This dissertation is regarded to investigate the system of infinite time-delay and non-instantaneous impulsive to fractional evolution equations containing an infinitesimal generator operator. It turns out that its mild solution is existed and is unique. Our model is built using a fractional Caputo approach of order lies between 1 and 2. To get the mild solution, the families associated with cosine and sine which are linear strongly continuous bounded operators, are provided. It is common to use Krasnoselskii's theorem and the Banach contraction mapping principle to prove the existence and uniqueness of the mild solution. To confirm that our results are applicable, an illustrative example is introduced.
Citation: Ahmed Salem, Kholoud N. Alharbi. Fractional infinite time-delay evolution equations with non-instantaneous impulsive[J]. AIMS Mathematics, 2023, 8(6): 12943-12963. doi: 10.3934/math.2023652
[1] | A.G. Ibrahim, A.A. Elmandouh . Existence and stability of solutions of ψ-Hilfer fractional functional differential inclusions with non-instantaneous impulses. AIMS Mathematics, 2021, 6(10): 10802-10832. doi: 10.3934/math.2021628 |
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[6] | Zhilin Li, Guoping Chen, Weiwei Long, Xinyuan Pan . Variational approach to p-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses. AIMS Mathematics, 2022, 7(9): 16986-17000. doi: 10.3934/math.2022933 |
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[9] | Noorah Mshary, Hamdy M. Ahmed, Ahmed S. Ghanem . Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps. AIMS Mathematics, 2024, 9(4): 9746-9769. doi: 10.3934/math.2024477 |
[10] | Ye Li, Biao Qu . Mild solutions for fractional non-instantaneous impulses integro-differential equations with nonlocal conditions. AIMS Mathematics, 2024, 9(5): 12057-12071. doi: 10.3934/math.2024589 |
This dissertation is regarded to investigate the system of infinite time-delay and non-instantaneous impulsive to fractional evolution equations containing an infinitesimal generator operator. It turns out that its mild solution is existed and is unique. Our model is built using a fractional Caputo approach of order lies between 1 and 2. To get the mild solution, the families associated with cosine and sine which are linear strongly continuous bounded operators, are provided. It is common to use Krasnoselskii's theorem and the Banach contraction mapping principle to prove the existence and uniqueness of the mild solution. To confirm that our results are applicable, an illustrative example is introduced.
Fractional differential equations are becoming a considerably more important and popular topic. In order to specify the so-called fractional differential equations, the conventional integer order derivative is generalized to arbitrary order. Fractional differential equations have been extensively employed to explain a variety of physical processes because of the effective memory function of the fractional derivative, such as seepage flow in porous media, fluid dynamic, and traffic models. Also, there are several applications of fractional differential equations in control theory, polymer rheology, aerodynamics, physics, chemistry, biology, and other exciting conceptual advancements (see [1,2,3,4] and their references).
In the real world, there are numerous processes and phenomena that are influenced by transient external factors as they evolve. When compared to the whole duration of the occurrences and processes being researched, their duration is tiny. As a result, it is reasonable to believe that these exterior impacts are "instantaneous", or take the shape of impulses. Differential equations including the impulse effect, or impulsive differential equations, seem to be a plausible explanation of the known evolution processes of various real-world issues. The impulsive differential equations have been studied as the subject of numerous excellent monographs [5,6].
Differential equations are used as representations for many processes in applied sciences research. There are a variety of classical mechanics that experience abrupt changes in their states, such as biological systems (heartbeats and blood flow), mechanical systems with impact, radiophysics, pharmacokinetics, population dynamics, mathematical economics, ecology, industrial robotics, biotechnology processes, etc [7,8]. The systems of differential equations with impulses are suitable mathematical models for such phenomena. Impulsive differential equations essentially have three parts: An impulse equation simulates an impulsive leap that is described by a jump function at the moment of impulse occurs, a continuous-time differential equation determines the state of the system between impulses, and jump criteria identifies a set of jump occurrences [9,10,11].
Furthermore, there are various models that have been developed in many fields like biology, economics, and materials science where the rate of change at time t depends not only on the system's current state but also on its history over a period of time [t−τ,t] [12,13,14,15]. These models have evolutionary equations with delay, which describe them mathematically. Equations with infinite delay are produced by the more generic if we take τ=∞.
Physics provides a compelling justification for studying the nonlocal partial differential equation. Fractional derivatives in space and time are used in abstract partial differential equations such as fractional diffusion equations. They may be used to simulate anomalous diffusion, in which a particle plume spreads differently than the traditional diffusion equation would suggest. By replacing the second-order space derivative in the classic diffusion equation by an infinitesimal generator operator of strongly continuous C0 semi-group or cosine functions, the time fractional evolution equation is derived [16,17,18].
In [19], Kumar and Pandey attempted to examine the results of the existence of a solution to a class of FDEs (the fractional calculus due to Atangana-Baleanu) of the sort
{ABCDρv(r)=Fv(r)+h(r,v(r)),r∈∪mj=0(sj,rj+1],v(r)=δj(r,v(r)),r∈∪mj=1(rj,sj],v(0)=v0−f(v), |
for all v∈D(F) (the domain of F), where ABCDρ is the ABC fractional derivative of order ρ∈(0,1), F:D(F)⊂X→X is a generator of ρ-resolvent operator {Sρ(r)}r≥0 on a Banach space (X,‖⋅‖), J=[0,d], 0=r0<r1<r2<⋯<rm<rm+1 and sj∈(rj,rj+1) for all j=1,2,…,m;m∈N. The functions h:∪mk=0(sj,rj+1]×X→X and f:X→X are given continuous functions and δj:(rj,sj]×X→X are non-instantaneous impulsive functions for each j=1,2,…,m;m∈N and vo∈X.
Recently, Kavitha et al. [20] examined the existence of solutions for a class of non-instantaneous impulses and infinite delay of fractional differential equations within the Mittag-Leffler kernel of the kind
{ABCDvp(ξ)=Bp(ξ)+F(ξ,pξ),ξ∈∪ml=0(sl,ξl+1],p(ξ)=κl(ξ,pξ),ξ∈∪ml=1(ξl,sl],p(ξ)=ϕ(ξ),ξ∈(−∞,0], |
where the fractional order v∈(0,1), B:D(B)⊂E→E is infinitesimal generator of an ρ-resolvent operator {Sρ(ξ)}ξ≥0 on a Banach space (E,‖⋅‖, K=[0,b], 0=ξ0=s0<ξ1≤s1<ξ2<⋯<ξq<ξq+1=b and sl∈(ξl,ξl+1) for all l=1,2,…,q;q∈N. The function F:∪ql=0(sl,ξl+1]×A→E satisfies Caratheodory conditions and the functions κl:(ξl,sl]×A→E are non-instantaneous impulsive functions for each l=1,2,…,q. They considered that pξ:(−∞,0]→E such that pξ(x)=p(ξ+x) for all x≤0 and ϕ∈A where A is an abstract phase space.
In light of the foregoing, in this publication, we examine the existence results for a class of fractional-order non-instantaneous impulses functional evolution equations.
Consider the fractional semilinear evolution of the form
{u(t)=ϕ(t),t∈(−∞,0],cDαtu(t)=Au(t)+h(t,ut),t∈∪mk=0(sk,tk+1],u(t)=μk(t,u(t)),t∈∪mk=1(tk,sk],u′(t)=ξk(t,u(t)),t∈∪mk=1(tk,sk],u′(0)=u0, | (1.1) |
where cDαt is the fractional derivative due to Caputo of order 1<α<2 and J=[0,a] is operational interval. Here, h:J×Ph→X is a given function satisfying some assumptions that will be determined later, where Ph is an abstract phase space and X is a Banach space. The functions μk,ξk∈C((tk,sk]×X;X) for all k=1,2,…m;m∈N reflect the impulsive circumstances and 0=t0=s0<t1≤s1≤t2<⋯<tm≤sm≤tm+1=a are pre-fixed numbers. The history function ut:(−∞,0]→X is an element of Ph and defined by ut(θ)=u(t+θ),θ∈(−∞,0].
The closed operator A is an infinitesimal generator of a uniformly bounded family of strongly continuous cosine operators {R(t)}t∈R, which is defined on a Banach space X. The Banach space of continuous and bounded functions from (−∞,a] into X provided with the topology of uniform convergence is denoted by C=Ca((−∞,a],X) with the norm
‖u‖C=supt∈(−∞,a]|u(t)|. |
As {R(t)}t∈R is a cosine family on X, then there exists ϖ≥1 [21] such that
‖R(t)‖≤ϖ. | (1.2) |
The rest of the text is structured as follows. In section 2, we give some fundamental concepts and lemmas related to our study. In section 3, we formulate the mild solution of (1.1) by considering that operator A is an infinitesimal generator of strongly continuous cosine functions {R(t)}t∈R. By using the fixed-point theorem, Section 4 presents our study outcomes. An instance is provided in Section 5 to be an application.
In this section, we present some concepts and definitions related to the components of the research paper such as fractional calculus, cosine and sine families operators, and abstract phase space. Also, some lemmas that give helpful results to prove the main results of this contribution, are provided.
The definitions of R-L integral and Caputo derivative and the important related lemma are introduced as follow.
Definition 2.1. (Caputo derivative [22]) Let ℓ−1<q≤ℓ;ℓ∈N and x:[a,b]→R (−∞<a<b<∞) be nth continuously differentiable function. Then, the left derivative of fractional order q due to Caputo is presented as
cDqax(s)=1Γ(ℓ−q)∫sa(s−t)ℓ−q−1x(ℓ)(t)dt,s∈[a,b]. |
Definition 2.2. (Riemann-Liouville fractional integral [22]) The left R-L fractional integral of the integrable function x over the interval [a,b] is derived as
Iqax(s)=1Γ(q)∫sa(s−t)q−1x(t)dt,q>0,s∈[a,b]. |
Lemma 2.1. [23] Let ℓ∈N,ℓ−1<q≤ℓ and x(s) be nth continuously differentiable function over the interval [a,b]. Then,
IqacDqax(s)=x(s)+a0+a1(s−a)+⋯+aℓ−1(s−a)ℓ−1,s∈[a,b]. |
Definition 2.3. (Atangana-Baleanu fractional derivative in Caputo sense [19]) ABC-derivative for the order α∈[0,1] and x(s)∈H1(a,b),a<b is given by
ABCDαx(s)=M(α)(1−α)∫saEα[−α(s−r)α1−α]x′(r)dr, |
where Eα(⋅) and H1 are the Mittag-Leffler function and the non-typical Banach space defined, respectively, as
Eα(z)=∞∑i=0ziΓ(αi+1),ℜ(α)>0,z∈C,H1(Ω)={η(s)|η(s),Dρη(s)∈L2(Ω),∀ρ≤1}. |
Let A be an infinitesimal generator of a uniformly bounded family of strongly continuous cosine operators {R(t)}t∈R which is defined on a Banach space X. We collect some basic properties of a cosine family and its relations with the operator A and the associated sine family.
Definition 2.4. [24] Consider {R(t)}t∈R is a one parameter family of bounded linear operators mapping the Banach space X→X. It is referred to a strongly continuous cosine family if and only if
(ⅰ) R(0)=I;
(ⅱ) R(s+t)+R(s−t)=2R(s)R(t) for all s,t∈R;
(ⅲ) The function t↦R(t)x is a continuous on R for any x∈X.
The sine family {T(t)}t∈R is correlated to the strongly continuous cosine family {R(t)}t∈R, it is characterized by
T(t)x=∫t0R(s)xds,x∈X,t∈R. |
Lemma 2.2. [24] Consider A is an infinitesimal generator of a strongly continuous cosine family {R(t)}t∈R on Banach space X such that ‖R(t)‖≤Meξ|t|,t∈R. Then, for λ>ξ and (ξ2,∞)⊂ρ(A) (the resolvent set of A), we have
λR(λ2;A)x=∫∞0e−λtR(t)xdt,R(λ2;A)x=∫∞0e−λtT(t)xdt,x∈X, |
where the operator R(λ;A)=(λI−A)−1 is the resolvent of the operator A and λ∈ρ(A).
In this case, the operator A is defined by
Ax=limt→0d2dt2R(t)x,∀x∈D(A), |
where D(A)={x∈X:R(t)x∈C2(R,X)} is the domain of the operator A. Clearly the infinitesimal generator A is densely defined operator in X and closed.
In the sequel to present our results, we need the following:
Definition 2.5. Suppose that τ>0, the Mainardi's Wright-type function is defined as
Mϱ(τ)=∞∑n=0(−τ)nn!Γ(1−ϱ(n+1)),ϱ∈(0,1),τ∈C, |
and achieves
Mϱ(τ)≥0,∫∞0θξMϱ(θ)dθ=Γ(1+ξ)Γ(1+ϱξ),ξ>−1. |
The abstract phase space Ph is demonstrated by convenient way [25,26]. Let h=C((−∞,0],[0,∞)) with ∫0−∞h(t)dt<∞. Then, for any c>0, we can define the set
P={A:[−c,0]→X,A is bounded and measurable}, |
and establish the space P with the norm
‖A‖P=sups∈[−c,0]|A(s)|,for allA∈P. |
Let us define the space
Ph={A:(−∞,0]→Xsuch that for any c>0,A|[−c,0]∈Pand∫0−∞h(t)supt≤s≤0A(s)dt<∞}. |
If Ph is furnished with the norm
‖A‖Ph=∫0−∞h(t)supt≤s≤0‖A(s)‖dt,∀A∈Ph, |
then (Ph,‖⋅‖Ph) is a Banach space. Next, we introduce the available space
¯Ph={v:(−∞,a]→Xsuch thatv|[0,a]∈C((tk,tk+1],X),v|(−∞,0]=ϕ∈Ph}, |
which has the norm
‖x‖¯Ph=sups∈[0,a]‖v(s)‖+‖ϕ‖Ph,x∈¯Ph. |
Definition 2.6. [27] If v:(−∞,a]→X,a>0, such that ϕ∈Ph. The situations listed below are accurate for all τ∈[0,a],
1) vτ∈Ph;
2) Two functions, ζ1(τ),ζ2(τ)>0, are such that ζ1(τ):[0,∞)→[0,∞) is a continuous function and ζ2(τ):[0,∞)→[0,∞) is a locally bounded function which are independent to v(⋅) whereas
‖vτ‖Ph≤ζ1(τ)sup0<s<τ‖v(s)‖+ζ2(τ)‖ϕ‖Ph; |
3) ‖v(τ)‖≤H‖vτ‖Ph, where H>0 is a constant.
Before introducing the mild solution of evolution Eq (1.1), we have to establish the following Lemmas.
Lemma 3.1. Let Iαs be the left R-L integral of order α and f(t) is integrable function defined for t≥s≥0. Then,
∫∞se−λtIαsf(t)dt=λ−α∫∞se−λtf(t)dt. |
Proof. From the Definition 2.2 and the rule of converting double integral to single integral, we get
∫∞se−λtIαsf(t)dt=∫∞se−λt∫ts(t−τ)α−1f(τ)dτdt=1Γ(α)∫∞sf(τ)dτ∫∞τe−λt(t−τ)α−1dt=1Γ(α)∫∞sf(τ)e−λτdτ∫∞0tα−1e−λtdt=λ−α∫∞se−λtf(τ)dτ. |
The proof is over.
Lemma 3.2. Let 1<α≤2 and h:J→X be an integrable function. Then, the mild solution to our problem (1.1) possess the form
u(t)={ϕ(t),t∈(−∞,0],Rq(t)ϕ(0)+∫t0Rq(s)uods+∫t0(t−s)q−1Tq(t,s)h(s)ds,t∈[0,t1],μk(t,u(t)),t∈∪mk=1(tk,sk],Rq(t−sk)μk(sk,u(sk))+∫tskRq(y−sk)ξk(sk,u(sk))dy+∫tsk(t−y)q−1Tq(t−y)h(y)dy,t∈∪mk=1(sk,tk+1], |
where 1/2<q=α2≤1,
Rq(t)=∫∞0Mq(θ)R(tqθ)dθ,Tq(t,s)=q∫∞0θMq(θ)T((t−s)qθ)dθ, |
and Mq is a probability density function defined by Definition 2.5.
Proof. Using Lemma 2.1 with operating by Iαsr on both sides to the fractional differential equation in (1.1), we arrive at
u(t)=Iαsk[Au(t)+h(t)]+c1,k(t−sk)+c0,k, | (3.1) |
where c1,k,c0,k∈R,k=0,1,…,m are constants to be determined.
● For t∈[0,t1]: By taking ρ→1 to the results given in Lemma 5 in [28], we have
u(t)=Rq(t)ϕ(0)+∫t0Rq(s)uods+∫t0(t−s)q−1Tq(t,s)h(s)ds. |
● For t∈(t1,s1]: We obtain
u(t)=μ1(t,u(t)) andu′(t)=ξ1(t,u(t)). |
● For t∈(s1,t2]: The problem (1.1) becomes
cDαs1u(t)=Au(t)+h(t),u(s1)=μ1(s1,u(s1)),u′(s1)=ξ1(s1,u(s1)). |
In this interval, Eq (3.1) becomes
u(t)=Iαs1[Au(t)+h(t)]+c1,1(t−s1)+c0,1. |
Considering the past impulsive conditions, we get
c0,1=μ1(s1,u(s1))andc1,1=ξ1(s1,u(s1)), |
which imply that
u(t)=Iαs1[Au(t)+h(t)]+ξ1(s1,u(s1))(t−s1)+μ1(s1,u(s1)). |
Multiplying both sides by e−λt followed by integrating from s1 to ∞, we achieve
U(λ)=λ−α{AU(λ)+H(λ)}+λ−1e−λs1μ1(s1,u(s1))+λ−2e−λs1ξ1(s1,u(s1)), |
where
U(λ)=∫∞s1u(t)e−λtdtandH(λ)=∫∞s1h(t)e−λtdt. |
Given that (λαI−A)−1 exists, then λα∈ρ(A). We obtain
U(λ)=(λαI−A)−1{λα−1e−λs1μ1(s1,u(s1))+λα−2e−λs1ξ1(s1,u(s1))+H(λ)}=λq−1e−λs1∫∞0e−λqtR(t)μ1(s1,u(s1))dt+λq−2e−λs1∫∞0e−λqtR(t)ξ1(s1,u(s1))dt+∫∞0e−λqtT(t)H(λ)dt. |
Let Ψq(θ)=qθq+1Mq(θ−q) be defined for θ∈(0,∞) and q∈(12,1). Then,
∫∞0e−pθΨq(θ)dθ=e−pq, |
which can be used to calculate the first term with replacing t by sq as
λq−1e−λs1∫∞0e−λqtR(t)μ1(s1,u(s1))dt=q∫∞0(λs)q−1e−(λs)qR(sq)e−λs1(μ1s1,u(s1))ds=−1λ∫∞0dds(e−(λs)q)R(sq)e−λs1(μ1s1,u(s1))ds=∫∞0∫∞0θΨq(θ)e−λsθR(sq)e−λs1(μ1s1,u(s1))dθds=∫∞0e−λ(x+s1){∫∞0Ψq(θ)R((xθ)q)μ1(s1,u(s1))dθ}dx=∫∞0e−λ(x+s1){∫∞0Mq(θ)R(xqθ)μ1(s1,u(s1))dθ}dx=∫∞0e−λ(x+s1)Rq(x)μ1(s1,u(s1))dx=∫∞s1e−λtRq(t−s1)μ1(s1,u(s1))dt. |
By using Lemma 3.1 with α=1, we get
λq−2∫∞0e−λqtRq(t)e−λs1ξ1(s1,u(s1))dt=∫∞s1e−λt{∫ts1Rq(y−s1)ξ1(s1,u(s1))dy}dt. |
Finally, we can write
∫∞0e−λqtT(t)H(λ)dt=q∫∞0e−(λs)qT(sq)sq−1H(λ)ds=q∫∞0∫∞0e−λsθΨq(θ)T(sq)sq−1H(λ)dθds=q∫∞0∫∞s1∫∞0θ−qe−λxΨq(θ)T((xθ)q)xq−1e−λyh(y)dθdydx=q∫∞0∫∞s1∫∞0e−λ(x+y)θMq(θ)T(xqθ)xq−1h(y)dθdydx=q∫∞s1∫∞y∫∞0e−λtθMq(θ)T((t−y)qθ)(t−y)q−1h(y)dθdtdy=∫∞s1∫∞ye−λt(t−y)q−1Tq(t−y)h(y)dtdy=∫∞s1e−λt{∫ts1(t−y)q−1Tq(t−y)h(y)dy}dt. |
In conclusion, we can write
∫∞s1e−λtu(t)dt=∫∞s1e−λt{Rq(t−s1)μ1(s1,u(s1))+∫ts1Rq(y−s1)ξ1(s1,u(s1))dy+∫ts1(t−y)q−1Tq(t−y)h(y)dy}dt. |
Therefore, by taking the inverse Laplace transform, we have
u(t)=Rq(t−s1)μ1(s1,u(s1))+∫ts1Rq(y−s1)ξ1(s1,u(s1))dy+∫ts1(t−y)q−1Tq(t−y)h(y)dy. |
● For t∈(sk,tk+1],k=2,3,…,m: In a similar manner, we can write
u(t)=Rq(t−sk)μk(sk,u(sk))+∫tskRq(y−sk)ξk(sk,u(sk))dy+∫tsk(t−y)q−1Tq(t−y)h(y)dy. |
Consequently, we get the solution from the earlier (1.1). Direct calculations show that the opposite results are true. The proof is completed.
Remark 3.1. [28] From linearity of R(t) and T(t) for all t≥0, it is clearly to deduce that Rq(t) and Tq(t,s) are also linear operators where 0<s<t. Therefore, the proofs of all next Lemmas are same when taking ρ approaches 1.
Lemma 3.3. [28] The following estimates for Rq(t) and Tq(t,s) are verified for any fixed t≥0 and 0<s<t
|Rq(t)x|≤ϖ|x|and|Tq(t,s)x|≤ϖaqΓ(2q)|x|. |
Lemma 3.4. [28] The operators Rq(t) and Tq(s,t) are strongly continuous for every 0<s<t and t>0.
Lemma 3.5. [28] Assume that R(t) and T(t,s) are compact for every 0<s<t. Then, the operators Rq(t) and Tq(s,t) are compact for every 0<s<t.
Define the operator N:¯Ph→¯Ph as follows
N(u)(t)={ϕ(t),t∈(−∞,0],Rq(t)ϕ(0)+∫t0Rq(y)uody+∫t0(t−y)q−1Tq(t,y)h(y,uy)dy,t∈[0,t1],μk(t,u(t)),t∈∪mk=1(tk,sk],Rq(t−sk)μk(sk,u(sk))+∫tskRq(y−sk)ξk(sk,u(sk))dy,+∫tsk(t−y)q−1Tq(t−y)h(y,uy)dy,t∈∪mk=1(sk,tk+1]. |
Let ϰ(⋅):(−∞,a]→X be the function denoted by
ϰ(t)={ϕ(t),t∈(−∞,0],0,t∈(0,a]. |
Plainly, ϰ(0)=ϕ(0). For each z∈C([0,a],X) with z(0)=0, we indicate by ϑ to the function defined as
ϑ(t)={0,t∈(−∞,0],z(t),t∈[0,a]. |
If u(⋅) satisfies that u(t)=N(u)(t) for all t∈(−∞,a], we can decompose that u(t)=ϑ(t)+ϰ(t), t∈(−∞,a], it denotes ut=ϑt+ϰt for every t∈(−∞,a] and the function z(⋅) satisfies
z(t)={Rq(t)ϕ(0)+∫t0Rq(y)uody+∫t0(t−y)q−1Tq(t,y)h(y,ϑy+ϰy)dy,t∈[0,t1],μk(t,ϑ+x),t∈∪mk=1(tk,sk],Rq(t−sk)μk(sk,ϑ+ϰ)+∫tskRq(y−sk)ξk(sk,ϑ+ϰ)dy,+∫tsk(t−y)q−1Tq(t−y)h(ϑy+ϰy)dy,t∈∪mk=1(sk,tk+1]. |
Set the space Υ={z∈C([0,a],X),z(0)=0} equipped the norm
‖z‖Υ=supt∈[0,a]‖z(t)‖. |
Therefore, (Υ,‖⋅‖Υ) is a Banach space. Assume that the operator G:Υ→Υ is formulated as follows:
G(z)(t)={Rq(t)ϕ(0)+∫t0Rq(y)uody+∫t0(t−y)q−1Tq(t,y)h(y,ϑy+ϰy)dy,t∈[0,t1],μk(t,ϑ+x),t∈∪mk=1(tk,sk],Rq(t−sk)μk(sk,ϑ+ϰ)+∫tskRq(y−sk)ξk(sk,ϑ+ϰ)dy,+∫tsk(t−y)q−1Tq(t−y)h(ϑy+ϰy)dy,t∈∪mk=1(sk,tk+1]. |
The operator N seems to have a fixed point is equivalent to G has a fixed point. Thus, we proceed to prove that G has a fixed point.
Now, we make the following assumptions:
(E1) The function h:[0,a]×Ph→X is a continuous and μk,ξk:[tk,sk]×X→X are continuous functions for all k=1,2,…,m;m∈N.
(E2) There is a constant Ω>0 satisfying
‖h(t,ut)−h(t,vt)‖≤Ω‖ut−vt‖Ph. |
(E3) There exist δk,δ∗k>0;k=1,2,…,m;m∈N such that
‖μk(t,u)‖≤δkand‖ξk(t,u)‖≤δ∗k. |
(E4) There are positive constants Dk,D∗k,k=1,2,…,m;m∈N such that
‖μk(t,u1)−μk(t,u2)‖≤Dk‖u1−u2‖, |
‖ξk(t,u1)−ξk(t,u2)‖≤D∗k‖u1−u2‖. |
(E5) There exists a continuous function g(t):[0,a]→[0,∞) such that, for any (t,ut)∈[0,a]×Ph, it satisfies
‖h(t,ut)‖≤g(t)‖ut‖Ph. |
The brief constants that will be utilized later to streamline handling, are listed as follow
E(q)=tq+11qΓ(2q),Ek(q)=a(a−sk)qqΓ(2q),B=ϖΩζ∗1,¯B=ϖgζ∗1E(q),Bk=ϖHζ∗1[Dk+D∗k(a−sk)],¯Bk=ϖgζ∗1Ek(q),O=ϖ(‖ϕ(0)‖+‖uo‖t1+E(q)Ωζ∗2‖ϕ‖Ph+E(q)c),Ok=ϖ(δk+δ∗k(a−sk)+Ek(q)Ωζ∗2‖ϕ‖Ph+Ek(q)c),Q=ϖ(‖ϕ(0)‖+‖uo‖t1+E(q)ζ∗2g‖ϕ‖Ph),Qk=ϖ(δk+δ∗k(a−sk)+Ek(q)ζ∗2g‖ϕ‖Ph) |
where k=1,2,…,m;m∈N.
Lemma 4.1. Assume that the requirement (E2) is met by c=maxt∈[0,a]|h(t,0)|. Ponder about the expressions ζ∗1=supt∈[0,a]ζ1(t) and ζ∗2=supt∈[0,a]ζ2(t) where ζ1(⋅) and ζ2(⋅) are established in Definition 2.6. Then,
‖h(t,ϑt+ϰt)‖≤Ω(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)+c. |
Proof. Regarding Definition 2.6 and the presumption (E2). Then,
‖h(t,ϑt+ϰt)‖=‖h(t,ϑt+ϰt)−h(t,0)+h(t,0)‖≤‖h(t,ϑt+ϰt)−h(t,0)‖+‖h(t,0)‖≤Ω‖ϑt+ϰt‖Υ+c≤Ω(ζ1(t)supt∈[0,a]‖ϑ(t)‖+ζ2(t)‖ϕ‖Ph)+c≤Ω(ζ1(t)‖z‖Υ+ζ2(t)‖ϕ‖Ph)+c≤Ω(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)+c. |
This ends the proof.
Lemma 4.2. Suppose that the statement (E5) is satisfied with g=supt∈[0,a]g(t). Let ζ∗1=supt∈[0,a]ζ1(t) and ζ∗2=supt∈[0,a]ζ2(t) where ζ1(⋅) and ζ2(⋅) are outlined in Definition 6. Then,
‖h(t,ϑt+ϰt)‖≤ℓ(t)≤ℓ, |
where
ℓ=supt∈[0,a]ℓ(t)=supt∈[0,a]{g(t)(ζ1(t)‖z‖Υ+ζ2(t)‖ϕ‖Ph)}=g(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph). |
Proof. By the same way in Lemma 4.1, we can easily reach the desired result.
Theorem 4.1. Consider the assertions (E1)−(E4) hold and
Λ=maxk{BE(q),DkHζ∗1,Bk+BEk(q)}. |
Then, the fractional evolution equation with non-instantaneous impulsive (1.1) has a unique mild solution on (−∞,a] if Λ<1.
Proof. To show that the operator G maps bounded subset of Υ into bounded subset in Υ, we set
Υr={z∈Υ:‖z‖Υ≤r}, |
where
r≥maxk{O1−BE(q),δk,Ok1−BEk(q)}. |
Then, for any z∈Υr and in spite of (E2) and (E3) and Lemma 4.1. Correspondingly, three situations are taken into consideration.
● Case Ⅰ. Whenever t∈[0,t1], we have
‖G(z)(t)‖Υ≤ϖ(‖ϕ(0)‖+‖uo‖t1)+ϖt1Γ(2q)∫t0(t−s)q−1‖h(y,ϑy+ϰy)‖dy≤ϖ[‖ϕ(0)‖+‖uo‖t1+tqt1qΓ(2q){Ω(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)+c}]≤ϖ[‖ϕ(0)‖+‖uo‖t1+E(q){Ω(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)+c}]≤O+E(q)B‖z‖Υ≤O+E(q)Br≤r. |
● Case Ⅱ. Whenever t∈(tk,sk],k=1,…,m;m∈N, we have
‖G(z)(t)‖Υ=‖μk(t,ϑ+ϰ)‖≤δk. |
● Case Ⅲ. Whenever t∈(sk,sk+1],k=1,…,m;m∈N, we have
‖G(z)(t)‖Υ≤ϖ[δk+δ∗k(t−sk)+aΓ(2q)∫tsk(t−y)q−1‖h(y,ϑy+ϰy)‖dy]≤ϖ[δk+δ∗k(a−sk)+a(a−sk)qqΓ(2q){Ω(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)+c}]≤ϖ[δk+δ∗k(a−sk)+Ek(q){Ω(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)+c}]≤Ok+Ek(q)Br≤r. |
For the aforementioned, we acquire ‖G(z)(t)‖Υ≤r. Thus, the operator G maps bounded subset into bounded subset in Υ.
Now, we prove that the operator G is a contraction mapping. Certainly, consider z,z∗∈Υ. Then, there still are the subsequent situations.
● Case Ⅰ. For any t∈[0,t1], we have
‖G(z)(t)−G(z∗)(t)‖≤ϖt1Γ(2q)∫t0(t−s)q−1‖h(y,ϑy+ϰy)−h(y,ϑ∗y+ϰy)‖dy≤Ωϖt1Γ(2q)∫t0(t−s)q−1‖ϑy−ϑ∗y‖Phdy≤Ωϖt1Γ(2q)ζ∗1‖z−z∗‖Υ∫t0(t−s)q−1dy≤Ωϖtq+11qΓ(2q)ζ∗1‖z−z∗‖Υ=BE(q)‖ϑ−ϑ∗‖Υ. |
● Case Ⅱ. For any t∈(tk,sk],k=1,…,m;m∈N, we have
‖G(z)(t)−G(z∗)(t)‖=‖μk(t,ϑ+ϰ)−μk(t,ϑ∗+ϰ)‖≤Dk‖ϑ−ϑ∗‖Υ≤DkH‖zt−z∗t‖Ph≤DkHζ∗1‖z−z∗‖Υ=DkHζ∗1‖ϑ−ϑ∗‖Υ. |
● Case Ⅲ. For any t∈(sk,sk+1],k=1,…,m;m∈N, we have
‖G(z)(t)−G(z∗)(t)‖≤‖Rq(t−sk)‖‖μk(sk,ϑ+ϰ)−μk(sk,ϑ∗+ϰ)‖+∫tsk‖Rq(y−sk)‖‖ξk(sk,ϑ+ϰ)−ξk(sk,ϑ∗+ϰ)‖dy+∫tsk(t−y)q−1‖Tq(t−y)‖‖h(ϑy+ϰy)−h(ϑ∗y+ϰy)‖dy≤ϖDk‖ϑ−ϑ∗‖Υ+ϖD∗k∫tsk‖ϑ(y)−ϑ∗(y)‖Υdy+ϖaΓ(2q)Ω∫tsk(t−y)q−1‖ϑy−ϑ∗y‖Phdy≤ϖDkH‖zy−z∗y‖Ph+ϖD∗kH∫tsk‖zy−z∗y‖Phdy+ϖaΓ(2q)Ω∫tsk(t−y)q−1‖ϑy−ϑ∗y‖Phdy≤ϖHζ∗1[Dk+D∗kEk(q)]‖z−z∗‖Υ+ϖaΓ(2q)Ωζ∗1∫tsk(t−y)q−1‖z−z∗‖Υdy≤ϖHζ∗1[Dk+D∗k(a−sk)+a(a−sk)qΩ qΓ(2q)H]‖z−z∗‖Υ=(Bk+BEk(q))‖z−z∗‖Υ=(Bk+BEk(q))‖ϑ−ϑ∗‖Υ. |
For the aforementioned, we may write
‖G(z)(t)−G(z∗)(t)‖Υ≤Λ‖ϑ−ϑ∗‖Υ. |
Amid the existing circumstances, Λ<1 shows that the operator G is a contraction. This suggests that the problem (1.1) has a unique solution on (−∞,a] relying on the Banach contraction mapping principle.
Remark 4.1. In viewing our problem, it is very difficult to obtain the exact solution and so it is useful to investigate some properties of the solutions, especially the uniqueness. The previous theorem shaw that the mild solution of the problem (1.1) is unique under the assumptions (E1)−(E4) and Λ<1. This enables us to apply our results to real-life problem or phenomena as in the last section.
Assume that the operator G is divided as a sum of the two operators Gi:Υ→Υ,i=1,2 as
G=G1(z)+G2(z) | (4.1) |
where,
G1(z)(t)={Rq(t)ϕ(0)+∫t0Rq(y)uody+∫t0(t−y)q−1Tq(t,y)h(y,ϑy+ϰy)dy,t∈[0,t1],0,t∈∪mk=1(tk,sk],∫tsk(t−y)q−1Tq(t−y)h(ϑy+ϰy)dy,t∈∪mk=1(sk,tk+1] |
and
G2(z)(t)={0,t∈[0,t1],μk(t,ϑ+x),t∈∪mk=1(tk,sk],Rq(t−sk)μk(sk,ϑ+ϰ)+∫tskRq(y−sk)ξk(sk,ϑ+ϰ)dy,t∈∪mk=1(sk,tk+1]. |
Theorem 4.2. Suppose the hypotheses (E1)and(E3)−(E5) are correct. Then the fractional evolution equation with non-instantaneous impulsive (1.1) has at least one mild solution on (−∞,a] if Δ<1 where Δ is given by
Δ=maxk{¯B,¯Bk}. |
Proof. Let the operators G1 and G2 be defined as (4.1). Setting g=supt∈[0,a]|g(t)|. Let us define the closed ball Υρ={z∈Υ:‖z‖Υ≤ρ} with radius
ρ≥maxk{Q1−¯B,δk,Qk1−¯Bk}. |
Then, for u,v∈Υρ, we claim that ‖G1(z)(u)+G2(z)(v)‖≤ρ which concludes that G1(u)+G1(v)∈Υρ. To verify our claiming, we show that G maps bounded sets of Υ into bounded sets in Υ, for any ρ≥0. Then for any z∈Υρ and in light of (E3),(E5) and Lemma 4.2, we have three cases
● Case Ⅰ. For any t∈[0,t1], we have
‖G(z)(t)‖Υ≤ϖ(‖ϕ(0)‖+‖uo‖t1)+ϖt1Γ(2q)∫t0(t−s)q−1‖h(y,ϑy+ϰy)‖dy≤ϖ[‖ϕ(0)‖+‖uo‖t1+t1Γ(2q)∫t0(t−s)q−1{g(y)(ζ1(y)‖z‖Υ+ζ2(y)‖ϕ‖Ph)}dy]≤ϖ[‖ϕ(0)‖+‖uo‖t1+E(q){g(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)}]≤Q+E(q)¯B‖z‖Υ≤Q+¯Bρ≤ρ. |
● Case Ⅱ. For any t∈(tk,sk],k=1,…,m;m∈N, we have
‖G(z)(t)‖Υ=‖μk(t,ϑ+ϰ)‖≤δk≤ρ. |
● Case Ⅲ. For any t∈(sk,sk+1],k=1,…,m;m∈N, we have
‖G(z)(t)‖Υ≤ϖ[δk+δ∗k(t−sk)+aΓ(2q)∫tsk(t−y)q−1‖h(y,ϑy+ϰy)‖dy]≤ϖ[δk+δ∗k(a−sk)+aΓ(2q)∫tsk(t−y)q−1{g(y)(ζ1(y)‖z‖Υ+ζ2(y)‖ϕ‖Ph)}dy]≤ϖ[δk+δ∗k(a−sk)+a(a−sk)qqΓ(2q){g(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)}]≤ϖ[δk+δ∗k(a−sk)+Ek(q){g(ζ∗1‖z‖Υ+ζ∗2‖ϕ‖Ph)}]≤Qk+¯Bkρ≤ρ. |
By virtue of the above, we obtain ‖G(z)(t)‖Υ≤ρ. Thus the operator G maps bounded sets into bounded sets in Υ.
The following step is to confirm that the operator G2 maps bounded sets into equicontinuous sets in Υ. In light of the situation (E1), G2 is continuous. The following scenarios are therefore possible.
● Case Ⅰ. For each tk≤γ1<γ2≤sk and z∈Υρ, we have
‖G2(z)(γ2)−G2(z)(γ1)‖≤‖μk(γ2,ϑ+ϰ)−μk(γ1,ϑ+ϰ)‖. |
Due to the continuity of μ(t,u(t)). It is clear that the above inequality approaches zero when letting γ2→γ1.
● Case Ⅱ. For any sk≤γ1<γ2≤tk+1,k=1,…,m;m∈N and z∈Υρ, we have
‖G2(z)(γ2)−G2(z)(γ1)‖≤δk‖Rq(γ2−sk)−Rq(γ1−sk)‖+δ∗k∫γ2γ1‖Rq(y−sk)‖dy≤δk‖Rq(γ2−sk)−Rq(γ1−sk)‖+δ∗kϖ(γ2−γ1). |
Due to compactness of operator Rq(y) and Tq(t,y) (see Lemma 3.5), we infer that ‖G2(z)(γ1)−G2(z)(γ2)‖→0 as γ2→γ1. Thus, G2 is a relatively compact on Υ. By Arezela Ascoli Theorem the operator G2 is completely continuous on Υρ. The only thing left to do is provide evidence that G1 is a contraction mapping. Thus, two cases are thought about.
● Case Ⅰ. For any t∈[0,t1],k=1,…,m;m∈N and z∈Υρ, we have
‖G1(z)(t)−G1(z∗)(t)‖≤ϖt1Γ(2q)∫t0(t−y)q−1‖h(y,ϑy+ϰy)−h(y,ϑ∗y+ϰy)‖dy≤ϖt1Γ(2q)∫t0(t−y)q−1g(y)‖ϑy−ϑ∗y‖Phdy≤gϖt1Γ(2q)ζ∗1‖z−z∗‖Υ∫t0(t−s)q−1dy≤gϖtq+11qΓ(2q)ζ∗1‖z−z∗‖Υ=¯B‖ϑ−ϑ∗‖Υ. |
● Case Ⅱ. For any t∈(sk,tk+1],k=1,…,m;m∈N, we have
‖G1(z)(t)−G1(z∗)(t)‖≤∫tsk(t−y)q−1‖Tq(t−y)‖‖h(ϑy+ϰy)−h(ϑ∗y+ϰy)‖dy≤ϖaΓ(2q)∫tsk(t−y)q−1g(y)‖ϑy−ϑ∗y‖Phdy≤aΓ(2q)ϖgζ∗1‖z−z∗‖Υ∫tsk(t−y)q−1dy≤a(a−sk)q qΓ(2q)ϖgζ∗1‖z−z∗‖Υ=¯Bk‖z−z∗‖Υ. |
As a sense, the fractional evolution equation with non-instantaneous impulsive (1.1) has at least one mild solution on Υ, according to the Krasnoselskii Theorem. The evidence is now complete.
Remark 4.2. Also, it is useful to investigate the existence of the solution instead of the its uniqueness. The theorem above shaw that the mild solution of the problem (1.1) exists under the assumptions (E1)−(E3) and (E5) with Δ<1.
Presume the following fractional wave equation with impulsive effect and infinite delay
{u(t,x)=13sint,t∈(−∞,0],x∈[0,π],cDαtu(t,x)=∂2∂x2u(t,x)+h(t,ut),t∈(0,25]∪(45,1],x∈[0,π],u(t,x)=17t32+15sinu(t),t∈(25,45]x∈[0,π],u′(t)=314t12+18cosu(t),t∈(25,45]x∈[0,π],u(t,0)=u(t,π)=0,t∈[0,1],u′(0,x)=32e−x3,x∈[0,π]. |
Consider that
J=[0,1],0=t0=s0<t1=25<s1=45<t2<1=a,α=32⇒q=34,u0=32e−x3,A=∂2∂x2,x∈[0,π],H=116.Whileζ1(t)=35t32⇒ζ∗1=35{supt∈(0,1]t32}≤ζ1(1)=35.If we takeϖ=1⇒‖Tq(t,s)‖≤1Γ(34),0<s<t≤1. |
Case Ⅰ. Banach fixed point theorem.
In order to explain Theorem 4.1, we obtain:
h(t,ut)=t38√t+1+19ut. | (5.1) |
Clearly, h:[0,1]×Ph→R is continuous and satisfying, for ut,vt∈Ph, that
‖h(t,ut)−h(t,vt)‖≤19‖ut−vt‖Ph, |
it suggests that Ω=19. For all t∈(25,45] and u,v∈R, we get
‖μ(t,u)−μ(t,v)‖≤15‖sinu−sinv‖≤15‖u−v‖,‖ξ(t,u)−ξ(t,v)‖≤18‖cosu−cosv‖≤18‖u−v‖. |
As you can see, the Theorem 4.1 condition (E4) is satisfied with
Dk=15andD∗k=18. |
In summary, we have
Λ=maxk{BE(q),DkHζ∗1,Bk+BEk(q)}={0.0202,0.0075,0.0384}=0.0384<1. |
Thus all assumptions of this theorem are verified. Therefore, the problem (1.1) has a unique mild solution on (−∞,1].
Case Ⅱ. Krasnoselskii's theorem.
To realize Theorem 4.2, take h(t,ut) as given in (5.1). Therefore, g(t)=t38√t+1 is increasing function which admits the hypothesis (E5) with
‖g‖≤g(1)=18√2. |
These calculate that
Δ=maxk{¯B,¯Bk}=maxk{0.0161,0.0239}=0.0239<1. |
Since every requirements of Theorem 4.2 are met, it follows that there exists at least one mild solution of (1.1) on (−∞,1].
We analyzed a set of impulsive fractional evolution equations with infinite delay in the current work. Current functional analysis methodologies serve as the foundation for our conclusions. By using the unbounded operator A as the generator of the strongly continuous cosine family, we were able to suggest a mild solution for the suggested problem. In the instance of problem (1.1), we had two successful outcomes: While the second argument focuses on whether there are solutions for the given problem, the first argument concentrated on the existence and uniqueness of the solution.
The first result, which is built on a Banach fixed point theorem, provides criteria for ensuring that the problem at hand has no prior solutions by requiring the usage of h(t,ut) to satisfy the classic Lipschitz condition.
The second argument was based on a Krasnoselskii's theorem, which allows h(t,ut) to behave as ‖h(t,ut)‖≤g(t)‖ut‖Ph. The instruments used by fixed point theory in the scenario with simple assumptions. Finally, a numerical example that examines a function that satisfies all the prerequisites was provided to illustrate our conclusion.
In the next paper, we will study the controllability of mild solution to fractional evolution equations with an infinite time-delay and nonlocal condition by applying Krasnoselskii's theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness.
The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia has funded this project, under grant no. (KEP-PhD: 34-130-1443).
The authors declare that they have no conflicts of interest.
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