1 Introduction

Let be an ordered Banach space, whose positive cone is a normal cone with normal constant is the zero element of E. In this paper, we discuss the positive S-asymptotically ω-periodic mild solutions for the following fractional evolution equation:

where is a Caputo fractional derivative of the order with the lower limits zero, is a closed linear (not necessarily bounded) operator, and generates a C₀-semigroup is a given continuous function, which will be specified later.

In the past decades, in view of the wide practical background and application prospects of fractional calculus in physics, chemistry, engineering, biology, financial sciences, and other applied disciplines, numerous scholars pay more attention to fractional differential equations and have found that in many practical applications, fractional differential equations can more truthfully describe the process and phenomena of things’ motion development than integer differential equations (see [1, 30, 40] and references therein). Since fractional evolution equations are abstract models from many practical applications, the study for fractional evolution equations has attracted more and more attention of mathematicians (see [6, 12, 17, 37, 38, 41] and references therein).

Recently, the periodicity problems or asymptotic periodicity problems have extensive physical background and realistic mathematical model, hence, it has been considerably developed and many properties of its solutions have been studied (see [3, 11, 13, 14, 18, 20, 21, 23-25, 31, 36, 39] and references therein). On the other hand, because fractionalderivative has genetic or memory properties, the solutions of periodic boundary valueproblems for fractional differential equations cannot be extended periodically to timet in R+. Specially, the nonexistence of nontrivial periodic solutions of fractional evolutionequations had been shown in [32]. In 2008, Henríquez et al. [15] formally introduced the concept of S-asymptotically ω-periodic function, which is a more general approximate period function. Since then, the S-asymptotically periodic functions have beenwidely studied in fractional evolution equations, and the existence and uniqueness ofS-asymptoticallyω-periodic solutions have been well studied (see [3, 9, 10, 19, 29, 32, 33, 35]). It is not difficult to find that in most of the above work, the Lipschitz-type condi-tions for nonlinear functions are necessary. In fact, for equations arising in complicated reaction–diffusion processes, the nonlinear function represents the source of material orpopulation, which depends on time in diversified manners in many contexts.

It is well known that in many practice models, such as heat conduction equations, neutron transport equations, reaction diffusion equations, etc., only positive solutions are significant. But as far as we know, only a few scholars are concerned about the existence of positive solutions for fractional evolution equations on infinite interval (see [7, 8, 35]). In [7, 8], by means of the monotone iterative method Chen presented the existence and uniqueness of the positive mild solutions for the abstract fractional evolution equations under certain initial conditions. In [35], Shu studied a class of semilinear neutral fractional evolution equations with delay and obtained the existence and uniqueness of the positive S-asymptotically ω-periodic mild solutions by using contraction mapping principle in positive cone.

Inspired by the above literature, we will use a completely different method to improve and extend the results mentioned above, which will make up the research in this area blank. In Section 3, we investigate the positive S-asymptotically ω-periodic mild solutions for problem (1) under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solution are obtained by using monotone iterative method and fixed point theorem. It is worth noting that we no longer require nonlinear functions to satisfy Lipschitz condition, which makes our results more widely applicable. Thus, our conclusions are new in some respects. In Section 2, some notions, definitions, and preliminary facts are introduced, and at last, an example of time-fractional partial differential equation is given to illustrate the application of our results.

2 Preliminaries

In this paper, we always assume that is an ordered Banach space, whose positive cone is a normal cone with normal constant is the zero element of E.

Let be a continuous and nondecreasing function such that for all and . Thus, we can define a Banach space

with the norm . For the Banach space , we have the following result.

Lemma 1.(See [16].) A set is relatively compact in if and only if

(i)B is equicontinuous;

(ii)uniformly for ;

(iii)The set is relatively compact in E for every .

Next, we introduce a standard definition of S-asymptotically ω-periodic function. Let denote the Banach space of bounded and continuous functions from to E with the norm .

Definition 1. (See [15].) A function is called S-asymptotically ω-periodic if there exists such that . Thus, ω is called an asymptotic period of u.

Let SAP be the subspace of consisting of all the E-valued S-as- ymptotically ω-periodic functions equipped with norm Then SAP is a Banach space [15].

Define a positive cone by

Thus, is an ordered Banach space, whose partial order relation is induced by the cone K_h.

Let be a closed linear operator and generate a positive C₀-semigroup in E. For a general C₀-semigroup , there exist and such that (see [28]).

Specially, C₀-semigroup is called to be uniformly bounded if

The growth exponent of is defined by

If , then is said to be exponentially stable. Clearly, the exponentially stable C₀-semigroup is uniformly bounded. If C₀-semigroup is continuous in the uniform operator topology for every in E, it is well known that ν₀ can also be determined by (the resolvent set of A)

where is the infinitesimal generator of C₀-semigroup . We know that is continuous in the uniform operator topology for if is a compact semigroup. For more details about positive C0-semigroups and compact semigroups, we can refer to [5, 22, 27, 34].

For the definition of Caputo fractional derivation, we can refer to many references (see [6, 37] and so on), so we will not repeat it here. Next, we define operator families and in E as following

where

is a probability density function satisfying

Lemma 2.The operator families and defined by (2) have the following properties:

(i) and are strongly continuous operators, i.e., for any E and ,

(ii) If is uniformly bounded , then U(t) and V(t) are linear bounded operators for any fixed t , i.e.,

(iii)If is compact, then U(t) and V(t) are compact operators for every .

(iv)If is equicontinuous, then U(t) and V(t) are uniformly continuous for .

(v) Ifis positive, then U(t) and V(t) are positive operators.

(vi)If is exponentially stable with the growth exponent , then

for every , where Eq(·) and Eq,q(·) are the Mittag-Leffler functions.

Remark. The proof of statements (i)–(v) can be found in [6, 12, 37, 41], while the last one was proved in [4].

Definition 2. A function is said to be a mild solution of problem (1) if and satisfies

for all . Moreover, if for al l, then it is said to be a positive mild solution of problem (1).

In the proof, we also need the following lemma.

Lemma 3.(See [26].) Let D be a convex, bounded and closed subset of a Banach space is a condensing map, then has a fixed poind in D.

3 Main results

Theorem 1.Let E be an ordered Banach space, whose positive cone K is normal cone, let be a closed linear operator and generate an exponentially stable, positive, and compact semigroup in E, whose growth exponent denotes . Assume that E is a continuous function and the following conditions hold:

(H1) There exist nonnegative constants and such that

(H2) F is nondecreasing with respect to the second variable, i.e., for ,

(H3) There exists 0 such that

Then there exists a minimal positive S-asymptotically ω-periodic mild solution of problem (1).

Proof. Define an operator on by

It is easy to find is well defined. In fact, for every and , we have . Hence, from (H1), (3), and (5) it follows that for ,

which implies that is well defined.

It is easy to show that is continuous. Let and in , that is, for arbitrary , there exists sufficiently large n such that for. For above ε, by the continuity of F it is easy to see

and the dominated convergence theorem ensures

Hence, we conclude that is continuous from to . Specially, by (8) one can find that for every

where and . Therefore, from Definition 2 it follows that the fixed points of are mild solutions to problem (1).

Next, we will prove that . Choose , then for any , there exists a constant such that for all . Thus, by continuity of F we have

On the other hand, by (H3) there is a sufficiently large constant such that for

Hence, for , from (7) it follows that

According to (6), let

then

Thus, by (13) and (H1) one can find that

and

Hence, we deduce that tend to 0 as . By (13), (11), and (H1) one can obtain that

which implies that tends to 0 as . Similarly, by (13), (12), and (H1) we can get that tends to 0 as . Thus, from the above results we can deduce that

namely, , which implies that . Now, we prove the existence of positive solutions by monotone iterative technique.

For any with , by (H2), (7), the positivity of , and , one can find that for all ,

Thus is monotone increasing.

Let . Clearly, . Now, define a sequence by

From the definition and properties of , (14), (10) one can find and

Since from (15) it follows that

thus, the sequence is uniformly bounded. Next, we prove that the sequence is uniformly convergent.

Firstly, is relatively compact on E for . Let and . Obviously, for . For arbitrary r, one can obtain that is relatively compact on E for . In fact, for all and for all , we define a set by

where

By the compactness of U(t) and the set is relatively compact in E. Moreover, for every and , one has

Hence, the set is relatively compact, which implies that is relatively compact on E for . Therefore, by the definition of and the arbitrariness of r₀ we can obtain that is relatively compact on E for .

Secondly, is equicontinuous in . In general, let . For any , by the definition of one can see

Next, we check if tend to 0 as independently of . From (4) it follows that . By (H1), (13), and (16) we can obtain

If and , then it easy to see that . For and small enough, by (H1), (13), (16), and Lemma 2(iv) we get that

As a result, tends to 0 as independently of , thus is equicontinuous.

Thirdly, for every , by (8) and (16) one can find

which implies that as uniformly for . Therefore, Lemma 1 allows us to deduce that is relatively compact in , thus there is convergent subsequence in . From the monotonicity of sequence and the normality of cone we can obtain that itself is uniformly convergent, which means that there is such that .

Namely . In general, , i = 1,2,... . Taking the limit in (17) as, we have , which means that is the minimal positive S-asymptotically ω-periodic mild solution of problem (1).

Next, we always assume that the positive cone K is regeneration cone. By the characteristic of positive semigroups (see [22]), for sufficiently large , we have that has positive bounded inverse operator . Since , the spectral radius

By the famous Krein–Rutmann theorem A has the smallest eigenvalue , which has a positive eigenfunction e₁, and

which implies that . Hence, by Theorem 1 we have the following results.

Corollary 1.Let E be an ordered Banach space, whose positive cone K is a regeneration cone, let be a closed linear operator and generate an exponentially stable, positive, and compact semigroup in E, . Assume that is a continuous function, and let conditions (H2), (H3), and

(H10) there exist nonnegative constants and such that

hold. Then problem (1) has a minimal positive S-asymptotically ω-periodic mild solution .

Theorem 2. Let E be an ordered Banach space, whose positive cone K is regeneration cone, let be a closed linear operator and generate an exponentially stable, positive, and compact semigroup in E. Assume that is a continuous function, and let conditions (H10), (H3),

(H4) for any with , there is a constant such that

hold and . Then problem (1) has at least one positive S-asymptotically ω-periodic mild solution.

Proof. Let be defined by (7). From the proof of Theorem 1 it follows

Since , we can choose . Denote

Then is a nonempty bounded convex closed set. Hence, for any and , from (H10), (3), and (5) the following holds:

Let . Then for any , and

By the positivity of semigroup , condition (H4), and (7), for any and , one can obtain that

Thus, and for any and .

Next, we show that is completely continuous. From assumptions (H10) and (H3) there is a constant M₁ such that for all ,

Thus, it is easy to verify that the set

is relatively compact on E for any . In order to do this, we define a set

where , and

By the compactness of U(t) and the set is relatively compact in E. Right now, for every and , one has

Hence, the set is relatively compact on E for . Therefore, by the definition of and the arbitrariness of r₀ we can deduce that is relatively compact on E for . Moreover, it is easy to prove that set is equicontinuous by using the method similar to Theorem 1. Also, for every , by (8) and (19) one can find

which implies that as uniformly for . Now, we can assert that is relatively compact by Lemma 1 in C_h. Hence, is completely continuous.

From above proof

is completely continuous, which implies that is a condensing mapping from into . It follows from Lemma 3 that has a fixed point .

Finally, we show that . Let converge to . Then, according to the continuity ofQ and (18), we can find that converges to uniformly in and , which implies that is a positive mild solution of problem (1). This completes the proof of Theorem 2.

4 Example

We consider the following semilinear fractional parabolic equation initial boundary value problem:

where is the Caputo fractional partial derivative of order with the lower limits zero, , are constants, is a continuous function.

To treat this system in the abstract form (1), we choose the space equipped with the L²-norm . Let , thus, E is an ordered Banach space, and positive cone K is a normal regeneration cone.

Define operator by

From [2] we know that is a self-adjoint operator in E and generates an exponentially stable analytic semigroup ,which is contractive in E. Hence, 1 for every . Moveover, A has a discrete spectrum with eigenvalues of the form n², , and the associated normalized eigenfunctions are given by for . On the other hand, by the maximum principle of the parabolic type it is easy to find that is a positive semigroup. Since the operator A has compact resolvent in thus, is a compact semigroup (see [28]),which implies that the growth exponent of the semigroup satisfies .

For , we set and

It is easy to verify that is a continuous function. From the assumptions of problem (20) and (21) one can deduce that F satisfies the monotonicity condition (H2) and the asymptotically periodic condition (H3).

Let , then

Combining (22) and , one can find that condition (H10) holds. Therefore, from Corollary 1 it follows that problem (20) has a minimal positive time S-asymptotically ω-periodic mild solution ). On the other hand, it is clear that for ,

which implies that condition (H4) holds. Hence, if , from Theorem 2 one can obtain that for problem (20), there exists at least one positive time S-asymptotically ω- periodic mild solution u with ) for every and .

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