Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(A,B)$\end{document}(A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.


Introduction
Fractional differential equations have been successfully applied to various fields, for example, physics, engineering, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [-]. In [-], the authors discussed the existence of solutions for various nonlinear differential equations or partial differential equations by measures of noncompactness and fixed point theorems, whereas in [-], the authors investigated the existence of solutions for the evolution equations by the monotone iterative method.
On the other hand, Hilfer [] proposed a generalized Riemann-Liouville fractional derivative (for short, the Hilfer fractional derivative), which includes the Riemann-Liouville and Caputo fractional derivatives. Furati et al. [] considered an initial value problem for a class of nonlinear fractional differential equations involving the Hilfer fractional derivative. Very recently, Gu and Trujillo [] investigated a class of evolution equations involving the Hilfer fractional derivatives by using Mittag-Leffler functions. To the best of our knowledge, there are no results about nonlinear fractional nonautonomous evolution equations of Sobolev type with delay. Motivated by the above discussion, in this paper, we use the fixed point theorems combined with the theory of propagation family to discuss the existence of mild solutions for nonlinear fractional nonautonomous evolution equations of Sobolev type with delay of the form ⎧ ⎨ ⎩ D ν,μ + Bu(t) = Au(t) + Bf (t, u(τ  (t)), . . . , u(τ m (t))), t ∈ J, I (-ν)(-μ) + where D ν,μ + is the Hilfer fractional derivative, which will be given in next section,  ≤ ν ≤ ,  < μ < , the state u ( Furthermore, we study problem (.) without assuming that B has bounded (or compact) inverse and without any assumption on the relation between D(A) and D(B). Our purpose is to introduce the theory of propagation family {W (t)} t≥ (an operator family generated by the operator pair (A, B); see Definition .) from Jin Liang and Ti-Jun Xiao [] and probability density function and then to give a proper definition of mild solutions for nonlinear fractional nonautonomous evolution equations (.), which plays a key role in our discussion. The existence of a mild solution for problem (.) is obtained under certain assumptions on the nonlinear term f by using the Hilfer fractional derivative, measure of noncompactness, and fixed point theorem. At last, as an application, we also obtain the existence of mild solutions for the nonlinear time fractional reaction-diffusion equation introduced by Ouyang [] and Zhu, Liu, and Wu [], where is the Laplace operator, ∈ R m is a bounded domain with a sufficiently smooth boundary ∂ , f : J × R m → R is a nonlinear function, and ϕ ∈ L  ( ). The rest of this paper is organized as follows: In Section , we recall some basic known results and introduce some notations. In Section , we discuss the existence theorems of mild solutions for problem (.). At last, two examples are presented to illustrate the main results.
with norm · ν,μ defined by For completeness, we recall the following definitions from fractional calculus.
Definition . The Riemann-Liouville fractional integral of order α of a function f : provided that the right-hand side is pointwise defined on (, ∞).
Definition . The Riemann-Liouville derivative of order α with the lower limit zero for a function f : [, ∞) → R can be written as Definition . The Caputo fractional derivative of order α for a function f : [, ∞) → R can be written as where n = [α] + , and [α] denotes the integer part of α.
If u is an abstract function with values in E, then the integrals appearing in Definitions . and . are taken in Bochner's sense.
Definition . (Hilfer fractional derivative; see []) The generalized Riemann-Liouville fractional derivative of order  ≤ ν ≤  and  < μ <  with lower limit a is defined as for functions such that the expression on the right-hand side exists.

Remark .
(i) When ν = ,  < μ < , and a = , the Hilfer fractional derivative corresponds to the classical Riemann-Liouville fractional derivative: (ii) When ν = ,  < μ < , and a = , the Hilfer fractional derivative corresponds to the classical Caputo fractional derivative: Now, we recall the basic definitions and properties of the Kuratowski measure of noncompactness.

Definition . ([]
) Let E be a Banach space, and let E be the bounded subsets of E. The Kuratowski measure of noncompactness is the map α :  Then α( (t)) is the Lebesgue integral on J, and

Lemma . ([]) Let E be a Banach space. Assume that D ⊂ E is a bounded closed convex set on E and that the operator Q : D → D is k-set-contractive. Then Q has at least one fixed point in D.
We recall the abstract degenerate Cauchy problem []: Banach space E such that {W (t)} t≥ is exponentially bounded, which means that, for any u ∈ D(B), there exist a >  and M >  such that is called an exponentially bounded propagation family for (.) if for λ > a, In this case, we also say that (.) has an exponentially bounded propagation family Moreover, if (.) holds, we also say that the pair (A, B) generates an exponentially bounded propagation family {W (t)} t≥ .

Lemma . ([]) Problem (.) is equivalent to the integral equation
Proof Let λ > . Applying the Laplace transform Then and thus provided that the integral in (.) exists, where I is the identity operator on E.
We consider the following one-sided stable probability density in []: whose Laplace transform is given by Thus, it follows from (.), (.), and (.) that, for t ∈ J, Since the Laplace inverse transform of λ ν(μ-) is where δ(t) is the delta function, we invert the last Laplace transform to obtain where ξ μ is the probability density function defined on (, ∞) by This completes the proof.
Based on Lemma ., we give the following definition of a mild solution of problem (.).
Definition . By a mild solution of problem (.) we mean a function u ∈ C(J , E) that satisfies (ii) When ν = , the fractional equation (.) simplifies to the classical Riemann-Liouville fractional equation studied by Zhou et al. []. In this case, (iii) When ν = , the fractional equation (.) simplifies to the classical Caputo fractional equation studied by Zhou and Jiao []. In this case, .
Hence, we have For t ∈ J and u ∈ E, we have This completes the proof. Proof For any u ∈ E and  < t  < t  ≤ b, we have Since W (t) is a norm-continuous family for t > , we have that is, {S ν,μ } t> is strongly continuous. This completes the proof.

Main results
In this section, we will state and prove our main results. First of all, we introduce the following assumptions: (H) {W (t)} t≥ is a norm-continuous family for t >  and uniformly bounded, that is, there exists M >  such that W (t) ≤ M. (H) For some r > , there exist a constant ρ >  and functions h r ∈ L p (J, R + ) (p >  μ > ) such that, for any t ∈ J and u k ∈ E satisfying u k ≤ r for k = , , . . . , m, and Proof We consider the operator Q : By direct calculation we know that the operator Q is well-defined. From Definition . it is easy to verify that the mild solution of problem (.) is equivalent to the fixed point of the operator Q defined by (.). In the following, we will prove that the operator Q : C ν,μ (J, E) → C ν,μ (J, E) has at least one fixed point by applying the fixed point theorem with respect to a k-set-contractive operator. Our proof will be divided into four steps.
Then B r is a closed and convex subset of C ν,μ (J, E). Observe that, for all u ∈ B r , .
Define t (-ν)(-μ) (Qu)(t) as follows: Step . We show that there exists r >  such that QB r ⊂ B r . Suppose this is not true. Then for each r > , there exists u r (·) ∈ B r such that (Qu r )(t) > r for some t ∈ J. Combining Lemma ., assumptions (H) and (H), and the Hölder inequality, we get that Dividing both sides of (.) by r and taking the lower limit as r → +∞, by (.) we get which is a contradiction. Therefore Q(B r ) ⊂ B r for some r > .
Step . Now we show that Q is continuous from B r into B r . To show this, for any u, u n ∈ B r , n = , , . . . , with lim n→∞ u nu ν,μ = , we get For t ∈ J and u n , u ∈ B r , we have which implies that Qu n → Qu uniformly on J as n → ∞, and so Q : B r → B r is a continuous operator.
Step . We will prove that {Qu : u ∈ B r } is an equicontinuous family of functions. For any u ∈ B r and  ≤ t  < t  ≤ b, by (.) and assumptions (H) and (H) we get that where . . , u τ m (s) ds , In conclusion, as t  → t  , which means that the operator Q : B r → B r is equicontinuous. Let H = coQ(B r ). Then it is easy to verify that Q maps H into itself and H ⊂ B r is equicontinuous.
Step . Now, we prove that Q : H → H is a condensing operator. For any D ⊂ H, by Lemma . there exists a countable set D  = {u n } ⊂ D such that By the equicontinuity of H we know that D  ⊂ D is also equicontinuous.
For t ∈ J, by the definition of Q and (H) we have Since Q(D  ) ⊂ H is bounded and equicontinuous, we know from Lemma . that Therefore we have Thus, Q : B r → B r is a k-set-contractive operator. It follows from Lemma . that Q has at least one fixed point u ∈ B r , which is just a mild solution of problem (.) on the interval J.
We further present two special cases. Case . When B = I, then D(B) = E. We assume that A-generate a norm-continuous semigroup {W (t)} t≥ of uniformly bounded linear operators on E. Then from the proof of Theorem . we have the following theorem.

Applications
In this section, we present two examples, which illustrate the applicability of our main results.
Example . We consider the following fractional diffusion equations of Sobolev type with delay: x ∈ ∂ , t ∈ J, where D ν,μ + is the Hilfer fractional derivative,  ≤ ν ≤ ,  < μ < , τ k : J → J are continuous functions such that  ≤ τ k (t) < t, k = , , . . . , m, ⊂ R m is a bounded domain with a sufficiently smooth boundary ∂ , and f : J × R m → R is continuous.
Let E = L  ( ) be the Banach space with the L  -norm ·  . We define where H  ( ) is the completion of the space C  ( ) with respect to the norm Then equation (.) can be rewritten in the abstract form as (.).
Example . We consider the initial boundary value problem to the following nonlinear time fractional reaction-diffusion equation with delay introduced in [, ]: where D ν,μ + is the Hilfer fractional derivative,  ≤ ν ≤ ,  < μ < , J = [, b], m is a positive integer number, the diffusion coefficient a(t) is continuous on J and |a(t  )a(t  )| ≤ C|t t  | γ ,  < γ ≤ , t  , t  ∈ J, C is a positive constant independent of t  and t  , is the Laplace operator, τ k : J → J are continuous function such that  ≤ τ k (t) < t, k = , , . . . , m, ⊂ R m is a bounded domain with a sufficiently smooth boundary ∂ , f : J × R m → R is continuous, and ϕ ∈ L  ( ).
Let E = L  ( ) be the Banach space with the L  -norm ·  . We define where H  ( ) is the completion of the space C  ( ) with respect to the norm Proof By assumptions (i)-(ii) we can easily verify that conditions (H)-(H) are satisfied with L k = l k (k = , , . . . , m). Furthermore, also from assumptions (i)-(ii) combined with assumption (.) we know that (.) are satisfied. Therefore, our Theorem . follows.

Conclusions
In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using the Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair (A, B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to a k-set-contractive operator, we obtain a new result on the existence of mild solutions with the assumption that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions on this topic.