AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation96873510.1155/2011/968735968735Research ArticlePerturbation Results and Monotone Iterative Technique for Fractional Evolution EquationsMuJiaPerez GarciaVictor M.Department of Mathematics, Northwest Normal UniversityLanzhou, Gansu 730000Chinanwnu.edu.cn2011258201120111804201117052011040720112011Copyright © 2011 Jia Mu.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.

We mainly study the fractional evolution equation in an ordered Banach space XDC0a+u(t)+Au(t)=f(t,u(t),Gu(t)) , 1<α<2, u(0)=xX, u(0)=θ. Using the monotone iterative technique based on lower and upper solutions, the existence and uniqueness results are obtained. The necessary perturbation results for accomplishing this approach are also developed.

1. Introduction

In this paper, we use the perturbation theory and the monotone iterative technique based on lower and upper solutions to investigate the existence and uniqueness of mild solutions for the fractional evolution equation in an ordered Banach space X:CD0+αu(t)+Au(t)=f(t,u(t),Gu(t)),tI=[0,T],u(0)=xX,u(0)=θ, where CD0+α is the Caputo fractional derivative, 1<α<2, A:D(A)XX is a linear closed densely defined operator, f:I×X×XX is continuous, θ is the zero element of X, andGu(t)=0tK(t,s)u(s)ds is a Volterra integral operator with integral kernel KC(Δ,+), Δ={(t,s)0stT}.

In particular, when f(t,u(t),Gu(t))=f(t,u(t)), we study the existence and uniqueness of mild solutions for the fractional evolution equation in an ordered Banach space X:CD0+αu(t)+Au(t)=f(t,u(t)),tI,u(0)=xX,u(0)=θ, where CD0+α is the Caputo fractional derivative, 1<α<2, A:D(A)XX is a linear closed densely defined operator, f:I×XX is continuous, and θ is the zero element of X.

The fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) goes back to Newton and Leibnitz in the seventieth century. It has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields such as physics, chemistry, aerodynamics, viscoelasticity, porous media, electrodynamics of complex medium, and electrochemistry, control, electromagnetic. For instance, fractional calculus concepts have been used in the modeling of transmission lines , neurons , viscoelastic materials , and electrical capacitors . Other examples from fractional order dynamics can be found in [5, 6] and the references therein.

One of the branches of fractional calculus is the theory of fractional evolution equations, that is evolution equations where the integer derivative with respect to time is replaced by a derivative of any order. Also, in recent years, fractional evolution equations have attracted increasing attention; see .

The monotone iterative technique based on lower and upper solutions is an effective and a flexible mechanism that offers theoretical as well as constructive existence results in a closed set. It yields monotone sequences of lower and upper approximate solutions that converge to the minimal and maximal solutions between the lower and upper solutions. Since under suitable conditions each member of the sequences happens to be the unique solution of a certain nonlinear problem, the advantage and importance of the technique is remarkable. For differential equations of integer order, many papers used the monotone iterative technique based on lower and upper solutions; see  and the references therein. Recently, there have been some papers which deal with the existence of the solutions of initial value problems or boundary value problems for fractional ordinary differential equations by using this method; see . They mainly involve Riemann-Liouville fractional derivatives.

However, to the best of the authors' knowledge, no results yet exist for the fractional evolution equations by using the monotone iterative technique based on lower and upper solutions. Our results can be considered as a contribution to this emerging field.

In comparison with fractional ordinary differential equations, we have great difficulty in using the monotone iterative technique for the fractional evolution equations. Firstly, how to introduce a suitable concept of a mild solution for fractional evolution equations based on the corresponding solution operator? A pioneering work has been reported by El-Borai [10, 11]. Later on, some authors introduced the definitions of mild solutions for fractional evolution equations. Wang and Zhou , Wang et al. [16, 18], and Zhou and Jiao [20, 21] also introduced a suitable definition of the mild solutions based on the well-known theory of Laplace transform and probability density functions. Moreover, Hernández et al.  used an approach to treat abstract equations with fractional derivatives based on the well-developed theory of resolvent operators for integral equations. Shu et al.  give the definition of a mild solution by investigating the classical solutions of the corresponding system. Secondly, do the solution operators for fractional evolution equations have the perturbation properties analogous to those for the C0-semigroup? For evolution equations of integer order, perturbation properties play a significant role in monotone iterative technique; see .

Our paper copes with the above difficulties, and the new features of this paper mainly include the following aspects. We firstly introduce a new concept of a mild solution based on the well-known theory of Laplace transform, and the form is very easy. Secondly, we discuss the perturbation properties for the corresponding solution operators. Thirdly, by the monotone iterative technique based on lower and upper solutions, we obtain results on the existence and uniqueness of mild solutions for problem (1.1) and (1.3).

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Definition 2.1 (see [<xref ref-type="bibr" rid="B13">5</xref>]).

The Riemann-Liouville fractional integral operator of order α>0 of function fL1(+) is defined as I0+αf(t)=1Γ(α)0t(t-s)α-1f(s)ds, where Γ(·) is the Euler gamma function.

Definition 2.2 (see [<xref ref-type="bibr" rid="B13">5</xref>]).

The Caputo fractional derivative of order α>0, n-1<α<n, is defined as CD0+αf(t)=1Γ(n-α)0t(t-s)n-α-1f(s)ds, where the function f(t) has absolutely continuous derivatives up to order n-1. If f is an abstract function with values in X, then the integrals and derivatives which appear in (2.1) and (2.2) are taken in Bochner sense.

Proposition 2.3.

For α,β>0 and f as a suitable function (e.g., ) one has the following:

I0+αI0+βf(t)=I0+α+βf(t);

I0+αI0+βf(t)=I0+βI0+αf(t);

I0+α(f(t)+g(t))=I0+αf(t)+I0+αg(t);

CD0+αI0+αf(t)=f(t);

CD0+αCD0+βf(t)CD0+α+βf(t);

CD0+αCD0+βf(t)CD0+βCD0+αf(t).

We observe from the above that the Caputo fractional differential operators do not possess neither semigroup nor commutative properties, which are inherent to the derivatives on integer order. For basic facts about fractional integrals and fractional derivatives one can refer to the books [5, 3234].

Let X be an ordered Banach space with norm · and partial order ≤, whose positive cone P={yXyθ} (θ is the zero element of X) is normal with normal constant N. Let C(I,X) be the Banach space of all continuous X-value functions on interval I with norm uc=maxtI(t). For u,vC(I,X), uvu(t)v(t) for all tI. For v,wC(I,X), denote the ordered interval [v,w]={uC(I,X)vuw}, and [v(t),w(t)]={yXv(t)yw(t)}, tI. By (X) we denote the space of all bounded linear operators from X to X.

Definition 2.4.

If CD0+αv0,Av0,v0C(I,X), and v0 satisfies CD0+αv0(t)+Av0(t)f(t,v0(t),Gv0(t)),tI,v0xX,v0(0)θ, then ṽ0 is called a lower solution of problem (1.1); if all inequalities of (2.3) are inverse, we call it an upper solution of problem (1.1).

Similarly, we give the definitions of lower and upper solutions of problem (1.3).

Definition 2.5.

If CD0+αṽ0,Aṽ0,ṽ0C(I,X), and ṽ0 satisfy CD0+αṽ0(t)+Aṽ0(t)f(t,ṽ0(t)),tI,  ṽ0(0)xX,ṽ0(0)θ, then ṽ0 is called a lower solution of problem (1.3); if all inequalities of (2.4) are inverse, we call it an upper solution of problem (1.3).

Consider the following problem: CD0+αu(t)+Au(t)=θ,tI,u(0)=x,u(0)=θ.

Definition 2.6 (see [<xref ref-type="bibr" rid="B3">9</xref>]).

A family {Sα(t)}t0(X) is called a solution operator for (2.5) if the following conditions are satisfied:

Sα(t) is strongly continuous for t0 and Sα(0)=I;

Sα(t)D(A)D(A) and ASα(t)x=Sα(t)Ax for all xD(A), t0;

Sα(t) is a solution of u(t)=x-1Γ(α)0t(t-s)α-1Au(s)ds, for all xD(A), t0.

In this case, -A is called the generator of the solution operator Sα(t) and Sα(t) is called the solution operator generated by -A.

Definition 2.7 (see [<xref ref-type="bibr" rid="B3">9</xref>]).

The solution operator Sα(t) is called exponentially bounded if there are constants M1 and ω0 such that Sα(t)Meωt,t0.

An operator -A is said to belong to 𝒞(X;M,ω), or 𝒞α(M,ω) for short, if problem (2.5) has a solution operator Sα(t) satisfying (2.7). Denote 𝒞α(ω)={𝒞α(M,ω)M1}, 𝒞α={𝒞α(ω)ω0}. In these notations 𝒞1 and 𝒞2 are the sets of all infinitesimal generators of C0-emigroups and cosine operator families (COF), respectively. Next, we give a characterization of 𝒞α(M,ω).

Lemma 2.8 (see [<xref ref-type="bibr" rid="B3">9</xref>]).

Let 1<α<2, -ACα(M,ω) and let Sα(t) be the corresponding solution operator. Then for λ>ω, one has λαρ(-A) and λα-1R(λα,-A)x=0+e-λtSα(t)xdt,xX.

Lemma 2.9 (see [<xref ref-type="bibr" rid="B3">9</xref>]).

Let 1<α<2 and -ACα. Then the corresponding solution operator is given by Sα(t)x=limn1n!k=1n+1bk,n+1α(I+(tn)αA)-kx=limn1n!k=1n+1bk,n+1α[(nt)αR((nt)α,-A)]kx, where bk,nα are given by the recurrence relations:

b1,1α=1,

The convergence is uniform on bounded subsets of [0,+) for any fixed xX.

Lemma 2.10 (see [<xref ref-type="bibr" rid="B3">9</xref>]).

Let 1<α<2. Then -ACα(M,ω) if and only if (ωα,)ρ(-A) and nλn(λα-1R(λα,-A))Mn!(λ-ω)n+1λ>ω,  n=0,1,.

Lemma 2.11.

Assume hC(I,X). For the linear Cauchy problem CD0+αu(t)+Au(t)=h(t),tI,u(0)=xX,u(0)=θ,u(t) has the form u(t)=Sα(t)x+0tTα(t-s)h(s)ds, where Sα(t) is the solution operator generated by -A, and Tα(t)=I0+α-1Sα(t).

Proof.

For λ>ω, applying the Laplace transform to (2.11), we have that λαLu(λ)-λα-1u(0)-λα-2u(0)+ALu(λ)=λαLu(λ)-λα-1x+ALu(λ)=Lh(λ). By Lemma 2.8, λαρ(-A), from the above equation, we obtain Lu(λ)=λα-1(λαI+A)-1x+λ1-αλα-1(λαI+A)-1Lh(λ). Since [tα-2/Γ(α-1)](λ)=λ1-α, by Lemma 2.8 and the inverse Laplace transform, we have that u(t)=Sα(t)x+0tTα(t-s)h(s)ds, where Tα(t)=1Γ(α-1)0t(t-s)α-2Sα(s)ds=I0+α-1Sα(t).

Remark 2.12.

If A=a (a is a constant), we know that u(t)=Eα(-atα)x+0t(t-s)α-1Eα,α(-a(t-s)α)h(s)ds is the solution of (2.11) by [5, Example  4.10], where Eα(-atα) and Eα,α(-a(t-s)α) are the Mittag-Leffler functions. We also find that Eα(-atα)x is the solution of the problem (2.5), and tα-1Eα,α(-atα)=I0+α-1Eα(-atα); see .

Definition 2.13.

A function u:IX is called a mild solution of (2.11) if uC(I,X) and satisfies the following equation: u(t)=Sα(t)x+0tTα(t-s)h(s)ds, where Sα(t) is the solution operator generated by -A, and Tα(t)=I0+α-1Sα(t).

Remark 2.14.

It is easy to verify that a classical solution of (2.11) is a mild solution of the same system.

Lemma 2.15.

If 1<α<2, -ACα(M,ω), Sα(t) is the solution operator generated by -A, and Tα(t)=I0+α-1Sα(t), then one has that Tα(t)MΓ(α)eωttα-1,t0.

Proof.

By (2.7), for s0, we have that Sα(s)Meωs. Thus, Tα(t)=1Γ(α-1)0t(t-s)α-2Sα(s)dsM1Γ(α-1)0t(t-s)α-2eωsds=Meωt1Γ(α-1)0t(t-s)α-2ds=Meωttα-1Γ(α).

Now, we discuss the perturbation properties of the solution operators.

Definition 2.16.

An operator S(t):XX(t0) is called a positive operator in X if uP and t0 such that S(t)  uθ.

From Definition 2.1, we can easily obtain the following result.

Lemma 2.17.

Assume that Sα(t) is the solution operator generated by -A and Tα(t)=I0+α-1Sα(t). Then Sα(t)(t0) is a positive operator if and only if Tα(t)(t0) is a positive operator.

By Lemmas 2.8 and 2.9 and the closedness of the positive cone, we can obtain the following result.

Lemma 2.18.

Assume that -ACα(M,ω) and Sα(t) is the solution operator generated by -A. The following results are true.

If Sα(t)(t0) is a positive solution operator, then for any λ>ω and uP, we have R(λα,-A)uθ.

If there is a λ0>ω, for any λ>λ0 and uP such that R(λα,-A)uθ, then Sα(t)(t0) is a positive solution operator.

Lemma 2.19.

Assume C>0, 1<α<2, -ACα(M,ω); then the following results hold.

-(A+CI)Cα(MEα(MCTα),ω), where Eα(MCTα) is the Mittag-Leffler function.

If λ>ω and uP such that R(λα,-A)uθ, then for λ>ω+CMω1-α and uP, one has R(λα,-(A+CI))uθ.

Proof.

(i) If S̃α(t) is the solution operator generated by -(A+CI), in view of [9, Theorem 2.26], we have that S̃α(t)MEα(MCTα)eωt,t0. That is, -(A+CI)𝒞α(MEα(MCTα),ω).

(ii) If λ>ω, by (i) and Lemma 2.8, we have λαρ(-A), λαρ(-(A+CI)). Then by Lemma 2.10, λα-1R(λα,-A)Mλ-ω. When λ>ω+CMω1-α, CR(λα,-A)CMλ1-αλ-ωCMω1-αλ-ω<1. Therefore, for such λ the operator I+CR(λα,-A) is invertible and R(λα,-(A+CI))=R(λα,-A)(I+CR(λα,-A))-1=R(λα,-A)n=0(-1)n(CR(λα,-A))n. For any uP, in view of R(λα,-A)uθ, C>0 and (2.22), then R(λα,-(A+CI))uθ.

Remark 2.20.

If α(0,1), Lemma 2.19 (i) is not true; see [9, Example  2.24]. However, a classical perturbation result for 𝒞1 or 𝒞2 (see [35, 36]) is as follows: if A is the generator of C0-semigroup (or COF) and B(X), then A+B is again a generator of a C0-semigroup (or COF).

By Definition 2.13, we can obtain the following result.

Lemma 2.21.

The linear Cauchy problem CD0+αu(t)+Au(t)+Cu(t)=h(t),tI,u(0)=xX,u(0)=θ, where C>0, hC(I,X), has the unique mild solution given by u(t)=S̃α(t)x+0tT̃α(t-s)h(s)ds, where S̃α(t) is the solution operator generated by -(A+CI), and T̃α(t)=I0+α-1S̃α(t).

By Lemmas 2.17, 2.18, and 2.19, the following result holds.

Lemma 2.22.

Assume that Sα(t) and S̃α(t) are the solution operators generated by -A and -(A+CI), respectively, and T̃α(t)=I0+α-1S̃α(t). Then Sα(t) is a positive operator S̃α(t) and T̃α(t) are positive operators.

Now, we recall some properties of the measure of noncompactness that will be used later. Let μ(·) denote the Kuratowski measure of noncompactness of the bounded set. For the 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}. If B is bounded in C(I,X), then B(t) is bounded in X, and μ(B(t))μ(B).

Lemma 2.23 (see [<xref ref-type="bibr" rid="B9">38</xref>]).

Let B={un}C(I,X)(n=1,2,) be a bounded and countable set. Then μ(B(t)) is Lebesgue integral on I, and μ({Iun(t)dtn=1,2,})2Iμ(B(t))dt.

3. Main Results

Theorem 3.1.

Let X be an ordered Banach space, whose positive coneP is normal with normal constant N, f:I×X×XX is continuous, and A:D(A)XX is a linear closed densely defined operator. Assume that -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.1) has a lower solution v0C(I,X) and an upper solution w0C(I,X) with v0w0, and the following conditions are satisfied.

There exists a constant C0 such that f(t,x2,y2)-f(t,x1,y1)-C(x2-x1), for any tI, v0(t)x1x2w0(t), and Gv0(t)y1y2Gw0(t).

There exists a constant L0 such that μ({f(t,xn,yn)})L(μ({xn})+μ({yn})), for any tI, and increasing or decreasing monotonic sequences {xn}[v0(t),w0(t)] and {yn}[Gv0(t),Gw0(t)].

Then the Cauchy problem (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.

Since -ACα(M,ω), by Lemmas 2.15 and 2.19, we have that S̃α(t)MEα(MCTα)eωt,t0,T̃α(t)MΓ(α)Eα(MCTα)eωttα-1,t0. Set M̃T=maxtIT̃α(t). Since Sα(t) is the positive solution operator generated by -A, by Lemma 2.22, S̃α(t) and T̃α(t) are positive operators.

Let D=[v0,w0]; we define a mapping Q:DC(I,X) by Qu(t)=S̃α(t)x+0tT̃α(t-s)(f(s,u(s),Gu(s))+Cu(s))ds,tI. By Lemma 2.21, uD is a mild solution of problem (1.1) if and only if u=Qu. By (H1), for u1,u2D and u1u2, we have that Qu1Qu2. That is, Q is an increasing monotonic operator. Now, we show that v0Qv0, Qw0w0.

Let σ(t)CD0+αv0(t)+Av0(t)+Cv0(t); by Definition 2.1, Lemma 2.21, and the positivity of operators S̃α(t) and T̃α(t), we have that v0(t)=S̃α(t)v0(0)+0tT̃α(t-s)σ(s)dsS̃α(t)x+0tT̃α(t-s)(f(s,v0(s),Gv0(s))+Cv0(s))ds=Qv0(t),tI, namely, v0Qv0. Similarly, we can show that Qw0w0. For uD, in view of (3.7), then v0Qv0QuQw0w0. Thus, Q:DD. We can now define the sequences vn=Qvn-1,wn=Qwn-1,n=1,2,, and it follows from (3.7) that v0v1vnwnw1w0. For convenience, by (1.2), we can denote K0=max(t,s)ΔK(t,s). Let B={vn}(n=1,2,) and B0={vn-1}(n=1,2,). It follows from B0=B{v0} that μ(B(t))=μ(B0(t)) for tI. Let φ(t)=μ(B(t))=μ(B0(t)),tI. In view of (3.10), since the positive cone P is normal, then B0 and B are bounded in C(I,X). By Lemma 2.23 and (3.12), φ(t) is Lebesgue integrable on I. For tI, by (3.11) and Lemma 2.23, we have that μ(GB0(t))=μ({0tK(t,s)vn-1(s)n=1,2,})2K00tφ(s)ds, and therefore, 0tμ(GB0(s))ds2TK00tφ(s)ds. For tI, from Lemma 2.23, (H2), (3.4), (3.5), (3.9), (3.12), (3.14), and the positivity of operator T̃α(t), we have that φ(t)=μ(B(t))=μ(QB0(t))=μ({0tT̃α(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))dsn=1,2,})20tμ({T̃α(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))n=1,2,})ds2M̃T0tL(μ(B0(s))+μ(GB0(s)))+Cμ(B0(s))ds=2M̃T(L+2LTK0+C)0tφ(s)ds. By (3.15) and the Gronwall inequality, we obtain that φ(t)0 on I. This means that vn(t)(n=1,2,) is precompact in X for every tI. So, vn(t) has a convergent subsequence in X. In view of (3.7), we can easily prove that vn(t) itself is convergent in X. That is, there exist u̲(t)X such that vn(t)u̲(t) as n for every tI. By (3.5) and (3.9), for any tI, we have that vn(t)=S̃α(t)x+0tT̃α(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))ds. Let n; then by Lebesgue-dominated convergence theorem, for any tI, we have that u̲(t)=S̃α(t)x+0tT̃α(t-s)(f(s,u̲(s),Gu̲(s))+Cu̲(s))ds, and u̲C(I,X). Then u̲=Qu̲. Similarly, we can prove that there exists u¯C(I,X) such that u¯=Qu¯. By (3.7), if uD, and u is a fixed point of Q, then v1=Qv0Qu=uQw0=w1. By induction, vnuwn. By (3.10) and taking the limit as n, we conclude that v0u̲uu¯w0. That 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 the Cauchy problem (1.1) on [v0,w0], respectively.

Remark 3.2.

Even if A=0, our results are also new.

Corollary 3.3.

Let X be an ordered Banach space, whose positive coneP is regular, f:I×X×XX is continuous, and A:D(A)XX is a linear closed densely defined operator. Assume that -ACα(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.1) has a lower solution v0C(I,X) and an upper solution w0C(I,X) with v0w0, and (H1) holds. Then the Cauchy problem (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.

Since (H1) is satisfied, then (3.10) holds. In regular positive coneP, any monotonic and ordered-bounded sequence is convergent. Then there exist u̲C(I,E), u¯C(I,E) and limnvn=u̲, limnwn=u¯. Then by the proof of Theorem 3.1, the proof is then complete.

Corollary 3.4.

Let X be an ordered and weakly sequentially complete Banach space, whose positive coneP is normal with normal constant N, f:I×X×XX is continuous, and A:D(A)XX is a linear closed densely defined operator. Assume that -ACα(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.1) has a lower solution v0C(I,X) and an upper solution w0C(I,X) with v0w0, and (H1) holds. Then the Cauchy problem (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.

Since X is an ordered and weakly sequentially complete Banach space, then the assumption (H2) holds. In fact, by [39, Theorem  2.2], any monotonic and ordered-bounded sequence is precompact. Let xn and yn be two increasing or decreasing sequences. By (H1), {f(t,xn,yn)+Cxn} is monotonic and ordered-bounded sequence. Then, by the properties of the measure of noncompactness, we have μ({f(t,xn,yn)})μ(f(t,xn,yn)+Cxn)+Cμ({xn})=0. So, (H2) holds. By Theorem 3.1, the proof is then complete.

Theorem 3.5.

Let X be an ordered Banach space, whose positive coneP is normal with normal constant N, f:I×X×XX is continuous, A:D(A)XX is a linear closed densely defined operator. Assume -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.1) has a lower solution v0C(I,X) and an upper solution w0C(I,X) with v0w0, (H1) holds, and the following condition is satisfied:

(H3) There are constants S1,S20 such that f(t,x2,y2)-f(t,x1,y1)S1(x2-x1)+S2(y2-y1), for any tI, v0(t)x1x2w0(t), and Gv0(t)y1y2Gw0(t).

Then the Cauchy problem (1.1) has the unique mild solution between v0 and w0, which can be obtained by a monotone iterative procedure starting from v0 or w0.

Proof.

We can find that (H1) and (H3) imply (H2). In fact, for tI, let {xn}[v0(t),w0(t)] and {yn}[Gv0(t),Gw0(t)] be two increasing or decreasing monotonic sequence. For m,n=1,2, with m>n, by (H1) and (H3), we have that θf(t,xm,ym)-f(t,xn,yn)+C(xm-xn)(S1+C)(xm-xn)+S2(ym-yn). By (3.20) and the normality of positive cone P, we have f(t,xm,ym)-f(t,xn,yn)(NS1+NC+C)xm-xn+NS2ym-yn. From (3.21) and the definition of the measure of noncompactness, we have that μ({f(t,xn,yn)})(NS1+NC+C)μ({xn})+NS2μ({yn})L(μ({xn})+μ({yn})), where L=NS1+NC+C+NS2. Hence, (H2) holds.

Therefore, by Theorem 3.1, the Cauchy problem (1.1) has the minimal solution u̲ and the maximal solution u¯ on D=[v0,w0]. In view of the proof of Theorem 3.1, we show that u̲=u¯. For tI, by (3.4), (3.5), (3.6), (3.11), (H3), and the positivity of operator T̃α(t), we have that θu¯(t)-u̲(t)=Qu¯(t)-Qu̲(t)=0tT̃α(t-s)[f(s,u¯(s),Gu¯(s))-f(s,u̲(s),Gu̲(s))+C(u¯(s)-u̲(s))]ds0tT̃α(t-s)[(S1+C)(u¯(s)-u̲(s))+S2(Gu¯(s)-Gu̲(s))]dsM̃T(S1+C+S2K0T)0tu¯(s)-u̲(s)ds. By (3.23) and the normality of the positive cone P, for tI, we obtain that u¯(s)-u̲(s)NM̃T(S1+C+S2K0T)0tu¯(s)-u̲(s)ds. By the Gronwall inequality, then u̲(t)u¯(t) on I. Hence u̲=u¯ is the the unique mild solution of the Cauchy problem (1.1) on [v0,w0]. By the proof of Theorem 3.1, we know that it can be obtained by a monotone iterative procedure starting from v0 or w0.

By Corollaries 3.3 and 3.4, and Theorem 3.5, we have the following results.

Corollary 3.6.

Let f:I×X×XX be continuous, and let A:D(A)XX be a linear closed densely defined operator. Assume that -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.1) has a lower solution v0C(I,X) and an upper solution w0C(I,X) with v0w0, (H1) and (H3) hold, and one of the following conditions is satisfied:

X is an ordered Banach space, whose positive coneP is regular;

X is an ordered and weakly sequentially complete Banach space, whose positive coneP is normal with normal constant N.

Then the Cauchy problem (1.1) has the unique mild solution between v0 and w0, which can be obtained by a monotone iterative procedure starting from v0 or w0.

Next, we consider the existence and uniqueness results of the Cauchy problem (1.3). Substituting f(t,,u(t)) for f(t,u(t),Gu(t)) in Theorem 3.1, Corollaries 3.3 and 3.4, and Theorem 3.5, we can obtain the following results.

Corollary 3.7.

Let X be an ordered Banach space, whose positive coneP is normal with normal constant N, f:I×XX is continuous, A:D(A)XX is a linear closed densely defined operator. Assume -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.3) has a lower solution ṽ0C(I,X) and an upper solution w̃0C(I,X) with ṽ0w̃0, and the following conditions are satisfied.

There exists a constant C̃0 such that

f(t,x2)-f(t,x1)-C̃(x2-x1), for any tI, and ṽ0(t)x1x2w̃0(t);

There exists a constant L̃0 such that μ({f(t,xn)})L̃μ({xn}), for any tI, and increasing or decreasing monotonic sequence {xn}[ṽ0(t),w̃0(t)].

Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between ṽ0 and w̃0, which can be obtained by a monotone iterative procedure starting from ṽ0 and w̃0, respectively.

Corollary 3.8.

Let X be an ordered Banach space, whose positive coneP is regular, f:I×XX is continuous, and A:D(A)XX is a linear closed densely defined operator. Assume that -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.3) has a lower solution ṽ0C(I,X) and an upper solution w̃0C(I,X) with ṽ0w̃0, and (H̃1) holds. Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between ṽ0 and w̃0, which can be obtained by a monotone iterative procedure starting from ṽ0 and w̃0, respectively.

Corollary 3.9.

Let X be an ordered and weakly sequentially complete Banach space, whose positive coneP is normal with normal constant N, f:I×XX is continuous, and A:D(A)XX is a linear closed densely defined operator. Assume that -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.3) has a lower solution ṽ0C(I,X) and an upper solution w̃0C(I,X) with ṽ0w̃0, and (H̃1) holds. Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between ṽ0 and w̃0, which can be obtained by a monotone iterative procedure starting from ṽ0 and w̃0, respectively.

Corollary 3.10.

Let X be an ordered Banach space, whose positive coneP is normal with normal constant N, f:I×XX is continuous, and A:D(A)XX is a linear closed densely defined operator. Assume that -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.3) has a lower solution ṽ0C(I,X) and an upper solution w̃0C(I,X) with ṽ0w̃0, (H̃1) holds, and the following condition is satisfied.

There exists a constant S̃10 such that f(t,x2)-f(t,x1)S̃1(x2-x1), for any tI, and ṽ0(t)x1x2w̃0(t).

Then the Cauchy problem (1.3) has the unique mild solution between ṽ0 and w̃0, which can be obtained by a monotone iterative procedure starting from ṽ0 or w̃0.

By Corollaries 3.8, 3.9, and 3.10, we have the following results.

Corollary 3.11.

Let f:I×XX be continuous, and let A:D(A)XX be a linear closed densely defined operator. Assume that -A𝒞α(M,ω), Sα(t) is the positive solution operator generated by -A, the Cauchy problem (1.3) has a lower solution ṽ0C(I,X) and an upper solution w̃0C(I,X) with ṽ0w̃0, (H̃1) and (H̃3) hold, and one of the following conditions is satisfied:

X is an ordered Banach space, whose positive coneP is regular;

X is an ordered and weakly sequentially complete Banach space, whose positive coneP is normal with normal constant N.

Then the Cauchy problem (1.3) has the unique mild solution between ṽ0 and w̃0, which can be obtained by a monotone iterative procedure starting from ṽ0 or w̃0.

4. Examples

Example 4.1.

In order to illustrate our main results, we consider the Cauchy problem in X=n (n-dimensional Euclidean space and y=(i=1nyi2)1/2): CD0+αu(t)+Au(t)=f(t,u(t),Gu(t)),tI=[0,T],u(0)=xX,u=θ, where CD0+α is the Caputo fractional derivative, 1<α<2, A=(aij)n×n(aij0) is a real matrix, f:I×X×XX is continuous, θ=(0,0,,0) is the zero element of X, and Gu(t)=0tK(t,s)u(s)ds is a Volterra integral operator with integral kernel KC(Δ,+), Δ={(t,s)0stT}.

For y=(y1,y2,,yn), and z=(z1,z2,,zn), we define the partial order yzyizi(i=1,2,,n). Set P={yXyθ}; then P is a normal conein X and normal constant N=1. It is easy to verify that -A generates a uniformly continuous positive cosine operator family 𝒮2(t): S2(t)=n=0t2n(-A)n2n!,t0. By [9, Theorem  3.1], there exist M1 and ω0 such that -ACα(M,ω2/α), and the corresponding solution operator is Sα(t)=0φt,α/2(s)S2(s)ds,t>0, where φt,α/2(s)=t-α/2Φα/2(st-α/2), Φα/2(τ) is a probability density function, Φα/2(τ)0, τ>0, and 0Φα/2(τ)dτ=1. Thus, 𝒮α(t) is the positive solution operator generated by -A. In order to solve the problem (4.1), we give the following assumptions.

xθ, f(t,θ,θ)θ for tI.

There exist xx¯X such that Ax¯f(t,x¯,Gx¯) for tI.

The partial derivative fu(t,u,v) is continuous on any bounded domain and fv(t,u,v) has upper bound.

Theorem 4.2.

If (O1), (O2), and (O3) are satisfied, then the problem (4.1) has the unique mild solution u(t), and θux¯.

Proof.

From (O1) and (O2), we obtain that θ is a lower solution of (4.1), and x¯ is an upper solution of (4.1). Form (O3), it is easy to verify that (H1) and (H3) are satisfied. Therefore, by Theorem 3.5, the problem (4.1) has the unique solution u(t), and θux¯.

Acknowledgments

The author is grateful to the anonymous referee for his/her valuable suggestions and comments. This research supported by the Project of NWNU-KJCXGC-3-47.

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