Positive Solutions for the Initial Value Problem of Fractional Evolution Equations
By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.
The differential equations involving fractional derivatives in time have recently been studied extensively. One can see, for instance, the monographs [1–5] and the survey [6–8]. In particular, there has been a significant development in fractional evolution equations. Existence of solutions for fractional evolution equations has been studied by many authors during recent years. Many excellent results are obtained in this field; see [9–19] and the references therein. In [9, 10], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then the author investigated the existence, uniqueness, and regularity of solutions of fractional integrodifferential equations in [11, 12]. Recently, this theory was developed by Zhou et al. [13–16]. Particularly, they studied the existence and controllability of mild solution of fractional delay integrodifferential equations with a compact analytic semigroup in . In [17–19], the authors studied the existence of mild solutions of fractional impulsive delay or impulsive evolution equations. But as far as we know, there are no results on the existence of positive solutions of fractional evolution equations.
In this paper, by using the fixed point theorems combined with the theory of analytic semigroup, we investigate the existence of positive mild solutions for the initial value problem (IVP) of fractional evolution equations in Banach space as where denotes the Caputo fractional derivative of order with the lower limits zero, is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators, and is the nonlinear term and will be specified later.
The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of the analytic semigroup and the definition of mild solutions of IVP(1). In Section 3, we study the existence of positive mild solutions for the IVP(1). In Section 4, an example is given to illustrate the applicability of abstract results obtained in Section 3.
In this section, we introduce some basic facts about the fractional power of the generator of analytic semigroup and the fractional calculus that are used throughout this paper.
Let be a Banach space with norm . Throughout this paper, we assume that is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operator in ; that is, there exists such that for all . Without loss of generality, let , where is the resolvent set of . Then for any , we can define by Then can be defined by because is one to one. It can be shown that each has dense domain and that for . Moreover, for every and with , where is the identity in (for proofs of these facts we refer to the literature [20–22]).
We denote by the Banach space of equipped with norm for , which is equivalent to the graph norm of . Then we have for (with ), and the embedding is continuous. Moreover, has the following basic properties.
Lemma 1 (see ). has the following properties.(i) for each and . (ii) for each and . (iii)For every , is bounded in and there exists such that
Let be a closed interval on . In the following we denote by the Banach space of all continuous functions from into endowed with supnorm given by for . For any , denote by the restriction of to . From Lemma 1(i) and (ii), for any , we have as . Therefore, is a strongly continuous semigroup in , and for all . To prove our main results, the following lemma is also needed.
Lemma 2 (see ). If is a compact semigroup in , then is a compact semigroup in , and hence it is norm continuous.
Definition 3. The fractional integral of order with the lower limits zero for a function is defined by
where is the gamma function.
The Riemann-Liouville fractional derivative of order with the lower limits zero for a function can be written as Also the Caputo fractional derivative of order with the lower limits zero for a function can be written as
Remark 4. The Caputo derivative of a constant is equal to zero.
If is an abstract function with values in , then integrals which appear in Definition 3 are taken in Bochner's sense.
Lemma 5 (see ). A measurable function is Bochner integrable if is Lebesgue integrable.
For , we define two families and of operators by where where is a probability density function defined on , which has properties for all and . It is not difficult to verify (see ) that for , we have Clearly, if the semigroup is positive, then, by the definitions, the operators and are also positive for all .
Lemma 6. The operators and have the following properties.(i)For any fixed and any , one has (ii)The operators and are strongly continuous for all . (iii)If the semigroup is compact, then and are compact operators in for .(iv)If the semigroup is norm continuous, then the restriction of to and the restriction of to are uniformly continuous for .
Definition 7 (see [25, 26]). Let be a bounded set of a real Banach space . Set : can be expressed as the union of a finite number of sets such that the diameter of each set does not exceed ; that is, with . is called the Kuratowski measure of noncompactness of set .
It is clear that . For the Kuratowski measure of noncompactness, we have the following well-known results.
Lemma 8 (see ). If is bounded and equicontinuous, then where .
Lemma 9 (see ). Let be a countable set of strongly measurable function such that there exists an such that for all . Then and
Lemma 10 (see  Mönch fixed point theorem). Let B be a closed and convex subset of and . Assume that the continuous operator has the following property: is countable, and is relatively compact. Then has a fixed point in .
Based on an overall observation of the previous related literature, in this paper we adopt the following definition of mild solution of IVP(1).
Definition 11. By a mild solution of the IVP(1), one means a function satisfying for all .
3. Existence of Positive Mild Solutions
In this section, we introduce the existence theorems of positive mild solutions of the IVP(1). The discussions are based on fractional calculus and fixed point theorems.
Let be the smallest positive real eigenvalue of the linear operator , and let be the positive eigenvector corresponding to . For any and , we write where is a constant. Our main results are as follows.
Theorem 12. Let be the infinitesimal generator of a positive and compact analytic semigroup of uniformly bounded linear operators. Assume that satisfies the following conditions.) For any , one has () maps bounded sets of into bounded sets of .If with and for some , then the IVP(1) has at least one positive mild solution . And if , one has
Proof. For any and with , we first prove that the initial value problem (IVP) of fractional evolution equations
has at least one positive mild solution on , where is a positive constant and will be given later.
Let . Denote Then is a nonempty bounded convex closed set. The assumption implies that there is a constant such that for any and .
Define an operator by By the continuity of , it is not difficult to prove that is continuous. By the positivity of the semigroup , the assumption (), and (20), we easily see that . Clearly, the positive mild solution of the IVP(17) on is equivalent to the fixed point of operator in . We will use Schauder fixed point theorem to prove that has fixed points in .
We first prove that is continuous. Let . For any and , by Lemma 6, (10), (19), and (20), we have
Let . Then for any and By the positivity of the semigroup , assumption (), and (20), for any , we have Thus, is continuous.
By using a similar argument as in the proof of Theorem 3.1 in , we can prove that is a compact operator. Hence by Schauder fixed point theorem, the operator has at least one fixed point in , which satisfies for all . Hence is a positive mild solution of the IVP(1) on .
Therefore, there exists such that the IVP(1) has at least one positive mild solution . Now, by the standard proof method of extension theorem of initial value problem, can be extended to a saturated solution of the IVP(1), whose existence interval is , and if , we have
For any and , define as in (15). If is increasing in , that is, satisfies the condition for any with for all , we have then we have for any and . On the other hand, if satisfies linear growth condition, then it maps the bounded sets into the bounded sets. Hence by Theorem 12, we have the following existence result.
Corollary 13. Let be the infinitesimal generator of a positive and compact analytic semigroup of uniformly bounded linear operators. Assume that satisfies condition ()* and ()* there exists a constant such that
for all and .
If for all , with and for some , then the IVP(1) has at least one positive mild solution . And if , one has
Since the analytic semigroup is norm continuous, it follows that we can delete the compactness condition on the analytic semigroup and obtain the following existence result.
Theorem 14. Assume that is the infinitesimal generator of a positive analytic semigroup of uniformly bounded linear operators, and that satisfies the condition () and() for any and , is relatively compact in for all , where is defined as in (15).If with and for some , then the IVP(1) has at least one positive mild solution . And if , one has
Proof. For any and with , we first prove that the IVP(17) has at least one positive mild solution on , where is a constant and will be specified later. Define an operator as in (20). Let . Write as in (18). The condition implies that is bounded for any , that is, there is a positive constant such that
Let . A similar argument as in the proof of Theorem 12 shows that is continuous and is equicontinuous.
Thus, for any , let . Since is equicontinuous and bounded, by Lemma 8, we have
Now, let with for some . It is obvious that Hence by Lemma 9 and (20), we have It follows that for all . By Lemma 8 and (27), we have . Thus, we have This implies that is relatively compact. Therefore, by Mönch fixed point theorem, the operator has at least one fixed point , which satisfies for all . Hence is a positive mild solution of the IVP(17) on .
Therefore, there exists such that the IVP(1) has at least one positive mild solution . can be extended to a saturated solution of IVP(1), whose existence interval is and when , we have .
4. Positive Mild Solutions of Parabolic Equations
Let be a bounded domain with a sufficiently smooth boundary . Let be a uniformly elliptic differential operator of divergence form in , where the coefficients and for some . We assume that is a positive define symmetric matric for every , and there exists a constant such that
Let on . We use to denote a generic point of , where and . Let be a continuous function. We discuss the existence of positive mild solutions for the parabolic initial boundary value problem (IBVP) where is a constant.
Let be the smallest positive real eigenvalue of elliptic operator under the Dirichlet boundary condition . It is well known (cf. Amann [22, 28]) that . Let be the positive eigenvector corresponding to . Assume that is continuous and satisfies the following conditions.() For any and , there exists a constant such that where with .() For any , there exists a constant such that
Let . Define an operator by It is well known (cf. Li ) that generates a compact analytic semigroup and . By the maximum principle of the equation of the parabolic type, it is easy to prove that is also a positive semigroup in . The assumptions () and () imply that the mapping defined by is continuous and satisfies the conditions () and (). Thus, the IBVP(33) can be rewritten into the abstract form of IVP(1). By Theorem 12, we have the following existence result for the IBVP(33).
Theorem 15. Assume that is continuous and satisfies conditions () and (). If with for any and , then the IBVP(33) has at least one positive mild solution that satisfies for any and . And if , one has
This research was supported by the NNSF of China (Grant no. 11261053), the Fundamental Research Funds for the Gansu Universities, and the Project of NWNU-LKQN-11-3.
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