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.
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:
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
Lemma 1 ()
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
Lemma 2 ()
Lemma 3The following properties are valid:
In this paper, we adopt the following definition of a mild solution of Eq. (1).
To prove our main results, we also need the following two lemmas.
Lemma 5 (Krasnoselskii’s fixed point theorem)
LetXbe a Banach space, letBbe a bounded closed and convex subset ofXand letandbe mappings fromBintoXsuch thatfor every pair. Ifis a contraction andis completely continuous, then the operator equationhas 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.
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.
then Eq. (1) has at least one mild solution onJ.
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
The next proof will be given in two steps.
This together with the Lebesgue dominated convergence theorem gives that
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.
hold, then Eq. (1) has a unique mild solution.
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
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
Lemma 6 ()
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 .
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