Mild solution of fractional order differential equations with not instantaneous impulses

Abstract In this paper, we investigate the boundary value problems of fractional order differential equations with not instantaneous impulse. By some fixed-point theorems, the existence results of mild solution are established. At last, one example is also given to illustrate the results.


Introduction
The impulsive differential equations arise from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, physics, engineering, etc. Due to their significance, many authors have established the solvability of impulsive differential equations. For the general theory and applications of such equations we refer the interested readers to see the papers [1][2][3][4] and references therein. However, in almost all the papers concerning impulsive differential equations, the impulses are all instantaneous impulses, and the classical models with instantaneous impulses can not characterize many practical problems, for example, the dynamics of evolution processes in pharmacotherapy. Let us consider the hemodynamic equilibrium of a person. The introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous process. In fact, this situation should be characterized by a new case of impulsive action, which starts at an arbitrary fixed point t i and stays active on a finite time interval OEt i ; s i . To this end, Hernandez and O'Regan [5] initially offered to study a new class of abstract semilinear impulsive differential equations with not instantaneous impulses in a PC-normed Banach space. In [5], the authors discussed the following problems.

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: ; N and f W OE0; a X ! X is a suitable function. Meanwhile, Pierri et al. [6] continue the work and development in [5] in a P C˛-normed Banach space.
On the one hand, the absorption of drugs has a memory effect, thus, the new class of impulsive conditions introduced by [5] may not explain this phenomenon very well. On the other hand, fractional calculus provide a powerful tool for the description of hereditary properties of various materials and memory processes [7][8]. Fractional differential equations have recently proved to be strong tools in the modeling of medical, physics, economics and technical sciences. For more details on fractional calculus theory, one can see the monographs of Diethelm [9], Kilbas et al. [10], Lakshmikantham et al. [11], Miller and Ross [12], Podlubny [13] and Tarasov [14]. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions (see for examples [7,8,[15][16][17][18][19]). In [20], the authors considered the following problem: where c D q t is the Caputo fractional derivative of order q 2 .0; 1/ with the lower limit zero, f W J R ! R is jointly continuous and t k satisfy represent the right and left limits of u.t / at t D t k ; I k 2 C.R; R/; and a; b; c are real constants with a Cb ¤ 0. Obviously, the impulses in (2) are instantaneous. Motivated by the work in [5][6]20], in this article, we consider the following impulsive fractional differential equations which impulses are not instantaneous.
where c D q t is the Caputo fractional derivative of order q 2 .0; 1/ with the lower limit zero, ; N ,f W OE0; T R ! R is a continuous, and a; b; c are real constants with a C b ¤ 0. The rest of this paper is organized as follows. In Section 2, some lemmas which are essential to prove our main results are stated. In Section 3, we give the main results. In Section 4, one examples is offered to demonstrate the application of our main results.

Preliminaries
At first, we present the necessary definitions for the fractional calculus theory.

Definition 2.1 ([10]
). The Riemann-Liouville fractional integral of order˛> 0 of a function y W .0; 1/ ! R is given by where the right side is pointwise defined on .0; C1/: where n D OE˛ C 1; OE˛ denotes the integer part of number˛, the right side is pointwise defined on .0; C1/.
where c i 2 R; i D 0; 1; ; n 1; n D OE˛ C 1: where c i 2 R; i D 0; 1; ; n 1; n D OE˛ C 1: Lemma 2.5 (Krasnoselskii's fixed point theorem [21]). Let M be a closed convex and nonempty subset of a Banach space X . Let A and B be two operators such that: A is compact and continuous; 3. B is a contraction mapping. Then there exists z 2 M such that z D Az C Bz.
We define ; N:g: Obviously, P C.J; R/ is a Banach space with the norm kxk P C D sup t2J jx.t /j. If u 2 P C.J; R/ satisfies impulsive problem (3), if t 2 OE0; t 1 , then integrating the first equation in (3) from 0 to t by virtue of the Definition 2.1, one can obtain

Main results
This section deals with the existence of mild solutions for problem (3). Before stating and proving the main results, we make the following hypotheses.
. ; N: Clearly, A is well defined. Next we show that A is contraction on B r 2 P C.J; R/, where B r D fx 2 P C.J; R/ W kxk P C Ä rg W Hence we can prove that kAx.t / Ay.t /k Ä nkx yk P C which implies that A is a contraction mapping and there exists a unique mild solution of (3).
In order to get the second main result, we give assumption .H 3 /. .
where d D maxf1; b a g; e D max iD1; N kg i .t; 0/k, then the problem (3) has a mild solution.
Proof. Let Au.t / be the map introduced in the proof of Theorem 3.1. We introduce the decomposition Au.t/ D if t … .s i ; t iC1 ; i 1; In order to use the Krasnoselski fixed-point theorem (Lemma 2.5), we divide our proof into three steps.
Step 1. First we show that A 1 x C A 2 y 2 B r whenever x; y 2 B r : Let, 8x 2 B r ; we have 8y 2 B r ,if t 2 .t i ; s i ; i 1; one can get Proceeding as above, we obtain that A 2 i y.t / Ä Lr C e, 8y 2 B r , if t 2 .s i ; t iC1 ; i 1. 8y 2 B r ,if t 2 OE0; t 1 ; we have Lr C e/: Then, 8x; y 2 B r ; we find that Step 2. We show A 2 D P N iD1 A 2 i is a contraction mapping. From the definition of A 2 u.t /; A 2 i u.t / and the assumption .H 2 /, we can easily get which implies that A 2 is a contraction mapping.
Step 3.Next we will prove that A 1 is compact and continuous. We also divide the proof into 3 steps.
I. A 1 is continuous. Let fx n g be a sequence such that x n ! x in P C.J; R/: Then 8t 2 J; by the definition of A 1 u.t /; A 1 i u.t / , we have which shows the operator A 1 is continuous.
II. A 1 maps bounded set into bounded sets in P C.J; R/. Indeed, it is enough to show that for any R > 0, there exists a R 0 > 0 such that for each x 2 B R D fu 2 P C.J; R/ W kuk P C Ä Rg; we have A 1 x P C Ä R 0 .
From the definition of A 1 u.t /; A 2 i u.t / and the assumption .H 3 /; 8t 2 J , one can obtain Then we conclude that A 1 maps bounded set into bounded sets in P C.J; R/.
OE2.l 2 l 1 / q Cˇ.l 1 s i / q .l 2 s i / qˇ : As l 1 ! l 2 , the right-hand side of the above inequality tends to zero, therefore A 1 is equicontinuous on interval .s i ; t iC1 ; i 1.
Proceeding as above, we can similarly prove that A 1 is equicontinuous for the time interval OE0; t 1 .And, is it is also easily to see that A 1 is equicontinuous for the time interval .t i ; s i ; i 1.
As a consequence of step I-III together with the PC-type Arzela-Ascoli Theorem, we can conclude that A 1 is continuous and compact.
Then we complete the proof of Steps 1-3. As a consequence of Lemma 2.5, we deduce that the operator A has a fixed point z 2 B r which is a mild solution of the problem (3).

Example
In this section we give an example to illustrate the usefulness of our main result. Consider the following impulsive system of fractional differential equations.
First we prove that with m f D 1 32 2 L 1 .J I R C /, h f .kuk/ D kuk 2 C.OE0; 1/I R C / is nondecreasing. Then the assumption .H 3 / is satisfied.
Also by (5) Hence, it is easy to choose r > 0 which satisfies the inequality (4). Thus, all the assumptions in Theorem 3.2 hold, our results can be applied to the Example 4.1, i.e., Example 4.1 has at least one mild solution.