# Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces

Authors: He Yang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach spaceE:u′(t)+Au(t)=f(t,u(t),Gu(t)), t∈[0,a], t≠tk, Δu|t=tk=Ik(u(tk)), 0<t1<t2<⋯<tm<a, u(0)=u0, where A:D(A)⊂E→E is a closed linear operator, and f:[0,a]×E×E→E is a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearity f, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.


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## Body

## 1. Introduction and Main Results

Differential equations involving impulse effects occur in many applications: physics, population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, and so forth. Therefore, it has been an object of intensive investigation in recent years; see, for instance, the monographs [1–5]. Correspondingly, the existence of mild solutions of impulsive evolution differential equations has also been studied by several authors; see [6–8]. However, the theory still remains to be developed.In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions for the initial value problem (IVP) of first-order nonlinear impulsive evolution equations(1.1)u′(t)+Au(t)=f(t,u(t),Gu(t)),t∈J,t≠tk,Δu∣t=tk=Ik(u(tk)),k=1,2,…,m,u(0)=u0
in an ordered Banach space E, where A:D(A)⊂E→E is a closed linear operator, -A generates a C0-semigroup T(t)(t≥0) in E, f:J×E×E→E is a nonlinear mapping, J=[0,a],a>0 is a constant, 0=t0<t1<t2<⋯<tm<tm+1=a, Ik:E→E is an impulsive function, k=1,2,…,m, u0∈E, and (1.2)Gu(t)=∫0tK(t,s)u(s)ds
is a Volterra integral operator with integral kernel K∈C(Δ,ℝ+),Δ={(t,s)∣0≤s≤t≤a}, Δu∣t=tk denotes the jump of u(t) at t=tk, that is, Δu∣t=tk=u(tk+)-u(tk-), where u(tk+) and u(tk-) represent the right and left limits of u(t) at t=tk, respectively. Let PC(J,E)={u:J→E∣u(t) is continuous at t≠tk, and left continuous at t=tk, and u(tk+) exists, k=1,2,…,m}. Evidently, PC(J,E) is a Banach space with norm ∥u∥PC=supt∈J∥u(t)∥. Let J'=J∖{t1,t2,…,tm}. An abstract function u∈PC(J,E)∩C1(J′,E)∩C(J',E1) (E1 is a Banach space with norm ∥x∥1=∥x∥+∥Ax∥) is called a solution of IVP(1.1), if u(t) satisfies all the equalities of (1.1).LetE be an ordered Banach space with norm ∥·∥ and partial order ≤, whose positive cone P={x∈E∣x≥0} is normal with a normal constant N. If an abstract function v∈PC(J,E)∩C1(J',E)∩C(J',E1) satisfies (1.3)v'(t)+Av(t)≤f(t,v(t),Gv(t)),t∈J,t≠tk,Δv∣t=tk≤Ik(v(tk)),k=1,2,…,m,v(0)≤u0,
we call it a lower solution of IVP(1.1). If all the inequalities of (1.3) are inverse, we call it an upper solution of IVP(1.1).In 1999, Liu [6], by means of the semigroup theory, has proved the existence and uniqueness of mild solutions for IVP(1.1) when f=f(t,u). He demands that the nonlinear term f and the impulsive function Ik satisfy the following conditions: (1.4)∥f(t,u)-f(t,v)∥≤C∥u-v∥,t∈J,u,v∈E,(1.5)∥Ik(u)-Ik(v)∥≤hk∥u-v∥,u,v∈E,
where C and hk(k=1,2,…,m) are positive constants and satisfy (1.6)M*(Ca+∑k=1mhk)<1,
where M*=maxt∈J∥T(t)∥. Inequality (1.6) is a strongly restricted condition, and it is difficult to satisfy in applications.Recently, Anguraj and Arjunan [7], under similar assumptions of [6], have obtained a unique mild solution for IVP(1.1) when f=f(t,u,Gu,Su). Cardinali and Rubbioni [8] have discussed the existence of mild solutions for the impulsive evolution differential inclusions under the measure of noncompactness conditions on every bounded set D⊂E. However the assumptions in these papers are also difficult to satisfy in applications.The purpose of this paper is to improve and extend the above mentioned results. We will delete the Lipschitz condition (1.5) for impulsive function Ik and the restriction condition (1.6) and improve condition (1.4) for nonlinear term f. Our main results are as follows.Theorem 1.1.
LetE be an ordered Banach space, whose positive cone  P is normal,  A:D(A)⊂E→E be a closed linear operator,  -A generate a positive  C0-semigroup  T(t)(t≥0),  f∈C(J×E×E,E), and  Ik∈C(E,E),  k=1,2,…,m. If IVP(1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0 and the following conditions are satisfied: (H1)
there exists a positive constantC such that
(1.7)f(t,x2,y2)-f(t,x1,y1)≥-C(x2-x1),
for any t∈J, v0(t)≤x1≤x2≤w0(t), and Gv0(t)≤y1≤y2≤Gw0(t),(H2)
for anyx1,x2∈E with v0(tk)≤x1≤x2≤w0(tk), k=1,2,…,m, one has
(1.8)Ik(x1)≤Ik(x2),(H3)
there exists a positive constantL such that
(1.9)α({f(t,xn,yn)})≤L(α({xn})+α({yn})),
for any t∈J, and increasing or decreasing monotonic sequences {xn}⊂[v0(t),w0(t)] and {yn}∈[Gv0(t),Gw0(t)].Then IVP(1.1) has minimal and maximal mild solutions between v0 and w0, which can be obtained by a monotone iterative procedure starting from v0 and w0, respectively.Clearly, condition(H3) greatly improves the measure of noncompactness condition in [8]. Therefore, Theorem 1.1 greatly improves the main results in [6–8]. In Theorem 1.1, if Banach space E is weakly sequentially complete, condition (H3) holds automatically; see [9, Theorem 2.2]. Hence, from Theorem 1.1, we have the following.Corollary 1.2.
LetE be an ordered and weakly sequentially complete Banach space, whose positive cone  P is normal,  A:D(A)⊂E→E be a closed linear operator,  -A generate a positive  C0-semigroup  T(t)(t≥0),  f∈C(J×E×E,E), and  Ik∈C(E,E),  k=1,2,…,m. If IVP(1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0, and the conditions  (H1) and  (H2) are satisfied, then IVP(1.1) has minimal and maximal mild solutions between  v0 and  w0, which can be obtained by a monotone iterative procedure starting from  v0 and  w0, respectively.The proof of Theorem1.1 will be shown in the next section. In Section 2, we also discuss the uniqueness of mild solutions for IVP(1.1) between the lower solution and upper solution (see Theorem 2.4).


## 2. Proof of the Main Results

LetC(J,E) denote the Banach space of all continuous E-value functions on interval J with norm ∥u∥C=maxt∈J∥u(t)∥ and let C1(J,E) denote the Banach space of all continuously differentiable E-value functions on interval J with norm ∥u∥C1=max{∥u∥C,∥u′∥C}. Consider the initial value problem (IVP) of linear evolution equation without impulse
(2.1)u'(t)+Au(t)=h(t),t∈J,u(0)=u0.
It is well-known [10, chapter 4, Theorem 2.9], when u0∈D(A) and h∈C1(J,E), IVP(2.1) has a classical solution u∈C1(J,E)∩C(J,E1) expressed by (2.2)u(t)=T(t)u0+∫0tT(t-s)h(s)ds,t∈J.
Generally, when u0∈E and h∈C(J,E), the function u given by (2.2) belongs to C(J,E) and it is called a mild solution of IVP(2.1).Let us start by defining what we mean by a mild solution of problem(2.3)u'(t)+Au(t)=h(t),t∈J,t≠tk,Δu∣t=tk=Ik(u(tk)),k=1,2,…,m,u(0)=u0.Definition 2.1.
A functionu∈PC(J,E)is called a mild solution of IVP(2.3), ifuis a solution of integral equation(2.4)u(t)=T(t)u0+∫0tT(t-s)h(s)ds+∑0<tk<tT(t-tk)Ik(u(tk)),t∈J.To prove Theorem1.1, for any h∈PC(J,E), we consider the linear initial value problem (LIVP) of impulsive evolution equation (2.5)u'(t)+Au(t)+Cu(t)=h(t),t∈J,t≠tk,Δu∣t=tk=yk,k=1,2,…,m,u(0)=x,
where C≥0, x∈E, and yk∈E, k=1,2,…,m.Lemma 2.2.
For anyh∈PC(J,E),  x∈E, and  yk∈E,k=1,2,…,m, LIVP(2.5) has a unique mild solution  u∈PC(J,E) given by
(2.6)u(t)=S(t)x+∫0tS(t-s)h(s)ds+∑0<tk<tS(t-tk)yk,t∈J,
where S(t)=e-CtT(t)(t≥0) is a C0-semigroup generated by -(A+CI).Proof.
Lety0=0. If u∈PC(J,E) is a mild solution of LIVP(2.5), then the restriction of u on (tk-1,tk] satisfies the initial value problem of linear evolution equation without impulse
(2.7)u'(t)+Au(t)+Cu(t)=h(t),tk-1<t≤tk,u(tk-1+)=u(tk-1)+yk-1.
Hence, on (tk-1,tk], u(t) can be expressed by
(2.8)u(t)=S(t-tk-1)u(tk-1)+S(t-tk-1)yk-1+∫tk-1tS(t-s)h(s)ds.
Iterating successively in the above equality with u(tj) for j=k-1,k-2,…,1,0, we see that u satisfies (2.6).
Inversely, we can verify directly that the functionu∈PC(J,E) defined by (2.6) satisfies all the equalities of LIVP(2.5).Letα(·) denote the Kuratowskii measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [11]. For any B⊂C(J,E) and t∈J, set B(t)={u(t)∣u∈B}⊂E. If B is bounded in C(J,E), then B(t) is bounded in E, and α(B(t))≤α(B). In the proof of Theorem 1.1 we need the following lemma.Lemma 2.3.
LetB={un}⊂PC(J,E) be a bounded and countable set. Then α(B(t)) is the Lebesgue integrable on J, and
(2.9)α({∫Jun(t)dt})≤2∫Jα(B(t))dt.This lemma can be found in [12].Evidently,PC(J,E) is also an ordered Banach space with the partial order “≤” reduced by the positive function cone KPC={u∈PC(J,E)∣u(t)≥0,t∈J}. KPC is also normal with the same normal constant N. For v,w∈PC(J,E) with v≤w, we use [v,w] to denote the ordered interval {u∈PC(J,E)∣v≤u≤w} in PC(J,E), and use [v(t),w(t)] to denote the ordered interval {x∈E∣v(t)≤x≤w(t)} in E.Proof of Theorem1.1.
LetD=[v0,w0]. We define a mapping Q:D→PC(J,E) by
(2.10)(Qu)(t)=S(t)u0+∫0tS(t-s)(f(s,u(s),Gu(s))+Cu(s))ds+∑0<tk<tS(t-tk)Ik(u(tk)),t∈J.
Clearly, Q:D→PC(J,E) is continuous. By Lemma 2.2, the mild solution of IVP(1.1) is equivalent to the fixed point of operator Q. By assumptions (H1) and (H2), Q is increasing in D and maps any bounded set in D into a bounded set.
We show thatv0≤Qv0,Qw0≤w0. Let h(t)≜v0'(t)+Av0(t)+Cv0(t). By the definition of the lower solution, we easily see that h∈PC(J,E) and h(t)≤f(t,v0(t),Gv0(t))+Cv0(t) for t∈J'. Because v0(t) is a solution of LIVP(2.5) for x=v0(0) and yk=Δv0∣t=tk,k=1,2,…,m, by Lemma 2.2 and the positivity of operator S(t)(t≥0), we have(2.11)v0(t)=S(t)v0(0)+∫0tS(t-s)h(s)ds+∑0<tk<tS(t-tk)Δv0∣t=tk≤S(t)u0+∫0tS(t-s)h(s)ds+∑0<tk<tS(t-tk)Ik(v0(tk))=S(t)u0+∑j=1k∫tj-1tjS(t-s)h(s)ds+∫tktS(t-s)h(s)ds+∑0<tk<tS(t-tk)Ik(v0(tk))≤S(t)u0+∑j=1k∫tj-1tjS(t-s)(f(s,v0(s),Gv0(s))+Cv0(s))ds+∫tktS(t-s)(f(s,v0(s),Gv0(s))+Cv0(s))ds+∑0<tk<tS(t-tk)Ik(v0(tk))=S(t)u0+∫0tS(t-s)(f(s,v0(s),Gv0(s))+Cv0(s))ds+∑0<tk<tS(t-tk)Ik(v0(tk))=(Qv0)(t),t∈J,
namely, v0≤Qv0. Similarly, it can be shown that Qw0≤w0. Combining these facts with the increasing property of Q in D, we see that Q maps D into itself, and Q:D→D is a continuously increasing operator.
Now, we define two sequences{vn} and {wn} in D by the iterative scheme(2.12)vn=Qvn-1,wn=Qwn-1,n=1,2,….
Then from the monotonicity of Q, it follows that
(2.13)v0≤v1≤⋯≤vn≤⋯≤wn≤⋯≤w1≤w0.
Next, we will show that {vn} and {wn} are uniformly convergent on J.
For convenience, we denoteM=maxt∈J∥S(t)∥,K0=max(t,s)∈ΔK(t,s). Let J1=[0,t1], Jk=(tk-1,tk], k=2,3,…,m+1 and let B={vn∣n∈ℕ} and B0={vn-1∣n∈ℕ}. From B0=B∪{v0}, it follows that α(B0(t))=α(B(t)) for t∈J. Let(2.14)φ(t):=α(B(t))=α(B0(t)),t∈J.
By Lemma 2.3, φ(t) is Lebesgue integrable on J. Going from J1 to Jm+1 interval by interval we show that φ(t)≡0 on J.
Fort∈J, there exists a Jk such that t∈Jk. By (1.2) and Lemma 2.3, we have that(2.15)α(G(B0)(t))=α({∫0tK(t,s)vn-1(s)ds∣n∈ℕ})≤∑j=1k-1α({∫tj-1tjK(t,s)vn-1(s)ds∣n∈ℕ})+α({∫tk-1tK(t,s)vn-1(s)ds∣n∈ℕ})≤2K0(∑j=1k-1∫tj-1tjα(B0(s))ds+∫tk-1tα(B0(s))ds)=2K0∫0tφ(s)ds,
and therefore,
(2.16)∫0tα(G(B0)(s))ds≤2aK0∫0tφ(s)ds.
Fort∈J1, from (2.10), (2.16), Lemma 2.3 and assumption (H3), we have(2.17)φ(t)=α(B(t))=α(Q(B0(t)))=α({∫0tS(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))ds∣n∈ℕ})≤2∫0tα({S(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))∣n∈ℕ})ds≤2M∫0t(L(α(B0(s))+α(G(B0)(s)))+Cα(B0(s)))ds=2M(L+C+2aK0L)∫0tφ(s)ds.
By this and the Gronwall-Bellman inequality, we obtain that φ(t)≡0 on J1. In particular, α(B0(t1))=φ(t1)=0, this means that B0(t1) is precompact in E. Combining this with the continuity of I1, it follows that I1(B0(t1)) is precompact in E, and α(I1(B0(t1)))=0.
Now, fort∈J2, by (2.10) and the above argument for t∈J1, we have(2.18)φ(t)=α(B(t))=α(Q(B0)(t))≤α({∫0tS(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))ds∣n∈ℕ})+α({S(t-t1)I1(vn-1(t1))∣n∈ℕ})≤2M(L+C+2aK0L)∫0tφ(s)ds+Mα(I1(B0(t1)))=2M(L+C+2aK0L)∫t1tφ(s)ds.
Again by the Gronwall-Bellman inequality, we obtain that φ(t)≡0 on J2, from which we obtain that α(B0(t2))=0 and α(I2(B0(t2)))=0.
Continuing such a process interval by interval up toJm+1, we can prove that φ(t)≡0 on every Jk, k=1,2,…,m+1. This means that {vn(t)} is precompact in E for every t∈Jk. Hence {vn(t)} has a convergent subsequence in E. Combining this fact with the monotonicity (2.13), we easily prove that {vn(t)} itself is convergent in E, that is, there exists u̲(t)∈E such that vn(t)→u̲(t) as n→∞ for every t∈Jk. On the other hand, for any t∈Jk, we have(2.19)vn(t)=S(t)u0+∫0tS(t-s)(f(s,vn-1(s),Gvn-1(s))+Cvn-1(s))ds+∑0<ti<tS(t-ti)Ii(vn-1(ti)).
Let n→∞, then by the Lebesgue-dominated convergence theorem, for t∈Jk, we have
(2.20)u̲(t)=S(t)u0+∫0tS(t-s)(f(s,u̲(s),Gu̲(s))+Cu̲(s))ds+∑0<ti<tS(t-ti)Ii(u̲(ti)),
and u̲∈C(Jk,E). Therefore, for any t∈J, we have
(2.21)u̲(t)=S(t)u0+∫0tS(t-s)(f(s,u̲(s),Gu̲(s))+Cu̲(s))ds+∑0<ti<tS(t-ti)Ii(u̲(ti)).
Namely, u̲∈PC(J,E), and u̲=Qu̲. Similarly, we can prove that there exists u¯∈PC(J,E) such that u¯=Qu¯. By the monotonicity of operator Q, it is easy to prove that u̲ and u¯ are the minimal and maximal fixed points of Q in [v0,w0], and they are the minimal and maximal mild solutions of IVP(1.1) in [v0,w0], respectively.Now we discuss the uniqueness of mild solutions for IVP(1.1) in [v0,w0]. If we replace assumption (H3) by the following assumption:(H4)
there exist positive constantsC1 and C2 such that(2.22)f(t,x2,y2)-f(t,x1,y1)≤C1(x2-x1)+C2(y2-y1),
for any t∈J, v0(t)≤x1≤x2≤w0(t),Gv0(t)≤y1≤y2≤Gw0(t), and we have the following existence result.Theorem 2.4.
LetE be an ordered Banach space, whose positive cone  P is normal,  A be a closed linear operator in  E,  -A generate a positive  C0-semigroup  T(t)(t≥0),  f∈C(J×E×E,E) and  Ik∈C(E,E),  k=1,2,…,m. If IVP(1.1) has a lower solution  v0 and an upper solution  w0 with  v0≤w0, such that the assumptions  (H1),(H2), and  (H4) are satisfied, then IVP(1.1) has a unique mild solution between  v0 and  w0, which can be obtained by a monotone iterative procedure starting from  v0 or  w0.Proof of Theorem2.4.
We firstly prove that(H1) and (H4) imply (H3). For t∈J, let {xn}⊂[v0(t),w0(t)] and {yn}⊂[Gv0(t),Gw0(t)] be two increasing sequences. For m,n∈ℕ with m>n, by (H1) and (H4), we have
(2.23)θ≤(f(t,xm,ym)-f(t,xn,yn))+C(xm-xn)≤(C1+C)(xm-xn)+C2(ym-yn).
By the normality of cone P, we have
(2.24)∥f(t,xm,ym)-f(t,xn,yn)∥≤(C+NC+NC1)∥xm-xn∥+NC2∥ym-yn∥.
From this inequality and the definition of the measure of noncompactness, it follows that
(2.25)α({f(t,xn,yn)})≤(C+NC+NC1)α(xn)+NC2α(yn)≤L(α({xn})+α({yn})),
where L=C+NC+NC1+NC2. If {xn} and {yn} are two decreasing sequences, the above inequality is also valid. Hence (H3) holds.
Therefore, by Theorem1.1, IVP(1.1) has minimal solution u̲ and maximal solution u¯ in [v0,w0]. By the proof of Theorem 1.1, (2.10)–(2.13) are valid. Going from J1 to Jm+1 interval by interval we show that u̲≡u¯ on every Jk.
Fort∈J1, by (2.10) and assumption (H4), we have(2.26)θ≤u¯(t)-u̲(t)=Qu¯(t)-Qu̲(t)=∫0tS(t-s)(f(s,u¯(s),Gu¯(s))-f(s,u̲(s),Gu̲(s))+C(u¯(s)-u̲(s)))ds≤∫0tS(t-s)((C+C1)(u¯(s)-u̲(s))+C2(Gu¯(s)-Gu̲(s)))ds≤M(C+C1+aC2K0)∫0t(u¯(s)-u̲(s))ds.
From this inequality and the normality of cone P, it follows that
(2.27)∥u¯(t)-u̲(t)∥≤NM(C+C1+aC2K0)∫0t∥u¯(s)-u̲(s)∥ds.
By the Gronwall-Bellman inequality, we obtain that u¯(t)≡u̲(t) on J1.
Fort∈J2, since I1(u¯(t1))=I1(u̲(t1)), using (2.10) and the same argument as above for t∈J1, we can prove that(2.28)∥u¯(t)-u̲(t)∥≤NM(C+C1+aC2K0)∫0t∥u¯(s)-u̲(s)∥ds=NM(C+C1+aC2K0)∫t1t∥u¯(s)-u̲(s)∥ds.
Again, by the Gronwall-Bellman inequality, we obtain that u¯(t)≡u̲(t) on J2.
Continuing such a process interval by interval up toJm+1, we see that u¯(t)≡u̲(t) over the whole J. Hence, u*:=u¯=u̲ is the unique mild solution of IVP(1.1) in [v0,w0], which can be obtained by a monotone iterative procedure starting from v0 or w0.



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