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.
1. Introduction
In this paper, we consider the periodic boundary value problem (PBVP) for semilinear fractional differential equation in an ordered Banach space X,
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,
∂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 Section 2, 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
‖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 order α with the lower limit zero for a function f∈AC[0,∞)is defined as
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 order α with the lower limit zero for a function f∈AC[0,∞) can be written as
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 order α for a function f∈AC[0,∞) can be written as
Dαf(t)=DαL(f(t)-f(0)),t>0,0<α<1.
Remark 2.4 (see [32]).
(i) If f∈C1[0,∞), then
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) If f 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, let X 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.
If v0∈Cα(I,X)∩C(I,X1) and satisfies
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]).
If h∈C(I,X), by the mild solution of LIVP,
Dαu(t)+Au(t)=h(t),t∈I,u(0)=x0∈X,
one means that the function u∈C(I,X) and satisfies
u(t)=U(t)x0+∫0t(t-s)α-1V(t-s)h(s)ds,
where
U(t)=∫0∞ζα(θ)T(tαθ)dθ,V(t)=α∫0∞θζα(θ)T(tαθ)dθ,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
∫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
‖T(t)‖≤Ne-δt,t≥0,
then the linear periodic boundary value problem (LPBVP)
Dαu(t)+Au(t)=h(t),t∈I,u(0)=u(ω)
has a unique mild solution
(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)
B(h)=(I-U(ω))-1∫0ω(ω-s)α-1V(ω-s)h(s)ds.
Proof.
In X, give equivalent norm |·| by
|x|=supt≥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,
|T(t)x|=sups≥0‖eδsT(s)T(t)x‖=e-δtsups≥0‖eδ(s+t)T(s+t)x‖=e-δtsupη≥t‖eδηT(η)x‖≤e-δt|x|.
Thus, |T(t)|≤e-δt. Then by Remark 2.7,
|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
(I-U(ω))-1=∑n=0∞(U(ω))n.
Set
x0=(I-U(ω))-1∫0ω(ω-s)α-1V(ω-s)h(s)ds,
then
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
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 stable C0-semigroup, one can see [49].
Definition 2.10.
A C0-semigroup {T(t)}t≥0 is called a compact semigroup if T(t) is compact for t>0.
Definition 2.11.
A C0-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.
A C0-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 Definition 2.13, 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 operator K on X is called to be positive if Kx≥θ for all x≥θ.
Lemma 2.16.
The operators U and V given by (2.9) have the following properties:
For any fixed t≥0, U(t) and V(t) are linear and bounded operators, that is, for any x∈X,
‖U(t)x‖≤M‖x‖,‖V(t)x‖≤αMΓ(1+α)‖x‖,
{U(t)}t≥0 and {V(t)}t≥0 are strongly continuous,
{U(t)}t≥0 and {V(t)}t≥0 are compact operators if {T(t)}t≥0 is a compact semigroup,
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,
U(t) and V(t) are positive for t≥0 if {T(t)}t≥0 is a positive semigroup,
(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.
For 0<t1≤t2, we have
‖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.
By Remark 2.7 (i), the proof is then complete.
By (v), (2.18), and (2.19), the proof is then complete.
3. Main ResultsCase 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.
There exists a constant C>0 such that
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
‖Φ(t)‖≤M,‖Ψ(t)‖≤αΓ(1+α)M,t≥0.
From Lemma 2.8, (I-Φ(ω)) has bounded inverse operator and
(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.
Let D=[v0,w0], then we define a mapping Q:D→C(I,X) by
Qu(t)=Φ(t)B1(u)+∫0t(t-s)α-1Ψ(t-s)[f(s,u(s))+Cu(s)]ds,t∈I,
where
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
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
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
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,
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
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),
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
vn=Qvn-1,wn=Qwn-1,n=1,2,…,
and it follows from (3.7) that
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
(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
(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
‖f(t,u(t))+Cu(t)‖≤M1,0≤t≤ω.
Thus, by (3.16) and Remark 2.7 (i), we have
‖∫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
‖(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, for 0≤t1<t2≤ω, by (3.16) and Lemma 2.16 (i) we have that
‖(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
u̲=limn→∞vn,u¯=limn→∞wn.
Let n→∞, by the continuity of Q and (3.12), we have
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:
There exists a constant C1>0 such that
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.
Let v0=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 cone P 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 Theorem 3.1, (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).
For 0≤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
u̲=limn→∞vn=limn→∞Qvn-1,u¯=limn→∞wn=limn→∞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.
Let X 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 cone P is regular. Then the proof is complete.
Corollary 3.5.
Let X 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 cone P is regular. Then the proof is complete.
By Theorem 3.3, 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:
X is an ordered Banach space, whose positive cone P is regular,
X is an ordered and weakly sequentially complete Banach space,
X is an ordered and reflective Banach space.
then PBVP (1.1) has positive mild solution u: θ≤u≤y.
4. ExamplesExample 4.1.
Consider the following periodic boundary value problem for fractional parabolic partial differential equations in X:
∂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
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.
Let X=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:
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:
Dαu(t)+Au(t)=f(t,u(t)),t∈I,u(0)=u(ω).
Theorem 4.2.
Let f(x,t,0)≥0. If there exists w0(x,t)∈C2,α(Ω×I) such that
∂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:
there exists a constant C2≥0 such that
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.
Set v0=0, by Theorem 3.3, PBVP (4.1) has the minimal and maximal solutions between 0 and w0.
Acknowledgments
This research was supported by NNSFs of China (nos. 10871160, 11061031) and Project of NWNU-KJCXGC-3-47.
PetrášI.A note on the fractional-order cellular neural networksthe International Joint Conference on Neural Networks (IJCNN '06)July 2006102110242-s2.0-40649111670DorcakL.PetrasI.KostialI.TerpakJ.Fractional-order state space modelsProceedings of the International Carpathian Control Conference2002193198CafagnaD.Past and present—fractional calculus: a mathematical tool from the past for present engineers20071235402-s2.0-3634897681110.1109/MIE.2007.901479BenchellalA.PoinotT.Thierry.Poinot@esip.univ-poitiers.frTrigeassouJ.-C.Fractional modelling and identification of a thermal process2the 2nd IFAC Workshop on Fractional Differentiation and Its Applications (FDA '06)July 2006248253JesusI. S.isj@isep.ipp.ptTenreiro MachadoJ. A.jtm@isep.ipp.ptBoaventura CunhaJ.jboavent@utad.ptFractional electrical dynamics in fruits and vegetables2the 2nd IFAC Workshop on Fractional Differentiation and its Applications (FDA '06)July 2006308313AhmadW. M.El-KhazaliR.Fractional-order dynamical models of love200733413671375231892210.1016/j.chaos.2006.01.098ZBL1133.91539OustaloupA.SabatierJ.MoreauX.From fractal robustness to the CRONE approach19985Paris, FaranceSoc. Math. Appl. Indust.177192ESAIM Proc.1665570PodlubnyI.The laplace transform method for linear differential equations of the fractional order1994Slovak Academy of Sciences, Institute of Experimental PhysicsPodlubnyI.1999198San Diego, Calif, USAAcademic Pressxxiv+340Mathematics in Science and Engineering1658022Tenreiro MacHadoJ. A.jtm@isep.ipp.ptSilvaM. F.mss@isep.ipp.ptBarbosaR. S.rsb@isep.ipp.ptJesusI. S.isj@isep.ipp.ptReisC. M.cmr@isep.ipp.ptMarcosM. G.mgm@isep.ipp.ptGalhanoA. F.amf@isep.ipp.ptSome applications of fractional calculus in engineering201020103463980110.1155/2010/639801OldhamK. B.SpanierJ.1974London, UKAcademic Pressxiii+234Mathematics in Science and Engineering, Vol. 110361633SilvaM. F.mfsilva@dee.isep.ipp.ptMachadoJ. A. T.jtm@dee.isep.ipp.ptLopesA. M.aml@fe.up.ptComparison of fractional and integer order control of an hexapod robot5Proceedings of the ASME Design Engineering Technical Conferences and Computers and Information in Engineering ConferenceSeptember 2003Chicago, Ill, USA667676GutiérrezR. E.rgutic@fem.unicamp.brRosárioJ. M.rosario@fem.unicamp.brTenreiro MacHadoJ.jtm@isep.ipp.ptFractional order calculus: basic concepts and engineering applications201020101937585810.1155/2010/375858DuarteF. B. M.Tenreiro MachadoJ. A.Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators2002291–431534210.1023/A:10165593147981926478ZBL1027.70011AgrawalO. P.A general formulation and solution scheme for fractional optimal control problems2004381–432333710.1023/B:NODY.0000045544.96418.bf2112177ZBL1121.70019EnghetaN.On fractional calculus and fractional multipoles in electromagnetism1996444554566138201710.1109/8.489308ZBL0944.78507MaginR. L.Fractional calculus models of complex dynamics in biological tissues201059515861593259593010.1016/j.camwa.2009.08.039ZBL1189.92007KulishV. V.LageJ. L.Application of fractional calculus to fluid mechanics200212438038062-s2.0-1084427044210.1115/1.1478062OldhamK. B.Fractional differential equations in electrochemistry20104119122-s2.0-7035056718310.1016/j.advengsoft.2008.12.012GafiychukV.DatskoB.MeleshkoV.Mathematical modeling of time fractional reaction-diffusion systems20082201-2215225244416610.1016/j.cam.2007.08.011ZBL1199.35152LedermanC.RoquejoffreJ.-M.WolanskiN.Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames20041832173239207547210.1007/s10231-003-0085-1BagleyR. L.TorvikP. J.A theoretical basis for the application of fractional calculus to viscoelasticity19832732012102-s2.0-002076520210.1122/1.549724MainardiF.CarpinteriA.MainardiF.Fractional calculus: some basic problems in continuum and statistical mechanics1997378Vienna, AustriaSpringer291348CISM Courses and Lectures1611587MeralF. C.RoystonT. J.MaginR.Fractional calculus in viscoelasticity: an experimental study2010154939945255700210.1016/j.cnsns.2009.05.004ZBL1221.74012Solteiro PiresE. J.Tenreiro MachadoJ. A.De Moura OliveiraP. B.Fractional order dynamics in a GA planner20038311237723862-s2.0-014189267810.1016/S0165-1684(03)00190-7HedrihK.khedrih@eunet.rsNikoli-StanojeviV.veranikolic1@gmail.comA model of gear transmission: fractional order system dynamics201020102397287310.1155/2010/972873CaoJ.MaC.XieH.JiangZ.Nonlinear dynamics of duffing system with fractional order damping20105460410122-s2.0-77953690360WangJ.ZhouY.WeiW.A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces2011161040494059280271110.1016/j.cnsns.2011.02.003ZBL1223.45007WangJ.ZhouY.A class of fractional evolution equations and optimal controls201112126227210.1016/j.nonrwa.2010.06.0132728679ZBL1214.34010WangJ.ZhouY.WeiW.XuH.Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls201162314271441282473010.1016/j.camwa.2011.02.040ZhouY.JiaoF.Existence of mild solutions for fractional neutral evolution equations2010593106310772579471ZBL1189.34154ZhouY.JiaoF.Nonlocal Cauchy problem for fractional evolution equations20101154465447510.1016/j.nonrwa.2010.05.0292683890El-BoraiM. M.Some probability densities and fundamental solutions of fractional evolution equations2002143433440190329510.1016/S0960-0779(01)00208-9ZBL1005.34051El-BoraiM. M.The fundamental solutions for fractional evolution equations of parabolic type20042004319721110.1155/S10489533043110202102832ZBL1081.34053El-BoraiM. M.DebboucheA.Almost periodic solutions of some nonlinear fractional differential equations2009425–28137313872604273ZBL1201.34009DebboucheA.El-BoraiM. M.Weak almost periodic and optimal mild solutions of fractional evolution equations200946182495851ZBL1171.34331LiY. X.Existence and uniqueness of positive periodic solutions for abstract semilinear evolution equations20052567207282203151ZBL1110.34328LiY. X.Existence of solutions to initial value problems for abstract semilinear evolution equations2005486108910942205049LiY. X.Periodic solutions of semilinear evolution equations in Banach spaces19984136296361640587LiY. X.Global solutions of initial value problems for abstract semilinear evolution equations2001343393471895672LiY. X.Positive solutions of abstract semilinear evolution equations and their applications19963956666721436036ZBL0870.47040KilbasA. A.SrivastavaH. M.TrujilloJ. J.2006204Amsterdam, The NetherlandsElsevierxvi+523North-Holland Mathematics Studies2218073MillerK. S.RossB.1993New York, NY, USAJohn Wiley & Sonsxvi+366A Wiley-Interscience Publication1219954SamkoS. G.KilbasA. A.MarichevO. I.1993Yverdon, SwitzerlandGordon and Breachxxxvi+9761347689MuJ.mujia88@163.comMonotone iterative technique for fractional evolution equations in banach spaces201120111376718610.1155/2011/767186FellerW.19712ndNew York, NY, USAJohn Wiley & Sonsxxiv+6690270403SchneiderW. R.WyssW.Fractional diffusion and wave equations198930113414497446410.1063/1.528578ZBL0692.45004WeiZ.DongW.CheJ.Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative201073103232323810.1016/j.na.2010.07.0032680017ZBL1202.26017HuangF. L.Spectral properties and stability of one-parameter semigroups19931041182195122412610.1006/jdeq.1993.1068ZBL0801.47026PazyA.198344New York, NY, USASpringerviii+279Applied Mathematical Sciences710486