) and (H1) holds. For any
, then the problem (1.1) has a unique mild solution u ∈ C 1−γ (J) given by Proof. By assumption (H1), we have By operator spectrum theorem, the operator Θ : exists and is bounded. Furthermore, by Neumann expression, we obtain According to Definition 2.5, a solution of system (2.2) can be expressed by Next, we substitute t = τ i into (2.13) and by applying λ i to both side of (2.13), we have Thus, we have has a bounded inverse operator Θ, which implies Submitting (2.16) to (2.14), we obtain that (2.13). It is imply that u is also a solution of the integral of Eq.(2.13) when u is a solution of system (2.12). The necessity has been proved. Next, we will prove its sufficiency. Applying I 1−γ 0+ to both side of (2.12), and by Lemma 2.7, we have Therefore, we have Then, we obtain It follows (2.16) and (2.17) that Next, by using D ν,µ 0+ to both sides of (2.12) and Lemma 2.9, we have Hence, This proof is completed. From Lemma 2.8, we adopt the following definition of mild solution of the problem (1.1). Definition 2.6. A function u ∈ C 1−γ (J, E) is said to be a mild solution of the problem (1.1), if it satisfies the operator equation where the operators S ν,µ (t) and K µ (t) are given by (2.3).
Since T (t)(t ≥ 0) is positive, by Remark 2.4, it is easy know that S(t)(t ≥ 0) is also positive. And by the definition of ξ µ (σ), the operators S * ν,µ (t) and K * µ (t) are also positive for all t ≥ 0.
To prove our main result, for any C > 0, we consider the following the system (F1) λ i > 0(i = 1, 2, . . . , m) and By assumption (F1), we have By operator spectrum theorem, the operator I − m i=1 λ i S * ν,µ (τ i )) has a bounded inverse operator Furthermore, by Neumann expression, Θ can be expressed by By the positivity of C 0 -semigroup S(t)(t ≥ 0), it is easy know that S * ν,µ (t) is positive, we have So, Θ is a positive operator, and In view of Lemma 2.8, we present the following lemma.

Lemma 2.9. Assume that (F0) and (F1) holds. For any
, then the problem (2.20) has a unique mild solution u ∈ C 1−γ (J) given by From Lemma 2.9 and Definition 2.7, we state the following definition of mild solution of the problem (2.20). E) is said to be a mild solution of the problem (2.20), if for any u ∈ C 1−γ (J, E), the integral equation is satisfied.
In the following, we will state some lemmas whose proofs are similar to those of the paper [11]. Here, we omit it.

Main results
For In this section, we will discuss the existence of extremal mild solutions for problem (1.1).

Definition 3.2. Let
are satisfied, then the problem (1.1) has minimal and maximal coupled mild Lquasi-solutions between v 0 and w 0 , which can be obtained by a monotone iterative procedure starting from v 0 and w 0 respectively.
Proof. Since C > 0, the problem (1.1) can be written as the system (2.20). By (2.21), we can define operator Q :

v(s))+(C + L)u(s)−Lv(s)]ds
Since f is continuous, it is easily see that the map Q :→ C 1−γ (J, E) is continuous. And by Lemma 2.9, the mild solutions of the problem (1.1) are equivalent to the fixed points of the operator Q. We will divide the proof in the following steps.

u(s)) + (C + L)u(s) − Lu(s)]ds
Thus, we have u(t), u(t) ∈ C 1−γ (J, E), and u = Qu, u = Qu. Combing this with monotonicity (3.5), we see that v 0 ≤ u ≤ u ≤ w 0 . By the monotonicity of Q, it is easy to see that u and u are the minimal and maximal coupled fixed points of Q in [v 0 , w 0 ]. Therefore, u and u are the minimal and maximal coupled mild L-quasi-solutions of the problem (1.1) in [v 0 , w 0 ], respectively..

Remark 3.1.
If we replace positive cone P is normal by positive cone P is regular. Then the conclusion in Theorem 3.1 is also valid. For more detail, see [6].
As a supplement to Theorem 3.1, we further discuss the existence of mild solutions for the problem (1.1) in weakly sequentially complete Banach space, we only need to verify the conditions (F1) and (F2) are satisfied. Proof. In view of Theorem 3.1, if E is weakly sequentially complete, the condition (F3) and (F4) holds automatically. And by Theorem 2.2 in [8], any monotonic and order bounded sequence is precompact. By the monotonicity (3.3), it is east to see that v n (t) and w n (t) are convergent on J. Thus, v n (0) and w n (0) are convergent, i.e. condition (F4) holds. For t ∈ J, let {u n } and {v n } be increasing or decreasing sequences obeying condition (F3), then by condition (F1), {f (t, u n , v n ) + Cu n − Lv n } is a monotone and order-bounded sequence. By the property of measure of noncompactness, we have and (F3) holds and by Theorem 3.1, our conclusion is valid. Now, we discuss the exists of mild solution to the problem (1.1) between the minimal and maximal coupled mild L-quasi-solutions u and u. If we replace the assumptions (F3) by the following assumptions: (F3)* The exists a L 1 > 0 such that We have the following results.
Proof. It is easy to see that (F 3) * ⇒ (H3). Hence, by Theorem 3.1, the problem (1.1) has minimal and maximal coupled mild L-quasi-solutions u and u between v 0 and w 0 . Next, we prove the existence of the mild solution of the equation between v 0 and w 0 . Let Au = Q(u, u), clearly, we know that A : [v 0 , w 0 ] → [v 0 , w 0 ] is continuous and the mild solution of the problem (1.1) is equivalent to fixed point of operator A. First, we will prove that A : is a equicontinuous C 0 -semigroup, and S(t)(t ≥ 0) is also a equicontiuous C 0 -semigroup. By the normality of the cone P , there exists M > 0 such that For 0 < t 1 < t 2 ≤ b, by (3.1), we get that For J 3 , by Lemma 2.10, we have For J 4 , by Lemma 2.10, we have For J 5 , by Lemma 2.10, we have Noting that It is easy to see that lim t2→t1 J 5 = 0. For J 6 , by Lemma 2.10, we have In conclusion, For t ∈ J, by the definition of the operator Q, we have , u n (s) + Cu n (s)))]ds Since A(D 0 ) is bounded and equicontinuous, we know from Lemma 2.3 that And by (3.8), we have .

Examples
In this section, we present an example, which illustrate the applicability of our main results.
Example 4.1. We consider the following fractional partial differential equation Then E is a Banach space, P is a normal cone of E, and −A generates a positive C 0semigroup T (t)(t ≥ 0) in E (see [25]). Let f (t, u(t), u(t)) = f (t, x, u(t, x), u(t, x)), u 0 = u 0 (·), then the problem (4.3) can be written as the abstract (1.1).
Proof. Assumption (H5) implies that v 0 ≡ 0 and w 0 ≡ w(x, t) are lower and upper solutions of the problem (4.3), respectively, and from (H6)-(H8), it is easy to verify that all conditions (F1)-(F3) are satisfied. So our conclusion follows from Theorem 3.1.