# Periodic Boundary Value Problems for Semilinear Fractional Differential Equations

Authors: Jia Mu; Yongxiang Li
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We study the periodic boundary value problem for semilinear fractional differential equations in an ordered Banach space. The method of upper and lower solutions is then extended. The results on the existence of minimal and maximal mild solutions are obtained by using the characteristics of positive operators semigroup and the monotone iterative scheme. The results are illustrated by means of a fractional parabolic partial differential equations.


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## Body

## 1. Introduction

In this paper, we consider the periodic boundary value problem (PBVP) for semilinear fractional differential equation in an ordered Banach spaceX,
(1.1)Dαu(t)+Au(t)=f(t,u(t)),t∈I,u(0)=u(ω),
where Dα is the Caputo fractional derivative of order 0<α<1, I=[0,ω], -A:D(A)⊂X→X is the infinitesimal generator of a C0-semigroup (i.e., strongly continuous semigroup) {T(t)}t≥0 of uniformly bounded linear operators on X, and f:I×X→X is a continuous function.Fractional calculus is an old mathematical concept dating back to the 17th century and involves integration and differentiation of arbitrary order. In a later dated 30th of September 1695, L’Hospital wrote to Leibniz asking him about the differentiation of order 1/2. Leibniz’ response was “an apparent paradox from which one day useful consequences will be drawn.” In the following centuries, fractional calculus developed significantly within pure mathematics. However, the applications of fractional calculus just emerged in the last few decades. The advantage of fractional calculus becomes apparent in science and engineering. In recent years, fractional calculus attracted engineers’ attention, because it can describe the behavior of real dynamical systems in compact expressions, taking into account nonlocal characteristics like infinite memory [1–3]. Some instances are thermal diffusion phenomenon [4], botanical electrical impedances [5], model of love between humans [6], the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water [7], and so forth. On the other hand, directing the behavior of a process with fractional-order controllers would be an advantage, because the responses are not restricted to a sum of exponential functions; therefore, a wide range of responses neglected by integer-order calculus would be approached [8]. For other advantages of fractional calculus, we can see real materials [9–13], control engineering [14, 15], electromagnetism [16], biosciences [17], fluid mechanics [18], electrochemistry [19], diffusion processes [20], dynamic of viscoelastic materials [21], viscoelastic systems [22], continuum and statistical mechanics [23], propagation of spherical flames [24], robotic manipulators [25], gear transmissions [26], and vibration systems [27]. It is well known that the fractional-order differential and integral operators are nonlocal operators. This is one reason why fractional differential operators provide an excellent instrument for description of memory and hereditary properties of various physical processes.In recent years, there have been some works on the existence of solutions (or mild solutions) for semilinear fractional differential equations, see [28–36]. They use mainly Krasnoselskii’s fixed-point theorem, Leray-Schauder fixed-point theorem, or contraction mapping principle. They established various criteria on the existence and uniqueness of solutions (or mild solutions) for the semilinear fractional differential equations by considering an integral equation which is given in terms of probability density functions and operator semigroups. Many partial differential equations involving time-variable t can turn to semilinear fractional differential equations in Banach spaces; they always generate an unbounded closed operator term A, such as the time fractional diffusion equation of order α∈(0,1), namely,
(1.2)∂tαu(y,t)=Au(y,t),t≥0,y∈R,
where A may be linear fractional partial differential operator. So, (1.1) has the extensive application value.However, to the authors’ knowledge, no studies considered the periodic boundary value problems for the abstract semilinear fractional differential equations involving the operator semigroup generator-A. Our results can be considered as a contribution to this emerging field. We use the method of upper and lower solutions coupled with monotone iterative technique and the characteristics of positive operators semigroup.The method of upper and lower solutions has been effectively used for proving the existence results for a wide variety of nonlinear problems. When coupled with monotone iterative technique, one obtains the solutions of the nonlinear problems besides enabling the study of the qualitative properties of the solutions. The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences, and these sequences converge monotonically to the maximal and minimal solutions. In some papers, some existence results for minimal and maximal solutions are obtained by establishing comparison principles and using the method of upper and lower solutions and the monotone iterative technique. The method requires establishing comparison theorems which play an important role in the proof of existence of minimal and maximal solutions. In abstract semilinear fractional differential equations, positive operators semigroup can play this role, see Li [37–41].In Section2, we introduce some useful preliminaries. In Section 3, in two cases: T(t) is compact or noncompact, we establish various criteria on existence of the minimal and maximal mild solutions of PBVP (1.1). The method of upper and lower solutions coupled with monotone iterative technique, and the characteristics of positive operators semigroup are applied effectively. In Section 4, we give also an example to illustrate the applications of the abstract results.


## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.If-A is the infinitesimal generator of a C0-semigroup in a Banach space, then -(A+qI) generates a uniformly bounded C0-semigroup for q>0 large enough. This allows us to reduce the general case in which -A is the infinitesimal generator of a C0-semigroup to the case in which the semigroup is uniformly bounded. Hence, for convenience, throughout this paper, we suppose that -A is the infinitesimal generator of a uniformly bounded C0-semigroup {T(t)}t≥0. This means that there exists M≥1 such that
(2.1)‖T(t)‖≤M,t≥0.We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.Definition 2.1 (see [9, 32]).
The fractional integral of orderα with the lower limit zero for a function  f∈AC[0,∞)is defined as
(2.2)Iαf(t)=1Γ(α)∫0tf(s)(t-s)1-αds,t>0,0<α<1,
provided the right side is pointwise defined on [0,∞), where Γ(·) is the gamma function.Definition 2.2 (see [9, 32]).
The Riemann-Liouville derivative of orderα with the lower limit zero for a function f∈AC[0,∞) can be written as
(2.3)DαLf(t)=1Γ(1-α)ddt∫0tf(s)(t-s)αds,t>0,0<α<1.Definition 2.3 (see [9, 32]).
The Caputo fractional derivative of orderα for a function f∈AC[0,∞) can be written as
(2.4)Dαf(t)=DαL(f(t)-f(0)),t>0,0<α<1.Remark 2.4 (see [32]).
(i) Iff∈C1[0,∞), then
(2.5)Dαf(t)=1Γ(1-α)∫0tf′(s)(t-s)αds,t>0,0<α<1.
(ii) The Caputo derivative of a constant is equal to zero.
(iii) Iff is an abstract function with values in X, then the integrals and derivatives which appear in Definitions 2.1–2.3 are taken in Bochner’s sense.For more fractional theories, one can refer to the books [9, 42–44].Throughout this paper, letX be an ordered Banach space with norm ∥·∥ and partial order ≤, whose positive cone P={y∈X∣y≥θ} (θ is the zero element of X) is normal with normal constant N. X1 denotes the Banach space D(A) with the graph norm ∥·∥1=∥·∥+∥A·∥. Let C(I,X) be the Banach space of all continuous X-value functions on interval I with norm ∥u∥C=maxt∈I∥u(t)∥. For u,v∈C(I,X), u≤v if u(t)≤v(t) for all t∈I. For v,w∈C(I,X), denote the ordered interval [v,w]={u∈C(I,X)∣v≤u≤w} and [v(t),w(t)]={y∈X∣v(t)≤y≤w(t)}, t∈I. Set Cα(I,X)={u∈C(I,X)∣Dαu exists and Dαu∈C(I,X)}.Definition 2.5.
Ifv0∈Cα(I,X)∩C(I,X1)  and satisfies
(2.6)Dαv0(t)+Av0(t)≤f(t,v0(t)),t∈I,v0(0)≤v(ω),
then v0 is called a lower solution of PBVP (1.1); if all inequalities of (2.6) are inverse, one calls it an upper solution of PBVP (1.1).Definition 2.6 (see [29, 45]).
Ifh∈C(I,X),  by the mild solution of LIVP,
(2.7)Dαu(t)+Au(t)=h(t),t∈I,u(0)=x0∈X,
one means that the function u∈C(I,X) and satisfies
(2.8)u(t)=U(t)x0+∫0t(t-s)α-1V(t-s)h(s)ds,
where
(2.9)U(t)=∫0∞ζα(θ)T(tαθ)dθ,V(t)=α∫0∞θζα(θ)T(tαθ)dθ,(2.10)ζα(θ)=1αθ-1-1/αρα(θ-1/α),ρα(θ)=1π∑n=0∞(-1)n-1θ-αn-1Γ(nα+1)n!sin⁡(nπα),θ∈(0,∞),
and ζα(θ) is a probability density function defined on (0,∞).Remark 2.7.
(i) [29–31] ζα(θ)≥0,θ∈(0,∞), ∫0∞ζα(θ)dθ=1, and ∫0∞θζα(θ)dθ=1/Γ(1+α).
(ii) [33, 34, 46, 47] The Laplace transform of ζα is given by
(2.11)∫0∞e-pθζα(θ)dθ=∑n=0∞(-p)nΓ(1+nα)=Eα(-p),
where Eα(·) is Mittag-Leffler function (see [42]).
(iii) [48] For p<0, 0<Eα(p)<Eα(0)=1.Lemma 2.8.
If{T(t)}t≥0 is an exponentially stable  C0-semigroup, there are constants  N≥1 and  δ>0, such that
(2.12)‖T(t)‖≤Ne-δt,t≥0,
then the linear periodic boundary value problem (LPBVP)
(2.13)Dαu(t)+Au(t)=h(t),t∈I,u(0)=u(ω)
has a unique mild solution
(2.14)(Ph)(t)=U(t)B(h)+∫0t(t-s)α-1V(t-s)h(s)ds,
where U(t) and V(t) are given by (2.9)
(2.15)B(h)=(I-U(ω))-1∫0ω(ω-s)α-1V(ω-s)h(s)ds.Proof.
InX, give equivalent norm |·| by
(2.16)|x|=sup⁡t≥0‖eδtT(t)x‖,
then ∥x∥≤|x|≤N∥x∥. By |T(t)|, we denote the norm of T(t) in (X,|·|), then for t≥0,
(2.17)|T(t)x|=sup⁡s≥0‖eδsT(s)T(t)x‖=e-δtsup⁡s≥0‖eδ(s+t)T(s+t)x‖=e-δtsup⁡η≥t‖eδηT(η)x‖≤e-δt|x|.
Thus, |T(t)|≤e-δt. Then by Remark 2.7,
(2.18)|U(ω)|=|∫0∞ζα(θ)T(ωαθ)dθ|≤∫0∞ζα(θ)e-δωαθdθ=Eα(-δωα)<1.
Therefore, I-U(ω) has bounded inverse operator and
(2.19)(I-U(ω))-1=∑n=0∞(U(ω))n.
Set
(2.20)x0=(I-U(ω))-1∫0ω(ω-s)α-1V(ω-s)h(s)ds,
then
(2.21)u(t)=U(t)x0+∫0t(t-s)α-1V(t-s)h(s)ds
is the unique mild solution of LIVP (2.7) and satisfies u(0)=u(ω). So set
(2.22)B(h)=(I-U(ω))-1∫0ω(ω-s)α-1V(ω-s)h(s)ds,(Ph)(t)=U(t)B(h)+∫0t(t-s)α-1V(t-s)h(s)ds,
then Ph is the unique mild solution of LPBVP (2.13).Remark 2.9.
For sufficient conditions of exponentially stableC0-semigroup, one can see [49].Definition 2.10.
AC0-semigroup {T(t)}t≥0 is called a compact semigroup if T(t) is compact for t>0.Definition 2.11.
AC0-semigroup {T(t)}t≥0 is called an equicontinuous semigroup if T(t) is continuous in the uniform operator topology (i.e., uniformly continuous) for t>0.Remark 2.12.
Compact semigroups, differential semigroups, and analytic semigroups are equicontinuous semigroups, see [50]. In the applications of partial differential equations, such as parabolic and strongly damped wave equations, the corresponding solution semigroups are analytic semigroups.Definition 2.13.
AC0-semigroup {T(t)}t≥0 is called a positive semigroup if T(t)x≥θ for all x≥θ and t≥0.Remark 2.14.
From Definition2.13, if h≥θ, x0≥θ, and T(t)(t≥0) is a positive C0-semigroup generated by -A, the mild solution u∈C(I,X) given by (2.8) satisfies u≥θ.For the applications of positive operators semigroup, we can see [37–41]. It is easy to see that positive operators semigroup can play the role as the comparison principles.Definition 2.15.
A bounded linear operatorK on X is called to be positive if Kx≥θ for all x≥θ.Lemma 2.16.
The operatorsU  and V given by (2.9) have the following properties: (i)
For any fixedt≥0, U(t) and V(t) are linear and bounded operators, that is, for any x∈X,
(2.23)‖U(t)x‖≤M‖x‖,‖V(t)x‖≤αMΓ(1+α)‖x‖,(ii)
{U(t)}t≥0 and {V(t)}t≥0 are strongly continuous,(iii)
{U(t)}t≥0 and {V(t)}t≥0 are compact operators if {T(t)}t≥0 is a compact semigroup,(iv)
U(t) and V(t) are continuous in the uniform operator topology (i.e., uniformly continuous) for t>0 if {T(t)}t≥0 is an equicontinuous semigroup,(v)
U(t) and V(t) are positive for t≥0 if {T(t)}t≥0 is a positive semigroup,(vi)
(I-U(ω))-1 is a positive operator if {T(t)}t≥0 is an exponentially and positive semigroup.Proof.
For the proof of (i)–(iii), one can refer to [29, 31]. We only check (iv), (v), and (vi) as follows.(iv)
For0<t1≤t2, we have
(2.24)‖U(t2)-U(t1)‖≤∫0∞ζα(θ)‖T(t2αθ)-T(t1αθ)‖dθ,‖V(t2)-V(t1)‖≤α∫0∞θζα(θ)‖T(t2αθ)-T(t1αθ)‖dθ.
Since T(t) is continuous in the uniform operator topology for t>0, by Lebesque-dominated convergence theorem and Remark 2.7 (i), U(t) and V(t) are continuous in the uniform operator topology for t>0.(v)
By Remark2.7 (i), the proof is then complete.(vi)
By(v), (2.18), and (2.19), the proof is then complete.


## 3. Main Results

Case 1.
{T(t)}t≥0 is compact.Theorem 3.1.
Assume that{T(t)}t≥0 is a compact and positive semigroup in  X,  PBVP (1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0 and satisfies the following. (H)
There exists a constantC>0 such that
(3.1)f(t,x2)-f(t,x1)≥-C(x2-x1),
for any t∈I, and v0(t)≤x1≤x2≤w0(t), that is, f(t,x)+Cx is increasing in x for x∈[v0(t),w0(t)].
Then PBVP (1.1) has the minimal and maximal mild solutions between v0 and w0, which can be obtained by a monotone iterative procedure starting from v0 and w0, respectively.Proof.
It is easy to see that-(A+CI) generates an exponentially stable and positive compact semigroup S(t)=e-CtT(t). By (2.1), ∥S(t)∥≤M. Let Φ(t)=∫0∞ζα(θ)S(tαθ)dθ,Ψ(t)=α∫0∞θζα(θ)S(tαθ)dθ. By Remark 2.7 (i), we have that
(3.2)‖Φ(t)‖≤M,‖Ψ(t)‖≤αΓ(1+α)M,t≥0.
From Lemma 2.8, (I-Φ(ω)) has bounded inverse operator and
(3.3)(I-Φ(ω))-1=∑n=0∞(Φ(ω))n.
By Lemma 2.16 (v) and (vi), Φ(t) and Ψ(t) are positive for t≥0, and (I-Φ(ω))-1 is positive.
LetD=[v0,w0], then we define a mapping Q:D→C(I,X) by
(3.4)Qu(t)=Φ(t)B1(u)+∫0t(t-s)α-1Ψ(t-s)[f(s,u(s))+Cu(s)]ds,t∈I,
where
(3.5)B1(u)=(I-Φ(ω))-1∫0ω(ω-s)α-1Ψ(ω-s)[f(s,u(s))+Cu(s)]ds.
By the continuity of f and Lemma 2.16 (ii), Q:D→C(I,X) is continuous. By Lemma 2.8, u∈D is a mild solution of PBVP (1.1) if and only if
(3.6)u=Qu.
For u1,u2∈D and u1≤u2, from (H), the positivity of operators (I-Φ(ω))-1, Φ(t), and Ψ(t), we have that
(3.7)Qu1≤Qu2.
Now, we show that v0≤Qv0, Qw0≤w0. Let Dαv0(t)+Av0(t)+Cv0(t)≜σ(t), by Definition 2.5, the positivity of operator Ψ(t), we have that
(3.8)v0(t)=Φ(t)v0(0)+∫0t(t-s)α-1Ψ(t-s)σ(s)ds≤Φ(t)v0(0)+∫0t(t-s)α-1Ψ(t-s)[f(s,v0(s))+Cv0(s)]ds,t∈I.
In particular,
(3.9)v0(ω)≤Φ(ω)v0(0)+∫0ω(ω-s)α-1Ψ(ω-s)[f(s,v0(s))+Cv0(s)]ds.
By Definition 2.5, v0(0)≤v(ω), and by the positivity of operator (I-Φ(ω))-1, we have that
(3.10)v0(0)≤(I-Φ(ω))-1∫0ω(ω-s)α-1Ψ(ω-s)[f(s,v0(s))+Cv0(s)]ds=B1(v0).
Then by (3.8) and the positivity of operator Φ(t),
(3.11)v0(t)≤Φ(t)B1(v0)+∫0t(t-s)α-1Ψ(t-s)[f(s,v0(s))+Cv0(s)]ds=(Qv0)(t),t∈I,
namely, v0≤Qv0. Similarly, we can show that Qw0≤w0. For u∈D, in view of (3.7), then v0≤Qv0≤Qu≤Qw0≤w0. Thus, Q:D→D is a continuous increasing monotonic operator. We can now define the sequences
(3.12)vn=Qvn-1,wn=Qwn-1,n=1,2,…,
and it follows from (3.7) that
(3.13)v0≤v1≤⋯vn≤⋯≤wn≤⋯≤w1≤w0.
In the following, we prove that{vn} and {wn} are convergent in C(I,X). First, we show that QD={Qu∣u∈D} is precompact in C(I,X). Let
(3.14)(Wu)(t)=∫0t(t-s)α-1Ψ(t-s)[f(s,u(s))+Cu(s)]ds,t∈I,then we prove that for all 0<t≤ω, (WD)(t)={(Wu)(t)∣u∈D} is precompact in X. For 0<ε<t, let
(3.15)(Wεu)(t)=∫0t-ε(t-s)α-1Ψ(t-s)[f(s,u(s))+Cu(s)]ds=∫0t-ε(t-s)α-1(α∫0∞θζα(θ)S((t-s)αθ)dθ)[f(s,u(s))+Cu(s)]ds=S(ε)∫0t-ε(t-s)α-1(α∫0∞θζα(θ)S((t-s)αθ-ε)dθ)[f(s,u(s))+Cu(s)]ds.
For u∈D, by (H), f(t,v0(t))+Cv0(t)≤f(t,u(t))+Cu(t)≤f(t,w0(t))+Cw0(t) for 0≤t≤ω. By the normality of the cone P, there is M1>0 such that
(3.16)‖f(t,u(t))+Cu(t)‖≤M1,0≤t≤ω.
Thus, by (3.16) and Remark 2.7 (i), we have
(3.17)‖∫0t-ε(t-s)α-1(α∫0∞θζα(θ)S((t-s)αθ-ε)dθ)[f(s,u(s))+Cu(s)]ds‖≤M1∫0t-ε(t-s)α-1(α∫0∞θζα(θ)‖S((t-s)αθ-ε)‖dθ)ds≤MM1∫0t-ε(t-s)α-1(α∫0∞θζα(θ)dθ)ds=MM1αΓ(1+α)∫0t-ε(t-s)α-1ds=MM1(tα-εα)Γ(1+α),0<t≤ω.
Then by (3.15), (3.17) and the compactness of S(ε), for 0<t≤ω, (WεD)(t)={(Wεu)(t)∣u∈D} is precompact in X. Furthermore, by (3.16) and Lemma 2.16 (i), we have
(3.18)‖(Wu)(t)-(Wεu)(t)‖=‖∫t-εt(t-s)α-1Ψ(t-s)[f(s,u(s))+Cu(s)]ds‖≤MM1αΓ(1+α)∫t-εt(t-s)α-1ds=MM1εαΓ(1+α).
Therefore, for 0<t≤ω, (WD)(t) is precompact in X. In particular, (WD)(ω) is precompact in X, and then B1(D)=(I-Φ(ω))-1(WD)(ω) is precompact. Then in view of Lemma 2.16 (i), (QD)(t)={(Qu(t))∣u∈D}=Φ(t)B1(D)+(WD)(t) is precompact in X for 0≤t≤ω.
Furthermore, for0≤t1<t2≤ω, by (3.16) and Lemma 2.16 (i) we have that
(3.19)‖(Wu)(t2)-(Wu)(t1)‖=‖∫0t2(t2-s)α-1Ψ(t2-s)[f(s,u(s))+Cu(s)]ds-∫0t1(t1-s)α-1Ψ(t1-s)[f(s,u(s))+Cu(s)]ds‖≤M1∫0t1‖(t2-s)α-1Ψ(t2-s)-(t1-s)α-1Ψ(t1-s)‖ds+MM1αΓ(1+α)∫t1t2(t2-s)α-1ds≤M1∫0t1(t2-s)α-1‖Ψ(t2-s)-Ψ(t1-s)‖ds+M1∫0t1‖[(t2-s)α-1-(t1-s)α-1]Ψ(t1-s)‖ds+MM1Γ(1+α)(t2-t1)α≤M1(t2-t1)α-1∫0t1‖Ψ(t2-s)-Ψ(t1-s)‖ds+MM1Γ(1+α)|t1α+(t2-t1)α-t2α|+MM1Γ(1+α)(t2-t1)α≤M1(t2-t1)α-1∫0t1‖Ψ(t2-s)-Ψ(t1-s)‖ds+2MM1Γ(1+α)(t2-t1)α+MM1Γ(1+α)(t2α-t1α).
By Remark 2.12 and Lemma 2.16 (iv), Ψ(t) is continuous in the uniform operator topology for t>0. Then by Lebesque-dominated convergence theorem, WD is equicontinuous in C(I,X). By Lemma 2.16 (ii), {Ψ(t)}t≥0 is strongly continuous. So, QD is equicontinuous in C(I,X).
Then by Ascoli-Arzela’s theorem,QD={Qu∣u∈D} is precompact in C(I,X). By (3.12) and (3.13), {vn} has a convergent subsequence in C(I,X). Combining this with the monotonicity of {vn}, it is itself convergent in C(I,X). Using a similar argument to that for {vn}, we can prove that {wn} is also convergent in C(I,X). Set
(3.20)u̲=lim⁡n→∞vn,u¯=lim⁡n→∞wn.
Let n→∞, by the continuity of Q and (3.12), we have
(3.21)u̲=Qu̲,u¯=Qu¯.
By (3.7), if u∈D is a fixed-point of Q, then v1=Qv0≤Qu=u≤Qw0=w1. By induction, vn≤u≤wn. By (3.13) and taking the limit as n→∞, we conclude that v0≤u̲≤u≤u¯≤w0. This means that u̲,u¯ are the minimal and maximal fixed-points of Q on [v0,w0], respectively. By (3.6), they are the minimal and maximal mild solutions of PBVP (1.1) on [v0,w0], respectively.Theorem 3.2.
Assume that{T(t)}t≥0 is a compact and positive semigroup in  X,  f(t,θ)≥θ for  t∈I. If there is  y∈X such that  y≥θ,  Ay≥f(t,y) for  t∈I, and  f satisfies the following: (H1)
There exists a constantC1>0 such that
(3.22)f(t,x2)-f(t,x1)≥-C1(x2-x1),
for any t∈I, and θ≤x1≤x2≤y, that is, f(t,x)+C1x is increasing in x for x∈[θ,y].
Then PBVP (1.1) has a positive mild solution u: θ≤u≤y.Proof.
Letv0=θ and w0=y, by Theorem 3.1, PBVP (1.1) has mild solution on [v0,w0].Case 2.
{T(t)}t≥0 is noncompact.Theorem 3.3.
Assume that the positive coneP is regular,  {T(t)}t≥0 is an equicontinuous and positive semigroup in  X, PBVP (1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0, and  (H) holds, then PBVP (1.1) has the minimal and maximal mild solutions between  v0 and  w0, which can be obtained by a monotone iterative procedure starting from  v0 and  w0, respectively.Proof.
By the proof of Theorem3.1, (3.2)–(3.13) and (3.19) are valid. By Lemma 2.16 (iv), Ψ(t) is continuous in the uniform operator topology for t>0. Then by Lebesque-dominated convergence theorem, WD is equicontinuous in C(I,X). From Lemma 2.16 (ii), {Ψ(t)}t≥0 is strongly continuous. So, QD is equicontinuous in C(I,X). Thus, {Qvn} is equicontinuous in C(I,X).
For0≤t≤ω, by (3.7) and (3.13), {(Qvn)(t)} is monotone in X. Since the cone P is regular, then {(Qvn)(t)} is convergent in X.
By Ascoli-Arzela’s theorem,{Qvn} is precompact in C(I,X) and {Qvn} has a convergent subsequence in C(I,X). Combining this with the monotonicity of {Qvn}, it is itself convergent in C(I,X). Using a similar argument to that for {Qwn}, we can prove that {Qwn} is also convergent in C(I,X). Let
(3.23)u̲=lim⁡n→∞vn=lim⁡n→∞Qvn-1,u¯=lim⁡n→∞wn=lim⁡n→∞Qwn-1,
then it is similar to the proof of Theorem 3.1 that u̲ and u¯ are the minimal and maximal mild solutions of PBVP (1.1) on [v0,w0], respectively.Corollary 3.4.
LetX be an ordered and weakly sequentially complete Banach space. Assume that  {T(t)}t≥0 is an equicontinuous and positive semigroup in  X, PBVP (1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0, and  (H) holds, then PBVP (1.1) has the minimal and maximal mild solutions between  v0 and  w0, which can be obtained by a monotone iterative procedure starting from  v0 and  w0, respectively.Proof.
In an ordered and weakly sequentially complete Banach space, the normal coneP is regular. Then the proof is complete.Corollary 3.5.
LetX be an ordered and reflective Banach space. Assume that  {T(t)}t≥0 is an equicontinuous and positive semigroup in  X, PBVP (1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0, and  (H) holds, then PBVP (1.1) has the minimal and maximal mild solutions between  v0 and  w0, which can be obtained by a monotone iterative procedure starting from  v0 and  w0, respectively.Proof.
In an ordered and reflective Banach space, the normal coneP is regular. Then the proof is complete.By Theorem3.3, Corollaries 3.4 and 3.5, we have the following.Corollary 3.6.
Assume that{T(t)}t≥0 is an equicontinuous and positive semigroup in  X,  f(t,θ)≥θ for  t∈I. If there is  y∈X such that  y≥θ,  Ay≥f(t,y) for  t∈I,  f satisfies  (H1) and one of the following conditions: (i)
X is an ordered Banach space, whose positive cone P is regular,(ii)
X is an ordered and weakly sequentially complete Banach space,(iii)
X is an ordered and reflective Banach space.
then PBVP (1.1) has positive mild solution u: θ≤u≤y.


## 4. Examples

Example 4.1.
Consider the following periodic boundary value problem for fractional parabolic partial differential equations inX:
(4.1)∂tαu+A(x,D)u=g(x,t,u),(x,t)∈Ω×I,Bu=0,(x,t)∈∂Ω×I,u(x,0)=u(x,ω),x∈Ω,
where ∂tα is the Caputo fractional partial derivative with order 0<α<1, I=[0,ω], Ω⊂ℝN is a bounded domain with a sufficiently smooth boundary ∂Ω, g:Ω¯×I×ℝ→ℝ is continuous, Bu=b0(x)u+δ(∂u/∂n) is a regular boundary operator on ∂Ω, and
(4.2)A(x,D)u=-∑i=1N∑j=1N∂∂xi(aij(x)∂u∂yi)
is a symmetrical strong elliptic operator of second order, whose coefficient functions are Hölder continuous in Ω.
LetX=Lp(Ω)(p≥2), P={v∣v∈Lp(Ω),v(x)≥0a.e.x∈Ω}, then X is a Banach space, and P is a regular cone in X. Define the operator A as follows:
(4.3)D(A)={u∈W2,p(Ω)∣Bu=0},Au=A(x,D)u.
Then -A generates a uniformly bounded analytic semigroup T(t)(t≥0) in X (see [39]). By the maximum principle, we can easily find that T(t)(t≥0) is positive (see [39]). Let u(t)=u(·,t), f(t,u)=g(·,t,u(·,t)), then the problem (4.1) can be transformed into the following problem:
(4.4)Dαu(t)+Au(t)=f(t,u(t)),t∈I,u(0)=u(ω).Theorem 4.2.
Letf(x,t,0)≥0. If there exists  w0(x,t)∈C2,α(Ω×I) such that
(4.5)∂tαw0+A(x,D)w0≥g(x,t,w0),(x,t)∈Ω×I,Bw=0,(x,t)∈∂Ω×I,w0(x,0)≥w0(x,ω),x∈Ω,
and g satisfies the following:(H4)
there exists a constantC2≥0 such that
(4.6)g(x,t,ξ2)-g(x,t,ξ1)≥-C2(ξ2-ξ1),
for any t∈I, and 0≤ξ1≤ξ2≤w0
Then PBVP (4.1) has a mild solution u:0≤u≤w0.Proof.
Setv0=0, by Theorem 3.3, PBVP (4.1) has the minimal and maximal solutions between 0 and w0.



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*Source: 746872-2012-01-27.xml*