Abstract
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.
Keywords:
impulsive fractional evolution equations; nonlocal initial conditions; extremal mild solutions; monotone iterative technique1 Introduction
In this article, we use the monotone iterative technique to investigate the existence of extremal mild solutions of the impulsive fractional evolution equation with nonlocal initial conditions in an ordered Banach space X
where
Fractional calculus is a generalization of ordinary differentiation and integration
to arbitrary real or complex order. The subject is as old as differential calculus,
and goes back to the time when Leibnitz and Newton invented differential calculus.
Fractional derivatives have been extensively applied in many fields which have been
seen an overwhelming growth in the last three decades. Examples abound: models admitting
backgrounds of heat transfer, viscoelasticity, electrical circuits, electrochemistry,
economics, polymer physics, and even biology are always concerned with fractional
derivative [16]. Fractional evolution equations have attracted many researchers in recent years,
for example, see [714]. A strong motivation for investigating the problem (1.1) comes form physics. For
example, fractional diffusion equations are abstract partial differential equations
that involve fractional derivatives in space and time. The time fractional diffusion
equation is obtained from the standard diffusion equation by replacing the firstorder
time derivative with a fractional derivative of order
we can take
The existence results to evolution equations with nonlocal conditions in Banach space
was studied first by Byszewski [15,16]. Deng [17] indicated that, using the nonlocal condition
where
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 [1921]. 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 [8], 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
2 Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this article.
Definition 2.1[22]
The fractional integral of order α with the lower limit zero for a function
provided the right side is pointwise defined on
Definition 2.2[22]
The RiemannLiouville derivative of order α with the lower limit zero for a function
Definition 2.3[22]
The Caputo fractional derivative of order α for a function
Remark 2.4
(i) If
(ii) The Caputo derivative of a constant is equal to zero.
(iii) If f is an abstract function with values in X, then the integrals and derivatives which appear in Definitions 2.12.3 are taken in Bochner’s sense.
Let X be an ordered Banach space with norm
Definition 2.5 If
then
Lemma 2.6[7]
Ifhsatisfies a uniform H
is given by
where
Remark 2.8[10]
Definition 2.9 If
Form Definition 2.9, we can easily obtain the following result.
Lemma 2.10For any
has the unique mild solution
where
Remark 2.11 We note that
Definition 2.12 A
Definition 2.13 A bounded linear operator K on X is called to be positive, if
Remark 2.14 By (2.9) and Remark 2.8,
Remark 2.15 From Remark 2.14, if
Now, we recall some properties of the measure of noncompactness will be used later.
Let
Lemma 2.16[27]
Let
In order to prove our results, we also need a generalized Gronwall inequality for fractional differential equation.
Lemma 2.17[28]
Suppose
on this interval; then
3 Main results
Theorem 3.1LetXbe an ordered Banach space, whose positive conePis normal with normal constantN. Assume that
(H_{1}) There exists a constant
for any
(H_{2})
(H_{3})
(H_{4}) There exists a constant
for any
(H_{5})
Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between
Proof It is easy to see that
Let
Clearly,
For
Now, we show that
For
Continuing such a process interval by interval to
and it follows from (3.6) that
Let
From (H_{4}), (H_{5}), (3.3), (3.4), (3.9), (3.11), Lemma 2.16 and the positivity of operator
By (3.12) and Lemma 2.17, we obtain that
By (3.13) and Lemma 2.17,
Let
Then
Corollary 3.2LetXbe an ordered Banach space, whose positive conePis regular. Assume that
Proof Since P is regular, any orderedmonotonic and orderedbounded sequence in X is convergent. For
So, (H_{4}) holds. Let
Corollary 3.3LetXbe an ordered and weakly sequentially complete Banach space, whose positive conePis normal with normal constantN. Assume that
Proof In an ordered and weakly sequentially complete Banach space, the normal cone P is regular. Then the proof is complete. □
Corollary 3.4LetXbe an ordered and reflective Banach space, whose positive conePis normal with normal constantN. Assume that
Proof In an ordered and reflective Banach space, the normal cone P is regular. Then the proof is complete. □
4 Examples
Example 4.1 In order to illustrate our results, we consider the following impulsive fractional partial differential equation with nonlocal initial condition
where
Let
then −A generate an analytic semigroup of uniformly bounded linear operators
(a)
(b) There exists w such that
where
(c)
(d) For any
Theorem 4.2If (a)(d) are satisfied, then the system (4.1) has the minimal and maximal mild solutions between 0 andw.
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 (H_{1}) are satisfied. (d) implies that (H_{3}) are satisfied. Then by Corollary 3.2, the system (4.1) has the minimal and maximal mild solutions between 0 and w. □
Competing interests
The author declares that she has no competing interests.
Acknowledgements
This research was supported by Talent Introduction Scientific Research Foundation of Northwest University for Nationalities.
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