Existence of positive S -asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

. In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation 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 solutions are obtained by using monotone iterative method and ﬁxed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.


Introduction
Let (E, · ) be an ordered Banach space, whose positive cone K := {x ∈ E: x θ} is a normal cone with normal constant N , θ 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: c D q t u(t) + Au(t) = F t, u(t) , t 0, where c D q t is a Caputo fractional derivative of the order q ∈ (0, 1) with the lower limits zero, A : D(A) ⊂ E → E is a closed linear (not necessarily bounded) operator, and −A generates a C 0 -semigroup T (t) (t 0) in E, F : R + × E → E 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 fractional derivative has genetic or memory properties, the solutions of periodic boundary value problems for fractional differential equations cannot be extended periodically to time t in R + . Specially, the nonexistence of nontrivial periodic solutions of fractional evolution equations 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 been widely studied in fractional evolution equations, and the existence and uniqueness of S-asymptotically ω-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 conditions for nonlinear functions are necessary. In fact, for equations arising in complicated reaction-diffusion processes, the nonlinear function represents the source of material or population, 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.

Preliminaries
In this paper, we always assume that (E, · ) is an ordered Banach space, whose positive cone K = {u ∈ E: u θ} is a normal cone with normal constant N , θ is the zero element of E.
Let h : R + → R + be a continuous and nondecreasing function such that h(t) 1 for all t ∈ R + and lim t→∞ h(t) = ∞. Thus, we can define a Banach space For the Banach space C h (E), we have the following result.
(iii) The set B(t) = {u(t): u ∈ B} is relatively compact in E for every t 0.
Next, we introduce a standard definition of S-asymptotically ω-periodic function. Let C b (R + , E) denote the Banach space of bounded and continuous functions from R + to E with the norm u C = sup t∈R + u(t) .
Let SAP ω (E) be the subspace of C b (R + , E) consisting of all the E-valued S-asymptotically ω-periodic functions equipped with norm · C . Then SAP ω (E) is a Banach space [15].
Define a positive cone K h ⊂ C h (E) by Thus, C h (E) is an ordered Banach space, whose partial order relation " " is induced by the cone K h . Let A : D(A) ⊂ E → E be a closed linear operator and −A generate a positive C 0 -semigroup T (t) (t 0) in E. For a general C 0 -semigroup T (t) (t 0), there exist M 1 and ν ∈ R such that (see [28]) M e νt , t 0.
https://www.journals.vu.lt/nonlinear-analysis Specially, C 0 -semigroup T (t)(t 0) is called to be uniformly bounded if The growth exponent of T (t) (t 0) is defined by If ν 0 < 0, then T (t) (t 0) is said to be exponentially stable. Clearly, the exponentially is continuous in the uniform operator topology for every t > 0 in E, it is well known that ν 0 can also be determined by σ(A) (the resolvent set of A) For more details about positive C 0 -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 U (t) (t 0) and V (t) (t 0) in E as following: where is a probability density function satisfying Lemma 2. The operator families U (t) (t 0) and V (t) (t 0) defined by (2) have the following properties: are strongly continuous operators, i.e., for any x ∈ E and 0 t 1 t 2 , is uniformly bounded , then U (t) and V (t) are linear bounded operators for any fixed t ∈ R + , i.e., for every t 0, where E q (·) and E q,q (·) are the Mittag-Leffler functions.
θ for all t 0, 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 E. If Q : D → D is a condensing map, then Q has a fixed poind in D.

Main results
Theorem 1. Let E be an ordered Banach space, whose positive cone K is normal cone, let A : D(A) ⊂ E → E be a closed linear operator and −A generate an exponentially stable, positive, and compact semigroup T (t) (t 0) in E, whose growth exponent denotes ν 0 < 0, u 0 θ. Assume that F : R + × E → E is a continuous function and the following conditions hold: (H1) There exist nonnegative constants a ∈ (0, |ν 0 |/M ) and b 0 such that (H2) F is nondecreasing with respect to the second variable, i.e., for x 2 x 1 θ, (H3) There exists ω > 0 such that Then there exists a minimal positive S-asymptotically ω-periodic mild solution u * of problem (1).
Proof. Define an operator Q on C h (E) by It is easy to find Q : (3), and (5) it follows that for t 0, as n → ∞, that is, for arbitrary ε > 0, there exists sufficiently large n such that u n − u h < ε for. For above ε, by the continuity of F it is easy to see and the dominated convergence theorem ensures Hence, we conclude that Q is continuous from C h (E) to C h (E). Specially, by (8) one can find that for every u ∈ C h (E), where γ = M u 0 + M b/|ν 0 | and β = M a/|ν 0 | < 1. Therefore, from Definition 2 it follows that the fixed points of Q are mild solutions to problem (1). Next, we will prove that Q(SAP ω (E)) ⊂ SAP ω (E). Choose u ∈ SAP ω (E), then for any ε > 0, there exists a constant t ε,1 > 0 such that ε for all t t ε,1 . Thus, by continuity of F we have On the other hand, by (H3) there is a sufficiently large constant t ε,2 such that for t t ε,2 , Hence, for t > t ε := max{t ε,1 , t ε,2 }, from (7) it follows that According to (6), let Thus, by (13) and (H1) one can find that Hence, we deduce that I 1 , I 2 tend to 0 as t → ∞. By (13), (11), and (H1) one can obtain that which implies that I 3 (t) tends to 0 as t → ∞. Similarly, by (13), (12), and (H1) we can get that I 4 (t) tends to 0 as t → ∞. Thus, from the above results we can deduce that namely, Qu ∈ SAP ω (E), which implies that Q(SAP ω (E)) ⊂ SAP ω (E). Now, we prove the existence of positive solutions by monotone iterative technique. For any u, v ∈ K h with u v, by (H2), (7), the positivity of U (t), V (t), and u 0 θ, one can find that for all t ∈ [0, ∞), θ Qu(t) Qv(t).
Thus Q is monotone increasing. Let From the definition and properties of Q, (14), (10) one can find Since thus, the sequence {v i } is uniformly bounded. Next, we prove that the sequence {v i } is uniformly convergent.
. For arbitrary r 0 ∈ [0, ∞), one can obtain that {v i (t)} is relatively compact on E for t ∈ [0, r 0 ]. In fact, for all ε ∈ (0, t) and for all δ > 0, we define a set Q ε,δ V 0 (t) by By the compactness of U (t) and T (ε q δ) the set Q ε,δ V 0 (t) is relatively compact in E. Moreover, for every v i ∈ V 0 and t ∈ [0, r 0 ], one has Hence, the set (QV 0 )(t) is relatively compact, which implies that {v i (t)} is relatively compact on E for t ∈ [0, r 0 ]. Therefore, by the definition of Q and the arbitrariness of r 0 we can obtain that {v i (t)} is relatively compact on E for t ∈ [0, ∞).
In general, let 0 t 1 < t 2 . For any u ∈ {v i }, by the definition of Q one can see Next, we check if J i (i = 1, 2, 3, 4) tend to 0 as t 2 − t 1 → 0 independently of u ∈ {v i }. From (4) it follows that J 1 → 0 as t 2 − t 1 → 0. By (H1), (13), and (16) we can obtain If t 1 = 0 and t 2 > 0, then it easy to see that J 3 = 0. For t 1 > 0 and > 0 small enough, by (H1), (13), (16), and Lemma 2(iv) we get that As a result, Qu(t 2 ) − Qu(t 1 ) tends to 0 as t 2 − t 1 → 0 independently of u ∈ {v i }, thus {v i } is equicontinuous. Thirdly, for every u ∈ {v i }, by (8) and (16) one can find , t 0, which implies that Qu(t) /h(t) → 0 as t → ∞ uniformly for u ∈ {v i }. Therefore, Lemma 1 allows us to deduce that {v i } is relatively compact in K h ∩ SAP ω (E), thus there is convergent subsequence in {v i }. From the monotonicity of sequence and the normality of cone we can obtain that {v i } itself is uniformly convergent, which means that there is u * ∈ K h ∩ SAP ω (E) such that lim i→∞ v i = u * . Taking the limit in (14), one can see u * = Qu * . Thus u * ∈ K h ∩ SAP ω (E) is fixed point of Q, which is a positive S-asymptotically ω-periodic mild solution of problem (1). It is easy to prove that u * is the minimal positive mild solution. To this end, let u ∈ K h ∩ SAP ω (E) be a positive S-asymptotically ω-periodic mild solution of problem (1), namely, u(t) = Q u(t) for every t ∈ [0, ∞). Obviously, u(t) v 0 , and from the monotonicity of Q it follows that namely, u v 1 . In general, u v i , i = 1, 2, . . . . Taking the limit in (17) as i → ∞, we have u u * , which means that u * 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 λ 0 > − inf{Re λ: λ ∈ σ(A)}, we have that λ 0 I + A has positive bounded inverse operator (λ 0 I + A) −1 . Since σ(A) = ∅, the spectral radius By the famous Krein-Rutmann theorem A has the smallest eigenvalue λ 1 > 0, which has a positive eigenfunction e 1 , and which implies that ν 0 = −λ 1 . 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 A : D(A) ⊂ E → E be a closed linear operator and −A generate an exponentially stable, positive, and compact semigroup  hold. Then problem (1) has a minimal positive S-asymptotically ω-periodic mild solution u * .
Theorem 2. Let E be an ordered Banach space, whose positive cone K is regeneration cone, let A : D(A) ⊂ E → E be a closed linear operator and −A generate an exponentially stable, positive, and compact semigroup T (t) (t 0) in E. Assume that F : R + × E → E is a continuous function, and let conditions (H1 ), (H3), hold and u 0 σe 1 . Then problem (1) has at least one positive S-asymptotically ω-periodic mild solution.
Next, we show that Q is completely continuous. From assumptions (H1 ) and (H3) there is a constant M 1 such that for all u ∈ Ω R0 , Thus, it is easy to verify that the set is relatively compact on E for any r 0 ∈ (0, ∞). In order to do this, we define a set where ε ∈ (0, t), δ > 0, and By the compactness of U (t) and T (ε q δ) the set Q ε,δ (Ω R0,r0 )(t) is relatively compact in E. Right now, for every u ∈ Ω R0 and t ∈ [0, r 0 ], one has Hence, the set Q(Ω R0,r0 )(t) is relatively compact on E for t ∈ [0, r 0 ]. Therefore, by the definition of Q and the arbitrariness of r 0 we can deduce that Q(Ω R0 )(t) is relatively compact on E for t ∈ [0, ∞). Moreover, it is easy to prove that set Q(Ω R0 ) is equicontinuous by using the method similar to Theorem 1. Also, for every u ∈ Ω R0 , by (8) and (19) one can find which implies that Qu(t) /h(t) → 0 as t → ∞ uniformly for u ∈ Ω R0 . Now, we can assert that Q(Ω R0 ) is relatively compact by Lemma 1 in C h . Hence, Q is completely continuous.
From above proof is completely continuous, which implies that Q is a condensing mapping from SAP ω (E) ∩ Ω R0 into SAP ω (E) ∩ Ω R0 . It follows from Lemma 3 that Q has a fixed point u * ∈ SAP ω (E) ∩ Ω R0 . Finally, we show that u * ∈ SAP ω (E). Let {u n } ⊂ SAP ω ∩ Ω R0 converge to u * . Then, according to the continuity of Q and (18), we can find that {Qu n } converges to Qu * = u * uniformly in [0, ∞) and u * σe 1 , which implies that u * ∈ SAP ω (E) is a positive mild solution of problem (1). This completes the proof of Theorem 2.

Example
We consider the following semilinear fractional parabolic equation initial boundary value problem: where ∂ q /∂t q is the Caputo fractional partial derivative of order q ∈ (0, 1) with the lower limits zero, a ∈ (0, 1), b > 0 are constants, u 0 : [0, π] → R + is a continuous function. To treat this system in the abstract form (1), we choose the space E = L 2 [0, π] equipped with the L 2 -norm · . Let K = {u ∈ E: u(ξ) 0 a.e. ξ ∈ [0, π]}, thus, E is an ordered Banach space, and positive cone K is a normal regeneration cone.
From [2] we know that −A is a self-adjoint operator in E and generates an exponentially stable analytic semigroup T (t) (t 0), which is contractive in E. Hence, T (t) M := 1 for every t 0. Moveover, A has a discrete spectrum with eigenvalues of the form n 2 , n ∈ N, and the associated normalized eigenfunctions are given by e n (ξ) = 2/π sin(nξ) for ξ ∈ [0, π]. On the other hand, by the maximum principle of the parabolic type it is easy to find that T (t) (t 0) is a positive semigroup. Since the operator A has compact resolvent in L 2 [0, π], thus, T (t) (t 0) is a compact semigroup (see [28]), which implies that the growth exponent of the semigroup T (t) (t 0) satisfies ν 0 = −1.
Let h(t) = e t , then F t, e t x a sin 2 t e t e t x + b e 1 a x + b.