On a new class of impulsive fractional evolution equations

This paper is concerned with the existence of PC-mild solutions for Cauchy and nonlocal problems of impulsive fractional evolution equations for which the impulses are not instantaneous. By using the theory of operator semigroups, probability density functions, and some suitable fixed point theorems, we establish some existence results for these types of problems.


Introduction
Recently, much attention has been paid to the study of fractional differential equations due to the fact that they have been proved to be valuable tools in the mathematical modeling of many phenomena in physics, biology, mechanics, etc. (see [-]).
Impulsive differential equations of integer order have found extensive applications in realistic mathematical modeling of a wide variety of practical situations, such as biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, and frequency modulated systems. For the general theory and relevant developments of impulsive differential equations, please see [-] and the references therein. Usually the impulses of the evolution process described by impulsive differential equations are assumed to be abrupt and instantaneous. That is to say, the perturbations (impulses) start abruptly and the duration of them is negligible in comparison with the duration of the process.
However, in [], the authors introduced a new class of abstract impulsive differential equations for which the impulses are not instantaneous. Specifically, they studied the existence of solutions for the following impulsive problem: , t ∈ (s i , t i+ ], i = , , , . . . , N, u(t) = g i (t, u(t)), t ∈ (t i , s i ], i = , , . . . , N, where A : D(A) ⊂ X → X is the generator of a C  -semigroup of bounded linear operators {T(t)} t≥ defined on a Banach space (X, · ), x  ∈ X,  = t  = s  < t  ≤ s  < t  ≤ s  < · · · < t N ≤ s N < t N+ = a are pre-fixed numbers, g i ∈ C((t i , s i ] × X; X) for i = , , . . . , N and f : [, a] × X → X is a suitable function. The impulses start abruptly at the points t i and their action continues on the interval [t i , s i ]. As a motivation for the study of such systems, see [], where an example of the hemodynamical equilibrium of a person was given.
Impulsive differential equations of fractional order have been studied by some authors, for example [-]. As for the study of impulsive fractional evolution equations, to the best of our knowledge, there are few papers [-] on this topic.
Motivated by [], in this paper we consider a class of impulsive fractional evolution equations of the form where c D α is the Caputo fractional derivative of order α ∈ (, ) with the lower limit zero, A : D(A) ⊂ X → X is the generator of a C  -semigroup of bounded linear operators The impulses in problem () start abruptly at the points t i and their action continues on the interval [t i , s i ]. To be precise, the function x takes an abrupt impulse at t i and follows different rules in the two subintervals (t i , s i ] and (s i , t i+ ] of the interval (t i , t i+ ]. At the point s i , the function x is continuous. The term I i (x(t i )) means that the impulses are also related to the value of x(t i ) = x(ti ). From the results obtained in the papers [-], we know that the definition of mild solutions for fractional evolution equations is more involved than integer order evolution equations. Moreover, to construct solutions for impulsive fractional differential equations, we should properly handle the fractional derivative and impulsive conditions due to the memory property of fractional calculus (see [-]).
We remark that if t i = s i and the second equation of () takes the form of representing the right and left limits of x(t) at t = t i , problem () reduces to the case considered in [] (with the fixed impulses).
We also study the nonlocal Cauchy problems for impulsive fractional evolution equa- where A, f , I i , g i are the same as above, b is a given function; this constitutes a nonlocal Cauchy problem. It is well known that the nonlocal condition has a better effect on the solution and is more precise for physical measurements than the classical initial condition alone. The rest of the paper is organized as follows. In Section  we present the notations, definitions and preliminary results needed in the following sections. In Section , a suitable concept of PC-mild solutions for our problems is introduced. Section  is concerned with the existence results of problems () and (). An example is given in Section  to illustrate the results. () W is a uniformly bounded subset of PC(J, X);

Preliminaries
. . , m, are relatively compact subsets of X. Then W is a relatively compact subset of PC(J, X).
Let us recall the following well-known definitions.

Definition . ([])
The Riemann-Liouville fractional integral of order q with the lower limit zero for a function f is defined as provided the integral exists, where (·) is the gamma function.

Definition . ([])
The Riemann-Liouville derivative of order q with the lower limit zero for a function f : [, ∞) → R can be written as (b) If f is an abstract function with values in X, then the integrals in Definitions . and . are taken in Bochner's sense.
Let us recall the following definition of mild solutions for fractional evolution equations involving the Caputo fractional derivative.

Definition . ([, ])
A function x ∈ C(J, X) is said to be a mild solution of the following problem: if it satisfies the integral equation Here and ξ α is a probability density function defined on (, ∞) [], that is, It is not difficult to verify that Remark . By applying the Laplace transform and probability density functions, Zhou and Jiao [, ] introduced the above definition of mild solutions for fractional evolution equations. For pioneering work on Caputo fractional evolution equations, we refer the readers to [, ].
We make the following assumption on A in the whole paper.

H(A): The operator A generators a strongly continuous semigroup
hold, then the operators P α and Q α have the following properties: () For any fixed t ≥ , P α (t) and Q α (t) are linear and bounded operators, and for any x ∈ X, , t ≥ } are strongly continuous; () for every t > , P α (t) and Q α (t) are compact operators.
Finally we recall a fixed point theorem which will be needed in the sequel.

The construction of mild solutions
Let y ∈ PC(J, X). We first consider the following fractional impulsive problem: From the property of the Caputo derivative, a general solution of problem () can be written as where d i , i = , , . . . , m, are elements of X. By () and the function x is continuous at the points s i , we have, for i = , , , . . . , m, Now we follow the idea used in the papers [, ] and apply the Laplace transformation for () to obtain Then where I is the identity operator defined on X. Note that the Laplace transform of α (θ ) defined by () is given by Then by the same computations in [, ] and the properties of Laplace transform (translation theorem and linearity of the inverse Laplace transform), we obtain Here P α and Q α are given by (). Hence we get Now it is time to determine the values of d i , i = , , . . . , m. Using the fact that x is continuous at the points s i , we have So we obtain Therefore, a mild solution of problem () is given by where, for i = , , . . . , m, Next, by using the above results, we introduce the following definition of the mild solution for problem ().
Definition . A function x ∈ PC(J, X) is said to be a PC-mild solution of problem () if it satisfies the following relation: where, for i = , , . . . , m, Remark . For treating the mild solutions for abstract fractional differential equations, we can also refer to [].

Existence results
This section deals with the existence results for problems () and (). From Definition ., we define an operator S : PC(J, X) → PC(J, X) as with d i , i = , , . . . , m, defined by ().
To prove our first existence result we introduce the following assumptions.

Then there exists a unique PC-mild solution of problem ().
Proof From the assumptions it is easy to show that the operator S is well defined on PC(J, X).
xy PC .
From the above we can deduce that (since xy PC . Then it follows from condition () that S is a contraction on the space PC(J, X). Hence by the Banach contraction mapping principle, S has a unique fixed point x ∈ PC(J, X) which is just the unique PC-mild solution of problem (). The proof is now complete.
The next result is based on the Krasnoselskii fixed point theorem.
for all x ∈ X and t ∈ J. H(Ig)  : There exist m g ∈ C(J, R + ) and ϕ I , ϕ g ∈ C([, ∞), R + ) nondecreasing such that, for all x ∈ X, i = , , . . . , m, and there exists a constant r >  such that Then there exists a PC-mild solution of problem ().
Proof We define two operators S  , S  : PC(J, X) → PC(J, X) as for i = , , . . . , m. Since P α () = I, it is easy to verify that for any x ∈ PC(J, X), S  x, S  x ∈ PC(J, X), hence they are well defined. We have Let r >  satisfy condition (). We set Then M is a closed, convex, and nonempty subset of the Banach space PC(J, X).
Next we will show that the operators S  , S  satisfy the requirements of Theorem ., i.e. S  is a contraction, S  is compact and continuous and S  x + S  y ∈ M for all x, y ∈ M.
Then we get .
It follows from () that S  x + S  y ∈ M for all x, y ∈ M.
Step : S  is a contraction. Let x, y ∈ PC(J, X). From H(I), H(g), and Lemma ., we have, for t ∈ [, t  ], Therefore we deduce that In view of (), the operator S  is a contraction on PC(J, X).
Step : S  is compact and continuous. Firstly, we will prove that S  is continuous. Let x n → x in PC(J, X) as n → ∞. We can assume without any loss of generality that x n PC ≤ R for some R >  and n ≥ . By H(f )  , we have and, for t ∈ [s i , t i+ ], i = , , . . . , m, Then from (), (), and by means of the Lebesgue dominated convergence theorem, we obtain This means that S  is continuous.
Next we shall show that S  maps bounded set into relatively compact set in PC(J, X). Let B be any bounded subset of PC(J, X) such that for x ∈ B, x PC ≤ R for some R > , it suffices to show that the set of functions S  (B) = {S  x : x ∈ B} satisfies the conditions of Lemma ..
For the same reasons as in Step , the set S  (B) is uniformly bounded. For Repeating the discussion in [] (see p. of it), we find that I  , I  , I  tend to zero as t → t independently of x ∈ B. If t i < t < t ≤ t i+ , i = , , . . . , m, we have the following.
Since H(A) and the proof of Lemma . in [] imply that the continuity of P α (t) and Q α (t) (t > ) in t is in the uniform operator topology, we deduce that the right-hand side of () tends to zero independently of x ∈ B, as t → t .
Case : x ∈ B}, t ∈ J, we shall prove that S  (B)(t) is relatively compact in X. Clearly, S  (B)() = {} is compact.
Case  :  < t ≤ t  . For each h ∈ (, t) and δ > , we define a set By the Hölder inequality and H(f )  , we get Therefore from the property of the probability density function ξ α and (), we obtain This means that there are relatively compact sets arbitrarily close to the set S  (B)(t). Hence the set S  (B)(t) is also relatively compact in X. Case  : t i < t ≤ s i , i = , , . . . , m. In such a case, Case  : s i < t ≤ t i+ , i = , , . . . , m, By the same argument as in Case  and P α (ts i ) is a compact operator (see Lemma .), we know S  (B)(t) is relatively compact. Therefore it follows from Lemma . that S  is compact and continuous. As a consequence of Steps -, we know that S  + S  satisfies all conditions of Krasnoselskii fixed point theorem (Theorem .). Hence the operator S has a fixed point in PC(J, X) which is a PC-mild solution of problem (). The proof is complete.
Finally in this section, we extend the results obtained above to nonlocal problems for impulsive fractional evolution equations. Specifically, we show study the existence and uniqueness of the mild solutions for problem (). Here we only state the existence results for problem () without proofs since these are similar to the ones obtained for problem () above.
Definition . A function x ∈ PC(J, X) is said to be a PC-mild solution of problem () if it satisfies the following relation: Then there exists a unique PC-mild solution of problem ().