JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 781404 10.1155/2013/781404 781404 Research Article Positive Solutions for the Initial Value Problems of Fractional Evolution Equation Liang Yue Ma Yu Gao Xiaoyan Mohiuddine S. A. Science College of Gansu Agricultural University Lanzhou 730070 China gsau.edu.cn 2013 4 11 2013 2013 27 07 2013 22 09 2013 2013 Copyright © 2013 Yue Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup Dqu(t)+Au(t)=f(t,u(t)), t0; u(0)=u0 in a Banach space X, where Dq denotes the Caputo fractional derivative of order q(0,1), A:D(A)XX is a closed linear operator, -A generates an equicontinuous C0 semigroup, and f:[0,)×XX 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.

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 . Correspondingly, the existence of solutions to fractional evolution equations in Banach space has also been studied by several authors; see . 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. . In , 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 .

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)),t0,u(0)=u0 in Banach space X, where Dq denotes the Caputo fractional derivative of order q(0,1), A:D(A)XX is a closed linear operator, -A generates a C0-semigroup S(t) (t0) in X, and f:+×XX is continuous and will be specified later, +=[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 Section 2, 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.

Let X be a Banach space with norm ·, let A:D(A)XX be a closed linear operator, and -A generates a C0-semigroup S(t) (t0) in X. It is well known that there exist M¯>0 and δ such that (2)S(t)M¯eδt,t0. 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, 1114, 16, 17] and the reference therein.

Definition 1.

The fractional integral of order σ>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 order n-1<σ<n with the lower limits zero for a function f can be written as (4)DLσf(t)=1Γ(n-σ)dndtn0t(t-s)n-σ-1f(s)ds,t>0,n. Also the Caputo fractional derivative of order n-1<σ<n with the lower limits zero for a function fCn[0,) can be written as (5)Dσf(t)=1Γ(n-σ)0t(t-s)n-σ-1f(n)(s)ds,t>0,n.

Remark 2.

(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 1 are taken in Bochner’s sense.

Lemma 3 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

A measurable function h:[0,T]X is Bochner integrable if h is Lebesgue integrable.

For xX, we define two families {U(t)}t0 and {V(t)}t0 of operators by (6)U(t)x=0ηq(θ)S(tqθ)xdθ,V(t)x=q0θηq(θ)S(tqθ)xdθ,0<q<1, where (7)ηq(θ)=1qθ-1-(1/q)ρq(θ-1/q),ρq(θ)=1πn=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 0ηq(θ)dθ=1. Clearly, if the semigroup S(t) (t0) is positive, then the operators U(t) and V(t) are also positive for all t0.

The following lemma is needed in the proof of the main results.

Lemma 4.

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

For any fixed t0 and any xX, one has (8)U(t)xMx,V(t)xqMΓ(1+q)x=MΓ(q)x.

The operators U(t) and V(t) are strongly continuous for all t0.

If S(t) (t0) 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 0t1<t2T, we have (9)U(t2)-U(t1)=0ηq(θ)S(t2qθ)-S(t1qθ)dθ,V(t2)-V(t1)=q0θηq(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-t10, which means that the operators U(t) and V(t) are equicontinuous in X for t>0.

Let t00 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 uC=maxtIu(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 0α(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 . For any BC(I,X) and tI, set B(t)={u(t):uB}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:BB 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 [<xref ref-type="bibr" rid="B19">22</xref>]).

Let BC(I,X) be bounded and equicontinuous. Then α(B(t)) is continuous on I and (10)α(B)=maxtIα(B(t))=α(B(I)), where B(I)={x(t):xB,tI}.

Lemma 6 (see [<xref ref-type="bibr" rid="B20">23</xref>]).

Let B={un}C(I,X) be countable. If there exists ψL1(I) such that un(t)ψ(t) a.e. tI, n=1,2,, then α(B(t)) is Lebesgue integral on I and (11)α({Iun(t)dt:n})2Iα(B(t))dt.

Lemma 7 (see [<xref ref-type="bibr" rid="B21">24</xref>]).

Let BC(I,X) be bounded. Then there exists a countable subset B0 of B such that α(B)2α(B0).

Lemma 8 (see [<xref ref-type="bibr" rid="B22">25</xref>] (Sadovskii’s fixed point theorem)).

Let X be a Banach space and let Ω be a nonempty bounded convex closed set in X. If Q:ΩΩ 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.

Suppose b0, β>0, and a(t) is a nonnegative function locally integrable on 0t<T (some T), and suppose u(t) is nonnegative and locally integrable on 0t<T with (12)u(t)a(t)+b0t(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,0t<T.

Remark 10.

In Lemma 9, if a(t)0 for all 0t<T, we easily see that u(t)=0.

For any x0X and hC(I,X), a function u is called the mild solution of the initial value problem (14)Dqu(t)+Au(t)=h(t),tI,u(t0)=x0, if uC(I,X) satisfies the integral equation: (15)u(t)=U(t-t0)x0+t0t(t-s)q-1V(t-s)h(s)ds,tI. Hence, for the IVP(1), we have the following definition.

Definition 11.

By a mild solution of the IVP(1), we mean a function uC(I,X) satisfying (16)u(t)=U(t)u0+0t(t-s)q-1V(t-s)f(s,u(s))ds for all t0.

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 e1D(A) be the positive eigenvector corresponding to λ1. Our main results are as follows.

Theorem 12.

Let X be a Banach space, let A:D(A)XX be a closed linear operator, and -A generate a positive and equicontinuous C0-semigroup S(t) (t0) in X. Assume that fC(+×X,X) and, for any T>0, f satisfies the following conditions.

There exist a,bL1/q1([0,T],+), q1(0,q) such that (17)f(t,x)a(t)x+b(t),t[0,T],xX.

For any uC([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.

For any bounded set DX, there exists a constant L>0 such that (19)α(f(t,D))Lα(D),t[0,T].

If f(t,σe1)λ1σe1 and u0X with u0σe1, then the IVP(1) has at least one positive and saturated mild solution uC([0,T),X). And if T<, one has limtT-u(t)=.

Proof.

For any t00 and x0X with x0σe1, 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,00000000000000000000000000000000000000tt0. 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.

Let Rt0:=2M(x0+1)+σe1>0. Denote (22)ΩRt0:={uC(J,X):u(t)Rt0,u(t)σe1,tJ}. Then ΩRt0C(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=aL1/q1([t0,t0+1],+), b0=bL1/q1([t0,t0+1],+). Then for any uΩRt0 and tJ, by Lemma 4(i), (H1), and (21), we have (23)(Qu)(t)U(t-t0)x0+t0t(t-s)q-1V(t-s)f(s,u(s))dsMx0+MΓ(q)t0t(t-s)q-1[a(s)u(s)+b(s)]ds=Mx0+MΓ(q)[Rt0t0t(t-s)q-1a(s)ds0000000000.00000+t0t(t-s)q-1b(s)ds]Mx0+M(Rt0a0+b0)Γ(q)(1+c)1-q1·ht0q-q1Rt0.

Let v0σe1. Then v0(t)=σe1 for any tJ and (24)ϕ(t)Dqv0(t)+Av0(t)=λ1σe1f(t,σe1),tJ. By the positivity of semigroup S(t) (t0), the assumption (H2) and (21), for any uΩRt0 and tJ, we have (25)σe1=v0(t)=U(t-t0)v0(t0)+t0t(t-s)q-1V(t-s)ϕ(s)dsU(t-t0)σe1+t0t(t-s)q-1V(t-s)f(s,σe1)dsU(t-t0)x0+t0t(t-s)q-1V(t-s)f(s,u(s))ds=(Qu)(t). Thus, Q:ΩRt0ΩRt0 is continuous and it implies that (Qσe1)(t)(Qu)(t) for any uΩRt0 and tJ.

Now, we prove that the set Q(ΩRt0):={Qu:uΩRt0} is equicontinuous in C(J,X). For any uΩRt0 and t0t1<t2t0+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))dsMU(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))dsI1+I2+I3+I4. By Lemma 4(ii), it is easy to see that I10 independently of uΩRt0 as t2-t10: (27)I2MΓ(q)[Rt0t1t2(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.

Hence I20 independently of uΩRt0 as t2-t10: (28)I3MΓ(q)[Rt0t0t1|(t2-s)q-1-(t1-s)q-1|×a(s)ds00000000+t0t1|(t2-s)q-1-(t1-s)q-1|b(s)dst0t1]MΓ(q)[Rt0(t0t1|(t2-s)q-10000000000000000.-(t1-s)q-1|1/(1-q1)dst0t1)1-q1000000.·aL1/q1([t0·t1],+)000000.+(t0t1|(t2-s)q-1-(t1-s)q-1|1/(1-q1)ds)1-q10000000·bL1/q1([t0·t1],+)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-q1M(Rt0a0+b0)Γ(q)(1+c)1-q1(t2-t1)q-q1. It follows that I30 independently of uΩRt0 as t2-t10. For t1=t0, t0<t2t0+ht0, it is easy to see that I4=0. Let t1>t0 and ϵ(0,t1-t0) be small enough, and we have (29)I4t0t1-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))dst0t1-ϵ(t1-s)q-1V(t2-s)-V(t1-s)0000·f(s,u(s))ds+2MΓ(q)t1-ϵt1(t1-s)q-1f(s,u(s))dst0t1-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)]dsRt0a0+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 I40 independently of uΩRt0 as t2-t10 and ϵ0. Thus, (Qu)(t2)-(Qu)(t1)0 independently of uΩRt0 as t2-t10, which means that the set Q(ΩRt0) is equicontinuous.

It remains to prove that Q:ΩRt0ΩRt0 is a condensing mapping. Let BΩRt0 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))=maxtJα(Q(B0)(t)). Thus, for any tJ, by (21), one has (30)α(Q(B0)(t))=α({U(t-t0)x0t0t000+t0t(t-s)q-1V(t-s)00000000t0t×f(s,un(s))ds:n})=α({t0t(t-s)q-1V(t-s)00000000×f(s,un(s))ds:nt0t})2t0t(t-s)q-1V(t-s)00·α(f(s,B0(s)))ds2qMLΓ(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))=2maxtJα(Q(B0)(t))(4MLht0q/Γ(q+1))α(B), which means that Q:ΩRt0ΩRt0 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 tJ. 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 uC([0,T),X) of the IVP(1), whose existence interval is [0,T), and if T<, one has limtT-u(t)=.

For any T>0 and r>0, define a set Ωr by (31)Ωr={r,u(t)σe1,t[0,T]uC([0,T],X):u(t)000r,u(t)σe1,t[0,T]}. If f(t,u) is increasing in Ωr, that is, f(t,u) satisfies the condition:

for any u1,u2Ωr 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 uΩr and t[0,T]. Hence by Theorem 12, we have the following existence result.

Corollary 13.

Let X be a Banach space, let A:D(A)XX be a closed linear operator, and -A generates a positive and equicontinuous C0-semigroup S(t) (t0) in X. Assume that fC(+×X,X) and, for any T>0, f satisfies the conditions (H1), (H3), and (H4). If f(t,σe1)λ1σe1 and u0X with u0σe1, then the IVP(1) has at least one positive and saturated mild solution uC([0,T),X). And if T<, one has limtT-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.

Let X be an ordered Banach space, whose positive cone K is normal, let A:D(A)XX be a closed linear operator, and -A generates a positive and equicontinuous C0-semigroup S(t) (t0) in X. Assume that fC(+×K,X) and for any T>0, f satisfies the conditions (H1), (H4), and

there exists a constant L1>0 such that (33)α(f(t,D(t)))L1α(D(t)),t[0,T]

for any increasing sequence D={xn}Ωr.

If f(t,σe1)λ1σe1 and u0X with u0σe1, then the IVP(1) has at least one positive and saturated mild solution uC([0,T),K). And if T<, one has limtT-u(t)=.

Proof.

For any t00 and x0X with x0σe1, 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:ΩRt0ΩRt0 is continuous and the set Q(ΩRt0) is equicontinuous. From the assumption (H4), it is easy to see that Q:ΩRt0ΩRt0 is an increasing operator.

Let v0σe1ΩRt0. Define a sequence {vn} by the iterative scheme (34)vn=Qvn-1,n=1,2,. Since v0=σe1Q(σe1)=Q(v0)=v1, by the increasing property of the operator Q, we have (35)v0v1v2vn. 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, let B={vn:n} and B0={vn-1:n}. From B0=B{v0}, it follows that α(B0(t))=α(B(t)) for any tJ. 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:nt0t  })=α({t0t(t-s)q-1V(t-s)000000×f(s,vn-1(s))ds:nt0t})2t0t(t-s)q-1V(t-s)00000·α(f(s,B0(s)))ds2ML1Γ(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 tJ. By Lemma 5, α(B)=maxtJα(B(t))=0, from which we obtain that the set {vn} is relatively compact. Thus, there is a subset {vnk}{vn} such that vnku*ΩRt0. Combining this with the monotonicity (35), we easily prove that {vn} itself is convergent in ΩRt0, that is, vnu*ΩRt0 as n.

Letting n in (34), by the continuity of the operator Q, we have u*=Qu* and u*(t)σe1>0 for all tJ. 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 uC([0,T),X) of the IVP(1), whose existence interval is [0,T), and if t<, then limtT-u(t)=.

In Theorem 14, 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.

Let X be an ordered and weakly sequentially complete Banach space, whose positive cone K is normal, let A:D(A)XX be a closed linear operator, and -A generates a positive and equicontinuous C0-semigroup S(t) (t0) in X. Assume that fC(+×K,X) and, for any T>0, f satisfies the conditions (H1) and (H4). If f(t,σe1)λ1σe1 and u0X with u0σe1, then the IVP(1) has at least one positive and saturated mild solution uC([0,T),K). And if T<, one has limtT-u(t)=.

4. Positive Mild Solutions of Parabolic Equations

Let ΩN be a bounded domain with a sufficiently smooth boundary Ω, F:Ω¯×+×. We consider the following problem of parabolic type: (38)qtqu(x,t)+Δu(x,t)=F(x,t,u(x,t))inΩ×+,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 f2=(Ω|f(x)|2dx)1/2 for any fX and the partial order “”. K:={uX:u(x)0  a.e.  xΩ} is the positive cone in X. Consider the operator A:D(A)XX defined by (39)D(A)=H2(Ω)H01(Ω),Au=-Δu. Then -A generates a positive and analytic semigroup S(t) (t0) 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 xΩ. For any T>0 and r>0, denote by (40)Pr={uC([0,T],L2(Ω)):u(x,t)200r,u(x,t)σe1(x),  t[0,T]uC([0,T],L2(Ω)):u(x,t)2}, where σ>0 is a constant. Assume that F:Ω¯×+× is continuous with F(x,t,σe1(x))λ1σe1(x), xΩ, t+ and satisfies the following conditions.

There exist a,bL1/q1([0,T],+), q1(0,q) such that (41)|F(x,t,u(x,t))|a(t)|u(x,t)|+b(t),xΩ¯,t[0,T].

For any u1,u2Pr with u1u2, we have (42)F(x,t,u1(x,t))F(x,t,u2(x,t)),xΩ¯,  t[0,T].

Let f:+×XX be defined by f(t,u(t))(·)=F(·,t,u(·,t)). Then fC(+×X,X) with f(t,σe1)λ1σe1 for t+ and satisfies the assumptions (H1) and (H4). Therefore, by Corollary 15, we have the following existence result for the problem (38).

Theorem 16.

Assume that FC(Ω¯×+×,) with F(x,t,σe1(x))λ1σe1(x) for xΩ,t+ and satisfies the assumptions (F1) and (F2). If φC(Ω¯) with φ(x)σe1(x) for any xΩ¯, then the problem (38) has at least one positive mild solution u, satisfies u(x,t)σe1(x) for any xΩ¯ and t[0,T]. And if T<+, one has limtT-|u(t)|=+.

Remark 17.

In Theorem 16, we do not use the property of compactness of the semigroup S(t) (t0), which is a key assumption in [79, 11, 12].

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