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IMPULSIVE EVOLUTION FRACTIONAL
DIFFERENTIAL EQUATIONS WITH SECTORIAL OPERATOR
Jorge Junior, José Vanterler, E Capelas de Oliveira
To cite this version:
Jorge Junior, José Vanterler, E Capelas de Oliveira. THE e-POSITIVE MILD SOLUTIONS FOR
IMPULSIVE EVOLUTION FRACTIONAL DIFFERENTIAL EQUATIONS WITH SECTORIAL
OPERATOR. 2020. �hal-02562927�
SECTORIAL OPERATOR Jorge F. Junior 1
J. Vanterler da C. Sousa 2,∗ and E. Capelas de Oliveira3
1,2,3 Department of Applied Mathematics, Imecc-State University of Campinas 13083-859, Campinas, SP, Brazil
(Communicated by the associate editor name)
ABSTRACT. In this paper, we will investigate the existence of globale- positive mild solutions to the initial value problem with nonlinear impulsive fractional evolution differential equation involving the theory of sectorial operators. To obtain the result, we used Kuratowski’s non-compactness measure theory, the Cauchy criterion and the Gronwall inequality.
1. Introduction. In this paper, we consider the initial value problem (IVP) with nonlinear impulsive fractional evolution differential equation, given by (1)
CDα0+ξ(t) +Aξ(t) = f t, ξ(t)
, t∈J∞, t6=tk,
∆ξ|t=tk = Ik(ξ(tk)), k∈N, ξ(0) = x0
where CDα0+(·) is Caputo fractional derivative of order 0< α <1,ξ:J →Ω,A: D(A)⊂Ω→Ω is a sectorial operator of type (M, θ, α, ρ) in Ω, f ∈C(J∞×Ω,Ω),
∆ξ|t=tk = ξ(t+k)−ξ(t−k) being that ξ(t+k) and ξ(t−k) represent the limits on the right and left of ξ(t) in t = tk, respectively, Ik : Ω → Ω (k ∈ N) are impulsive functions andx0 ∈Ω. Furthermore, be 0< t1 < t2 <· · ·< tm· · ·, tm → ∞with m → ∞, a partition in J∞, define J∞0 = J∞\{t1, t2, . . . , tm, . . .}, J0 = [0, t1] e Jk = (tk, tk+1] (k∈N). Further consider, λ1 the small positive real eigenvalue of the linear operatorAand be e1∈D(A) the corresponding positive eigenvector.
In 2012, Shu and Wang[30], considered the fractional semi-linear integrodiffer- ential equation in Banach spaceX, given by
(2)
D0+α u(t) = Au(t) +f(t, u(t)) + Z t
0
q(t−s)g(s, u(s))ds u(0) +m(u) = u0∈X
u0(0) +n(u) = u1∈X
where Dα0+(·) is Caputo fractional derivative with 1 < α < 2, A is a sectorial operator of type (M, θ, α, µ), defined from the domains D(A) ⊂ X in to X, the nonlinear mapf, gcontinuous functions defined from [0, T]×X→X,q: [0, T]→X
1991 AMS Mathematics subject classification. Primary: 26A33, 34A08; Secondary:
34A12,47H08.
Keywords and phrases. Fractional impulsive evolution equations, globale-positive mild solu- tions, Gronwall inequality, measure noncompactness.
The second author is supported by PNPD-CAPES grant no88882.305834/2018-01.
∗Corresponding author: vanterler@ime.unicamp.br.
1
is an integrable function on [0, T] and the nonlocal condition m : X → X, n:X →X are two continuous functions.
As is well known, a mild solution to system Eq.(2) satisfies the following equation u(t) = Sα(t) (u0−m(u)) +Kα(t) (u1−n(u))
+ Z t
0
Tα(t−s)
f(s, u(s)) + Z s
0
q(s−τ)g(τ, u(τ))dτ
ds.
In this sense, the authors investigated the existence and uniqueness of a mild solution for Eq.(2), using the Krasnoselskii theorem, Arzel`a-Ascoli theorem and the fixed point theorem.
The importance of fractional differential equations in the theoretical and ap- plication scope is notable. The number of works published in the area of frac- tional differential equations, comes in an important and interesting growth in the scientific community [2, 4, 6, 7, 8, 17, 10, 18, 19, 21]. The fact that nu- merous researchers justify that working with fractional operators (derivative and integral) it is possible to obtain better results when comparing with classical op- erators, when it comes to applications[1, 9, 13, 14, 15, 16, 29, 32, 33]. From a theoretical point of view, there is still a vast path to be explored, since the theory is being constructed in innumerable directions of the theory of fractional differential equations, especially involving sectorial and almost sectorial operators [12, 24, 25, 26, 28, 34, 35, 36, 40]. In addition, numerous questions still need to be answered, which will enrich the theory in general. Here, we highlight two relevant works in the theory of fractional differential equations involving sectorial and almost sectorial operators[3,5,36,39, 40, 41].
In 2013, Yang and Liang[39], using fixed point theorems and the analytical semi- group theory, investigated the presence of positive light solutions to the Cauchy problem of Caputo’s fractional evolution equations in Banach spaces. Examples were discussed, in order to validate the results obtained. In 2013 Wang et al. [35], performed out a study on optimal controls and listing of nonlinear fractional impul- sive evolution equations. In this work, they dedicated to investigating the existence of mild continuous solutions by parts and applications of fractional impulsive par- abolic control. In 2015, Wang et al. [32], investigated the existence of positive mild solutions of fractional evolution equations with nonlocal conditions of order 1< α <2, using Schauder’s fixed point theorem and the Krasnoselskii fixed point theorem. In the same year, Ding and Ahmad [5], dedicated themselves to inves- tigating the existence and uniqueness of mild solutions for equations of fractional evolution with almost sectorial operators. As highlighted above, numerous studies have been published, some important and relevant to the theory.
Motivated by the works above, and in order to contribute with new results for the theory of fractional differential equations, in particular, of impulsive evolution, in this present paper, we have as main objective, to investigate the existence ofe- positive mild solutions for a initial value problem with nonlinear impulsive fractional evolution differential equation involving the theory of sectorial operators. In order to obtain the result, we will make use of Kuratowski’s non-compactness measurement theory and Gronwall’s inequality.
The article is organized as follows: in section 2, we present the definitions of the ψ-Riemann-Liouville fractional integral and ψ-Hilfer fractional derivative, and two
particular cases, which were used to formulate the problem investigated. We present the Gronwall theorem (inequality) and its respective lemma. On the other hand, we discuss a brief part of the theory of sectorial operators and some fundamental results. Finally, we approach the concept of Kuratowski’s noncompactness measure, with some essential results in obtaining the main result of this paper. In section 3, we investigate the main result of this paper, that is, the existence of e-positive mild solutions for Eq.(1), through Kuratowski’s noncompactness measure, using Cauchy’s criterion and Gronwall’s inequality.
2. Preliminaries. In this section, we will present some fundamental concepts and results that will be of paramount importance in obtaining our main result.
Consider the Banach space (Ω,k · k), and the intervalJ= [a, b]⊂Rwithn∈N. The continuous functions space, given by[24,25]
C(J,Ω) :={f :J→Ω; f : continuous}, with norm
kfkC := sup
t∈J
|f(t)|.
On the other hand, we have the space of the continuously differentiable functions given by,
Cn(J,Ω) :=n
f :J →Ω; f(n)∈C(J,Ω)o , endowed with the norm
kfkCn := sup
t∈J
|f(n)(t)|.
Note that the spaces defined above are Banach spaces.
Now consider, the intervalJ∞ = [0,∞). The space of the continuous functions by parts given by[38]
P C(J∞,Ω) :=
ξ:J∞→Ω; ξ(t) be continuous int6=tk, continuous left in t=tk and there is the limit on the right , ξ(t+k), ∀k∈N
whose norm is given bykξkP C = max
k∈N
sup
t∈Jk
kξ(t)k
, is a Banach space.
Definition 2.1. [38] Let Ω be a real Banach space. A non-empty, closed and convex subset Ω+⊂Ω is considered a cone if it meets the following conditions:
(i) Ifx∈Ω+ andλ>0, thenλx∈Ω+. (ii) Ifx∈Ω+ and−x∈Ω+, thenx= 0.
Every cone Ω+⊂Ω induces an order in Ω given by: x6y ⇔ y−x∈Ω+. Let J = [a, b] ⊂ R be a interval with −∞ 6 a < b 6 ∞ and let ψ(x) a monotonous increasing and positive function at (a, b), with derivative ψ0(x) be continuous at (a, b). The leftψ-Riemann-Liouville fractional integrals with respect
to theψfunction of af function inJ of orderα >0 is defined, by[20, 23, 27]
(3) Ia+α;ψf(x) = 1 Γ(α)
Z x a
ψ0(t)
ψ(x)−ψ(t)α−1
f(t)dt.
Analogously, the rightψ-Riemann-Liouville fractional integral is defined.
In particular, forψ(x) =x, we have the Riemann-Liouville fractional integral to the left, given by
Iaα+f(x) = 1 Γ(α)
Z x a
(x−t)α−1f(t)dt, x > a.
(4)
On the other hand, let n ∈ N and J = [a, b] ⊂ R an interval such that
−∞ 6 a < b 6 ∞. Further consider, the functions f, ψ ∈ Cn(J; R) so that ψ be increasing andψ0(x)6= 0, for every x∈J. Theψ-Hilfer fractional derivative to the left off, of ordern−1< α < nand type 0≤β≤1 is defined by[20,23,27]
HDα,β;ψa+ f(x) =Ia+β(n−α);ψ 1
ψ0(x) d dx
n
I(1−β)(n−α);ψ
a+ f(x).
Analogously, toψ-Hilfer fractional derivative right is defined.
In particular, for ψ(x) = x and taking the limit β → 1, we have the Caputo fractional derivative of, given by
(5) CDa+α f(x) =Ia+n−α d
dx n
f(x) = 1 Γ(n−α)
Z x a
(x−t)n−α−1f(n)(t)dt.
For details on how to obtain other particular cases for derivatives and fractional integrals, we suggest the work[27].
In the following, we will present two fundamental results, Theorem 2.2 and Lemma 2.3. However, your demonstration will not be presented here, but can be obtained from article[22].
Theorem 2.2. [22]Letξandν be two integrable functions andgcontinuous, with domain J = [a, b]. Let ψ ∈ C1(J) be an increasing function such that ψ0(t) 6= 0,
∀t∈J. Suppose that
(1) ξandν are non-negative;
(2) g be non-negative and non-decreasing.
If
ξ(t)6ν(t) +g(t) Z t
a
ψ0(τ)
ψ(t)−ψ(τ)α−1
u(τ)dτ, then
ξ(t)6ν(t) + Z t
a
∞
X
k=1
[g(t)Γ(α)]k Γ(αk) ψ0(τ)
ψ(t)−ψ(τ)αk−1
v(τ)dτ.
Lemma 2.3. [22] Under the hypothesis of the Theorem 2.2, let v be a non- decreasing function inJ = [a, b]. So, yields
ξ(t)6ν(t)Eα
g(t)Γ(α)h
ψ(t)−ψ(τ)iα ,
whereEα(t) =
∞
P
k=0
tk
Γ(αk+ 1), with<(α)>0, is Mittag-Leffler the function.
In order to investigate our results, we will work with the initial value problem Eq.(1), using Caputo fractional derivative, defined by Eq.(5).
Definition 2.4. [38]Leta, α∈R. A functionf : [a,∞)→Ω belongs to spaceCa,α
if exist a real numberp > αand a functiong∈C([a,∞); Ω) such thatf(t) =tpg(t).
Also, we sayf ∈Ca,αm for some positive integermiff(m)∈Ca,α.
LetAa densely operator in Ω satisfying the following conditions[30, 37]:
(1) For some 0< θ < π2,ρ+Sθ={ρ+λα; λ∈C,|arg(−λα)|< θ};
(2) There exists a constantMsuch that k(λI− A)−1k6 M
|λ−ρ|, λ /∈ρ+Sθ.
Definition 2.5. [30,37]A closed linear operator A:D⊂Ω→Ω is considered a sectorial operator of the type (M, θ, α, ρ) if exist 0< θ < π2,M>0 andρ∈Rsuch that theα-resolvent of theAexists outside the sector,
ρ+Sθ={ρ+λα; λ∈C,|arg(−λα)|< θ}
and
k(λαI− A)−1k6 M
|λα−ρ|, λα∈/ ρ+Sθ.
IfAis a sectorial operator of type (M, θ, α, ρ) then it is not difficult to see thatA is the infinitesimal generator of aα-resolvent familykTα(t)kt≥0in a Banach space, whereTα(t) = 1
2πi Z
C
eλtR(λα,A)dλ. Analogously, we will make the estimates for kSα(t)kt≥0andkKα(t)kt≥0, as we will present below.
The existence of soft solutions and the qualitative theory of evolution fractional equations are researched through operator-solutions[30,37],
Sα(t) = 1 2πi
Z
C
eλtλα−1R(λα,A)dλ and
Kα(t) = 1 2πi
Z
C
eλtλα−2R(λα,A)dλ,
C being an appropriate path andAa sectorial operator of the type (M, θ, α, ρ).
We will present and highlight the following two Lemma2.6and Lemma 2.7 Lemma 2.6. [30, 37] Let A a sectorial operator of type (M, θ, α, ρ), then, for kSα(t)k andt >0, the following estimates are valid:
(i) if ρ>0 and φ∈
max{θ,(1−α)π}, π2(2−α) , then
kSα(t)k 6
K1Me[K1(1+ρtα)]
1 α h
K0α1 −1i
π(sinθ)1+α1 (1 +ρtα) + Γ(α)M
π(1 +ρtα)|cosπ−φα |αsinθsinφ, (6)
beingK0=K0(θ, φ) = 1+ sinφ
sin(φ−θ) and K1=K1(θ, φ) = max
1, sinθ sin(φ−θ)
. (ii) If ρ <0 and φ∈
max{π2,(1−α)π}, π2(2−α) , then
kSα(t)k6 M[(1 + sinφ)α1 −1]
π|cosφ|1+1α + Γ(α)M π|cosφ||cosπ−φα |α
! 1 1 +|ρ|tα.
Lemma 2.7. [30, 37] Let A a sectorial operator of type (M, θ, α, ρ) and t > 0, then the following estimates are valid:
(i) If ρ>0 and φ∈
max{θ,(1−α)π}, π2(2−α) , then
kTα(t)k6 Mh
K
1 α
0 −1i
πsinθ (1 +ρtα)α1 tα−1 e[K1(1+ρtα)]
1
α + Mtα−1
π(1 +ρtα)|cosπ−φα |αsinθsinφ and
kKα(t)k6 Mh
K
1 α
0 −1i K1
π(sinθ)α+2α
(1 +ρtα)α−1α tα−1 e[K1(1+ρtα)]
1
α + MαΓ(α)
π(1 +ρtα)|cosπ−φα |αsinθsinφ beingK0=K0(θ, φ) = 1+ sinφ
sin(φ−θ) and K1=K1(θ, φ) = max
1, sinθ sin(φ−θ)
. (ii) If ρ <0 and φ∈
max{π2,(1−α)π}, π2(2−α) , then
kTα(t)k6
eMh
(1 + sinφ)α1 −1i
π|cosφ| + M
π|cosφ||cosπ−φα |
1 1 +|ρ|tα, and
kKα(t)k6
eMh
(1 + sinφ)α1 −1i t
π|cosφ|α+2α + αΓ(α)M π|cosφ||cosπ−φα |
1 1 +|ρ|tα. Lemma 2.8. [30, 37]Let Aa sectorial operator of type(M, θ, α, ρ), then
Sα(t) = 1 2πi
Z
C
eλtλα−1R(λα,A)dλ=Eα,1(Atα) =
∞
X
k=0
(Atα)k Γ(1 +αk), (7)
Tα(t) = 1 2πi
Z
C
eλtR(λα,A)dλ=tα−1Eα,α(Atα) =tα−1
∞
X
k=0
(Atα)k Γ(α+αk), (8)
Kα(t) = 1 2πi
Z
C
eλtλα−2R(λα,A)dλ=tEα,2(Atα) =t
∞
X
k=0
(Atα)k Γ(2 +αk). (9)
whereC is an appropriate path belonging toΣθ,ω.
Lemma 2.9. [30, 37]Let Aa sectorial operator of type(M, θ, α, ρ), then d
dt(Kα(t)) =Sα(t) and d
dt(Sα(t)) =ATα(t).
Lemma 2.10. [30,37]LetAa sectorial operator of type(M, θ, α, ρ)andα∈(0,1), then
CDα0+[Sα(t)x0] =A[Sα(t)x0] and
CDα0+
Z t 0
Tα(t−θ)f(θ)dθ
=A Z t
0
Tα(t−θ)f(θ)dθ+f(t),
whereΓ(·)is an appropriate path belonging to Σθ,ω,Sα(·)andTα(·), are given by Eq.(7) andEq.(8), respectively.
Corollary 1. [30, 37]
CDαtk
Z t tk
Tα(t−θ)f(θ)dθ
=A Z t
tk
Tα(t−θ)f(θ)dθ+f(t), wheretk>0.
As we are working with fractional differential equations with impulses, it is important to note the following result that we present below.
Lemma 2.11. [30,37]
d dt
Z t 0
Tα(t−θ)f(θ)dθ
6= d dt
Z t tk
Tα(t−θ)f(θ)dθ
, wheretk>0.
Lemma 2.12. [30, 31, 37] Let A a sectorial operator of type (M, θ, α, ρ) and 0< α <1, then
CD0+α Sα(t−tk)Ik 6=ASα(t−tk)Ik and
CD0+α
Z t tk
Tα(t−θ)f(θ)dθ
6=A Z t
tk
Tα(t−θ)f(θ)dθ
.
Lemma 2.13. [30, 31, 37] Let A a sectorial operator of type (M, θ, α, ρ). If 0< α <1 andt > tk, then
CDtαkSα(t−tk)Ik=ASα(t−tk)Ik.
The following observation has the same objective as Lemma 2.11, that is, to present the difference between an integral calculated in the determined interval in relation to the partitioned interval from the choice ofk∈N.
Remark 1.
CDα0+
Z t 0
Tα(t−θ)f(θ)dθ
6=CDα0+
Z t tk
Tα(t−θ)f(θ)dθ
, wheretk>0.
In order to obtain the existence of an e-positive mild solution from Eq.(1), we present the concept of Kuratowski’s non-compactness measure and some important consequences of it.
Definition 2.14. [35, 38, 40] Let B be a limited set in a Banach space Ω and δ(B) the diameter of a setB. Kuratowski’s noncompactness measure µ(·) is given by
(10) µ(B) = inf (
ε >0; B=
m
[
i=1
Bi andδ(Bi)6ε, ∀i∈[1· · ·m]
) .
The Kuratowski’s noncompactness measure guarantees that every limited setB admits finite coverage, that is, Bcan be covered by a finite number of sets with a diameter not exceedingε >0.
Consider the interval J = [0, b] and the Banach space C(J,Ω), then for all B⊂C(J,Ω) andt∈J, define
B(t) :={u(t); u∈B} ⊂Ω.
IfBis limited inC(J,Ω), thenB(t) will be limited in Ω andµ B(t)
6µ(B).
Lemma 2.15. [35, 38, 40] Let B ⊂ C(J,Ω) limited and equicontinuous. So µ(B(t))is continuous in J,
µ(B) = max
t∈J µ(B(t)) and µ Z t
0
B(s)ds
6 Z t
0
µ B(s)
ds.
Lemma 2.16. [35,38,40]LetSandT be limited sets in a Banach spaceΩ, either S¯the closing ofS,co(S)the convex hull ofS andaa real number. So the measure of noncompactness has the following properties:
(1) S⊂T ⇒ µ(S)6µ(T);
(2) µ({x} ∪S) =µ(S), ∀x∈Ω, ∅ 6=S⊂Ω (3) µ(S) = 0 ⇐⇒ S¯for compact;
(4) µ(S+T)6µ(S) +µ(T), whereS+T ={x+y; x∈S, y∈T};
(5) µ(S∪T) = max{µ(S), µ(T)};
(6) µ(aS) =|a| µ(S);
(7) µ(S) =µ( ¯S) =µ(co(S));
For allW ⊂C(J; Ω), define Z t
0
W(s)ds= Z t
0
u(s)ds; u∈W
, t∈J.
Lemma 2.17. [35, 38,40]Let J = [a, b], W ⊂C(J; Ω)limited and equicontin- uous, so co(W)⊂C(J; Ω)is also limited and equicontinuous.
Lemma 2.18. [35,38,40]Let{ξn}∞n=1a sequence of Bochner-integrable functions, J = [a, b] in Ω, with kξn(t)k 6m(t), for almost every t∈J and alln>1, where m∈L(J;R+), then the function Φ(t) =µ({ξn(t)}∞n=1)∈L(J;R+)which satisfies
µ Z t
a
ξn(s)ds; n∈N
62 Z t
a
Φ(s)ds.
Lemma 2.19. [35, 38, 40] Let W limited, so for each ε >0, there is a sequence {un}∞n=1⊂W, such that
µ(W)6µ
{ξn}∞n=1 +ε.
3. Existence of e-positive mild solutions. In this section, we will investi- gate the existence of e-positive mild solutions for an initial value problem with impulsive evolution fractional differential equation of nonlinear in the Banach Ω space, through the Gronwall inequality, Cauchy’s criterion and Kuratowski’s non- compactness measure[22,35, 38, 40].
Consider the following initial value problem with linear impulsive evolution fractional equation in Ω, given by
(11)
CDα0+ξ(t) +Aξ(t) = ϕ(t), t∈J∞, t6=tk,
∆ξ|t=tk = Ik(ξ(tk)), k∈N, ξ(0) = x0
where CDα0+(·) is a Caputo fractional derivative of order 0 < α < 1, u: J → Ω, A : D(A) ⊂ Ω → Ω is a sectorial operator of type (M, θ, α, ρ) in Ω, ∆ξ|t=tk = ξ(t+k)−ξ(t−k) beingξ(t+k) andξ(t−k) represent the limits on the right and left ofξ(t) em t = tk, respectively, Ik : Ω→ Ω (k ∈N) are impulsive functions, x0 ∈ D(A) and ϕ ∈ C(J,Ω). Also, be 0 < t1 < t2 < · · · < tm· · ·, with tm → ∞ when m→ ∞, a partition inJ∞, define J∞0 =J∞\{t1, t2, . . . , tm, . . .}, J0 = [0, t1] and Jk = (tk, tk+1] (k∈N).
Definition 3.1. [30, 31, 37] An abstract function u ∈ P C(J∞,Ω) is a mild solution for Eq.(11) if it satisfies the following integral equation:
x(t) =eSα(t)x0+ Z t
0
Tα(t−s)ϕ(s)ds+eSα(t)
k
X
i=1
eS−1α (ti)Ii(xi).
with eSα(·) and eTα(·) given by Eq.(7) and Eq.(8), respectively. Besides that, eS−1α (·) denotes the inverse of the fractional solution operator eSα(·) at t = ti, i = 1,2,3, ..., m.
In addition, if there ise>0 andσ >0, so thatu(t)>σe fort ∈J∞, then we have ane-positive mild solution for Eq.(11).
Let (Ω,k · k) be a Banach space, A : D(A) ⊂ Ω→ Ω a closed linear operator and −A the infinitesimal generator of α-resolvent families {Sα(t); t > 0} and {Tα(t); t>0}. So, there areM >f 0 andδ >0 such that[17, 35,36]
kSα(t)kC 6M ef δt and kTα(t)kC 6M ef δt, t>0.
Through the results presented in the preliminary section, we are ready to attack the main result of this article, that is, the Theorem3.2.
Theorem 3.2. Let (Ω,k · k) be a Banach space with partial order“6”, whose positive coneΩ+ is normal, and−Ais the generator of positiveα-resolvent families Sα(t); t>0 and
Tα(t); t>0 . For a constantσ >0 andt∈J∞, letx0>σe1
and f(t, σe1) > λ1σe1. If the non-linearity of f ∈ C(J∞ ×Ω+,Ω) satisfy the following conditions:
(H1): For t ∈ J∞ and x ∈Ω+, there are functions a, b ∈ C(J∞,Ω+), such that
kf(t, x)k6a(t)kxk+b(t).
(H2): For allR >0 andT >0, existC=C(R, T)>0, such that f(t, x2)−f(t, x1)>−C(x2−x1),
for allt∈[0, T]and for 06x16x2 , with kx1k6R and kx2k6R (H3): For allR >0andT >0, existL=L(R, T)>0, such that any growing
monotonous sequence D={xn} ⊂Ω+∩B(0, R)satisfies µ
f(t, D)
6Lµ(D), ∀ t∈[0, T].
ThenEq.(1) have ane-positive mild solution in J∞.
Proof. The proof of this theorem will be divided into two parts.
(I)In this first part, we will prove the global existence ofe-positive mild solutions in the intervalJ0= [0, t1].
In this case, Eq.(1) is equivalent to Eq.(12) with the evolution fractional equation without impulse in Ω,
(12)
(C
D0+α ξ(t) +Aξ(t) =f t, ξ(t)
, t∈J0, ξ(0) =x0
.
(1) The local existence of soft solutions for the Eq.(12) in J0= [0, t1].
For all t0 > 0 and x0 ∈ Ω, we will prove that Eq.(13) below, with fractional evolution equation
(13)
(C
Dtα0+ξ(t) +Aξ(t) =f t, ξ(t)
, t > t0, ξ(t0) =x0,
have ane-positive mild solution inI= [t0, t0+ht0], whereht0 ∈(0,1) which will be presented according to Eq.(16).
Consider the interval I∗ = [0, t0 + 1], α ∈ (0,1), we introduced the following constants:
Mt0 = sup
(t−t0)1−αkSα(t)k; t∈I∗ , Mt0 = sup
(t−t0)1−αkTα(t)k; t∈I∗ ,
and
Rt0= (Mt0+Mt0)(kx0k+ 1) +σe1.
Letaandbbe functions in the condition (H1), such that at0 = max
t∈I∗
a(t) e bt0= max
t∈I∗
b(t).
On the other hand, the functions in the condition (H2) and (H3), given C=C(Rt0, t0+ 1) and L=L(Rt0, t0+ 1).
Adding the portionCξ(t) on both sides of the Eq.(13), we can rewrite it as
(14)
(CDtα0+ξ(t) + (A+CI)ξ(t) =f t, ξ(t)
+Cξ(t), t > t0, ξ(t0) =x0.
Consider the operators eSα(t) = e−CtSα(t) and Teα(t) = e−CtTα(t) belongs, the positiveα-resolvent families,{Sα(t);t>0}and{Tα(t);t>0}, both generated by−(A+CI), respectively. Consider the setQapplication to
(15) (Qu)(t) =eSα(t−t0)x0+ Z t
t0
Teα(t−s)h
f s, ξ(s)
+Cξ(s)i
ds, t∈I.
From continuity of f and the condition (H2), we have that function Q:C(I,Ω+)→C(I,Ω) is continuous and increasing. In addition, a fixed point ofQis also a solution of Eq.(14) inI.
Define the set Ω:
Λ :=n
u∈C(I,Ω+); kξ(t)kC 6Rt0, ξ(t)>σe1, t∈Io
Then, Λ⊂C(I,Ω+) is nonempty, bounded, convex and closed set. Let
(16) hαt0 6min
1, (kx0k+ 1)α (at0+C)Rt0+bt0
,
with 0< α <1, then by Eq.(15) and by condition (H1), for eachu∈Λ and
t∈I, yields k(Qξ)(t)k=
eSα(t−t0)x0+ Z t
t0
Teα(t−s)h
f s, ξ(s)
+Cξ(s)i ds
(17)
6keSα(t−t0)kkx0k+ Z t
t0
eTα(t−s)
f s, ξ(s)
+Cξ(s) ds 6Mt0kx0k+Mt0
Z t t0
(t−s)α−1h
a(s)kξ(s)k+b(s) +Ckξ(s)ki ds 6Mt0kx0k+Mt0
Z t t0
h
(at0+C)Rt0+bt0i
(t−s)α−1ds 6Mt0kx0k+Mt0
h
(at0+C)Rt0+bt0
i(t−t0)α
α .
From Eq.(17), yields k(Qξ)(t)k6Mt0kx0k+Mt0
[(at0+C)Rt0+bt0] α
(kx0k+ 1)α [(at0+C)Rt0+bt0] 6[Mt0 + Mt0] (kx0k+ 1)
6Rt0.
Letv0(t) =σe1, ∀t∈I, sov0∈Λ. Like this
(18) ϕ(t),CDα0+ν0(t) + (A+CI)ν0(t) =λ1σe1+Cσe16f(t, σe1) +Cσe1. AseSα(t) andTeα(t) are positiveα-resolvent operators andQis a increas- ing operator, then, from Eq.(15), yields
σe1=ν0(t) =eSα(t−t0)ν0(t0) + Z t
t0
Teα(t−s)ϕ(s)ds 6eSα(t−t0)x0+
Z t t0
eTα(t−s)h
f(s, σe1) +Cσe1i
ds= (Q(σe1))(t).
Note thatσe16u(t)∀t∈I, then
σe16(Q(σe1))(t)6(Qξ)(t), t∈I.
Thus,Q: Λ→Λ is continuous and increasing.
The setQ(Λ) is a family of equicontinuous functions inC(I,Ω+).
Letν0=σe1∈Ω and define a sequence on the interval{νn}by (19) νn=Qνn−1, n= 1,2,· · ·.
AsQis an increasing operator andν1=Qν0>ν0, yields (20) ν06ν16ν26· · ·6νn6· · ·
Therefore,{νn}={Qνn−1} ⊂Q(Λ)⊂Λ is bounded and equicontinuous.
Now, letB={νn; n∈N}andB0={νn−1; n∈N}, thenB0=B∪{ν0}.
By Lemma2.16(2), yields µ(B(t)) =µ(Q(B0)(t)) fort∈I.
SubstitutingQ(B0)(t), define by Eq.(15), yields (21)
µ(B(t)) =µ
eSα(t−t0)x0+ Z t
t0
Teα(t−s)h
f(s, νn−1(s)) +Cνn−1(s)i
ds; n∈N
.
By Lemma2.16(3), we haveµ
eSα(t−t0)x0
= 0, so Eq.(21), yields µ(B(t)) =µ
Z t t0
Teα(t−s)h
f(s, νn−1(s)) +Cνn−1(s)i
ds; n∈N
. Using the Lemma2.18, yields
µ(B(t))62 Z t
t0
µn
Teα(t−s)h
f(s, νn−1(s)) +Cνn−1(s)i
; n∈N o
ds 62
Z t t0
kTeα(t−s)kµn
f(s, νn−1(s)) +Cνn−1(s); n∈N o
ds 62Mt0
Z t t0
(t−s)α−1h µ
f(s,B0(s)) +µ
CB0(s)i ds.
By condition (H3), for allt∈I, yields µ
B(t)
62Mt0
Z t t0
(t−s)α−1[L+C]µ B0(s)
ds 62Mt0(L+C)
Z t t0
(t−s)α−1µ B0(s) ds.
By Gronwall inequality (see Lemma2.3withψ(t) =t), yields µ
B(t)
60·Eα
2M(L+C)Γ(α)(t−s)α
= 0.
So, µ(B(t)) ≡ 0 for t ∈ I. Using the Lemma 2.15, we have µ(B) = maxt∈Iµ(B(t)) = 0, that is, {νn} is relatively compact in C(I,Ω+).
Therefore, there exist a subsequence{νnk} ⊂ {νn}such thatνnk→ξ∗∈Λ, when k → ∞. Combining this with the sequence in Eq.(20) and the normality of the cone Ω+, it’s easy to see that νn → ξ∗, with n → ∞.
Taking the limitn→ ∞on both sides of Eq.(19), and the continuity of the operatorQ, we haveξ∗=Qξ∗ a fixed point. Therefore,ξ∗∈Λ⊂C(I,Ω+) is ane-positive mild solution of the Eq.(14).
(2) The global existence of mild solutions for the Eq.(12) onJ0= [0, t1].
In the item 1, we prove that Eq.(12) admit an e-positive mild solution ξ0∈C([0, h0],Ω+), given by
(22) ξ0(t) =eSα(t)x0+ Z t
0
Teα(t−s)h
f(s, ξ0(s)) +Cξ0(s)i ds.
By extension theorem[38], ξ0 can be extended to a saturated solution of the Eq.(12), which is also denoted byξ0∈C([0, T),Ω+), whose interval of existence is [0, T).
Next, we will show thatT > t1. Denote a= max
t∈[0,T+1]a(t), b= max
t∈[0,T+1]b(t), and
M1= sup
t∈[0,T+1]
k(t−T)1−αSα(t)k and M1= sup
t∈[0,T+1]
k(t−T)1−αTα(t)k.
SupposeT 6t1and taking a norm of the solutionu0(see Eq.(22)), yields kξ0(t)k6
eSα(t)x0 +
Z t 0
Teα(t−s)h
f(s, ξ0(s)) +Cξ0(s)i
ds 6keSα(t)kkx0k+
Z t 0
(t−s)α−1(t−s)1−α
eTα(t−s) h
f(s, ξ0(s)) +Cξ0(s)i ds 6M1kx0k+M1
Z t 0
(t−s)α−1h
kf(s, ξ0(s))k+kCξ0(s)ki ds 6M1kx0k+M1
Z t 0
(t−s)α−1h
b+ (a+C)kξ0(s)ki ds 6M1kx0k+M1b Tα
α +M1(a+C) Z t
0
(t−s)α−1kξ0(s)kds.
By Gr¨onwall inequality (see Lemma2.3withψ(t) =t), yields kξ0(t)k6
M1kx0k+M1b Tα α
Eα
M1(a+C)Γ(α)t 6
M1kx0k+M1b Tα α
Eα
M1(a+C)Γ(α)T ,M2. (23)
Now, we define the following constant
(24) N0:= supn
kf(t, x)k; t∈[0, T + 1] ekxk6M2o .
As eSα(t) is a continuous standard operator for t >0, for any 0< τ1<
τ2< T, consider the following functions:
ξ0(τ2) =eSα(τ2)x0+ Z τ2
0
Teα(τ2−s)
f(s, ξ0(s)) +Cξ0(s) ds (25)
and
ξ0(τ1) =eSα(τ1)x0+ Z τ1
0
Teα(τ1−s)
f(s, ξ0(s)) +Cξ0(s) ds.
(26)
Subtracting Eq.(26) from Eq.(25), and readjusting the integrals with respect to the integration limits, yields
ξ0(τ2)−ξ0(τ1) = Seα(τ2)x0−eSα(τ1)x0
+ Z τ1
0
h
Teα(τ2−s)−Teα(τ1−s)ih
f(s, ξ0(s)) +Cξ0(s)i ds +
Z τ2
τ1
Teα(τ2−s)h
f(s, ξ0(s)) +Cξ0(s)i ds.
Let’s draw the norm for this difference to determine a higher quota.
Then, making the following variable changes→τ1−s, using the Eq.(24),
Eq.(23) and the constantM1, yields kξ0(τ2)−ξ0(τ1)k 6 keSα(τ2)x0−eSα(τ1)x0k
+ Z τ1
0
kTeα(τ2−s)−Teα(τ1−s)kkf(s, ξ0(s)) +Cξ0(s)kds +
Z τ2 τ1
kTeα(τ2−s)kkf(s, ξ0(s)) +Cξ0(s)kds 6 eSα(τ2)x0−eSα(τ1)x0k
+ Z τ1
0
kTeα(τ2−τ1+s)−Teα(s)kkf(s, ξ0(s)) +Cξ0(s)kds +
Z τ2 τ1
(τ2−s)α−1(τ2−s)1−αkeTα(τ2−s)kkf(s, ξ0(s)) +Cξ0(s)kds 6 keSα(τ2)x0−eSα(τ1)x0k
+ (N0+CM2) Z τ1
0
kTeα(τ2−τ1+s)−Teα(s)kds + M1(N0+CM2)
Z τ2
τ1
(τ2−s)α−1ds
6 keSα(τ2)x0−eSα(τ1)x0k+M1(N0+CM2)(τ2−τ1)α α + (N0+CM2)
Z τ1
0
kTeα(τ2−τ1+s)−Teα(s)kds.
Whenτ1→T− andτ2→T−, yields
keS∗α(τ2)x0−eS∗α(τ1)x0k →0, (τ2−τ1)α
α →0
and
Z T 0
kTe∗α(τ2−τ1+s)−Te∗α(s)kds→0.
So,ku0(τ2)−u0(τ1)k ≡0. By Cauchy criteria, there existx∈Ω+ such that lim
t→T−u0(t) =x.
Now, consider the initial value problem with fractional evolution equation and without impulse in Ω, given by
(27)
(cD0+α ξ(t) + (A+CI)ξ(t) =f(t, u(t)) +Cξ(t), t > T, ξ(T) =x.
From item 1, we have that Eq.(27), has ane-positive mild solution v in [T, T +hT]. Let
u(t) =
(ξ0(t), t∈[0, T), ν(t), t∈[T, T +hT].
It is easy to see that ξ(t), is an e-positive mild solution of Eq.(12) in [0, T +hT]. As ξ(t) is an extension of the ξ0(t), that is, a contradiction.
Thus,T > t1, i.e., a globale-positive mild solutionξ0(t) of the Eq.(12) exist inJ0, which is also ane-positive mild solution from Eq.(1) inJ0.
Thus, we finalized the first part of the theorem.
(II) In this second part, we will prove the existence of global e-positive mild solutions in the intervalJ∞.
Initially, we will prove that Eq.(1) has a global e-positive mild solution in the intervalJ1= (t1, t2]. As in Eq.(27), here we also consider the initial value problem with evolution fractional equation without impulse inJ1, given by
(28)
(C
D0+α,βξ(t) + (A+CI)ξ(t) =f(t, ξ(t)) +Cξ(t), t∈J1, ξ(t+1) =ξ0(t1) +I1(ξ0(t1)).
Clearly, a globale-positive mild solution of Eq.(28) inJ1, is alsoe-positive mild solution of Eq.(1) inJ1. From the proof of item I, for t∈J0= [0, t1], yields (29) ξ0(t) =eSα(t)x0+
Z t 0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds.
By a similar argument to proof I, the Eq.(28) has an e−positive mild solution ξ1∈C(J1,Ω+) (J1= (t1, t2]), given by
(30) ξ1(t) =eSα(t)θ0+ Z t
0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds.
For the impulsive condition and Eq.(29) and Eq.(30), yields θ0=x0+eS−1α (t1)I1(ξ0(t1)). So, fort∈J1= (t1, t2], yields
(31)
ξ1(t) =Seα(t)x0+ Z t
0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds+eSα(t)eS−1α (t1)I1(ξ0(t1)). Now, considerJ2= (t2, t3] andξ2∈C(J2,Ω+),yields
(32) ξ2(t) =eSα(t)θ1+ Z t
0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds.
For the impulsive condition and from Eq.(31), Eq.(32), yields θ1=x0+eS−1α (t1)I1(ξ0(t1)) +eS−1α (t2)I2(ξ1(t2)). So, fort∈J2= (t2, t3], yields
ξ2(t) = eSα(t)x0+ Z t
0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds +eSα(t)eS−1α (t1)I1(ξ0(t1)) +eSα(t)eS−1α (t2)I2(ξ1(t2)).
Suppose that, for t ∈ Jk−1 (k = 4,5, . . .), the Eq.(1) has an e-positive mild solutionξk−1∈C(Jk−1,Ω+) (k= 4,5, . . .). So, fort∈Jk (k= 3,4, . . .), the IVP
with fractional evolution differential equations without impulsive in Ω, given by (33)
(C
Dα0+ξ(t) + (A+CI)ξ(t) =f(t, ξ(t)) +Cξ(t), t∈Jk, k= 3,4, . . . ξ(t+k) =ξk−1(tk) +Ik(ξk−1(tk))
has ane-positive mild solutionξk∈C(Jk,Ω+), given by ξk(t) = eSα(t)θk−1+
Z t 0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds
= eSα(t)
x0+eS−1α (t1)I1(ξ0(t1)) +eS−1α (t2)I2(ξ1(t2)) +· · ·+ +eSα−1(tk)Ik(ξk−1(tk)) +
Z t 0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds
= eSα(t)x0+ Z t
0
Tα(t−s) [f(s, ξ(s)) +Cξ(s)]ds+eSα(t)
k
X
j=1
eS−1α (tj)Ij(ξj−1(tj)). (34)
Now, we define aufunction as
(35) ξ(t) =
ξ0(t), t∈J0, ξ1(t), t∈J1,
· · ·
ξk(t), t∈Jk (k= 2,3, . . .) .
Of courseξ(t)∈P C(J∞,Ω+) is ane-positive mild solution of Eq.(1), satisfying ξ(t) =eSα(t)x0+
Z t 0
Teα(t−s)h
f(s, ξ(s)) +Cξ(s)i
ds+eSα(t)
k
X
j=1
eS−1α (tj)Ij(ξ(tj)). For the property of global existence ofξi(t) inJi,i∈N, a solutionξ(t) define by Eq.(35) is a globale-positive mild solution of Eq.(1) inJ∞. When Ω is a Banach space that is ordered and complete in a weak and sequential way, we exclude the condition (H3) of noncompactness measure from Theorem3.2 and obtain the following result:
Corollary 2. Let Ωa Banach space ordered and complete in a weak and sequential way, whose positive cone Ω+ be normal, −A be an infinitesimal generator of the positive α-resolvent family {Sα(t); t > 0} and {Tα(t); t > 0}. Let x0 > σe1, f(t, σe1)>λ1σe1forσ >0andt∈J∞. If a non-linearity of thef ∈C(J∞×Ω+,Ω) satisfy assumptions(H1)and(H2), then the Eq.(1)has ane-positive mild solution inJ∞.
Acknowledgment. JFJ acknowledges CAPES (number of process no88882.32909093/2019- 01) for financial support scholarship of the Postgraduate Program in Applied Math-
ematics of IMECC-Unicamp. JVCS acknowledges the financial support of a PNPD- CAPES (number of process no88882.305834/2018-01) scholarship of the Postgrad- uate Program in Applied Mathematics of IMECC-Unicamp.
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