Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions
© Mu; licensee Springer 2012
Received: 11 November 2011
Accepted: 20 February 2012
Published: 5 July 2012
In this article, the theory of positive semigroup of operators and the monotone iterative technique are extended for the impulsive fractional evolution equations with nonlocal initial conditions. The existence results of extremal mild solutions are obtained. As an application that illustrates the abstract results, an example is given.
Keywordsimpulsive fractional evolution equations nonlocal initial conditions extremal mild solutions monotone iterative technique
where is the Caputo fractional derivative of order , is a linear closed densely defined operator, −A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators (), , , , is continuous, is continuous ( will be defined in Section 2), the impulsive function is continuous, , where and represent the right and left limits of at , respectively.
we can take , for , or for , where , , are the fractional derivatives of order α, , , respectively.
where () are given constants and . On the other hand, the differential equations involving impulsive effects appear as a natural description of observed evolution phenomena introduction of the basic theory of impulsive differential equations, we refer the reader to  and the references therein. The study of impulsive evolution equations with nonlocal initial conditions has attracted a great deal of attention in fractional dynamics and its theory has been treated in several works [12–14]. They use the contraction mapping principle, the Krasnoselskii fixed point theorem and the Schaefer fixed point theorem.
To the authors’ knowledge, there are no studies on the existence of solutions for the impulsive fractional evolution equations with nonlocal initial conditions by using the monotone iterative technique in the presence of lower and upper solutions. Nevertheless, the monotone iterative technique concerning upper and lower solutions is a powerful tool to solve the differential equations with various kinds of boundary conditions, see [19–21]. This technique is that, for the considered problem, starting from a pair ordered lower and upper, one constructs two monotone sequences such that them uniformly converge to the extremal solutions between the lower and upper solutions. In this article, based on Mu , we obtained the existence of extremal mild solutions of the problem (1.1) by using the monotone iterative technique.
In following section, we introduce some preliminaries which are used throughout this article. In Section 3, by combining the theory of positive semigroup of linear operators and the monotone iterative technique coupled with the method of upper and lower solutions, we construct two groups of monotone iterative sequences, and then prove these sequences monotonically converge to the maximal and minimal mild solutions of the problem (1.1), respectively, under some monotone conditions and noncompactness measure conditions of f, g, and . In Section 4, in order to illustrate our results, an impulsive fractional partial differential equation with nonlocal initial condition is also considered.
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this article.
provided the right side is pointwise defined on , where is the gamma function.
- (i)If , then(2.4)
The Caputo derivative of a constant is equal to zero.
If f is an abstract function with values in X, then the integrals and derivatives which appear in Definitions 2.1-2.3 are taken in Bochner’s sense.
then is called a lower solution of problem (1.1); if all inequalities of (2.6) are inverse, we call it an upper solution of the problem (1.1).
is a probability density function defined on.
, , , .
Definition 2.9 If , by the mild solution of IVP (2.7), we mean that the function satisfying the integral Equation (2.8).
Form Definition 2.9, we can easily obtain the following result.
whereandare given by (2.9).
Remark 2.11 We note that and do not possess the semigroup properties. The mild solution of (2.12) can be expressed only by using piecewise functions.
Definition 2.12 A -semigroup is called a positive semigroup, if for all and .
Definition 2.13 A bounded linear operator K on X is called to be positive, if for all .
Remark 2.14 By (2.9) and Remark 2.8, and are positive, if is a positive semigroup.
Remark 2.15 From Remark 2.14, if () is a positive semigroup generated by −A, , and , , then the mild solution of (2.12) satisfies . For the applications of positive operators semigroup, one can refer to [23–25].
Now, we recall some properties of the measure of noncompactness will be used later. Let denotes the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see . For any and , set . If B is bounded in , then is bounded in X, and . If E is a precompact set in X, then .
In order to prove our results, we also need a generalized Gronwall inequality for fractional differential equation.
3 Main results
Theorem 3.1 Let X be an ordered Banach space, whose positive cone P is normal with normal constant N. Assume that () is positive, the Cauchy problem (1.1) has a lower solutionand an upper solutionwith, and the following conditions are satisfied:
for any, and. That is, is increasing in x for.
(H2) is decreasing in u for.
(H3) is increasing in x for.
for any, and increasing or decreasing monotonic sequence.
(H5) is precompact in X, for any increasing or decreasing monotonic sequence. That is, .
Then the Cauchy problem (1.1) has the minimal and maximal mild solutions betweenand, which can be obtained by a monotone iterative procedure starting fromand, respectively.
Then and . Similarly, we can prove that there exists such that . By (3.6), if , and u is a fixed point of Q, then . By induction, . By (3.10) and taking the limit as , we conclude that . That means that , are the minimal and maximal fixed points of Q on , respectively. By (3.5), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on , respectively. □
Corollary 3.2 Let X be an ordered Banach space, whose positive cone P is regular. Assume that () is positive, the Cauchy problem (1.1) has a lower solutionand an upper solutionwith, (H1), (H2), and (H3) hold. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions betweenand, which can be obtained by a monotone iterative procedure starting fromand, respectively.
So, (H4) holds. Let be an increasing or decreasing sequence in . By (H2), is an ordered-monotonic and ordered-bounded sequence in X. Then is precompact in X. Thus, (H5) holds. By Theorem 3.1, the proof is then complete. □
Corollary 3.3 Let X be an ordered and weakly sequentially complete Banach space, whose positive cone P is normal with normal constant N. Assume that () is positive, the Cauchy problem (1.1) has a lower solutionand an upper solutionwith, (H1), (H2), and (H3) hold. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions betweenand, which can be obtained by a monotone iterative procedure starting fromand, respectively.
Proof In an ordered and weakly sequentially complete Banach space, the normal cone P is regular. Then the proof is complete. □
Corollary 3.4 Let X be an ordered and reflective Banach space, whose positive cone P is normal with normal constant N. Assume that () is positive, the Cauchy problem (1.1) has a lower solutionand an upper solutionwith, (H1), (H2), and (H3) hold. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions betweenand, which can be obtained by a monotone iterative procedure starting fromand, respectively.
Proof In an ordered and reflective Banach space, the normal cone P is regular. Then the proof is complete. □
where , , (), is continuous, () is continuous, .
for , , for .
- (b)There exists w such that(4.2)
is continuous on any bounded and ordered interval.
- (d)For any , on a bounded and ordered interval, and , we have(4.3)
Theorem 4.2 If (a)-(d) are satisfied, then the system (4.1) has the minimal and maximal mild solutions between 0 and w.
Proof By (a) and (b), we know 0 and w are the lower and upper solutions of the problem (1.1), respectively. (c) implies that (H1) are satisfied. (d) implies that (H3) are satisfied. Then by Corollary 3.2, the system (4.1) has the minimal and maximal mild solutions between 0 and w. □
This research was supported by Talent Introduction Scientific Research Foundation of Northwest University for Nationalities.
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