# Existence of the Mild Solution for Impulsive Semilinear Differential Equation

Authors: Alka Chadha; Dwijendra N. Pandey
Journal: International Journal of Partial Differential Equations
 (2014)

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## Abstract

We study the existence of solutions of impulsive semilinear differential equation in a Banach spaceX in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator.


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## Body

## 1. Introduction

In a few decades, impulsive differential equations have received much attention of researchers mainly due to its demonstrated applications in widespread fields of science and engineering such as biology, physics, control theory, population dynamics, medicine and so on. The real world processes and phenomena which are subjected during their development to short-term external inuences can be modeled as impulsive differential equation. Their duration is negligible compared to the total duration of the entire process or phenomena. Impulsive differential equations are an appropriate model to hereditary phenomena for which a delay argument arises in the modelling equations. To further study on impulsive differential equations, we refer to books [1, 2] and papers [3–11].In this paper, our purpose is to establish the existence of a solution to the following differential equations with non instantaneous impulses(1)u′(t)=-Au(t)+F(t,u(t)),t∈(si,ti+1],i=0,1,…,δ,(2)u(t)=Gi(t,u(t)),t∈(ti,si],i=1,…,δ,δ∈N,(3)u(0)=x0,
where -A:D(A)⊂X→X is a closed and bounded linear operator with dense domain D(A)⊂X. We assume that -A is the infinitesimal generator of a strongly continuous semigroup {T(t),t≥0} in a Banach space (X,∥·∥). Here, x0∈X, 0=t0=s0<t1≤s1≤t2<⋯<tδ≤sδ<tδ+1=T0 and F:[0,T0]×X→X, Gi:(ti,si]×X→X for all i=1,…,δ are suitable functions to be specified later.In [4], authors have introduced a new class of abstract impulsive differential equations in which impulses are not instantaneous and established the existence of solutions to the problem (1)–(3) with the assumption that operator A generates a C0-semigroup of bounded linear operators. In this system of (1)–(3), the impulses begin all of a sudden at the points ti and their proceeding continues on a finite time interval [ti,si] [5]. To concern the hemodynamical harmony of an individual we think about the following simplified situation. One can recommend a few intravenous sedates (insulin) on account of a decompensation (e.g., high or low level of glucose). Since the presentation of the medications in the bloodstream and the ensuing retention for the form are progressive and continuous processes, we can depict this circumstance as an impulsive activity which begins abruptly and stays animated on a finite time interval.In [12], the generalization of the condensing operator as convex-power condensing operator has been introduced by Sun and Zhang and a new fixed point theorem for convex-power condensing operator has been established. The new fixed point theorem for convex-power condensing operator, defined by Sun and Zhang, is the generalization of the famous Schauder’s fixed point theorem and Sadovskii’s fixed point theorem. Sun and Zhang [12] have considered the following problem:
(4)u′(t)+Au(t)=f(t,u(t)),0≤t≤T0,u(0)=x0
in a Banach space and established existence theorems for positive mild solutions and global mild solutions to the problem (4) with noncompact semigroup. The nonlinear function f fulfills the suitable conditions on the measure of noncompactness as α(f(t,B(t)))≤Lα(B(t)) for any bounded set B⊂X, where α is the Kuratowski measure of noncompactness and L is a positive constant. For more details about measure of noncompactness, we refer to [12–20].In the present work, our aim is to obtain results concerning the existence of mild solutions to problem (1)–(3) by using convex-power condensing operator and fixed point theorem for convex-power condensing operator Sun and Zhang [12].The organization of the paper is as follows. We provide some basic definitions, Lemmas and theorems in Section2 as “preliminaries.” We prove the existence of a mild solution for system (1) in Section 3 as “existence of mild solution.” In the last section, we present an example to illustrate the application of the abstract results.


## 2. Preliminaries

In this section, we give some definitions, notations, theorems, and lemmas which will be used in later sections.Let(X,∥·∥) be a real Banach space. The symbol C([0,T0];X) stands for the Banach space of all the continuous functions from [0,T0] into X equipped with the norm ∥z∥=sup⁡{∥z(s)∥,s∈[0,T0]} and L1([0,T0],X) stands for the space of X-valued Bochner integrable functions on [0,T0] endowed with the norm ∥f∥L1=∫0T0∥f(t)∥‍dt, f∈L1([0,T0],X).The operator-A:D(A)⊂X→X is the infinitesimal generator of a uniformly continuous semigroup {T(t),t≥0} and D(A) denotes the domain of A, which is densely defined, endowed with the graph norm. A semigroup T(t) is said to be equicontinuous if {T(t)y:y∈K} is equicontinuous at arbitrary t, 0<t<∞, for any bounded subset K⊂X. Throughout the paper we assume that(H1)
the operator-A generates the equicontinuous semigroup {T(t);t≥0} and there exists a positive number M such that ∥T(t)∥≤M.For the study of impulsive differential equation, we define the spacePC([0,T0];X) which contains all the functions y:[0,T0]→X such that y(·) is continuous at t≠tj, y(tj-)=y(tj) and y(tj+) exists for all j=1,2,…,δ. Clearly, PC([0,T0];X) is a Banach space endowed with norm ∥y∥PC=sup⁡{y(t):t∈[0,T0]}. For a function y∈PC([0,T0];X) and i∈{0,1,…,δ}, we define the function yi~∈C([ti,ti+1],X) such that
(5)yi~(t)={y(t),fort∈(ti,ti+1],y(ti+),fort=ti.
For B⊂PC([0,T0];X), we have Bi~={yi~:y∈B} and we have following Accoli-Arzelà type criteria.Lemma 1 (see [4]).
A setB⊂PC([0,T0];X) is relatively compact in PC([0,T0];X) if and only if each set Bj~  (j=0,1,…,δ) is relatively compact in C([tj,tj+1],X)  (j=0,1,…,δ).Now we present the following definition of mild solution.Definition 2.
A piecewise continuous functionu∈PC([0,T0];X) is said to be a mild solution of the (1)–(3) if u(0)=x0, u(t)=Gi(t,u(t)), for all t∈(ti,si], i=1,…,δ, and
(6)u(t)=T(t)x0+∫0t‍T(t-ζ)F(ζ,u(ζ))dζ,
for all t∈[0,t1], and
(7)u(t)=T(t-si)Gi(si,u(si))+∫sit‍T(t-ζ)F(ζ,u(ζ))dζ,
for all t∈[si,ti+1] and every i=1,…,δ.Next, we give the definition of the Hausdorff measure of noncompactness (MNC).Definition 3 (see [14]).
The Hausdorff measure of noncompactnessβ of the set E in Banach space X is the greatest lower bound of those ε>0 for which the set E has in the space X a finite ε-net; that is,
(8)β(E)=inf⁡{ε>0:Ehasafiniteε-netinX},
for every bounded subset E in a Banach space X.Definition 4 (see [14]).
The Kuratowski measure of noncompactnessα defined on each bounded subset E of X as
(9)α(E)=inf⁡{ϵ>0:Eiscoveredbyafiniteϵ>0:Eiscoveredbyafinitenumberofsetswithdiameter≤ϵ}.The relation between Kuratowski measure of noncompactnessα and the Hausdorff measure of noncompactness β is given by
(10)β(E)≤α(E)≤2β(E).
To set the structure for our primary existence results, we review some essential properties about the Kuratowski and Hausdorff measure of noncompactness.Lemma 5.
For any bounded setU,V⊂Y, where Y is a Banach space. Then, we have following results: (i)
β(U)=0 if and only if  U is precompact;(ii)
β(U)=β(convU)=β(U¯), where convU and U¯ denote the convex hull and closure of  U, respectively;(iii)
β(U)⊂β(V), when U⊂V;(iv)
β(U+V)≤β(U)+β(V), where U+V={u+v:u∈U,v∈V};(v)
β(U∪V)≤max⁡{β(U),β(V)};(vi)
β(λU)=λ·β(U), for any λ∈R;(vii)
If the mapQ:D(Q)⊂Y→Z is continuous and satisfies the Lipschitsz condition with constant κ, then, we have that χZ(QU)≤κβ(U) for any bounded subset U⊂D(Q), where Y and Z are Banach space.Definition 6.
A continuous and bounded mapQ:D⊆X→X is called β-condensing if, for any noncompact bounded subset E⊂D,
(11)β(QE)<β(E),
where X is a Banach space.To avoid confusion, we denote byβ(·) the Hausdorff measure of noncompactness on set X, C([0,T0];X) and PC([0,T0];X).Lemma 7 (see [14], Darbo-Sadovskii).
LetD⊂X be bounded, closed, and convex. If the continuous map Q:D→D is a β-contraction, then the map Q has a fixed point in D.In [12], authors have presented the generalization of the condensing operator and suggested a new fixed point theorem for such operators. Firstly, we present some recognition. Let D⊂X be a bounded, closed, and convex set and let Q be a continuous map from D into itself with u0∈D. For every E⊂D, we set
(12)Q(1,u0)(E)=Q(E),Q(n,u0)E=Q(conv¯{Qn-1,u0E,u0}),forn=2,3,….Definition 8 (see [12]).
LetD⊂X be bounded, closed, and convex. A bounded and continuous operator Q:D→D is called a convex-power condensing operator if for any bounded nonprecompact subset E⊂D, there exist x0∈D and an integer n0>0 such that
(13)β(Q(n0,u0)(E))<β(E).Aβ-convex-power condensing operator is β-condensing for n0=1. Therefore, it is clear that convex-power condensing operator is a generalization of the condensing operator.Lemma 9 (see [12]).
LetD⊂X be bounded, closed, and convex set. If the continuous map Q:D→D is β-convex-power condensing, then there exists a fixed point of map Q in D.Lemma 10 (see [11, 14]).
IfE⊆C([0,T0];X) is bounded, then β(E(t))≤β(E), ∀t∈[0,T0], where E(t)={u(t);u∈E}⊆X. In addition, if E is equicontinuous on [0,T0], then β(E(t)) is continuous on [0,T0] and
(14)β(E)=sup⁡t∈[0,T0]{β(E(t))}.Lemma 11 (see [11]).
IfE⊆PC([0,T0];X) is bounded, then β(E(t))≤β(E), ∀t∈[0,T0]. Besides, suppose the following conditions are satisfied: (1)
E is equicontinuous on [0,t1] and each (ti,si], [si,ti+1], i=1,…,δ,(2)
E is equicontinuous at t=ti+, i=1,…,δ.Then, we have β(E)=sup⁡t∈[0,T0]β(E(t)).Lemma 12 (see [14]).
IfE⊂C([0,T0];X) is bounded and equicontinuous, then β(E(t)) is continuous and
(15)β(∫0t‍E(τ)dτ)≤∫0t‍β(E(τ))dτ,∀t∈[0,T0],
where ∫0t‍E(τ)dτ={∫0t‍w(τ)dτ,w∈E}.Lemma 13.
Let{un}n=1∞ be a sequence of functions in L1([0,T0];R+). Suppose that there exist γ(t)∈L1([0,T0];R+) satisfying ∥un(t)∥≤γ(t) for almost all t∈[0,T0] and every n≥1. Then, we have
(16)β({∫0t‍un(τ)dτ:n≥1})≤2∫0t‍β({un(τ)}n=1∞)dτ,∀t∈[0,T0].Lemma 14.
We assume that (H1) holds. Then the set{∫0t‍T(t-ζ)u(ζ)dζ, ∥u(ζ)∥≤η(ζ) for a.e. ζ∈[0,T0]} is equicontinuous for all t∈[0,T0].Proof.
Leth be a positive constant such that 0≤t<t+h≤T0. For u∈PC, we have
(17)∥∫0t+h‍T(t+h-ζ)u(ζ)dζ-∫0t‍T(t-ζ)u(ζ)dζ∥≤∥∫0t‍[T(t+h-ζ)-T(t-ζ)]u(ζ)dζ∥+∫tt+h∥T(t+h-ζ)u(ζ)dζ∥‍.
It is obvious for t=0. Let ϵ>0 be arbitary integer with 0<ϵ<t. For t>0, we have
(18)∥∫0t‍T(t+h-ζ)u(ζ)dζ-∫0t‍T(t-ζ)u(ζ)dζ∥≤∥T(h+ϵ)∫0t-ϵ‍T(t-ϵ-ζ)u(ζ)dζ-T(ϵ)×∫0t-ϵT(t-ϵ-ζ)u(ζ)dζ)∥+∥∫t-ϵt‍T(t+h-ζ)u(ζ)dζ∥+∥∫t-ϵt‍T(t-ζ)u(ζ)dζ∥.
Since T(t), t>0 is equicontinuous; therefore,
(19)∥[T(h+ϵ)-T(ϵ)]∫0t-ϵ‍T(t-ϵ-ζ)u(ζ)dζ∥⟶0,
as h→0, uniformly for u. Second and third terms of (18) tend to zero when ϵ→0 since ϵ is arbitrary small.
Then from (17), (18), and (19) and the absolute continuity of integrals, we obtain that {∫0t‍T(t-ζ)u(ζ)dζ, ∥u(ζ)∥≤η(ζ) for a.e. ζ∈[0,T0]} is equicontinuous for all t∈[0,T0].


## 3. Main Results

In this section, the existence of the mild solution of (1)–(3) under some specified conditions on Gi is established by using the measure of noncompactness and fixed point theorem for convex-power condensing operator.Now, we made the following hypothesis which will be useful for proving our results.(HG)
The functionsGi  (i=1,…,δ) are compact and continuous.(HF)
F:[0,T]×X→X is a nonlinear function such that F satisfies the Carathèodary condition; that is, we have the following.
(i)
F(τ,·):X→X is continuous for a.e. τ∈[0,T0].(ii)
For eachx∈X, F(·,x):[0,T0]→X is strongly measurable. For any k>0, there exists a function mk∈L1([0,T0],R+) such that for u∈X(20)∥F(s,u)∥≤mk(s),fora.e.s∈[0,T0].(iii)
There exists a constantL>0 such that for any bounded set B⊂X,
(21)β(F(τ,B))≤Lβ(B),fora.e.τ∈[0,T0].(HG1)
The functionsGi are continuous and there exist positive constants LGi such that
(22)∥Gi(τ,x)-Gi(τ,y)∥≤LGi∥x-y∥,for allx,y∈X, τ∈(ti,si], (i=1,…,δ).(Hk)
(23)Mmax⁡{i=1,…,δ}[∥mk∥L1([0,T0])+sup⁡t∈[0,T0]∥Gi(t,u(t))∥sup⁡t∈[0,T0]+∥x0∥]≤k.(H)
max⁡{i=1,…,δ}⁡M(LGi+4LT0)<1.Theorem 15.
Suppose that (H1), (HG), (HF), and (Hk) are satisfied. Then, there exists at least one mild solution on[0,T0] for the problem (1)–(3).Proof.
We define the operatorQ:PC([0,T0];X)→PC([0,T0];X) as Qu(0)=x0, Qu(t)=Gi(t,u(t)), for all t∈(ti,si] and
(24)Qu(t)=T(t)x0+∫0t‍T(t-ζ)F(ζ,u(ζ))dζ,t∈[0,t1],
for i=0,1,…,δ and for t∈[si,ti+1], where i=1,…,δ,
(25)Qu(t)=T(t-si)Gi(si,u(si))+∫sit‍T(t-ζ)F(ζ,u(ζ))dζ.
To prove the result, we show that the operator Q has a fixed point. Firstly, we show that the Q is continuous on PC([0,T0];X). Let {un}n=1∞ be a sequence in PC([0,T0];X) such that lim⁡n→∞un=u in PC([0,T0];X). For t∈[si,ti+1], we have
(26)∥(Qun)(t)-(Qu)(t)∥≤M∥Gi(si,un(si))-Gi(si,u(si))∥+M∫sit∥F(ζ,un(ζ))-F(ζ,u(ζ))∥dζ.‍
By the continuity of F and Gi  (i=1…,δ), we have
(27)lim⁡n→∞F(t,un(t))=F(t,u(t)),t∈[0,T0],(28)lim⁡n→∞Gi(ti,un(ti))=Gi(ti,u(ti)),t∈[0,T0].
Therefore from (26), (27), (28), and Lebesgue dominated convergence theorem, we get
(29)∥Qun(t)-Qu(t)∥⟶0,asn⟶∞,
which implies that Q is continuous on [si,ti+1]. For t∈[0,t1], we get
(30)∥(Qun)(t)-(Qu)(t)∥≤M∫0t∥F(ζ,un(ζ))-F(ζ,u(ζ))∥dζ.‍
From the (27), we get
(31)∥Qun(t)-Qu(t)∥⟶0,asn⟶∞;
thus, Q is continuous on [0,t1]. Hence, we conclude that Q is continuous on [0,T0].
Secondly, we claim thatQ(Bk)⊆Bk, where Bk(PC)=Bk={u∈PC([0,T0];X):∥u∥≤k}. For each u∈Bk⊂PC([0,T0];X) and t∈[0,t1], we get
(32)∥Qu(t)∥≤∥T(t)x0∥+∫0t‍∥T(t-ζ)∥·∥F(ζ,u(ζ))∥dζ≤M∥x0∥+M∫0t‍mk(ζ)dζ≤M(∥x0∥+∥mk∥L1[0,T0]).
For t∈(si,ti+1], (i=1,…,δ), we have
(33)∥Qu(t)∥≤∥T(t-si)∥·∥Gi(si,u(si))∥+M∫sit‍∥F(ζ,u(ζ))∥dζ≤M(∥Gi(si,u(si))∥+∥mk∥L1[si,ti+1])≤M(∥Gi(si,u(si))∥+∥mk∥L1[0,T0]),
which implies that ∥Qu∥C([0,T0],X)≤M(∥Gi(si,u(si))∥+∥mk∥L1[0,T0]), for all i=1,…,δ. On the other hand, by the property of Gi(·), we get
(34)∥Qu(t)∥≤∥Gi(t,u(t))∥≤sup⁡t∈(ti,si]∥Gi(t,u(t))∥,
for t∈(ti,si].   By the assumption (Hk), we have Mmax⁡{i=1,…,δ}[∥mk∥L1([0,T0])+sup⁡t∈[0,T0]∥Gi(t,u(t))∥+∥x0∥]≤k.
Therefore, we conclude that∥Qu∥PC≤k; that is, Q has values in Bk.
Now we show the equicontinuity ofQ(Bk) on [0,T0]. Since Gi(·) is compact, therefore it is obvious that QBk is equicontinuous on (ti,si]. Assume t∈[0,t1]. Let h>0 be a constant such that 0<t<t+h≤t1<T0. For u∈Bk, we get
(35)∥Qu(t+h)-Qu(t)∥≤∥[T(t+h)-T(t)]x0∥+∥∫0t+h‍T(t+h-ζ)F(ζ,u(ζ))dζ-∫0t‍T(t-ζ)F(ζ,u(ζ))dζ∥.
Using the semigroup property, we have
(36)∥T(t+h)x0-T(t)x0∥=∥T(t)[T(t+h-t)-T(0)]x0∥≤M∥[T(h)-T(0)]x0∥.
By the strong continuity of T(·) and Lemma 14, we conclude that QBk is equicontinuous on [0,t1].
Fort∈[si,ti+1] we have
(37)∥Qu(t+h)-Qu(t)∥≤∥[T(t+h-si)-T(t-si)]Gi(si,u(si))∥+∥∫0t+h‍T(t+h-ζ)F(ζ,u(ζ))dζ-∫0t‍T(t-ζ)F(ζ,u(ζ))dζ∥.
Since we have
(38)∥[T(t+h-si)-T(t-si)]Gi(si,u(si))∥≤M∥[T(h)-T(0)]Gi(si,u(si))∥,Gi(·) is compact and T(·) is strongly continuous, which implies that QBk is equicontinuous on [si,ti+1]. Hence, QBk is equicontinuous on each [0,T0].
SetB=conv¯Q(Bk), where conv and conv¯ denote the convex hull and closure of the convex hull, respectively. It can be shown easily that Q maps B into itself and B is equicontinuous on each [0,t1], (ti,si], [si,ti+1], i=1,2,…,δ. Next we prove that Q:B→B is a convex-power condensing operator. We take u0∈B and show that there exists a positive integer n0 such that
(39)β(Q(n0,u0)(H))<β(H),
for every nonprecompact bounded subset H⊂B. From (6), (15), and compactness of Gi, for t∈[si,ti+1], where i=1,…,δ, we have
(40)β(Q(1,u0)H(t))=β((QH)(t))≤β(T(t-si)Gi(si,H(si)))+β(∫sit‍T(t-ζ)F(ζ,H(ζ))dζ)≤4ML∫sit‍β(H(ζ))dζ≤4ML(t-si)β(H),
and similarly for t∈[0,t1], we have
(41)β(Q(1,u0)H(t))≤4MLtβ(H).
For t∈[ti,si], we get β(Q(1,u0)H(t))=β((QH)(t))=β(Gi(t,H(t)))=0 by the fact that Gi(·) are compact.
Further fort∈[si,ti+1], we have
(42)β((Q(2,u0)H)(t))=β((Qconv¯{Q(1,u0)H,u0})(t))≤β(T(t-si)Gi(si,conv¯{Q(1,u0)H,u0})(si))+β(∫sit‍T(t-ζ)F(ζ,conv¯{Q(1,u0)H,u0})(ζ))dζ≤4M∫sit‍β(F(ζ,conv¯{Q(1,u0)H,u0}))dζ≤4ML∫sit‍β(Q(1,u0)(H))(ζ)dζ≤4ML∫sit‍4ML(ζ-si)β(H)dζ≤(4ML)2(t-si)22β(H).
Proceeding with this iterative method, we get
(43)β((Q(n,u0)H)(t))≤4nMnLn(t-si)nn!β(H),
for t∈[si,ti+1] and similarly for t∈[0,t1], we get β((Q(n,u0)H)(t))≤((4nMnLntn)/n!)β(H). Thus, we obtain
(44)β((Q(n,u0)H)(t))≤4nMnLnT0nn!β(H),
for all t∈[0,T0].
We have thatQ(n,u0)(H) is equicontinuous on [0,T0] by Lemma 14. Therefore, from Lemma 11, we get
(45)β((Q(n,u0)H))=sup⁡t∈[0,T0]β((Q(n,u0)H)(t))≤4nMnLnT0nn!β(H)
since we have that (4nMnLnT0n/n!)→0 as n→∞, which infers that there exists a substantial enough positive integer n0 such that
(46)4n0Mn0Ln0T0n0n0!<1,
which means that Q:B→B is a convex-power condensing operator. Therefore, from Lemma 10, we get that Q has at least one fixed point in B which is just a mild solution to the problem (1)–(3). This completes the proof of the theorem.In the next result, the existence of the solution for problem (1)–(3) under Lipschitz conditions of Gi is established by using Darbo-Sadovskii’s fixed point theorem.Theorem 16.
Suppose that assumptions (H1), (HF), (HG1), and (H) hold. Then, the impulsive problem (1)–(3) has at least one mild solution on [0,T0].Proof.
Firstly, we decompose the mapQ such that Q=∑i=1δ‍Qi1+∑i=1δ‍Qi2, where Qij:PC([0,T0];X)→PC([0,T0];X), i=0,1,2,…,δ, j=1,2, are defined as
(47)Qi1u(t)={Gi(t,u(t)),fort∈(ti,si],i≥1,T(t-si)Gi(si,u(si)),fort∈(si,ti+1],i≥1,0,fort∉(ti,ti+1],i≥0,T(t)x0,fort∈[0,t1],i=0,(48)Qi2u(t)={∫sit‍T(t-ζ)F(ζ,u(ζ))dζ,fort∈(si,ti+1],i≥0,0,fort∉(si,ti+1],i≥0.
For t∈(ti,si] and u,v∈Bk, we have
(49)∥Qi1u(t)-Qi1v(t)∥=sup⁡t∈[0,T0][∥Gi(t,u(t))-Gi(t,v(t))∥]≤LGi∥u-v∥PC,
and for t∈(si,ti+1](50)∥Qi1u(t)-Qi1v(t)∥≤MLGi∥u-v∥PC;
therefore, for all t∈[0,T0] we conclude that
(51)∥Qi1u(t)-Qi1v(t)∥≤MLGi∥u-v∥PC.
Since β(H)=sup⁡t∈[0,T0]{β(H(t))}. By Lemma 5 (vii), we have that for any bounded set H⊂PC(52)β(Qi1H)≤MLGiβ(H).
For operator Qi2, we have
(53)β(Qi2Bk)=sup⁡t∈[si,ti+1]β(Qi2Bk(t))≤2sup⁡t∈[si,ti+1]∫sit‍β(T(t-ζ)F(ζ,Bk(ζ)))dζ≤4Msup⁡t∈[si,ti+1]∫sit‍Lβ(Bk(ζ))dζ≤4ML(t-si)β(Bk)≤4MLT0β(Bk).
From (51)–(53), we have
(54)β(QBk)≤β(Qi1Bk)+β(Qi2Bk)≤max⁡{i=1,…,δ}⁡(MLGi+4MLT0)β(Bk).
From the assumption (H), we get that β(QBk)<β(Bk), which implies that map Q is β-condensing in Bk. Therefore, by Darbo-Sadovskii’s fixed point theorem, the solution map Q has a fixed point in Bk which is a mild solution of the nonlocal problem (1)–(3). This completes the proof of the theorem.


## 4. Example

Let us consider the following impulsive problem:(55)∂∂tw(t,x)=∂2∂x2w(t,x)+f(t,w(t,x)),(t,x)∈∪i=1N[si,ti+1]×[0,π],w(t,0)=w(t,π)=0,fort∈[0,T0],w(0,x)=z(x),x∈[0,π],w(t,x)=gi(t,w(t,x)),t∈(ti,si],x∈[0,π],
where 0=t0=s0<t1≤s1<⋯<tN≤sN<tN+1=T0 are fixed numbers, z∈X, f∈C([0,T0]×R,R), and gi∈C((ti,si]×R,R), for all i=1,…,N.To convert problem (55) into the abstract form (1)–(3), where u(t)=w(t,·), that is, u(t)x=w(t,x), we introduced the functions F:[0,T0]×X→X and Gi:(ti,si]×X→X such that F(t,u(t))(x)=f(t,w(t,x)) and Gi(t,u(t))(x)=gi(t,w(t,x)). We consider X=L2[0,π] and the operator A defined as -Au=u′′ with the the domain
(56)D(A)={u∈X:u,u′areabsolutelycontinuousandu′′∈X,u(0)=u(π)=0u′′}.
It is well known (Pazy [21]) that -A is the infinitesimal generator of an analytic semigroup T(t), t≥0 and every analytic semigroup is equicontinuous. This means that A satisfies the assumption (H1).Next we have thatu∈PC([0,T0];X) is a mild solution of the problem (55).Case 1.
We take(57)f(t,w(t,x))=c0sin|w(t,x)|,t∈[0,T0],w∈[0,T0]×X,gi(t,u(t))(x)=∫tisi‍hi(τ)log⁡(1+|w(τ,x)|)dτ,w∈PC([0,T0]×X;X),t∈[ti,si],
and here hi:[si,ti]→R are continuous functions and ∫tisi‍|hi(s)|ds<∞. Now, we show that F satisfies the assumption (HF). Now, we have that for u∈X(58)∥F(t,u(t))∥=∥c0sin|(u(t))|∥≤c0∥u(t)∥,
and for u1,u2∈X(59)∥F(t,u1(t))-F(t,u2(t))∥=∥c0sin|u1(t)|-c0sin|u2(t)|∥≤c0[∥u1(t)-u2(t)∥].
Therefore, for any bounded sets D1⊂X, we obtain
(60)β(F(t,D1))≤Lβ(D1),a.e.t∈[0,T0],
where L=c0. For u∈X, we have
(61)∥Gi(t,u(t))∥≤∫siti‍|hi(τ)|∥log⁡(1+|u(τ)|)∥dτ≤∫siti‍|hi(τ)|dτ×∥u(t)∥.
Clearly, Gi, i=1,…,δ are compact and satisfy the assumption (HG). Then by Theorem 15, problem (55) has at least a mild solution.Case 2.
Takef(t,w(t,x))=f1(t)w(t,x) and gi(t,w(t,x))=(1/(5+αit3))[|w(t,x)|/(1+|w(t,x)|)], w∈PC([ti,si]×X;X); here, f1:[0,T0]→R is a bounded and continuous function and αi>0.
Fort∈(si,ti+1], i=1,…,δ and u,u1,u2∈X=R, we have
(62)∥F(t,u(t))x∥≤mk(t)∥u(t)∥,
where mk(t)=|f1(t)|. For any bounded set D1⊂X, we have
(63)β(F(t,D1))≤Lβ(D1),
where L=|f(t)|. Thus, (HF) holds. For u1,u2∈PC([ti,si];X), we have
(64)∥Gi(t,u1(t))(x)-Gi(t,u2(t))(x)∥≤15∥u1-u2∥.
Thus, (HG1) holds. Then by Theorem16, problem (55) has at least a mild solution.



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*Source: 640931-2014-05-18.xml*