Existence and Attractivity for Fractional Evolution Equations

Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering. In recent years, there has been a significant development on ordinary and partial differential equations involving fractional derivatives; see the monographs of Podlubny [1], Kilbas et al. [2], Diethelm [3], Tarasov [4], and Zhou [5, 6] and a series of papers [7–28] and the references cited therein. Recently, Zhou [7], Chen et al. [19], Losada et al. [20], and Bana ́ s and O’Regan [21] investigated the attractivity of solutions for fractional ordinary differential equations and integral equations. On the other hand, the existence theory of solutions for time fractional evolution equations has been investigated intensively by many authors; for example, see Kim et al. [16], Bazhlekova [22], Wang et al. [23], Zacher [24], and Zhou et al. [25]. However, to the best of our knowledge, there are no results on the attractivity of solutions for fractional evolution equations in the literature. Consider fractional evolution equation with RiemannLiouville derivative: D0+x (t) = Ax (t) + f (t, x (t)) , t ∈ [0,∞) , I1−α 0+ x (0) = x0, (1)


Introduction
Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering.In recent years, there has been a significant development on ordinary and partial differential equations involving fractional derivatives; see the monographs of Podlubny [1], Kilbas et al. [2], Diethelm [3], Tarasov [4], and Zhou [5,6] and a series of papers  and the references cited therein.
Recently, Zhou [7], Chen et al. [19], Losada et al. [20], and Bana ś and O'Regan [21] investigated the attractivity of solutions for fractional ordinary differential equations and integral equations.On the other hand, the existence theory of solutions for time fractional evolution equations has been investigated intensively by many authors; for example, see Kim et al. [16], Bazhlekova [22], Wang et al. [23], Zacher [24], and Zhou et al. [25].However, to the best of our knowledge, there are no results on the attractivity of solutions for fractional evolution equations in the literature.
In this paper, we initiate the question of the attractivity of solutions for Cauchy problem (1).We establish sufficient conditions for the global attractivity for mild solutions of (1) in the case that semigroup is compact.These results essentially reveal the characteristics of solutions for fractional evolution equations with Riemann-Liouville derivative.More precisely, integer order evolution equations do not have such attractivity.

Preliminaries
In this section, we firstly recall some concepts on fractional integrals and derivatives and then give some lemmas which are useful in next sections.
Let  ∈ (0, 1) and  ∈  1 ([0, ∞), ).The Riemann-Liouville fractional integral is defined by where * denotes the convolution, and in case  = 0, we set  0 () = (), the Dirac measure is concentrated at the origin.For  ∈ ([0,∞),), the Riemann-Liouville fractional derivative is defined by The Wright function   () is defined by It is known that   () satisfies the following equality: We give the following definition of the mild solution of (1).
Suppose that  is the infinitesimal generator of a  0semigroup {()} ≥0 of uniformly bounded linear operators on Banach space .This means that there exists  ≥ 1 such that where () be the space of all bounded linear operators from  to  with the norm ‖‖ () = sup{|()|: || = 1}, where  ∈ () and  ∈ .
For any  ∈   (, ), consider the operator  defined by where It is easy to see lim →0 + Then,  ∈   (, ).Define an operator F as follows: where In this section, we always suppose that the following condition holds: (H1) There exist  ≥ 0 and  1 ∈ (,1), such that Let  1 ∈ (0,  1 ).Then there exists  1 > 0 such that Let Before obtaining our main results, we firstly prove some lemmas as follows.
On the other hand, let  > 0 be given, fixed enough large  > 0 with replaced   such that (26) holds.Then, for  > , by virtue of ( 26) and ( 27), we have Therefore, it is obvious that ‖F  − F‖ → 0 as  → ∞.Combined with the above statement, It can imply that F  → F uniformly on [0, ∞) as  → ∞; that is, F is continuous.

Main Results
Theorem 11.Assume that () ( > 0) is compact, and the condition (H1) holds.Then the Cauchy problem (1) admits at least one attractive solution.
Proof.Obviously,  is a mild solution of ( Since () is compact for  > 0, by Proposition 5, we know that   () is compact.In addition, from the compactness of (  ), we obtain that the set { , ,  ∈  1 } is relatively compact in  for any  ∈ (0, ) and for any  > 0. For every  ∈ Therefore, the set () is closed to an arbitrary compact set.As a result, the set () is also relatively compact set in  for  ∈ [0, ∞).By Lemma 6, we know that F 1 is a relatively compact set.On the other hand, by Lemma 10, we know that F maps  1 into itself and F is continuous.Hence, F is a completely continuous operator.Therefore, according to Schauder fixed point theorem, there exists at least one fixed point  * ∈  1 such that  * = F * holds.Let  * () =  −1  * (); then and, therefore,  * is a mild solution of (1).
Then the Cauchy problem ( 1) admits at least one attractive solution.
Proof.Let  2 ∈ (0,1).Then there exist constants  = (,  2 , ) and  2 > 0 such that The remaining part of the proof is similar to that of Theorem 11, and we omit it.