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.

1. Introduction

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 [16]. 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 X as
(1)D0qu(t)+Au(t)=f(t,u(t)),t>0,u(0)=u0,
where D0q denotes the Caputo fractional derivative of order q∈(0,1) with the lower limits zero, -A:D(A)⊂X→X is the infinitesimal generator of an analytic semigroup S(t)(t≥0) of uniformly bounded linear operators, and f 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.

2. Preliminaries

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 X be a Banach space with norm ∥·∥. Throughout this paper, we assume that -A:D(A)⊂X→X is the infinitesimal generator of an analytic semigroup S(t)(t≥0) of uniformly bounded linear operator in X; that is, there exists M≥1 such that ∥S(t)∥≤M for all t≥0. Without loss of generality, let 0∈ρ(-A), where ρ(-A) is the resolvent set of -A. Then for any α>0, we can define A-α by
(2)A-α:=1Γ(α)∫0∞tα-1S(t)dt.
Then Aα can be defined by Aα:=(A-α)-1 because A-α is one to one. It can be shown that each Aα has dense domain and that D(Aβ)⊂D(Aα) for 0≤α≤β. Moreover, Aα+βx=AαAβx=AβAαx for every α,β∈ℝ and x∈D(Aμ) with μ:=max{α,β,α+β}, where A0=I,I is the identity in X (for proofs of these facts we refer to the literature [20–22]).

We denote by Xα the Banach space of D(Aα) equipped with norm ∥x∥α=∥Aαx∥ for x∈D(Aα), which is equivalent to the graph norm of Aα. Then we have Xβ↪Xα for 0≤α≤β≤1 (with X0=X), and the embedding is continuous. Moreover, Aα has the following basic properties.

Lemma 1 (see [<xref ref-type="bibr" rid="B24">23</xref>]).

Aα has the following properties.

S(t):X→Xα for each t>0 and α≥0.

AαS(t)x=S(t)Aαx for each x∈D(Aα) and t≥0.

For every t>0, AαS(t) is bounded in X and there exists Mα>0 such that
(3)∥AαS(t)∥≤Mαt-α.

Let J be a closed interval on ℝ+=[0,∞). In the following we denote by C(J,Xα) the Banach space of all continuous functions from J into Xα endowed with supnorm given by ∥u∥C=supt∈J∥u(t)∥α for u∈C(J,Xα). For any t≥0, denote by Sα(t) the restriction of S(t) to Xα. From Lemma 1(i) and (ii), for any x∈Xα, we have
(4)∥S(t)x∥α=∥Aα·S(t)x∥=∥S(t)·Aαx∥≤∥S(t)∥·∥Aαx∥=∥S(t)∥·∥x∥α,∥S(t)x-x∥α=∥Aα·S(t)x-Aαx∥=∥S(t)·Aαx-Aαx∥→0
as t→0. Therefore, S(t)(t≥0) is a strongly continuous semigroup in Xα, and ∥Sα(t)∥α≤∥S(t)∥ for all t≥0. To prove our main results, the following lemma is also needed.

Lemma 2 (see [<xref ref-type="bibr" rid="B25">24</xref>]).

If S(t)(t≥0) is a compact semigroup in X, then Sα(t)(t≥0) is a compact semigroup in Xα, and hence it is norm continuous.

Let us recall the following known definitions in fractional calculus. For more details, see [9, 13–16, 18, 19].

Definition 3.

The fractional integral of order σ>0 with the lower limits zero for a function f is defined by
(5)I0σf(t)=1Γ(σ)∫0t(t-s)σ-1f(s)ds,t>0,
where Γ is the gamma function.

The Riemann-Liouville fractional derivative of order n-1<σ<n with the lower limits zero for a function f can be written as
(6)LD0σf(t)=1Γ(n-σ)dndtn∫0t(t-s)n-σ-1f(s)ds,t>0,n∈ℕ.
Also the Caputo fractional derivative of order n-1<σ<n with the lower limits zero for a function f∈Cn[0,∞) can be written as
(7)D0σf(t)=1Γ(n-σ)∫0t(t-s)n-σ-1f(n)(s)ds,t>0,n∈ℕ.

Remark 4.

(1) The Caputo derivative of a constant is equal to zero.

(2) If f is an abstract function with values in X, then integrals which appear in Definition 3 are taken in Bochner's sense.

Lemma 5 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

A measurable function h:J→X is Bochner integrable if ∥h∥ is Lebesgue integrable.

For x∈X, we define two families {U(t)}t≥0 and {V(t)}t≥0 of operators by
(8)U(t)x=∫0∞ηq(θ)S(tqθ)xdθ,V(t)x=q∫0∞θηq(θ)S(tqθ)xdθ,0<q<1,
where
(9)ηq(θ)=1qθ-1-1/qρq(θ-1/q),ρq(θ)=1π∑n=1∞(-1)n-1θ-qn-1Γ(nq+1)n!sin(nπq),θ∈(0,∞),
where ηq is a probability density function defined on (0,∞), which has properties ηq(θ)≥0 for all θ∈(0,∞) and ∫0∞ηq(θ)dθ=1. It is not difficult to verify (see [14]) that for μ∈[0,1], we have
(10)∫0∞θμηq(θ)dθ=Γ(1+μ)Γ(1+qμ).
Clearly, if the semigroup S(t)(t≥0) is positive, then, by the definitions, the operators U(t) and V(t) are also positive for all t≥0.

The following lemma follows from the results in [14, Lemma 2.9] and [15, Lemmas 3.2–3.5].

Lemma 6.

The operators U(t) and V(t) have the following properties.

For any fixed t≥0 and any x∈Xα, one has
(11)∥U(t)x∥α≤M∥x∥α,∥V(t)x∥α≤qMΓ(1+q)∥x∥α=MΓ(q)∥x∥α.

The operators U(t) and V(t) are strongly continuous for all t≥0.

If the semigroup S(t)(t≥0) is compact, then U(t) and V(t) are compact operators in X for t>0.

If the semigroup Sα(t)(t≥0) is norm continuous, then the restriction of U(t) to Xα and the restriction of V(t) to Xα are uniformly continuous for t>0.

Definition 7 (see [<xref ref-type="bibr" rid="B26">25</xref>, <xref ref-type="bibr" rid="B27">26</xref>]).

Let B be a bounded set of a real Banach space E. Set β(B)=inf{δ>0: B can be expressed as the union of a finite number of sets such that the diameter of each set does not exceed δ; that is, B=∪i=1mBi with diam(Bi)≤δ,i=1,2,…,m}. β(B) is called the Kuratowski measure of noncompactness of set B.

It is clear that 0≤β(B)<∞. For the Kuratowski measure of noncompactness, we have the following well-known results.

Lemma 8 (see [<xref ref-type="bibr" rid="B27">26</xref>]).

If D∈C(J,E) is bounded and equicontinuous, then
(12)β(D)=β(D(J))=maxt∈Jβ(D(t)),
where D(J)={x(t):x∈D,t∈J}.

Lemma 9 (see [<xref ref-type="bibr" rid="B28">27</xref>]).

Let D be a countable set of strongly measurable function x:J→E such that there exists an M∈L(J,ℝ+) such that ∥x(t)∥≤M(t)a.e.,t∈J for all x∈D. Then β(D(t))∈L(J,ℝ+) and

(13)β({∫Jx(t)dt:x∈D})≤2∫Jβ(D(t))dt.

Lemma 10 (see [<xref ref-type="bibr" rid="B26">25</xref>] Mönch fixed point theorem).

Let B be a closed and convex subset of E and y0∈B. Assume that the continuous operator A:B→B has the following property: D⊂B is countable, and D⊂Co¯({y0}∪A(D))→D is relatively compact. Then A has a fixed point in B.

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 u∈C(J,Xα) satisfying
(14)u(t)=U(t)u0+∫0t(t-s)q-1V(t-s)f(s,u(s))ds
for all t∈J.

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 λ1 be the smallest positive real eigenvalue of the linear operator A, and let e1∈D(A) be the positive eigenvector corresponding to λ1. For any T>0 and r>0, we write
(15)Ωr:={u∈C([0,T],Xα):∥u(t)∥α≤r,u(t)≥σe1,t∈[0,T]},
where σ>0 is a constant. Our main results are as follows.

Theorem 12.

Let -A:D(A)⊂X→X be the infinitesimal generator of a positive and compact analytic semigroup S(t)(t≥0) of uniformly bounded linear operators. Assume that f∈C(ℝ+×Xα,X) satisfies the following conditions.

For any u∈Ωr, one has
(16)f(t,u(t))≥f(t,σe1)≥λ1σe1,t∈[0,T].

f maps bounded sets of ℝ+×Xα into bounded sets of X.

If u0∈Xα with u0≥σe1 and αq<1/2 for some 1/2<q<1, then the IVP(1) has at least one positive mild solution u∈C([0,T),Xα). And if T<∞, one has limt∈T-∥u(t)∥α=∞. Proof.

For any t0≥0 and x0∈Xα with x0≥σe1, we first prove that the initial value problem (IVP) of fractional evolution equations
(17)Dt0qu(t)+Au(t)=f(t,u(t)),t>t0,u(t0)=x0
has at least one positive mild solution on J=[t0,t0+ht0], where ht0 is a positive constant and will be given later.

Let Rt0:=2M(∥x0∥α+1)+σe1>0. Denote
(18)ΩRt0:={u∈C(J,Xα):∥u(t)∥α≤Rt0,u(t)≥σe1,t∈J}.
Then ΩRt0⊂C(J,Xα) is a nonempty bounded convex closed set. The assumption (H2) implies that there is a constant C=C(t0)>0 such that
(19)∥f(t,u)∥≤C
for any t∈J and u∈ΩRt0.

Define an operator Q by
(20)(Qu)(t)=U(t-t0)x0+∫t0t(t-s)q-1V(t-s)f(s,u(s))ds,t∈J.
By the continuity of f, it is not difficult to prove that Q:C(J,Xα)→C(J,Xα) is continuous. By the positivity of the semigroup S(t)(t≥0), the assumption (H1), and (20), we easily see that (Qu)(t)≥(Qσe1)(t). Clearly, the positive mild solution of the IVP(17) on J is equivalent to the fixed point of operator Q in ΩRt0. We will use Schauder fixed point theorem to prove that Q has fixed points in ΩRt0.

We first prove that Q:ΩRt0→ΩRt0 is continuous. Let ht0≤[M(1-α)Γ(1+q(1-α))(∥x0∥α+1)/MαCΓ(2-α)]1/q(1-α). For any u∈ΩRt0 and t∈J, by Lemma 6, (10), (19), and (20), we have
(21)∥(Qu)(t)∥α≤∥U(t-t0)x0∥α+∥∫t0t(t-s)q-1V(t-s)f(s,u(s))ds∥α≤M∥x0∥α+∫t0t(t-s)q-1∥AαV(t-s)∥·∥f(s,u(s))∥ds≤M∥x0∥α+qMαC∫t0t(t-s)q(1-α)-1ds·∫0∞θ1-αηq(θ)dθ=M∥x0∥α+MαCΓ(2-α)(1-α)Γ(1+q(1-α))ht0q(1-α)≤Rt0.

Let v0≡σe1. Then v0(t)=σe1 for any t∈J and
(22)ϕ(t)≜Dt0qv0(t)+Av0(t)=λ1σe1≤f(t,σe1),t∈J.
By the positivity of the semigroup S(t)(t≥0), assumption (H1), and (20), for any t∈J, we have
(23)σe1=v0(t)=U(t-t0)v0(t0)+∫t0t(t-s)q-1V(t-s)ϕ(s)ds≤U(t-t0)x0+∫t0t(t-s)q-1V(t-s)f(s,σe1)ds=(Qσe1)(t)≤(Qu)(t).
Thus, Q:ΩRt0→ΩRt0 is continuous.

By using a similar argument as in the proof of Theorem 3.1 in [14], we can prove that Q:ΩRt0→ΩRt0 is a compact operator. Hence by Schauder fixed point theorem, the operator Q has at least one fixed point u* in ΩRt0, which satisfies u*(t)≥σe1>0 for all t∈J. Hence u* is a positive mild solution of the IVP(1) on J.

Therefore, there exists [0,h0] such that the IVP(1) has at least one positive mild solution u*∈C([0,h0],Xα). Now, by the standard proof method of extension theorem of initial value problem, u* can be extended to a saturated solution u∈C([0,T),Xα) of the IVP(1), whose existence interval is [0,T), and if T<∞, we have limt→T-∥u(t)∥α=∞.

For any T>0 and r>0, define Ωr as in (15). If f(t,u) is increasing in Ωr, that is, f(t,u) satisfies the condition

for any u1,u2∈Ωr with u1(t)≤u2(t) for all t∈[0,T], we have
(24)f(t,u1(t))≤f(t,u2(t)),t∈[0,T],

then we have f(t,u(t))≥f(t,σe1) for any u∈Ωr and t∈[0,T]. On the other hand, if f(t,u) 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 -A:D(A)⊂X→X be the infinitesimal generator of a positive and compact analytic semigroup S(t)(t≥0) of uniformly bounded linear operators. Assume that f∈C(ℝ+×Xα,X) satisfies condition (H1)* and

(H2)* there exists a constant af>0 such that
(25)∥f(t,x)∥≤af(1+∥x∥α)

for all t∈[0,T] and x∈Xα.

If f(t,σe1)≥λ1σe1 for all t∈[0,T], u0∈Xα with u0≥σe1 and αq<1/2 for some 1/2<q<1, then the IVP(1) has at least one positive mild solution u∈C([0,T),Xα). And if t<∞, one has limt→T-∥u(t)∥α=∞.

Since the analytic semigroup is norm continuous, it follows that we can delete the compactness condition on the analytic semigroup S(t)(t≥0) and obtain the following existence result.

Theorem 14.

Assume that -A:D(A)⊂X→X is the infinitesimal generator of a positive analytic semigroup S(t)(t≥0) of uniformly bounded linear operators, and that f∈C(ℝ+×Xα,X) satisfies the condition (H1) and

for any T>0 and r>0, f(t,Ωr):={f(t,u):u∈Ωr} is relatively compact in Xα for all t∈[0,T], where Ωr is defined as in (15).

If u0∈Xα with u0≥σe1 and αq<1/2 for some 1/2<q<1, then the IVP(1) has at least one positive mild solution u∈C([0,T),Xα). And if t<∞, one has limt→T-∥u(t)∥α=∞. Proof.

For any t0≥0 and x0∈Xα with x0≥σe1, we first prove that the IVP(17) has at least one positive mild solution on J=[t0,t0+ht0], where ht0>0 is a constant and will be specified later. Define an operator Q as in (20). Let Rt0=2M(∥x0∥α+1)+σe1. Write ΩRt0 as in (18). The condition (H3) implies that f(t,ΩRt0) is bounded for any t∈J, that is, there is a positive constant C=C(t0) such that
(26)∥f(t,u(t))∥≤C,t∈J,u∈ΩRt0.
Let ht0≤[M(1-α)Γ(1+q(1-α))(∥x0∥α+1)/MαCΓ(2-α)]1/q(1-α). A similar argument as in the proof of Theorem 12 shows that Q:ΩRt0→ΩRt0 is continuous and QΩRt0 is equicontinuous.

Thus, for any D⊂ΩRt0, let D(t):={u(t):u∈D},t∈J. Since QD⊂QΩRt0⊂ΩRt0 is equicontinuous and bounded, by Lemma 8, we have
(27)β(QD)=maxt∈Jβ((QD)(t)).

Now, let D={um:m=1,2,…}⊂ΩRt0 with D⊂Co¯({y0}∪QD) for some y0∈ΩRt0. It is obvious that
(28)∥(t-s)q-1V(t-s)f(s,u(s))∥α≤CMαqΓ(2-α)Γ(1+q(1-α))(t-s)q-1∈L(J,ℝ+).
Hence by Lemma 9 and (20), we have
(29)β((QD)(t))=β(U(t-t0)x0+∫t0t(t-s)q-1V(t-s)f(s,D(s))ds)=β(∫t0t(t-s)q-1V(t-s)f(s,D(s))ds)≤2∫t0t(t-s)q-1∥AαV(t-s)∥·β(f(s,D(s)))ds≤2MαqΓ(2-α)Γ(1+q(1-α))∫t0t(t-s)q(1-α)-1·β(f(s,D(s)))ds=0.
It follows that β((QD)(t))=0 for all t∈J. By Lemma 8 and (27), we have β(QD)=maxt∈Jβ((QD)(t))=0. Thus, we have
(30)β(D)≤β(Co¯({y0}∪QD))=β({y0}∪QD)=β(QD)=0.
This implies that D is relatively compact. Therefore, by Mönch fixed point theorem, the operator Q has at least one fixed point u*∈ΩRt0, which satisfies u*(t)≥σe1>0 for all t∈J. Hence u* is a positive mild solution of the IVP(17) on J.

Therefore, there exists [0,h0] such that the IVP(1) has at least one positive mild solution u*∈C([0,h0],Xα). u* can be extended to a saturated solution u∈C([0,T),Xα) of IVP(1), whose existence interval is [0,T) and when t≤∞, we have limt→T-∥u(t)∥α=∞.

4. Positive Mild Solutions of Parabolic Equations

Let Ω⊂ℝN be a bounded domain with a sufficiently smooth boundary ∂Ω. Let
(31)A(x,D)u=-∑i,j=1N∂∂xi(aij(x)∂u∂xj)+a0(x)u
be a uniformly elliptic differential operator of divergence form in Ω¯, where the coefficients aij∈C1+μ(Ω¯)(i,j=1,2,…,N) and a0∈Cμ(Ω¯) for some μ∈(0,1). We assume that [aij(x)]N×N is a positive define symmetric matric for every x∈Ω¯, and there exists a constant ν>0 such that (32)∑i,j=1Naij(x)ξiξj≥ν|ξ|2,∀ξ=(ξ1,ξ2,…,ξN)∈ℝN,x∈Ω¯.

Let a0(x)≥0 on Ω¯. We use (x,t,η) to denote a generic point of Ω¯×ℝ+×ℝ, where ℝ=[0,+∞) and ℝ=(-∞,+∞). Let F:Ω¯×ℝ+×ℝ→ℝ be a continuous function. We discuss the existence of positive mild solutions for the parabolic initial boundary value problem (IBVP)
(33)∂q∂tqu(x,t)+A(x,D)u(x,t)=F(x,t,u(x,t))inΩ×ℝ+,u|∂Ω=0,u(x,0)=φ(x)inΩ,
where 0<q<1 is a constant.

Let λ1 be the smallest positive real eigenvalue of elliptic operator A(x,D) under the Dirichlet boundary condition u|∂Ω=0. It is well known (cf. Amann [22, 28]) that λ1>0. Let e1(x) be the positive eigenvector corresponding to λ1. Assume that F:Ω¯×ℝ+×ℝ→ℝ is continuous and satisfies the following conditions.

For any T>0 and r>0, there exists a constant σ>0 such that
(34)F(x,t,η)≥F(x,t,σe1(x))≥λ1σe1(x),t∈[0,T],x∈Ω¯,

where η∈ℝ with σe1(x)≤η≤r.

For any T>0, there exists a constant af>0 such that
(35)|F(x,t,η)|≤af(1+|η|),x∈Ω¯,t∈[0,T].

Let X=L2(Ω). Define an operator A:D(A)⊂X→X by
(36)D(A)=H2(Ω)∩H01(Ω),Au=A(x,D)u.
It is well known (cf. Li [29]) that -A generates a compact analytic semigroup S(t)(t≥0) and D(A1/2)=H01(Ω). By the maximum principle of the equation of the parabolic type, it is easy to prove that S(t)(t≥0) is also a positive semigroup in X. The assumptions (F1) and (F2) imply that the mapping f:ℝ+×X1/2→X defined by
(37)f(t,u)(·)=F(·,t,u(·)),t∈ℝ+,u∈H01(Ω),
is continuous and satisfies the conditions (H1) and (H2). 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 F:Ω¯×ℝ+×ℝ→ℝ is continuous and satisfies conditions (F1) and (F2). If φ∈H01(Ω) with φ(x)≥σe1(x) for any x∈Ω¯ and 1/2<q<1, then the IBVP(33) has at least one positive mild solution u that satisfies u(x,t)≥σe1(x) for any x∈Ω¯ and t∈[0,T]. And if T<+∞, one has limt→T-|u(t)|=+∞.

Acknowledgments

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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