Abstract
This paper deals with the existence and uniqueness of mild solutions for a class of fractional evolution equations with nonlocal initial conditions. We present some local growth conditions on a nonlinear part and a nonlocal term to guarantee the existence theorems. An example is given to illustrate the applicability of our results.
MSC: 34A12, 35F25.
Keywords:
fractional evolution equations; nonlocal initial conditions; existence; uniqueness1 Introduction
The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering and science. Indeed, we can find numerous applications in electrochemistry, control, porous media, electromagnetic processing etc. (see [15]). Hence, the research on fractional differential equations has become an object of extensive study during recent years; see [611] and references therein.
On the other hand, the nonlocal initial condition, in many cases, has much better effect in applications than the traditional initial condition. As remarked by Byszewski and Lakshmikantham (see [12,13]), the nonlocal initial value problems can be more useful than the standard initial value problems to describe many physical phenomena.
Let X be a Banach space with norm
where
In some existing articles, the fractional differential equations with nonlocal initial
conditions were treated under the hypothesis that the nonlocal term is completely
continuous or global Lipschitz continuous. It is obvious that these conditions are
not easy to verify in many cases. To make the things more applicable, in [6] the authors studied the existence and uniqueness of mild solutions of Eq. (1) under
the case
(F1) there exist a constant
(F2) there exists a constant
and some other conditions.
In this paper, we will improve the conditions (F1) and (F2). We only assume that f and h satisfy local growth conditions (see assumption (
The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of an analytic semigroup and on the mild solutions of Eq. (1). In Section 3, we study the existence and uniqueness of mild solutions of Eq. (1). In Section 4, we give an example to illustrate the applicability of our results.
2 Preliminaries
Let X be a Banach space with norm
Lemma 1 ([14])
(i)
(ii)
(iii)
(iv)
(v) If
Let
and
as
Lemma 2 ([15])
If
For
where
where
The following lemma follows from the results in [68,10].
Lemma 3The following properties are valid:
(i) For fixed
(ii) The operators
(iii) If
(iv) If
In this paper, we adopt the following definition of a mild solution of Eq. (1).
Definition 1 By a mild solution of Eq. (1), we mean a function
for all
To prove our main results, we also need the following two lemmas.
Lemma 4A measurable function
Lemma 5 (Krasnoselskii’s fixed point theorem)
LetXbe a Banach space, letBbe a bounded closed and convex subset ofXand let
Lemmas 4 and 5, which can be found in many books, are classical.
The following are the basic assumptions of this paper.
(
(
(i) For each
(ii) For each
(
and there are positive constants
(
for all
3 Main results
In this section, we introduce the existence and uniqueness theorems of mild solutions of Eq. (1). The discussions are based on fractional calculus and Krasnoselskii’s fixed point theorem. Our main results are as follows.
Theorem 1Suppose that the assumptions (
then Eq. (1) has at least one mild solution onJ.
Proof Define two operators
Obviously, u is a mild solution of Eq. (1) if and only if u is a solution of the operator equation
Dividing on both sides by r and taking the lower limit as
which contradicts (3). Hence, for some
The next proof will be given in two steps.
Step 1.
For any
which implies that
Step 2.
We first prove that
as
This together with the Lebesgue dominated convergence theorem gives that
as
Next, we will show that the set
For any
Hence, it is only necessary to consider
where
For any
It follows from Lemma 3 that
It remains to prove that for any
Obviously,
Then the sets
Therefore, there are relatively compact sets arbitrarily close to the set
Therefore, the set
The following existence and uniqueness theorem for Eq. (1) is based on the Banach contraction principle. We will also need the following assumptions.
(
(
for any
Theorem 2Let the assumptions (
hold, then Eq. (1) has a unique mild solution.
Proof From Lemma 4 and assumption (
According to the proof of Theorem 1, we know that
Thus,
which means that Q is a contraction according to (4). By applying the Banach contraction principle,
we know that Q has a unique fixed point on
4 An example
Let
where
Let the operator
Then −A generates a compact analytic semigroup
for each
Lemma 6 ([16])
If
Let
(P_{1}) The function
Define
Let
and
that f and h are functions from
Then for any
Thus, the system (5) has at least one mild solution due to Theorem 1 provided that
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Research was supported by the Fundamental Research Funds for the Gansu Universities.
References

Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process.. 5, 81–88 (1991). Publisher Full Text

Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of selfsimilar protein dynamics. Biophys. J.. 68, 46–53 (1995). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Metzler, F, Schick, W, Kilian, HG, Nonnenmacher, TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys.. 103, 7180–7186 (1995). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Mainardi, F: Fractional calculus some basic problems in continuum and statistical mechanics. In: Carpinteri A, Mainardi F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Vienna (1997)

Hilfer, R: Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000)

Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal.. 11, 4465–4475 (2010). Publisher Full Text

Wang, RN, Xiao, TJ, Liang, J: A note on the fractional Cauchy problems with nonlocal initial conditions. Appl. Math. Lett.. 24, 1435–1442 (2011). Publisher Full Text

ElBorai, MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals. 14, 433–440 (2002). Publisher Full Text

ElBorai, MM: Exact solutions for some nonlinear fractional parabolic partial differential equations. Appl. Math. Comput.. 206, 150–153 (2008). Publisher Full Text

Wang, JR, Zhou, Y: Existence of mild solutions for fractional delay evolution systems. Appl. Math. Comput.. 218, 357–367 (2011). Publisher Full Text

Lv, ZW, Liang, J, Xiao, TJ: Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order. Comput. Math. Appl.. 62, 1303–1311 (2011). Publisher Full Text

Byszewski, L: Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl.. 162, 494–505 (1991). Publisher Full Text

Byszewski, L, Lakshmikantham, V: Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal.. 40, 11–19 (1991). Publisher Full Text

Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983)

Liu, H, Chang, J: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal.. 70, 3076–3083 (2009). Publisher Full Text

Travis, CC, Webb, GF: Existence stability and compactness with αnorm for partial functional differential equations. Trans. Am. Math. Soc.. 240, 129–143 (1978)

Chang, J, Liu, H: Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the αnorm. Nonlinear Anal.. 71, 3759–3768 (2009). Publisher Full Text