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; uniqueness
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 [1-5]). Hence, the research on fractional differential equations has become an object of extensive study during recent years; see [6-11] 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 , and let be a constant. Consider the existence and uniqueness of mild solutions of fractional evolution equation with nonlocal condition in the form
where is the Caputo fractional derivative of order , the linear operator −A is the infinitesimal generator of an analytic semigroup in X, the functions f, h and g will be specified later. , where , . Throughout this paper, we always assume that .
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  the authors studied the existence and uniqueness of mild solutions of Eq. (1) under the case . In their main results, they did not assume the complete continuity of the nonlocal term, but they needed the following assumptions:
(F1) there exist a constant and such that for all and almost all ;
(F2) there exists a constant such that for ;
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 ( )) and g is local Lipschitz continuous (see assumption ( )). We will carry out our investigation in the Banach space , , where is the domain of the fractional power of A. Finally, an example is given to illustrate the applicability of our main results. We can see that the main results in  cannot be applied to our example.
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.
Let X be a Banach space with norm , and let be the infinitesimal generator of an analytic semigroup ( ) of a uniformly bounded linear operator in X, that is, there exists such that for all . Without loss of generality, let . Then for any , we can define by
is injective, and can be defined by with the domain . For , let .
Lemma 1 ()
has the following properties:
(i) is a Banach space with the norm for ;
(ii) for each ;
(iii) for each and ;
(iv) is a bounded linear operator onXwith ;
(v) If , then .
Let be the Banach space of endowed with the norm . Denote by the Banach space of all continuous functions from J into with the supnorm given by for . From Lemma 1(iv), since is a bounded linear operator for , we denote by the operator norm of in X, that is, . For any , denote by the restriction of to . From Lemma 1(ii) and (iii), for any , we have
as . Therefore, ( ) is a strongly continuous semigroup in , and for all . To prove our main results, the following lemma is needed.
Lemma 2 ()
If ( ) is a compact semigroup inX, then ( ) is an immediately compact semigroup in , and hence it is immediately norm-continuous.
For , define two families and of operators by
where is the probability density function defined on , which has properties for all and
Lemma 3The following properties are valid:
(i) For fixed and any , we have
(ii) The operators and are strongly continuous for all .
(iii) If ( ) is a compact semigroup inX, then and are norm-continuous inXfor .
(iv) If ( ) is a compact semigroup inX, then and are compact operators inXfor .
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 satisfying
for all .
To prove our main results, we also need the following two lemmas.
Lemma 4A measurable function is Bochner integrable if is Lebesgue integrable.
Lemma 5 (Krasnoselskii’s fixed point theorem)
LetXbe a Banach space, letBbe a bounded closed and convex subset ofXand let and be mappings fromBintoXsuch that for every pair . If is a contraction and is completely continuous, then the operator equation has a solution onB.
Lemmas 4 and 5, which can be found in many books, are classical.
The following are the basic assumptions of this paper.
( ) ( ) is a compact operator semigroup in X.
( ) There exists a constant such that the functions satisfy the following conditions:
(i) For each , the functions , are measurable.
(ii) For each , the functions , are continuous.
( ) For and , there exist positive functions satisfying and such that
and there are positive constants and such that
( ) and for , there exists a positive constant L such that
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 ( )-( ) hold. If and the following inequality holds:
then Eq. (1) has at least one mild solution onJ.
Proof Define two operators and as follows:
Obviously, u is a mild solution of Eq. (1) if and only if u is a solution of the operator equation on J. To prove the operator equation has solutions, we first show that there is a positive number r such that for every pair . If this were not the case, then for each , there would exist and such that . Thus, from Lemma 3, ( ) and ( ), we have
Dividing on both sides by r and taking the lower limit as , we have
which contradicts (3). Hence, for some , for every pair .
The next proof will be given in two steps.
Step 1. is a contraction on .
For any and , according to Lemma 3 and assumption ( ), we have
which implies that . It follows from (3) that , hence is a contraction on .
Step 2. is a completely continuous operator on .
We first prove that is continuous on . Let with as . Then for any , , by assumption ( ), we have
as , and from assumption ( ), we have
This together with the Lebesgue dominated convergence theorem gives that
as . Hence, . This means that is continuous on .
Next, we will show that the set is relatively compact. It suffices to show that the family of functions is uniformly bounded and equicontinuous, and for any , the set is relatively compact.
For any , we have for some , which means that is uniformly bounded. In what follows, we show that is a family of equicontinuous functions. For , we have
Hence, it is only necessary to consider . For , from Lemma 3 and assumption ( ), we have
For any , we have
It follows from Lemma 3 that as and independently of . From the expressions of and , it is clear that and as independently of . Therefore, we prove that is a family of equicontinuous functions.
It remains to prove that for any , the set is relatively compact.
Obviously, is relatively compact in . Let be fixed. For each , and , we define an operator by
Then the sets are relatively compact in since by Lemma 2, the operator is compact for in . Moreover, for every , from Lemma 3 and assumption ( ), we have
Therefore, there are relatively compact sets arbitrarily close to the set for and since it is compact at , we have the relative compactness of in for all .
Therefore, the set is relatively compact by the Ascoli-Arzela theorem. Thus, the continuity of and relative compactness of the set imply that is a completely continuous operator. Hence, Krasnoselskii’s fixed point theorem shows that the operator equation has a solution on . Therefore, Eq. (1) has at least one mild solution. The proof is completed. □
The following existence and uniqueness theorem for Eq. (1) is based on the Banach contraction principle. We will also need the following assumptions.
( ) There exists a constant such that the functions are strongly measurable.
( ) For , there exist functions such that
for any and .
Theorem 2Let the assumptions ( )-( ) be satisfied. If and the inequalities (3) and
hold, then Eq. (1) has a unique mild solution.
Proof From Lemma 4 and assumption ( ), it is easy to see that is Bochner integrable with respect to for all . For any , we define an operator Q by
According to the proof of Theorem 1, we know that for some . For any and , from Lemma 3, assumptions ( ) and ( ), we have
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 , which is the unique mild solution of Eq. (1). This completes the proof. □
4 An example
Let equip with its natural norm and inner product . Consider the following system:
where is a constant, , and will be specified later.
Let the operator be defined by
Then −A generates a compact analytic semigroup of uniformly bounded linear operators and for all . Hence, we take . Moreover, the eigenvalues of A are , and the corresponding normalized eigenvectors are , . The operator is given by
for each and .
Lemma 6 ()
If , thenξis absolutely continuous, and .
Let , where for all . Assume that
(P1) The function , , and the partial derivative belongs to .
Let , it follows from
that f and h are functions from into and they are continuous. Moreover, a similar computation of  together with Lemma 6 and assumption (P1) shows that whenever .
Then for any , we see that the assumptions ( )-( ) are satisfied with
Thus, the system (5) has at least one mild solution due to Theorem 1 provided that . And by Theorem 2, this mild solution of the system (5) is unique on .
The author declares that they have no competing interests.
Research was supported by the Fundamental Research Funds for the Gansu Universities.
Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process.. 5, 81–88 (1991). Publisher 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)
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
El-Borai, MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals. 14, 433–440 (2002). Publisher Full Text
El-Borai, 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 semi-linear 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
Liu, H, Chang, J: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal.. 70, 3076–3083 (2009). Publisher Full Text
Chang, J, Liu, H: Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α-norm. Nonlinear Anal.. 71, 3759–3768 (2009). Publisher Full Text