# Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

Authors: Yue Liang; Yu Ma; Xiaoyan Gao
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroupDqu(t)+Au(t)=f(t,u(t)), t≥0; u(0)=u0 in a Banach space X, where Dq denotes the Caputo fractional derivative of order q∈(0,1), A:D(A)⊂X→X is a closed linear operator, -A generates an equicontinuous C0 semigroup, and f:[0,∞)×X→X is continuous. In the case where f satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.


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## Body

## 1. Introduction

The theory of fractional differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model; see [1–6]. Correspondingly, the existence of solutions to fractional evolution equations in Banach space has also been studied by several authors; see [7–17]. In [7, 8], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then they investigated the existence, uniqueness, and regularity of solutions to fractional integrodifferential equations in [9, 10]. Recently, this theory was developed by Zhou et al. [11–14]. In [15–17], the authors studied the existence of mild solutions to fractional impulsive evolutions equations. But as far as we know, there are seldom results on the existence of positive solutions to the fractional evolution equations; see [18–20].In this paper, we use the Sadovskii’s fixed point theorem and monotone iterative technique to discuss the existence of positive and saturated mild solutions for the initial value problem (IVP) of fractional evolution equations:(1)Dqu(t)+Au(t)=f(t,u(t)),t≥0,u(0)=u0
in Banach space X, where Dq denotes the Caputo fractional derivative of order q∈(0,1), A:D(A)⊂X→X is a closed linear operator, -A generates a C0-semigroup S(t) (t≥0) in X, and f:ℝ+×X→X is continuous and will be specified later, ℝ+=[0,∞).In some existing articles, the fractional evolution equations were treated under the hypothesis that (I)-A generates a compact semigroup or (II) the nonlinearity f(t,u) is Lipschitz continuous in u on a bounded set. For the case (I), the continuity of nonlinearity f can guarantee the local existence of solutions. Hence it is convenient to apply to partial differential equations with compact resolvent. But for the case of noncompact semigroup, the condition (II) is not easy to verify sometimes. To make the things more applicable, in this work, we will prove the existence of mild solutions of the IVP(1) under the measure of noncompactness conditions. We will see that our conditions are weaker than the condition (II). In addition, we obtain the existence of positive mild solutions of the IVP(1) in this work, which is studied seldom before.The rest of this paper is organized as follows. In Section2, some preliminaries are given on the fractional calculus and the measure of noncompactness. In Section 3, we study the existence of positive and saturated mild solutions of the IVP(1). An example is given in Section 4 to illustrate the applicability of the abstract results obtained in Section 3.


## 2. Preliminaries

In this section, we introduce some basic facts about the fractional calculus and the measure of noncompactness that are used throughout this paper.LetX be a Banach space with norm ∥·∥, let A:D(A)⊂X→X be a closed linear operator, and -A generates a C0-semigroup S(t) (t≥0) in X. It is well known that there exist M¯>0 and δ∈ℝ such that
(2)∥S(t)∥≤M¯eδt,t≥0.
Let T>0 be a constant. If t∈[0,T], it follows from (2) that there exists a constant M>0 such that ∥S(t)∥≤M.Let us recall the following known definitions in fractional calculus. For more details, see [7, 8, 11–14, 16, 17] and the reference therein.Definition 1.
The fractional integral of orderσ>0 with the lower limits zero for a function f is defined by
(3)Iσf(t)=1Γ(σ)∫0t‍(t-s)σ-1f(s)ds,t>0,
where Γ is the gamma function.
The Riemann-Liouville fractional derivative of ordern-1<σ<n with the lower limits zero for a function f can be written as
(4)DLσf(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
(5)Dσf(t)=1Γ(n-σ)∫0t‍(t-s)n-σ-1f(n)(s)ds,t>0,n∈ℕ.Remark 2.
(1) The Caputo derivative of a constant is equal to zero.
(2) Iff is an abstract function with values in X, then integrals which appear in Definition 1 are taken in Bochner’s sense.Lemma 3 (see [12]).
A measurable functionh:[0,T]→X is Bochner integrable if ∥h∥ is Lebesgue integrable.Forx∈X, we define two families {U(t)}t≥0 and {V(t)}t≥0 of operators by
(6)U(t)x=∫0∞‍ηq(θ)S(tqθ)xdθ,V(t)x=q∫0∞‍θηq(θ)S(tqθ)xdθ,0<q<1,
where
(7)ηq(θ)=1qθ-1-(1/q)ρq(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. Clearly, if the semigroup S(t) (t≥0) is positive, then the operators U(t) and V(t) are also positive for all t≥0.The following lemma is needed in the proof of the main results.Lemma 4.
The operatorsU(t) and V(t) have the following properties.(i)
For any fixedt≥0 and any x∈X, one has
(8)∥U(t)x∥≤M∥x∥,∥V(t)x∥≤qMΓ(1+q)∥x∥=MΓ(q)∥x∥.(ii)
The operatorsU(t) and V(t) are strongly continuous for all t≥0.(iii)
IfS(t) (t≥0) is a equicontinuous semigroup, then U(t) and V(t) are equicontinuous in X for t>0.Proof.
(i) and (ii) can be found in [12, 13], and we only need to prove (iii). For any 0≤t1<t2≤T, we have
(9)∥U(t2)-U(t1)∥=∫0∞‍ηq(θ)∥S(t2qθ)-S(t1qθ)∥dθ,∥V(t2)-V(t1)∥=q∫0∞‍θηq(θ)∥S(t2qθ)-S(t1qθ)∥dθ.
According to the equicontinuity of S(t) for t>0, we see that ∥U(t2)-U(t1)∥ and ∥V(t2)-V(t1)∥ tend to zero as t2-t1→0, which means that the operators U(t) and V(t) are equicontinuous in X for t>0.Lett0≥0 be a constant and I=[t0,T]. We denote by C(I,X) the Banach space of all continuous X value functions on interval I with the norm ∥u∥C=maxt∈I∥u(t)∥. Let α(B) denote the Kuratowski measure of noncompactness of the bounded set B in X and C(I,X). It is clear that 0≤α(B)<∞. If α(B)=0, then the set B is relatively compact. For more details of the definition and properties of the measure of noncompactness; see [21]. For any B⊂C(I,X) and t∈I, set B(t)={u(t):u∈B}⊂X. If B is bounded in C(I,X), then B(t) is bounded in X, and α(B(t))≤α(B). A mapping Q:B→B is said to be condensing if α(Q(B))<α(B). The following Lemmas will be used in the proof of the main results.Lemma 5 (see [22]).
LetB∈C(I,X) be bounded and equicontinuous. Then α(B(t)) is continuous on I and
(10)α(B)=maxt∈Iα(B(t))=α(B(I)),
where B(I)={x(t):x∈B,t∈I}.Lemma 6 (see [23]).
LetB={un}⊂C(I,X) be countable. If there exists ψ∈L1(I) such that ∥un(t)∥≤ψ(t) a.e. t∈I, n=1,2,…, then α(B(t)) is Lebesgue integral on I and
(11)α({∫I‍un(t)dt:n∈ℕ})≤2∫I‍α(B(t))dt.Lemma 7 (see [24]).
LetB⊂C(I,X) be bounded. Then there exists a countable subset B0 of B such that α(B)≤2α(B0).Lemma 8 (see [25] (Sadovskii’s fixed point theorem)).
LetX be a Banach space and let Ω be a nonempty bounded convex closed set in X. If Q:Ω→Ω is a condensing mapping, then Q has a fixed point in Ω.In the proof of the main results, we also need the following generalized Gronwall-Bellman inequality, which can be found in [26, Page 188].Lemma 9.
Supposeb≥0, β>0, and a(t) is a nonnegative function locally integrable on 0≤t<T (some T≤∞), and suppose u(t) is nonnegative and locally integrable on 0≤t<T with
(12)u(t)≤a(t)+b∫0t‍(t-s)β-1u(s)ds
on this interval, and then
(13)u(t)≤a(t)+∫0t‍[∑n=1∞‍(bΓ(β))nΓ(nβ)(t-s)nβ-1a(s)]ds,0≤t<T.Remark 10.
In Lemma9, if a(t)≡0 for all 0≤t<T, we easily see that u(t)=0.For anyx0∈X and h∈C(I,X), a function u is called the mild solution of the initial value problem
(14)Dqu(t)+Au(t)=h(t),t∈I,u(t0)=x0,
if u∈C(I,X) satisfies the integral equation:
(15)u(t)=U(t-t0)x0+∫t0t‍(t-s)q-1V(t-s)h(s)ds,t∈I.
Hence, for the IVP(1), we have the following definition.Definition 11.
By a mild solution of the IVP(1), we mean a function u∈C(I,X) satisfying
(16)u(t)=U(t)u0+∫0t‍(t-s)q-1V(t-s)f(s,u(s))ds
for all t≥0.


## 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.Letλ1 be the smallest positive real eigenvalue of the linear operator A, and let e1∈D(A) be the positive eigenvector corresponding to λ1. Our main results are as follows.Theorem 12.
LetX be a Banach space, let A:D(A)⊂X→X be a closed linear operator, and -A generate a positive and equicontinuous C0-semigroup S(t) (t≥0) in X. Assume that f∈C(ℝ+×X,X) and, for any T>0, f satisfies the following conditions.(H1)
There exista,b∈L1/q1([0,T],ℝ+), q1∈(0,q) such that
(17)∥f(t,x)∥≤a(t)∥x∥+b(t),t∈[0,T],x∈X.(H2)
For anyu∈C([0,T],X) with u(t)≥σe1, t∈[0,T], we have
(18)f(t,u(t))≥f(t,σe1),t∈[0,T],
where σ>0 is a constant.(H3)
For any bounded setD⊂X, there exists a constant L>0 such that
(19)α(f(t,D))≤Lα(D),t∈[0,T].
Iff(t,σe1)≥λ1σe1 and u0∈X with u0≥σe1, then the IVP(1) has at least one positive and saturated mild solution u∈C([0,T),X). And if T<∞, one has limt∈T-∥u(t)∥=∞.Proof.
For anyt0≥0 and x0∈X with x0≥σe1, we first prove that there exists a constant ht0=h(t0,∥x0∥)>0 such that the initial value problem (IVP)
(20)Dqu(t)+Au(t)=f(t,u(t)),t>t0,u(t0)=x0
has at least one positive mild solution on J=[t0,t0+ht0]. For this purpose, we define an operator Q by
(21)(Qu)(t)=U(t-t0)x0+∫t0t‍(t-s)q-1V(t-s)f(s,u(s))ds,00000000000000000000000000000000000000t≥t0.
Then Q:C(J,X)→C(J,X) is continuous, and the mild solutions of the IVP(20) are equivalent to the fixed point of the operator Q.
LetRt0:=2M(∥x0∥+1)+σe1>0. Denote
(22)ΩRt0:={u∈C(J,X):∥u(t)∥≤Rt0,u(t)≥σe1,t∈J}.
Then ΩRt0⊂C(J,X) is a nonempty bounded convex closed set. Let ht0=h(t0,∥x0∥)=min{1,(Γ(q)(1+c)1-q1(∥x0∥+1)/(Rt0a0+b0))1/(q-q1),(Γ(q+1)/(4ML+1))1/q}, where c=((q-1)/(1-q1))∈(-1,0), a0=∥a∥L1/q1([t0,t0+1],ℝ+), b0=∥b∥L1/q1([t0,t0+1],ℝ+). Then for any u∈ΩRt0 and t∈J, by Lemma 4(i), (H1), and (21), we have
(23)∥(Qu)(t)∥≤∥U(t-t0)x0∥+∫t0t‍(t-s)q-1∥V(t-s)f(s,u(s))∥ds≤M∥x0∥+MΓ(q)∫t0t‍(t-s)q-1[a(s)∥u(s)∥+b(s)]ds=M∥x0∥+MΓ(q)[Rt0∫t0t‍(t-s)q-1a(s)ds0000000000.00000+∫t0t‍(t-s)q-1b(s)ds]≤M∥x0∥+M(Rt0a0+b0)Γ(q)(1+c)1-q1·ht0q-q1≤Rt0.
Letv0≡σe1. Then v0(t)=σe1 for any t∈J and
(24)ϕ(t)≜Dqv0(t)+Av0(t)=λ1σe1≤f(t,σe1),t∈J.
By the positivity of semigroup S(t) (t≥0), the assumption (H2) and (21), for any u∈ΩRt0 and t∈J, we have
(25)σe1=v0(t)=U(t-t0)v0(t0)+∫t0t‍(t-s)q-1V(t-s)ϕ(s)ds≤U(t-t0)σe1+∫t0t‍(t-s)q-1V(t-s)f(s,σe1)ds≤U(t-t0)x0+∫t0t‍(t-s)q-1V(t-s)f(s,u(s))ds=(Qu)(t).
Thus, Q:ΩRt0→ΩRt0 is continuous and it implies that (Qσe1)(t)≤(Qu)(t) for any u∈ΩRt0 and t∈J.
Now, we prove that the setQ(ΩRt0):={Qu:u∈ΩRt0} is equicontinuous in C(J,X). For any u∈ΩRt0 and t0≤t1<t2≤t0+ht0, it follows from assumption (H1) and (21) that
(26)∥(Qu)(t2)-(Qu)(t1)∥≤∥U(t2-t0)x0-U(t1-t0)x0∥+∫t1t2‍(t2-s)q-1×∥V(t2-s)f(s,u(s))∥ds+∫t0t1‍|(t2-s)q-1-(t1-s)q-1|·∥V(t2-s)f(s,u(s))∥ds+∫t0t1‍(t1-s)q-1×∥[V(t2-s)-V(t1-s)]00000000×f(s,u(s))∥ds≤M∥U(t2-t1)x0-x0∥+MΓ(q)∫t1t2‍(t2-s)q-1×[a(s)∥u(s)∥+b(s)]ds+MΓ(q)∫t0t1‍|(t2-s)q-1000000000000-(t1-s)q-1|·[a(s)∥u(s)∥+b(s)]ds+∫t0t1‍(t1-s)q-1×∥[V(t2-s)-V(t1-s)]0000000×f(s,u(s))∥ds≜I1+I2+I3+I4.
By Lemma 4(ii), it is easy to see that I1→0 independently of u∈ΩRt0 as t2-t1→0:
(27)I2≤MΓ(q)[Rt0∫t1t2‍(t2-s)q-1a(s)ds00000000+∫t1t2‍(t2-s)q-1b(s)ds]≤M(Rt0a0+b0)Γ(q)(1+c)1-q1(t2-t1)q-q1.
HenceI2→0 independently of u∈ΩRt0 as t2-t1→0:
(28)I3≤MΓ(q)[Rt0∫t0t1‍|(t2-s)q-1-(t1-s)q-1|×a(s)ds00000000+∫t0t1|(t2-s)q-1-(t1-s)q-1|b(s)ds∫t0t1‍]‍≤MΓ(q)[Rt0(∫t0t1‍|(t2-s)q-10000000000000000.-(t1-s)q-1|1/(1-q1)ds∫t0t1‍)1-q1000000.·∥a∥L1/q1([t0·t1],ℝ+)000000.+(∫t0t1‍|(t2-s)q-1-(t1-s)q-1|1/(1-q1)ds)1-q10000000·∥b∥L1/q1([t0·t1],ℝ+)∫t0t1‍]≤M(Rt0a0+b0)Γ(q)×(∫t0t1‍[(t1-s)c-(t2-s)c]ds)1-q1=M(Rt0a0+b0)Γ(q)(1+c)1-q1×((t1-t0)1+c-(t2-t0)1+c+(t2-t1)1+c)1-q1≤M(Rt0a0+b0)Γ(q)(1+c)1-q1(t2-t1)q-q1.
It follows that I3→0 independently of u∈ΩRt0 as t2-t1→0. For t1=t0, t0<t2≤t0+ht0, it is easy to see that I4=0. Let t1>t0 and ϵ∈(0,t1-t0) be small enough, and we have
(29)I4≤∫t0t1-ϵ‍(t1-s)q-1∥[V(t2-s)-V(t1-s)]00000000000000000×f(s,u(s))∥ds+∫t1-ϵt1‍(t1-s)q-1∥[V(t2-s)-V(t1-s)]000000000000000000×f(s,u(s))∥ds≤∫t0t1-ϵ‍(t1-s)q-1∥V(t2-s)-V(t1-s)∥0000·∥f(s,u(s))∥ds+2MΓ(q)∫t1-ϵt1‍(t1-s)q-1∥f(s,u(s))∥ds≤∫t0t1-ϵ‍(t1-s)q-1[a(s)∥u(s)∥+b(s)]ds·sups∈[t0,t1-ϵ]∥V(t2-s)-V(t1-s)∥+2MΓ(q)∫t1-ϵt1‍(t1-s)q-100000000×[a(s)∥u(s)∥+b(s)]ds≤Rt0a0+b0(1+c)1-q1((t1-t0)1+c-ϵ1+c)1-q1·sups∈[t0,t1-ϵ]∥V(t2-s)-V(t1-s)∥+2M(Rt0a0+b0)Γ(q)(1+c)1-q1ϵq-q1.
Since Lemma 4(iii) implies the continuity of V(t) for t>0 in the uniform operator topology, it is easy to see that I4→0 independently of u∈ΩRt0 as t2-t1→0 and ϵ→0. Thus, ∥(Qu)(t2)-(Qu)(t1)∥→0 independently of u∈ΩRt0 as t2-t1→0, which means that the set Q(ΩRt0) is equicontinuous.
It remains to prove thatQ:ΩRt0→ΩRt0 is a condensing mapping. Let B⊂ΩRt0 be a bounded set. By Lemma 7, there exists B0={un}⊂B such that α(Q(B))≤2α(Q(B0)). Since Q(B0)⊂Q(ΩRt0)⊂ΩRt0 is bounded and equicontinuous, by Lemma 5, it follows that α(Q(B0))=maxt∈Jα(Q(B0)(t)). Thus, for any t∈J, by (21), one has
(30)α(Q(B0)(t))=α({U(t-t0)x0∫t0t‍000+∫t0t‍(t-s)q-1V(t-s)00000000∫t0t‍×f(s,un(s))ds:n∈ℕ})=α({∫t0t‍(t-s)q-1V(t-s)00000000×f(s,un(s))ds:n∈ℕ∫t0t‍})≤2∫t0t‍(t-s)q-1∥V(t-s)∥00·α(f(s,B0(s)))ds≤2qMLΓ(q+1)∫t0t‍(t-s)q-1α(B0(s))ds=2MLht0qΓ(q+1)α(B0)≤2MLht0qΓ(q+1)α(B).
Thus, α(Q(B))≤2α(Q(B0))=2maxt∈Jα(Q(B0)(t))≤(4MLht0q/Γ(q+1))α(B), which means that Q:ΩRt0→ΩRt0 is a condensing mapping. By Lemma 8, the operator Q has at least one fixed point u* in ΩRt0, and u*(t)≥σe1>0 for all t∈J. Hence u*∈C(J,X) is a positive mild solution of the IVP(20).
Hence, for the IVP(1), there exists an interval [0,h0] such that the IVP(1) has a positive mild solution u on [0,h0]. Now, by the extension theorem of initial value problem, u can be extended to a saturated solution u∈C([0,T),X) of the IVP(1), whose existence interval is [0,T), and if T<∞, one has limt→T-∥u(t)∥=∞.For anyT>0 and r>0, define a set Ωr by
(31)Ωr={≤r,u(t)≥σe1,t∈[0,T]u∈C([0,T],X):∥u(t)∥000≤r,u(t)≥σe1,t∈[0,T]}.
If f(t,u) is increasing in Ωr, that is, f(t,u) satisfies the condition:(H4)
for anyu1,u2∈Ωr with u1(t)≤u2(t),t∈[0,T], we have
(32)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]. Hence by Theorem 12, we have the following existence result.Corollary 13.
LetX be a Banach space, let A:D(A)⊂X→X be a closed linear operator, and -A generates a positive and equicontinuous C0-semigroup S(t) (t≥0) in X. Assume that f∈C(ℝ+×X,X) and, for any T>0, f satisfies the conditions (H1), (H3), and (H4). If f(t,σe1)≥λ1σe1 and u0∈X with u0≥σe1, then the IVP(1) has at least one positive and saturated mild solution u∈C([0,T),X). And if T<∞, one has limt∈T-∥u(t)∥=∞.Noticing that the condition (H3) is not easy to verify in applications, we can weaken or delete the condition (H3) in ordered Banach space.Theorem 14.
LetX be an ordered Banach space, whose positive cone K is normal, let A:D(A)⊂X→X be a closed linear operator, and -A generates a positive and equicontinuous C0-semigroup S(t) (t≥0) in X. Assume that f∈C(ℝ+×K,X) and for any T>0, f satisfies the conditions (H1), (H4), and(H5)
there exists a constantL1>0 such that
(33)α(f(t,D(t)))≤L1α(D(t)),t∈[0,T]for any increasing sequence D={xn}⊂Ωr.
Iff(t,σe1)≥λ1σe1 and u0∈X with u0≥σe1, then the IVP(1) has at least one positive and saturated mild solution u∈C([0,T),K). And if T<∞, one has limt∈T-∥u(t)∥=∞.Proof.
For anyt0≥0 and x0∈X with x0≥σe1, we first prove that the IVP(20) has at least one positive mild solution on J=[t0,t0+ht0], where ht0=min{1,(Γ(q)(1+c)1-q1(∥x0∥+1)/(Rt0a0+b0))1/(q-q1)}. Define an operator Q as in (21). Let Rt0=2M(∥X0∥+1)+σe1. Write ΩRt0 as in (22). A similar argument as in the proof of Theorem 12 shows that Q:ΩRt0→ΩRt0 is continuous and the set Q(ΩRt0) is equicontinuous. From the assumption (H4), it is easy to see that Q:ΩRt0→ΩRt0 is an increasing operator.
Letv0≡σe1∈ΩRt0. Define a sequence {vn} by the iterative scheme
(34)vn=Qvn-1,n=1,2,….
Since v0=σe1≤Q(σe1)=Q(v0)=v1, by the increasing property of the operator Q, we have
(35)v0≤v1≤v2≤⋯≤vn≤⋯.
By the equicontinuity property of the set Q(ΩRt0), the set {vn}={Qvn-1}⊂Q(ΩRt0) is equicontinuous. Next, we prove that the set {vn} is uniformly convergent on J.
For convenience, letB={vn:n∈ℕ} and B0={vn-1:n∈ℕ}. From B0=B∪{v0}, it follows that α(B0(t))=α(B(t)) for any t∈J. Let φ(t):=α(B(t))=α(B0(t)). By Lemma 6, assumption (H5), and (21), we have
(36)φ(t)=α(B(t))=α({U(t-t0)x0+∫t0t‍(t-s)q-1V(t-s)0000000×f(s,vn-1(s))ds:n∈ℕ∫t0t‍})=α({∫t0t‍(t-s)q-1V(t-s)000000×f(s,vn-1(s))ds:n∈ℕ∫t0t‍})≤2∫t0t‍(t-s)q-1∥V(t-s)∥00000·α(f(s,B0(s)))ds≤2ML1Γ(q)∫t0t‍(t-s)q-1α(B0(s))ds=2ML1Γ(q)∫t0t‍(t-s)q-1φ(s)ds.
Hence by Lemma 9, α(B(t))=φ(t)≡0 for any t∈J. By Lemma 5, α(B)=maxt∈Jα(B(t))=0, from which we obtain that the set {vn} is relatively compact. Thus, there is a subset {vnk}⊂{vn} such that vnk→u*∈ΩRt0. Combining this with the monotonicity (35), we easily prove that {vn} itself is convergent in ΩRt0, that is, vn→u*∈ΩRt0 as n→∞.
Lettingn→∞ in (34), by the continuity of the operator Q, we have u*=Qu* and u*(t)≥σe1>0 for all t∈J. Hence u*∈C(J,X) is a positive mild solution of the IVP(20).
Hence, for the IVP(1), there exists an interval [0,h0] such that the IVP(1) has a positive mild solution u on [0,h0]. By the extension theorem of the initial value problem, u can be extended to a saturated solution u∈C([0,T),X) of the IVP(1), whose existence interval is [0,T), and if t<∞, then limt→T-∥u(t)∥=∞.In Theorem14, if X is weakly sequentially complete, the condition (H5) holds automatically. In fact, by [27, Theorem 2.2], any monotonic and order-bounded sequence is precompact. Let D={xn}⊂Ωr be an increasing sequence. Then by the conditions (H1) and (H4), {f(t,xn)} is a monotonic increasing and order-bounded sequence. By the property of the measure of noncompactness, we have
(37)α({f(t,xn)})=0.
Thus, the condition (H5) holds. From Theorem 14, we have the following.Corollary 15.
LetX be an ordered and weakly sequentially complete Banach space, whose positive cone K is normal, let A:D(A)⊂X→X be a closed linear operator, and -A generates a positive and equicontinuous C0-semigroup S(t) (t≥0) in X. Assume that f∈C(ℝ+×K,X) and, for any T>0, f satisfies the conditions (H1) and (H4). If f(t,σe1)≥λ1σe1 and u0∈X with u0≥σe1, then the IVP(1) has at least one positive and saturated mild solution u∈C([0,T),K). And if T<∞, one has limt∈T-∥u(t)∥=∞.


## 4. Positive Mild Solutions of Parabolic Equations

LetΩ⊂ℝN be a bounded domain with a sufficiently smooth boundary ∂Ω, F:Ω¯×ℝ+×ℝ→ℝ. We consider the following problem of parabolic type:
(38)∂q∂tqu(x,t)+Δu(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, and Δ is the Laplace operator. Let X:=L2(Ω). Then X is an ordered Banach space with the norm ∥f∥2=(∫Ω‍|f(x)|2dx)1/2 for any f∈X and the partial order “≤”. K:={u∈X:u(x)≥0a.e.x∈Ω} is the positive cone in X. Consider the operator A:D(A)⊂X→X defined by
(39)D(A)=H2(Ω)∩H01(Ω),Au=-Δu.
Then -A generates a positive and analytic semigroup S(t) (t≥0) in X (see [28, 29]). Let λ1 be the smallest positive real eigenvalue of operator A under the Dirichlet boundary condition u|∂Ω=0 and let e1(x) be the positive eigenvector corresponding to λ1. Then λ1>0 and e1(x)>0 for x∈Ω. For any T>0 and r>0, denote by
(40)Pr={u∈C([0,T],L2(Ω)):∥u(x,t)∥200≤r,u(x,t)≥σe1(x),t∈[0,T]u∈C([0,T],L2(Ω)):∥u(x,t)∥2},
where σ>0 is a constant. Assume that F:Ω¯×ℝ+×ℝ→ℝ is continuous with F(x,t,σe1(x))≥λ1σe1(x), x∈Ω, t∈ℝ+ and satisfies the following conditions.(F1)
There exista,b∈L1/q1([0,T],ℝ+), q1∈(0,q) such that
(41)|F(x,t,u(x,t))|≤a(t)|u(x,t)|+b(t),x∈Ω¯,t∈[0,T].(F2)
For anyu1,u2∈Pr with u1≤u2, we have
(42)F(x,t,u1(x,t))≤F(x,t,u2(x,t)),x∈Ω¯,t∈[0,T].Letf:ℝ+×X→X be defined by f(t,u(t))(·)=F(·,t,u(·,t)). Then f∈C(ℝ+×X,X) with f(t,σe1)≥λ1σe1 for t∈ℝ+ and satisfies the assumptions (H1) and (H4). Therefore, by Corollary 15, we have the following existence result for the problem (38).Theorem 16.
Assume thatF∈C(Ω¯×ℝ+×ℝ,ℝ) with F(x,t,σe1(x))≥λ1σe1(x) for x∈Ω,t∈ℝ+ and satisfies the assumptions (F1) and (F2). If φ∈C(Ω¯) with φ(x)≥σe1(x) for any x∈Ω¯, then the problem (38) has at least one positive mild solution u, satisfies u(x,t)≥σe1(x) for any x∈Ω¯ and t∈[0,T]. And if T<+∞, one has limt→T-|u(t)|=+∞.Remark 17.
In Theorem16, we do not use the property of compactness of the semigroup S(t) (t≥0), which is a key assumption in [7–9, 11, 12].



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