We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach space X. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.
1. Introduction
In recent years, fractional calculus has becomes an active area of research due to its demonstrated applications in widespread fields of science and engineering such as mechanics, electrical engineering, medicine, biology, ecology, and many others. The memory and hereditary properties of various materials and processes can be described by a differential equation with fractional order. The fractional differential equation also describes the efficiency of nonlinear oscillations of the earthquake. The details on the theory and its applications can be found in [1–4] and references given therein.
On the other hand, many real world processes and phenomena which are subjected during their development to short-term external influences can be modeled as impulsive differential equation with fractional order which have been used efficiently in modelling many practical problems. Their duration is negligible compared with the total duration of the entire process and phenomena. Such process is investigated in various fields such as biology, physics, control theory, population dynamics, economics, chemical technology, and medicine. In addition, the improvement of the hypothesis of the functional differential equation with infinite delay relies on a choice of phase space. There are various phase spaces which have been studied. Hale and Kato in [5] introduced a common phase space 𝒫. For more details on phase space, we refer to books by Hale and Kato [5], Hino et al. [6] and papers [7–10]. For the study of impulsive differential equation, we refer to papers [7, 8, 11–18] and references given therein.
The purpose of this work is to establish the existence of mild solution for impulsive fractional differential equation with infinite delay:
(1)Dtq[u(t)+g(t,ut)]=A[u(t)+g(t,ut)]+ℐt1-qf(t,ut),t∈J=[0,T],t≠tiu0=φ∈𝒫,Δu(ti)=Ii(uti),i=1,2,…,n,n∈ℕ,
where 0<T<∞, 0<q<1, A:D(A)⊂X→X is a closed and densely defined linear operator and infinitesimal generator of a solution (resolvent) operator {Sq(t)}t≥0 on Banach space X, and Dtq denotes the fractional derivative in Caputo sense and ℐt1-q denotes the Riemann-Liouville fractional integral operator. The history ut:(-∞,0]→X defined by ut(s)=u(t+s) for s∈(-∞,0] belongs to some abstract phase space 𝒫 defined axiomatically and Ii∈C(X,X)(i=1,…,n); 0≤t0<t1<⋯<tn≤tn+1=T are fixed numbers and Δu(t) denotes the jump of the function u at the point t, given by Δu(t)=u(t+)-u(t-). The functions f,g:J×𝒫→X are appropriate functions and satisfy some conditions to be specified later.
In [13], authors have considered the following impulsive fractional differential equation in a Banach space of the form
(2)Dtqu(t)=Au(t)+f(t,u(t)),q∈(0,1],t∈J=[0,T],t≠ti,u(0)=u0,u(ti+)=u(ti-)+yi,i=1,2,…,n,n∈ℕ,
where A:D(A)⊂X→X is the infinitesimal generator of a C0-semigroup {T(t):t≥0} on a Banach space, f:J×X→X is continuous, and u0, yi are the element of X. Authors have established some existence and uniqueness results for system (2) under the different assumptions on initial conditions.
In this work, we adopt the idea of Wang et al. [13] and establish the existence of a mild solution for the problem (1) by using the measure of noncompactness and solution operator. The tool of measure of noncompactness has been used in linear operator theory, theory of differential and integral equations, the fixed point theory, and many others. For an initial study of theory of the measure of noncompactness, we refer to book of Banaś and Goebel [19] and Akhmerov et al. [20] and papers [9, 21–25] and references given therein.
This paper is organized as follows: In Section 2 we recall some basic definitions, lemmas, and theorems. We will prove the existence of a mild solution for the system (1) in Section 3. In the last section, we shall discuss an example to illustrate the application of the abstract results.
2. Preliminaries
Now we provide some basic definitions, notations, theorems, lemmas, and preliminary facts which will be used throughout this paper.
Let X be a Banach space and let C([0,T];X) be the Banach space of continuous functions u(t) from [0,T] to X equipped with the norm ∥y∥C=supt∈[0,T]∥y(t)∥X and Lp((0,T);X) denotes the Banach space of all Bochner-measurable functions from (0,T) to X with the norm
(3)∥u∥Lp=(∫(0,T)∥u(s)∥Xpds)1/p.
Assume that 0∈ρ(A), that is, A is invertible. Then, this permits us to define the positive fractional power Aα as closed linear operator with domain D(Aα)⊆H for α∈(0,1]. Moreover, D(Aα) is dense in H with the norm
(4)∥y∥α=∥Aαy∥.
It is easy to see that D(Aα) which is dense in X is a Banach space. Henceforth, we use Xα as notation of D(Aα). Also, we have that Xκ↪Xα for 0<α<κ and, therefore, the embedding is continuous. Then, we define X-α=(Xα)*, for each α>0. The space X-α, standing for the dual space of Xα, is a Banach space with the norm ∥z∥-α=∥A-αz∥ for z∈X-α. For more details on the fractional powers of closed linear operators, we refer to book by Pazy [26].
To consider the mild solution for the impulsive problem, we propose that the set 𝒫𝒞([0,T];X)={u:[0,T]→X:uiscontinuousatt≠tiandleftcontinuousatt = tiandu(ti+)exists,foralli=1,…,m}. Clearly, 𝒫𝒞([0,T];X) is a Banach space endowing the norm ∥u∥𝒫𝒞=supt∈[0,T]∥u(s)∥. For a function u∈𝒫𝒞([0,T];X) and i∈{0,1,…,m}, we define the function ui~∈C([ti,ti+1],X) such that
(5)ui~(t)={u(t),fort∈(ti,ti+1],u(ti+),fort=ti.
For W⊂𝒫𝒞([0,T];X) and i∈{0,1,…,m}, we have Wi~={ui~:u∈W} and following Accoli-Arzelà type criteria.
Lemma 1.
A set B⊂𝒫𝒞([0,T];X) is relatively compact in 𝒫𝒞([0,T];X) if and only if each set Bj~(j=1,2,…,m) is relatively compact in C([tj,tj+1],X)(j=0,1,…,m).
For the differential equation with infinite delay, Hale and Kato [5] proposed the phase space 𝒫 satisfying certain fundamental axioms.
Definition 2 (see [6]).
A phase space 𝒫 is a linear space which contains all the functions mapping (-∞,0] into Banach space X with a seminorm ∥·∥𝒫. The fundamental axioms assumed on 𝒫 are the following,
If u:(-∞,a+T]→X, T>0 is a continuous function on [a,a+T] such that ua∈𝒫 and u|[a,a+T]∈𝒫∈𝒫𝒞([a,a+T];X), then for every t∈[a,a+T), the following conditions hold:
ut∈𝒫,
H∥ut∥𝒫≥∥u(t)∥,
∥ut∥𝒫≤N(t+a)∥ua∥𝒫+K(t-a)sup{∥u(s)∥:a≤s≤t}.
Where H is a positive constant, N,K:[0,∞)→[0,∞), N is a locally bounded, K is continuous, and H, N, K are independent of u(·).
For the function u in (A1), ut is a 𝒫-valued continuous function for t∈[a,a+T].
The space 𝒫 is complete.
Now, we state some basic definitions and properties of fractional calculus.
Mittag-Leffler. The definition of one parameter Mittag-Leffler function is given as
(6)Eα(z)=∑k=0∞zkΓ(αk+1),
and two-parameter Mittag-Leffler function is defined as
(7)Eα,β(z)=∑k=0∞zkΓ(αz+β)=12πi∫Cμα-βeμμa-zdμ,α,β>0,z∈ℂ,
where C is a contour which starts and ends at -∞ and encircles the disc |μ|≤|z|1/2 counter clockwise. The Laplace transform of the Mittag-leffler is defined as
(8)L(tβ-1Eα,β(-ραtα))=λα-βλα+ρα,Reλ>ρ1/α,ρ>0.
For more details we refer to [1].
Laplace transform of integer order derivatives is defined as
(9)L[Fn(t);λ]=λnL[F(t)]-∑k=0n-1λn-k-1Fk(0)=λnL[F(t)]-∑k=0n-1λkFn-k-1(0).
Definition 3.
The Riemann-Liouville fractional integral operator ℐ is defined as
(10)ℐtqF(t)=1Γ(q)∫0t(t-s)q-1F(s)ds,
where F∈L1((0,T);X) and q>0 is the order of the fractional integration.
Definition 4.
The Riemann-Liouville fractional derivative is given as
(11)DtqF(t)=Dtnℐtn-qF(t),n-1<q<n,n∈ℕ,
where Dtn=dn/dtn, F∈L1((0,T);X), and ℐtn-qF∈Wn,1((0,T);X). Here, the notation Wn,1((0,T);X) stands for the Sobolev space defined as
(12)Wn,1((0,T);X)={∑k=0n-1y∈X:∃z∈L1((0,T);X):y(t)=∑k=0n-1dktkk!+tn-1(n-1)!*z(t),t∈(0,T)}.
Note that z(t)=yn(t) and dk=yk(0).
Definition 5.
The Caputo fractional derivative is given as
(13)DtαF(t)=1Γ(n-α)∫0t(t-s)n-α-1Fn(s)ds,n-1<α<n,
where F∈Cn-1((0,T);X)∩L1((0,T);X) and the following holds
(14)ℐtq(DtqF(t))=F(t)-∑k=0n-1tkk!Fk(0).
Definition 6 (see [27]).
A family {Sq(t)}t≥0⊂ℒ(X) of bounded linear operators in X is called a resolvent (or solution operator) generating by A if the following conditions are fulfilled:
Sq(t) is strongly continuous on ℝ+ and Sq(0)=I;
for x∈D(A) and t≥0, Sq(t)D(A)⊂D(A) and ASq(t)x=Sq(t)Ax;
Sq(t)x is the solution of the equation
(15)u(t)=x+1Γ(q)∫0t(t-s)q-1Au(s)ds,∀x∈D(A),t≥0,
where ℒ(X) denotes the space of all bounded linear operators from X into X endowed with the norm of operators.
Also, the solution operator Sq(t) for (15) is defined as (see [27])
(16)λq-1(λqI-A)-1x=∫0∞e-λtSq(t)xdt,Reλ>ω,x∈X,
where ω≥0 and {λq:Reλ>ω}⊂ρ(A).
Let
(17)∑(ω,θ)={λ∈ℂ:|arg(λ-ω)|<θ}.
Definition 7 (see [27]).
A solution operator Sq(t) is said to be analytic if Sq(·):ℝ+→ℒ(X) admits analytic extension to a sector ∑(0,θ0) for some 0<θ0≤π/2. Furthermore, An analytic resolvent Sq(t) is said to be of analyticity type (ω0,θ0) if, for θ0>θ and ω0<ω, there exists M=M(ω,θ) such that ∥Sq(t)∥≤MeωRez for z∈∑(0,θ); here Rez means the real part of z.
In this work, we assume that solution operator {Sq(t)}t≥0 is analytic; that is, {Sq(t)}t≥0 satisfy the following property.
The map t↦Sq(t) is continuous from [0,T] to ℒ(X) endowed with the uniform operator norm ∥·∥ℒ(X).
Without loss of generality, we have that there exist a positive constant M such that ∥Sq(t)∥≤M, for t≥0.
Definition 8 (see [19]).
The Hausdorff measure of noncompactness χZ is defined as
(18)χZ(F)=inf{23ϵ>0:Fcanbecoveredbyfinitenumberofballswithradiusϵ23},
for bounded set F⊂Z, where Z is a Banach space.
Lemma 9 (see [19]).
For any bounded set U,V⊂Y, where Y is a Banach space. Then, the following properties are fulfilled:
χY(U)=0 if and only if U is pre-compact;
χY(U)=χY(convU)=χY(U¯), where convU and U¯ denotes the convex hull and closure of U, respectively;
χY(U)⊂χY(V), when U⊂V;
χY(U+V)≤χY(U)+χY(V), where U+V={u+v:u∈U,v∈V};
χY(U∪V)≤max{χY(U),χY(V)};
χY(λU)=λ·χY(U), for any λ∈ℝ;
if the map P:D(P)⊂Y→𝒵 is continuous and satisfies the Lipschitsz condition with constant κ, then we have that χ𝒵(PU)≤κχY(U) for any bounded subset U⊂D(P), where Y and 𝒵 are Banach space.
The details on the measure of noncompactness and its applications can be found in a book by Banaś and Goebel [19] and papers [9, 10, 21, 23, 24].
Lemma 10 (see [19]).
A bounded and continuous map Q:D⊂Z→Z is a χZ-contraction if there exists a constant 0<κ<1 such that χZ(Q(U))≤κχZ(U), for any bounded closed subset U⊂D, where Z is a Banach space.
Lemma 11 (see [28]).
Let D⊂Z be closed and convex with 0∈D and let the continuous map Q:D→D be a χZ-contraction. If the set {u∈D:u=λQu,for0<λ<1} is bounded, then the map Q has a fixed point in D.
Lemma 12 ((Darbo-Sadovskii) [19]).
Let D⊂Z be bounded, closed, and convex. If the continuous map Q:D→D is a χZ-contraction, then the map Q has a fixed point in D.
In this work, we consider that χ denotes the Hausdorff measure of noncompactness in X, χC denotes the Hausdorff measure in noncompactness of C([0,T];X) and χ𝒫𝒞 denotes the Hausdorff measure of noncompactness in 𝒫𝒞([0,T];X).
Lemma 13 (see [19, 21]).
If U is bounded subset of C([0,T];X). Then, one has that χ(U(t))≤χC(U), forallt∈[0,T], where U(t)={u(t);u∈U}⊆X. Furthermore, if U is equicontinuous on [0,T], then χ(U(t)) is continuous on the interval [0,T] and
(19)χC(U)=supt∈[0,T]{χ(U(t))}.
Lemma 14 (see [19]).
If U⊂C([0,T];X) is bounded and equicontinuous set, then χ(U(t)) is continuous and
(20)χ(∫0tU(s)ds)≤∫0tχ(U(s))ds,∀t∈[0,T],
where ∫0tU(s)ds={∫0tu(s)ds,u∈U}.
Lemma 15 (see [29]).
(1) If U⊂𝒫𝒞([0,T];X) is bounded, then χ(U(t))≤χ𝒫𝒞(U), forallt∈[0,T], where U(t)={u(t):u∈U}⊂X.
(2) If U is piecewise equicontinuous on [0,T], then χ(U(t)) is piecewise continuous for t∈[0,T] and
(21)χ𝒫𝒞(U)=sup{χ(U(t)):t∈[0,T]}.
(3) If U⊂𝒫𝒞([0,T];X) is bounded and equicontinuous, then χ(U(t)) is piecewise continuous for t∈[0,T] and
(22)χ(∫0tU(s)ds)≤∫0tχ(U(s))ds,∀t∈[0,T],
where ∫0tU(s)ds={∫0tu(s)ds:u∈U}.
3. Main Results
In this section, we will establish the existence results of solution for (1) by using solution operator and Hausdorff's measure of noncompactness.
From [13], we adopt the following concept of solution for impulsive differential problem (1).
Definition 16.
A piecewise continuous function u:(-∞,T]→X is said to be a mild solution for impulsive problem (1) if u0=φ, u(·)|J∈𝒫𝒞 and
(23)u(t)={Sq(t)[φ(0)+g(0,φ)]-g(t,ut)+∫0tSq(t-s)f(s,us)ds,t∈[0,t1],Sq(t)[φ(0)+g(0,φ)]-g(t,ut)+Sq(t-t1)I1(ut1)+Sq(t-t1)×[g(t1,ut1+I1(ut1))-g(t1,ut1)]+∫0tSq(t-s)f(s,us)ds,t∈(t1,t2],⋮Sq(t)[φ(0)+g(0,φ)]-g(t,ut)+∑i=1nSq(t-ti)I1(uti)+∑i=1nSq(t-ti)×[g(ti,uti+Ii(uti))-g(ti,uti)]+∫0tSq(t-s)f(s,us)ds,t∈(tn,T],
where
(24)Sq(t)=12πi∫Γeλtλq-1R(λq,A)dλ.
Now we list the following assumptions which are required to establish main results.
The function f:J×𝒫→X satisfies the following conditions:
the function f(·,u):J→X is strongly measurable for every u∈𝒫 and u0∈𝒫, u|J∈𝒫𝒞;
f(t,·):𝒫→X is a continuous function for each t∈J;
there exists an integrable function mf:[0,T]→[0,∞) and a nondecreasing continuous function Ω:[0,∞)→(0,∞) such that
(25)∥f(τ,x)∥≤mf(τ)Ω(∥x∥𝒫),τ∈J,x∈𝒫;
there exists an integrable function η:[0,T]→[0,∞) such that, for any bounded set B⊂𝒫, we have
(26)χ(Sq(τ)f(τ,B))≤η(τ)sup-∞≤θ≤0χ(B(θ));
for almost everywhere τ∈J, where B(θ)={u(θ):u∈B}.
(1) For 0<β<1, Aβg(·,·) is Lipschitz continuous function for all (t,v)∈J×𝒫 and there exist positive constants C1 and C2 such that
(27)∥Aβg(t,u)∥≤C1∥u∥𝒫+C2;
there exists a constant Lg>0 such that
(28)∥Aβg(t,u)-Aβg(t,v)∥≤Lg∥u-v∥𝒫,
for all u,v∈𝒫.
The functions Ii:X→X,(i=1,…,n) are continuous functions and satisfy the following conditions:
There is a constant LI>0 such that
(29)∥Ii(u)-Ii(v)∥≤LI∥u-v∥𝒫,∀u,v∈𝒫.
There exist positive constants Lj(j=1,2) such that
(30)∥Ii(u)∥=L1∥u∥𝒫+L2,
for all u∈𝒫.
(31)KTM1-ζ2∫0tmf(s)ds<∫d∞dsΩ(s),
where
(32)ζ1=(MKTH+NT+MC1∥A-β∥KT)∥φ∥𝒫+KTC2∥A-β∥(M+1)+KT(nML2(1+Lg)),ζ2=[∥A-β∥C1+nML1(1+Lg)]KT<1,d=ζ11-ζ2;
(33)KT[∥A-β∥Lg+nMLI+MnLg(2+LI)]+∫0tη(s)ds<1.
Now, let z:(-∞,T]→X be a function given by z0=φ and z(t)=Sq(t)φ(0) on J. It is easy to see that ∥zt∥𝒫≤(KTMH+NT)∥φ∥𝒫, where KT=sup0≤t≤TK(t), NT=sup0≤t≤TN(t).
Theorem 17.
Suppose that hypotheses (HA), (Hf), (Hg), (HI), and (H1)-(H2) are satisfied. Then, there exists a mild solution for the impulsive problem (1).
Proof.
Consider the space S(T)={u:(-∞,T]→X:u0=0,u|J∈𝒫} endowed with supremum norm ∥·∥𝒫. Define the operator Q:S(T)→S(T) by
(34)(Qx)(t)={0,t∈(-∞,0]Sq(t)g(0,φ)-g(t,ut+zt)+∫0tSq(t-s)f(s,us+zs)ds,t∈[0,t1],Sq(t)g(0,φ)-g(t,ut+zt)+Sq(t-t1)I1(ut1+zt1)+Sq(t-t1)×[g(t1,ut1+zt1+I1(ut1+zt1))-g(t1,ut1+zt1)]+∫0tSq(t-s)f(s,us+zs)ds,t∈(t1,t2],⋮Sq(t)g(0,φ)-g(t,ut+zt)+∑i=1nSq(t-ti)I1(uti+zti)+∑i=1nSq(t-ti)×[g(ti,uti+zti+Ii(uti+zti))-g(ti,uti+zti)]+∫0tSq(t-s)f(s,us+zs)ds,t∈(tn,T],
and we have that
(35)∥ut+zt∥𝒫≤∥ut∥𝒫+∥zt∥𝒫,≤K(t)sup{∥u(s)∥:0≤s≤t}+N(t)∥u0∥𝒫+K(t)sup{∥z(s)∥:0≤s≤t}+N(t)∥z0∥𝒫,≤K(t)∥u∥t+K(t)MH∥φ∥𝒫+N(t)∥φ∥𝒫,≤(KTMH+NT)∥φ∥𝒫+KT∥u∥t,
where ∥u∥t=sup0≤s≤t∥u(s)∥. Thus, Q is well defined and with the values in S(T) by our assumptions. By Lebesgue dominated convergence theorem, axioms of phase space and assumptions (Hf), (Hg), and (HI), it is clear that Q is continuous map. Furthermore, by uniformly continuity of the map t↦Sq(t) on (0,T], we obtain that set Q(S(T)) is equicontinuous. We prove the result in following steps.
Step 1. The set {u∈𝒫𝒞:u=λQu,for0<λ<1} is bounded.
Let uλ be a solution of u=λQu for 0<λ<1. Therefore, we have that
(36)∥uλt+zt∥𝒫≤(KTMH+NT)∥φ∥𝒫+KT∥uλ∥t.
Take vλ(t)=(KTMH+NT)∥φ∥𝒫+KT∥uλ∥t, for t∈J. Then, we get that for t∈[0,t1](37)∥uλ(t)∥=∥λQuλ(t)∥≤∥Quλ(t)∥,≤M∥A-β∥(C1∥φ∥𝒫+C2)+∥A-β∥(C1vλ(t)+C2)+M∫0tmf(s)Ω(vλ(s))ds.
For t∈(t1,t2], we have
(38)∥uλ(t)∥≤M∥A-β∥(C1∥φ∥𝒫+C2)+∥A-β∥(C1vλ(t)+C2)+M∥I1(vλ(t1))∥+M∥g(t1,vλ(t1)+I1(vλ(t1)))-g(t1,vλ(t1))∥+M∫0tmf(s)Ω(vλ(s))ds,≤M∥A-β∥(C1∥φ∥𝒫+C2)+∥A-β∥(C1vλ(t)+C2)+M(L1(vλ(t))+L2)+MLg(L1vλ(t)+L2)+M∫0tmf(s)Ω(vλ(s))ds.
For t∈(tn,T],
(39)∥uλ(t)∥≤M∥A-β∥(C1∥φ∥𝒫+C2)+∥A-β∥(C1vλ(t)+C2)+nM(L1(vλ(t))+L2)+nMLg(L1vλ(t)+L2)+M∫0tmf(s)Ω(vλ(s))ds.
Therefore, for all t∈[0,T]=J, we have
(40)∥uλ(t)∥≤M∥A-β∥(C1∥φ∥𝒫+C2)+∥A-β∥(C1vλ(t)+C2)+nML1(1+Lg)vλ(t)+nML2(1+Lg)+M∫0tmf(s)Ω(vλ(s))ds,≤M∥A-β∥(C1∥φ∥𝒫+C2)+∥A-β∥C2+nML2(1+Lg)+(∥A-β∥C1+nML1(1+Lg))vλ(t)+M∫0tmf(s)Ω(vλ(s))ds.
From vλ(t)=(KTMH+NT)∥φ∥𝒫+KT∥uλ∥t, it implies that
(41)vλ(t)≤(MKTH+NT+M∥A-β∥C1KT)∥φ∥𝒫+KT∥A-β∥C2(M+1)+KT(nML2(1+Lg))+KT(∥A-β∥C1+nML1(1+Lg))vλ(t)+MKT∫0tmf(s)Ω(vλ(s))ds,
and consequently,
(42)vλ(t)≤ζ11-ζ2+MKT1-ζ2∫0tmf(s)Ω(vλ(s))ds,=d+MKT1-ζ2∫0tmf(s)Ω(vλ(s))ds,
where
(43)ζ1=(MKTH+NT+MC1∥A-β∥KT)∥φ∥𝒫+KT∥A-β∥C2(M+1)+nKTML2(1+Lg),ζ2=KT[∥A-β∥C1+nML1(1+Lg)].
Let ξλ(t)=d+(MKT/(1-ζ2))∫0tmf(s)Ω(ξλ(s))ds. Thus ξλ(0)=d. Therefore, we get
(44)ξλ′(t)Ω(ξλ(t))≤KTM1-ζ2mf(t).
Integrating above inequality we have that
(45)∫dξλ(t)dsΩ(s)≤KTM1-ζ2∫0tmf(s)ds<∫d∞dsΩ(s).
It gives that the functions ξλ(t) are bounded on interval J. Therefore, the functions vλ(t) are bounded and uλ(·) are also bounded on J.
Step 2. The map Q is a χ-contraction.
Firstly, we introduce decomposition of Q into Q=Q1+Q2, for t≥0 such that
(46)Q1u(t)={Sq(t)g(0,φ)-g(t,ut+zt),t∈[0,t1],Sq(t)g(0,φ)-g(t,ut+zt)+∑i=1mSq(t-ti)Ii(uti+zti)+∑i=1mSq(t-ti)×[g(ti,uti+zti+Ii(uti+zti))-g(ti,uti+zti)],t∈(tm,tm+1],m=1,2,3,…,n;Q2u(t)=∫0tSq(t-s)f(s,us+zs)ds,t∈[0,T].
To prove the result, we firstly show that Q1 is Lipschitz continuous. For x1,x2∈S(T) and t∈[0,t1], we have that
(47)∥Q1x1(t)-Q1x2(t)∥=∥g(t,x1t+zt)-g(t,x2t+zt)∥,≤∥A-β∥Lg∥x1t-x2t∥.
For t∈(tm,tm+1], m=1,2,…,n we have
(48)∥Q1x1(t)-Q1x2(t)∥≤∥A-β∥Lg∥x1t-x2t∥𝒫+mMLI∥x1t-x2t∥𝒫+MmLg(2+LI)∥x1t-x2t∥𝒫,=KT[∥A-β∥Lg+mMLI+MmLg(2+LI)]×∥x1-x2∥T.
Thus for t∈[0,T] we have
(49)∥Q1x1(t)-Q1x2(t)∥≤KT[∥A-β∥Lg+nMLI+MnLg(2+LI)]×∥x1-x2∥T.
Taking supremum on [0,T], we get
(50)∥Q1x1-Q1x2∥T≤KT[∥A-β∥Lg+nMLI+MnLg(2+LI)]×∥x1-x2∥T.
Hence, it implies that Q1 satisfies the Lipschitz condition with Lipschitz constant L, where L=KT[∥A-β∥Lg+nMLI+MnLg(2+LI)].
Therefore, from Lemma 9 (vii), we have that for any bounded set B∈𝒫𝒞(51)χ𝒫𝒞(Q1B)≤Lχ𝒫𝒞(B).
Next, we show that Q2 is a χ-contraction. Let B be an arbitrary bounded subset S(T). Since Sq(t) is equicontinuous solution operator, therefore Sq(t-s)f(s,us+zs) is piecewise continuous. From Lemma 9, we have that for any bounded set B∈𝒫𝒞,
(52)χ(Q2B(t))=χ(∫0tSq(t-s)f(s,Bs+zs)ds)≤∫0tη(s)sup-∞<ω≤0χ(B(s+ω)+z(s+ω))ds≤∫0tη(s)sup0≤τ≤sχ(B(τ))ds≤χ𝒫𝒞(B)∫0tη(s)ds,t∈[0,T].
Thus for any bounded set B∈𝒫𝒞(53)χ𝒫𝒞(QB)=χ𝒫𝒞(Q1B+Q2B)≤χ𝒫𝒞(Q1B)+χ𝒫𝒞(Q2B)≤(L+∫0tη(s)ds)χ𝒫𝒞(B).
By the assumption (H2), we obtain that χ𝒫𝒞(QB)<χ𝒫𝒞(B); that is, Q is a contraction. Therefore, Q has at least one fixed point in B by Darbo fixed point theorem. Let u be a fixed point of Q on S(T), then y=u+z is a mild solution for (1).
Theorem 18.
Suppose that (Hf), (Hg), and (HI) are satisfied and
(54)KT[∥A-β∥C1+nML1(1+Lg)]+∫0Tmf(s)dslimτ→∞supΩ(τ)τ<1.
Then, there exists a mild solution for the impulsive problem (1).
Proof.
Proceeding as in the proof of Theorem 17, we infer that Q defined by (34) is continuous from S(T) into S(T). Next we indicate that there exists r>0 such that Q(Br)⊂Br, where B is defined by Br={u∈S(T):∥u∥T≤r}. To this end, let us assume that assertion is false, then for any r>0 there exists ur∈Br and tr∈J such that r<∥Qur(tr)∥. Therefore, for tr∈[0,t1] and ur∈Br,
(55)r<∥Qur(tr)∥≤M∥g(t,φ)∥+∥g(t,urtr+ztr)∥+M∫0tmf(s)Ω(∥urs+zs∥)ds,≤M∥g(t,φ)∥+∥A-β∥(C1∥urtr+ztr∥𝒫+C2)+M∫0tmf(s)Ω(∥urs+zs∥)ds,≤M∥g(t,φ)∥+∥A-β∥×(C1(KTMH+NT)∥φ∥𝒫+C1KTr+C2)+M∫0tmf(s)dsΩ((KTMH+NT)∥φ∥𝒫+KTr)=r0.
For tr∈(t1,t2], we have
(56)r<∥Qur(tr)∥≤M∥g(t,φ)∥+∥A-β∥(C1∥urtr+ztr∥𝒫+C2)+M(L1∥urtr+ztr∥𝒫+L2)+MLg(L1∥urtr+ztr∥𝒫+L2)+M∫0tmf(s)Ω(∥urs+zs∥)ds,≤M∥g(t,φ)∥+∥A-β∥×(C1(KTMH+NT)∥φ∥𝒫+C1KTr+C2)+M(L1(KTMH+NT)∥φ∥𝒫+L1KTr+L2)+MLg(L1(KTMH+NT)∥φ∥𝒫+L1KTr+L2)+M∫0tmf(s)dsΩ((KTMH+NT)∥φ∥𝒫+KTr),=r1.
For tr∈(tn,T], we have
(57)r<∥Qur(tr)∥≤M∥g(t,φ)∥+∥A-β∥(C1(KTMH+NT)∥φ∥𝒫+C1KTr+C2)+nM(L1(KTMH+NT)∥φ∥𝒫+L1KTr+L2)+nMLg(L1(KTMH+NT)∥φ∥𝒫+L1KTr+L2)+M∫0tmf(s)dsΩ((KTMH+NT)∥φ∥𝒫+KTr),=rn,
which implies that r<max{r0,r1,…,rn}. Therefore, we conclude that
(58)r<M∥g(t,φ)∥+∥A-β∥×(C1(KTMH+NT)∥φ∥𝒫+C1KTr+C2)+nM(L1(KTMH+NT)∥φ∥𝒫+L1KTr+L2)+nMLg(L1(KTMH+NT)∥φ∥𝒫+L1KTr+L2)+M∫0tmf(s)dsΩ((KTMH+NT)∥φ∥𝒫+KTr).
We divide both the sides of (58) by r and letting r→∞, we obtain that
(59)1<KT[∥A-β∥C1+nML1+nMLgL1]+M∫0Tmf(s)ds×limr→∞supΩ((KTMH+NT)∥φ∥𝒫+KTr)r,<KT[∥A-β∥C1+nML1(1+Lg)]+∫0Tmf(s)dslimτ→∞supΩ(τ)τ.
This contradicts the inequality (54). Hence, there exists a positive constant r>0 such that Q(Br)⊂Br. Moreover, by uniform continuity of the map t↦Sq(t) on (0,T], we have that set Q(Br) is equicontinuous. As the proof of Theorem 17, we infer that system (1) has a mild solution.
4. Example
In this section, we consider an example to illustrate the application of the theory. Here we take the space C0×L2(h,X) as the phase space 𝒫 (see, [7]).
Now we study the following fractional differential equation with infinite delay:
(60)dqdtq[u(t,x)+∫-∞t∫0πb(t-s,μ,x)u(s,μ)dμds]=∂2dx2[u(t,x)+∫-∞t∫0πb(t-s,μ,x)u(s,μ)dμds]+ℐt1-q∫-∞tf(t,t-s,x,u(s,x))ds,t∈[0,T],x∈[0,π],u(t,0)=u(t,π)=0,0≤t≤T,u(τ,x)=φ(τ,x),0≥τ,x∈[0,π],Δu(tk,x)=∫-∞tkak(tk-s)u(s,x)ds,k=1,2,…,n,
where φ∈C0×L2(h;X) and 0<t1<⋯<tn<tn+1=T are prefixed numbers. Let X=L2[0,π] and consider the operator A:D(A)⊂X→X as Av=v′′ with the domain
(61)D(A)={59v∈X:v,v′areabsolutelycontinuousandv′′∈X,v(0)=0=v(π)59}.
Then,
(62)A=∑n=1∞n2(v,vn)vn,v∈D(A),
Here vn(x)=(2/π)sin(nx), n∈ℕ is the orthogonal set of eigenvectors of A. It is clear that A is the infinitesimal generator of an analytic semigroup {S(t)}t≥0 in X which is given by
(63)S(t)v=∑n=1∞e-n2t(v,vn)vn,∀v∈X,t>0.
By Theorem 3.1 in [27], we get that A is the infinitesimal generator of solution operator {Sq(t)}t≥0 and there exists a positive constant M such that ∥Sq(t)∥ℒ(X)≤M for t∈[0,T]. The functions b,f,ak satisfy the following conditions:
the functions b(s,μ,x), (∂b/∂x)(s,μ,x) are measurable such that b(s,μ,π)=b(s,μ,0)=0 and
(64)Lg=max{(∫0π∫-∞0∫0π1h(s)(∂ib(s,μ,x)∂xi)2dμdsdx)1/2:i=0,1(∫0π∫-∞0∫0π1h(s)(∂ib(s,μ,x)∂xi)2dμdsdx)1/2}<∞;
f:ℝ4→ℝ is continuous and there exist a η:C(ℝ2,ℝ) such that
(65)∥f(t1,t2,x,u)∥≤η(t1,t2)∥u∥,(t1,t2,x,u)∈ℝ4;
the functions ak:[0,∞)→ℝ, k=1,2,…,n are continuous and
(66)LI=(∫-∞0(ak(s))2h(s)ds)1/2<∞.
The impulsive system (60) might be reformulated as the abstract impulsive Cauchy problem (1) where
(67)g(t,ϕ)(x)=∫-∞0∫0πb(s,y,x)ϕ(s,y)dyds,F(t,ϕ)(x)=∫-∞0f(t,s,x,ϕ(s,x))ds,Ik(ϕ)(x)=∫-∞0ak(s)ϕ(s,x)ds,k=1,…,n.
It may be verified that g, F satisfy the assumptions (Hf), (Hg), and (HI); that is, g(t,·) and Ik, k=1,…,n are bounded linear operators and the range of g(·) is contained in X1/2, ∥A1/2g(t,·)∥≤Lg, ∥Ik∥≤LI, k=1,…,n, and ∥F(t,ϕ)∥≤mF(t)∥ϕ∥𝒫 for every t∈[0,T], where mF(t) is defined as mF(t)=(∫-∞0η(t,s)2/h(s)ds)1/2. Applying Theorem 17, we obtain that problem (60) has a mild solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the UGC (University Grants Commission, India) under Grant no. (6405-11-061).nt.
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