#### Existence of positive mild solutions for a class of fractional evolution equations on the half line

Chen et al. Advances in Difference Equations
half line
Yi Chen 0 1 2 5
Zhanmei Lv 0 1 4
Liang Zhang 0 1 3
0 Technology , Xuzhou, 221116 , China
1 University of Mining
2 School of Mathematics , China
3 School of Mathematical Sciences, University of Jinan , Jinan, 250022 , China
4 School of Business, Central South University , Changsha, 410083 , China
5 School of Mathematics, China University of Mining and Technology , Xuzhou, 221116 , China
the half-line. iterative method The equivalent integral equation of a new form for a class of fractional evolution equations is obtained by the method of Laplace transform, which is different from those given in the existing literature. By the monotone iterative method without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional evolution equations on
fractional order; abstract evolution equations; mild solutions; monotone
1 Introduction
Banach space E:
In this paper, we are concerned with the following fractional evolution equation in the
⎧⎨ CDα+u(t) = Au(t) + f (t, u(t)), t ∈ (, +∞),
⎩ u() = μu(β),
(.)
where CDα+ is the Caputo fractional derivative, < α < , μ > , β > , A is the infinitesimal
generator of a C semigroup {T (t)}t≥ of operators on Banach E, and f : [, +∞) × E → E
satisfies certain conditions.
Fractional calculus, a generalization of the ordinary differentiation and integration, has
played a significant role in science, economy, engineering, and other fields (see [–]).
Today there is a large number of papers dealing with the fractional differential equations (see
[–]) due to their various applications. One of the branches is the research on the
theory about the evolution equations of fractional order, which comes from physics. Recently,
fractional evolution equations have attracted increasing attention around the world, see
[–] and the references therein. Among the existing literature, most of them are focused
on the existence of the solutions on the finite interval, see [–].
(.)
(.)
In [], El-Sayed investigated the Cauchy problem in a Banach space for a class of
fractional evolution equations
⎧⎨ ddααut = Au(t) + B(t)u(t), t ∈ [, T ],
⎩ u() = u,
where < α ≤ , T > . The existence and uniqueness of the solution for the above Cauchy
problem were studied for a wide class of the family of operators {B(t) : t ≥ }.
As far as we know, for the first time, the equivalent integral equation of the above
equation was given in terms of some probability densities by the method of Laplace transform.
And since then, most of the research in this direction has been based on this paper.
However, many of the previous papers about the existence of solutions of fractional evolution
equations are only on the finite interval, and those presenting the existence results on the
half-line are still few.
Motivated by [, , ], in this paper, we study the differential equation (.) under certain
conditions on the unbounded domains. Here, by a method similar to that used in [, ],
we give a corrected form of the equivalent integral equation of the main problem (.),
which is different from those obtained in the existing literature. Employing the monotone
iterative method, without the assumption of lower and upper solutions, we present some
new results on the existence of positive mild solutions for the abstract evolution equations
of fractional order. And to our best knowledge, there is not any paper to deal with the
abstract problems of fractional order on the unbounded domains.
The rest of the paper is organized as follows. In Section , we introduce the definitions
of fractional integral and fractional derivative, some results about fractional differential
equations and some useful preliminaries. In Section , we obtain the existence result of
the solution for problem (.) by the monotone iterative method. Then an example is given
in Section to demonstrate the application of our result.
2 Preliminaries
First of all, we present some fundamental facts on the fractional calculus theory which we
will use in the next section.
Definition . ([–]) The Riemann-Liouville fractional integral of order ν > of a
function h : (, ∞) → R is given by
Iν+h(t) = D–+νh(t) =
(t – s)ν–h(s) ds,
(ν)
t
provided that the right-hand side is pointwise defined on (, ∞).
Definition . ([–]) The Caputo fractional derivative of order ν > of a continuous
function h : (, ∞) → R is given by
CDν+h(t) =
(n – ν)
t
(t – s)n–ν–hn(s) ds,
where n = [ν] + , provided that the right-hand side is pointwise defined on (, ∞).
Lemma . ([, ]) Assume that CDν+h(t) ∈ L(, +∞), ν > . Then we have
Iν+CDν+h(t) = h(t) + C + Ct + · · · + CN tN–, t > ,
(.)
for some Ci ∈ R, i = , , . . . , N , where N is the smallest integer greater than or equal to ν.
If h is an abstract function with values in the Banach space E, then the integrals
appearing in Definition ., Definition . and Lemma . are taken in Bochner’s sense. And a
measurable function h is Bochner integrable if the norm of h is Lebesgue integrable.
Now let us recall some definitions and standard facts about the cone.
Let P be a cone in the ordered Banach space E, which defines a partial order on E by
x ≤ y if and only if y – x ∈ P. P is normal if there exists a positive constant N such that
θ ≤ x ≤ y implies x ≤ N y , where θ is the zero element of the Banach space E. The
infimum of all N with the property above is called the normal constant of P. For more
details on the cone P, we refer readers to [, ].
Throughout the paper, we set E to be an ordered Banach space with the norm · and
the partial order ‘≤’. Let P = {x ∈ E | x ≥ θ } be a positive cone, which is normal with normal
constant N . Let J = [, +∞). Set
BC(J, E) = u(t) | u(t) is continuous and bounded on J .
Obviously, BC(J, E) is a Banach space with the norm u BC = supt∈J u(t) . Let
PBC = u ∈ BC(J, E) | u(t) ≥ θ , t ∈ J .
It is easy to see that PBC is also normal with the same normal constant N of the cone P.
Besides, BC(J, E) is also an ordered Banach space with the partial order ‘≤’ induced by the
positive cone PBC (without confusion, we denote by ‘≤’ the partial order on both E and
BC(J, E)).
We denote by [v, w] the order interval {u ∈ PBC | v ≤ u ≤ w, v, w ∈ BC(J, E)} on BC(J, E),
and use [v(t), w(t)] to denote the order interval {z ∈ E | v(t) ≤ z ≤ w(t)} on E for t ∈ J .
Next, we give some facts about the semigroups of linear operators. These results can be
found in [, ].
For a strongly continuous semigroup (i.e., C-semigroup) {T (t)}t≥, the infinitesimal
generator of {T (t)}t≥ is defined by
Ax = tl→im+ T (t)tx – x , x ∈ E.
We denote by D(A) the domain of A, that is,
D(A) = x ∈ E tl→im+ T (t)tx – x exists .
Lemma . ([, ]) Let {T (t)}t≥ be a C-semigroup, then there exist constants C ≥
and ω ∈ R such that T (t) ≤ Ceωt, t ≥ .
Lemma . ([, ]) A linear operator A is the infinitesimal generator of a C-semigroup
{T (t)}t≥ if and only if
(i) A is closed and D(A) = E.
(ii) The resolvent set ρ(A) of A contains R+ and, for every λ > , we have
where
R(λ, A) ≤ λ ,
R(λ, A) := (λI – A)– =
+∞ e–λtT (t)x dt, x ∈ E.
Definition . ([, ]) A C-semigroup {T (t)}t≥ is said to be uniformly exponentially
stable if ω < , where ω is the growth bound of {T (t)}t≥, which is defined by
ω = inf ω ∈ R | ∃C ≥ such that T (t) ≤ Ceωt, t ≥ .
Definition . ([]) A C-semigroup {T (t)}t≥ is said to be positive on E if order
inequality T (t)x ≥ θ , x ∈ E and t ≥ .
According to Lemma . and Definition ., if {T (t)}t≥ is a uniformly exponentially
stable C-semigroup with the growth bound ω, there exists a constant C ≥ such that
T (t) ≤ Ceωt, t ≥ , for any ω ∈ (, |ω|]. Now, we define a norm in E by
x ω = sup eωtT (t)x .
t≥
T (t) ω ≤ e–ωt, t ≥ .
Evidently, x ≤ x ω ≤ C x , that is to say, the norms · ω and · are equivalent. We
denote by T (t) ω the norm of T (t) induced by the norm · ω, then
Also, we can define the equivalent norm on BC(J, E) by
u BCω = sup u(t) ω,
t∈J
u ∈ BC(J, E).
3 Main results
In this section, we present the existence theorem for the abstract fractional differential
equation on the half-line. In order to prove our main result, we need the following facts
and lemmas.
Consider the one-sided stable probability density [, , ]
∞
ψα(θ ) = π n=
θ –αn– (nα + ) sin(nπ α), θ ∈ (, ∞),
n!
where < α < .
(.)
From [, , ], the Laplace transform of the one-sided stable probability density ψα(θ )
is given by
L ψα(θ ) =
∞ e–λθ ψα(θ ) dθ = e–λα ,
< α < .
By Remark . in [], for ≤ γ ≤ , one has
∞
θ –αγ ψα(θ ) dθ =
( + γ )
( + αγ ) .
In the following, we assume that {T (t)}t≥ is a uniformly exponentially stable
Csemigroup with the growth bound ω, and ω ∈ (, |ω|].
Lemma . Define a linear operator V : E −→ E as
Vx = μ
∞
Then V is bounded and V ω ≤ μ. Besides, if < μ < , then (I – V )– is a linear bounded
operator and
(I – V )– ω ≤ – μ .
Vx ω ≤ μ
∞
∞
≤ μ
α
≤ μ ( – α)
≤ μ x ω.
Hence, V is bounded and V ω ≤ μ.
Lemma . Set
Then Q : BC(J, E) −→ BC(J, E) and
(Qh)(t) ω ≤ ω (α + ) h BCω ;
(α– α) ψθα(αθ ) τ –α( – τ )α–e–ω βα(θα–τ)α dθ dτ
∞ ψα(θ ) dθ dτ
x ω
x ω dθ dτ
ω
x ω
t
∞
α ψθα(αθ ) (t – s)α–T
Proof Since
∞
∞
(t – s)α
θ α
α ψθα(αθ ) tα( – τ )α–T
α ψθα(αθ ) tα( – τ )α– T
tα( – τ )α
θ α
α ψθα(αθ ) tα( – τ )α–e–ω tα(θ–ατ)α
∞
∞
e–ω( tα(θ–ατ)α ) d –ω
α ψα(θ ) dθ
– e–ω θtα
θ α
≤ ω (α + ) h BCω .
ω
h(tτ ) ω dθ dτ
h(tτ ) ω dθ dτ
ψθα(αθ ) dθ
Therefore,
⎧⎨ CDα+u(t) = Au(t) + h(t), t ∈ (, +∞),
⎩ u() = u,
has a unique solution u ∈ BC(J, E) of the following form:
u(t) =
=
t
∞
∞
( – α) ψθα(αθ ) τ –α( – τ )α–T
α
Proof In view of Definitions ., . and Lemma ., equation (.) can be rewritten by
the equivalent integral equation as follows:
Denote by U(λ) and H(λ) the Laplace transforms of u(t) and h(t), respectively, using a
similar method as that in [, ], with the Laplace transform, then we can rewrite the above
equation as
Hence,
U(λ) = λ u + λα AU(λ) + λα H(λ),
λ > .
λαI – A U(λ) = λα–u + H(λ).
By virtue of (.) and Lemma ., we obtain
U(λ) = λαI – A –λα–u + λαI – A –H(λ)
= λα–
= λα–
= λα–
+
+
∞
∞
∞ e–λt
∞ e–λt
∞ e–λαsT (s)u ds +
∞ e–λαsT (s)H(λ) ds
∞ e–λs/αθ ψα(θ )T (s)u dθ ds
∞ e–λs/αθ ψα(θ )T (s)H(λ) dθ ds
t
∞ tα–
α θ α ψα(θ )T
tα
θ α u dθ dt
∞ α (t –θsα)α–
ψα(θ )T
By the definition of Laplace transforms and the convolution theorem, applying
Lemma . and the inverse Laplace transforms, one can derive that
u(t) = L– λα– ∗ L–
+ L–
t–α
= ( – α) ∗
t
=
=
=
+
t
∞
t
∞
∞
∞ e–λt
∞ α (t –θsα)α–
∞ e–λt
t
∞ tα–
α θ α ψα(θ )T
∞ α (t –θsα)α–
tα
θ α u dθ
α (t – s)–α sα–
( – α) θ α ψα(θ )T
α s–α (t – s)α–
( – α) θ α ψα(θ )T
( – α) ψθα(αθ ) τ –α( – τ )α–T
α
sα
θ α u dθ ds + (Qh)(t)
τ –α( – τ )α– ψθα(αθ ) T
τ –α( – τ )α– ψα(θ ) e–ω( tα(θ–ατ)α ) u ω dθ dτ + (Qh)(t) ω
θ α
α
≤ ( – α) u ω
≤ u ω + ω (α+ ) h BCω .
Therefore, u ∈ BC(J, E). Then we complete the proof.
Lemma . Let h ∈ BC(J, E) and u ∈ D(A). Let < μ < . Then the linear fractional
evolution equation
⎧⎨ CDα+u(t) = Au(t) + h(t), t ∈ (, +∞),
⎩ u() = μu(β),
has a unique solution u ∈ BC(J, E) of the following form:
u(t) = (LAh)(t)
t ∞
:=
=
+ h(s)
∞
α (t –θsα)α–
dθ ds
( – α) (I – V )–(Qh)(β)
(α– α) ψθα(αθ ) τ –α( – τ )α–T tα ( –θ ατ )α
(I – V )–(Qh)(β) dθ dτ
(.)
(.)
Also, LA is a linear operator on the Banach space BC(J, E) and
κ
LA BCω ≤ ω ,
where
κ :=
(α+ ) – μ .
Proof In view of Lemma ., one can obtain
u(β) =
β
∞
α (β –θ αs)α–
Therefore,
(I – V )u() = μ(Qh)(β).
( – α) ψθα(αθ ) s–α(β – s)α–T
α
(β – s)α
θ α
So, we obtain
u() = μ(I – V )–(Qh)(β).
Then (.) is followed.
By (.), one has
(LAh)(t) ω ≤
∞
(α– α) ψθα(αθ ) τ –α( – τ )α– T tα ( –θ ατ )α
ω
≤
×
(I – V )–μ(Qh)(β) ω dθ dτ + (Qh)(t) ω
∞
( – α) ψθα(αθ ) τ –α( – τ )α–e–ω tα(–τ)α
α θα
× (I – V )– ω μ(Qh)(β) ω dθ dτ + (Qh)(t) ω
μ
≤ – μ ω (α + ) h BCω + ω (α + ) h BCω
= ω (α + ) – μ h BCω
κ
= ω h BCω .
Therefore,
κ
(LAh) BCω ≤ ω
h BCω .
Now, we state the main result on the existence of the positive solutions to problem (.)
in the following.
Theorem . Let E be a Banach space, and P is its positive normal with N as the normal
constant. Let {T (t)}t≥ be a uniformly exponentially stable C-semigroup with the growth
bound ω (ω < ), and A is the infinitesimal generator of {T (t)}t≥. Let < μ < . Provided
that f (t, u) : J × E −→ E is continuous and f (t, θ ) ≥ θ is bounded on J . If f (t, u) satisfies the
following conditions:
(a) There exists a constant K < –ω such that for
f (t, y) – f (t, x) ≤ K(y – x), θ ≤ x ≤ y.
(b) There exists a constant K > max{–K, ω} such that for
f (t, y) – f (t, x) ≥ –K(y – x), θ ≤ x ≤ y.
(c)
< KK +– Kω < κ .
Then problem (.) has a unique positive mild solution in BC(J, E).
Proof For simplicity of notation, we denote f(t) = f (t, θ ), then we have f(t) ∈ BC(J, E) and
f(t) ≥ θ , t ∈ J .
In the following, similar to the methods used in [], we deduce the result of the theorem
in four steps.
Step : Consider the abstract fractional differential equation
⎧⎨ CDα+u(t) = Au(t) + Ku(t) + f(t), t ∈ (, +∞),
By virtue of the theory of semigroups, we can get that {eKtT (t)}t≥ is a uniformly
exponentially stable C-semigroup on Banach E generated by A + KI. Besides, the semigroup
is positive with the growth bound K + ω (K + ω < ). In view of Lemma ., equation
(.) has a unique mild solution ϑ ∈ BC(J, E) and ϑ ≥ θ as a result of f(t) ≥ θ , t ∈ J .
Step : For a given function g ∈ BC(J, E), consider the abstract fractional differential
equation
⎧⎨ CDα+u(t) + Ku(t) = Au(t) + g(t), t ∈ (, +∞),
It is obvious that A – KI generates a uniformly exponentially stable C-semigroup
{e–KtT (t)}t≥ on Banach E. Also, it is positive with the growth bound –K +ω (–K +ω <
).
Based on Lemma ., the unique mild solution of (.) is given by u = LA–KI g, where
LA–KI : BC(J, E) −→ BC(J, E) is a positive bounded linear operator (similar to the operator
LA) with the property that
LA–KI BCω ≤ K – ω
κ
, for ω = K – ω.
ϑ = LA–KI (f + Kϑ + Kϑ).
Combined with the first step, one can notice that ϑ is the mild solution of problem
(.) for g = f + Kϑ + Kϑ, so
Step : Take G(u) = f (t, u) + Ku. Evidently, G(θ ) = f (t, θ ) = f(t) ≥ θ and G : BC(J, E) −→
BC(J, E) is continuous due to conditions (a), (b) and the normality of the cone PBC.
By condition (b), for θ ≤ x ≤ y, one can obtain
G(y) – G(x) = f (t, y) + Ky – f (t, x) – Kx = f (t, y) – f (t, x) + K(y – x) ≥ θ ,
that is, G is an increasing operator on the positive cone PBC.
Let υ = θ . Taking account of a composition operator defined by F = LA–KI ◦ G on
the order interval [θ , ϑ], it is easy to see that the fixed point of F is the mild solution of
problem (.). Now, our task is to demonstrate that the operator F has at least one fixed
point.
Consider the following two sequences:
ϑn = F (ϑn–),
n = , , , . . . ,
and
then
So, we can get
By condition (a), we have
υn = F (υn–),
n = , , , . . . .
f t, ϑ(t) – f (t, θ ) ≤ Kϑ(t),
f t, ϑ(t) ≤ Kϑ(t) + f(t).
G(ϑ) = f t, ϑ(t) + Kϑ(t) ≤ Kϑ(t) + Kϑ(t) + f(t).
By the fact that G is an increasing operator on the cone P, therefore, we can obtain
θ ≤ f(t) = G(θ ) ≤ G(ϑ) ≤ Kϑ + Kϑ + f.
Combining (.), (.) and the positivity of the linear bounded operator LA–KI , one
can get
θ ≤ LA–KI ◦ G(θ ) = F (θ ) ≤ LA–KI ◦ G(ϑ) = F (ϑ) ≤ LA–KI (Kϑ + Kϑ + f) = ϑ,
that is,
θ = υ ≤ ϑ ≤ ϑ.
As F is an increasing operator on the order interval [θ , ϑ], by the definition of F and
(.), we can get two sequences {ϑn} and {υn} (n = , , , , . . .) satisfying
θ = υ ≤ υ ≤ υ ≤ · · · ≤ υn ≤ · · · ≤ ϑn ≤ · · · ≤ ϑ ≤ ϑ ≤ ϑ.
According to the above facts, by condition (a), one can get that
θ ≤ ϑn – υn = F (ϑn–) – F (υn–)
= LA–KI ◦ G(ϑn–) – LA–KI ◦ G(υn–)
= LA–KI f (·, ϑn–) + Kϑn– – f (·, υn–) – Kυn–
≤ (K + K)LA–KI (ϑn– – υn–).
Thus, by induction, we have
θ ≤ ϑn – υn ≤ (K + K)nLA–KI (ϑ – υ) = (K + K)nLnA–KI (ϑ).
On account of the fact that the cone PBC is normal with the normal constant N , by virtue
of condition (c), we get
ϑn – υn BCω ≤ N (K + K)n LnA–KI (ϑ) BCω
≤ N (K + K)n LnA–KI BCω ϑ BCω
≤ N (K + K)n LA–KI BCω ϑ BCω
n
n
≤ N (K + K)n
ϑ BCω
κ
K – ω
= N
κ(K + K) n
K – ω
ϑ BCω → , n → +∞.
(.)
Thus, from (.), analogous to the nested interval method, there exists a unique u∗ ∈
n∞=[υn, ϑn] such that u∗ = limn→∞ ϑn = limn→∞ υn.
By (.) and (.), taking limit of n → ∞, we obtain that
u∗ = F u∗ ,
namely, u∗ is a fixed point of the operator F . Thus, u∗ is a mild positive solution of problem
(.).
Step : In this part, we certify the uniqueness of the mild solution for problem (.).
By using reduction to absurdity, suppose that u∗ and u∗ are two different positive mild
solutions for the fractional evolution equation (.), thus, u∗ – u∗ BCω > .
Replace ϑ by u∗ and u∗ in (.), respectively. Following the same steps as above, for
each ui∗ (i = , ), we can get that ui∗ = F (ui∗), ui∗ – υn BCω → (n → ∞) and ϑn = ui∗ for
each n ∈ N (i = , ). Therefore,
< u∗ – u∗ BCω ≤ u∗ – υn BCω + u∗ – υn BCω → , n → ∞,
which is a contradiction.
Hence, problem (.) has a unique positive solution. The proof is completed.
4 Examples
To illustrate our main result, we present an example. Consider the following partial
fractional differential equation.
Example .
⎧⎪ ∂tαz(t, x) = ∂xz(t, x) + F(t, z(t, x)), t ∈ [, +∞),
⎪⎨ z(t, ) = z(t, π ) = , t ∈ [, +∞),
⎪⎪⎩ z(, x) = μz(β, x), x ∈ [, π ],
(.)
where ∂tα is the Caputo fractional partial derivative of order α ∈ (, ).
Set E = L([, π ], R) and Az = ∂xz, according to [], then A : D(A) −→ E is a linear
operator with domain D(A) = {u ∈ E | u ∈ E, u() = u(π ) = }. Besides, the operator A
generates a uniformly exponentially stable C-semigroup {T (t)}t≥ with the growth bound
ω ≤ –.
Let u(t) = z(t, ·), f (t, u(t)) = F(t, z(t, ·)), then problem (.) can be written as
Take α = /, μ = /, β = , then we can get
κ =
(α + ) – μ = √π .
Consider the following function:
f (t, x) = –K + a(t) x,
where a ∈ C[, +∞) is bounded and
K = ω,
K = – + κ –
ω.
It is easy for us to certify that
< KK +– Kω = κ– < κ .
Since
–K – κω– = + κ –
ω – κω– ≤ ω = K,
then, for θ ≤ x ≤ y,
–K ≤ f (t, y) – f (t, x)
= –K + ( + a–(tω))(κ – ) (y – x)
≤ –K – κω– (y – x)
≤ K(y – x).
(.)
Noting that f (t, θ ) = θ . Thereby, f satisfies the conditions of Theorem ., we can
conclude that problem (.) has a unique positive mild solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All of the authors contributed equally in writing this paper. All authors read and approved the final manuscript.
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1. Kilbas , AA , Srivastava, HM , Trujillo, JJ : Theory and Applications of Fractional Differential Equations . Elsevier, Amsterdam ( 2006 )
2. Miller , KS , Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations . Wiley, New York ( 1993 )
3. Lakshmikantham , V , Leela , S , Vasundhara Devi, J: Theory of Fractional Dynamic Systems . Cambridge Academic Publishers, Cambridge ( 2009 )
4. Henderson , J , Luca, R: Systems of Riemann-Liouville fractional equations with multi-point boundary conditions . Appl. Math. Comput . 309 , 303 - 323 ( 2017 )
5. Cabada , A , Kisela, T: Existence of positive periodic solutions of some nonlinear fractional differential equations . Commun. Nonlinear Sci. Numer . Simul. 50 , 51 - 67 ( 2017 )
6. Ahmad , B , Ntouyas, BK , Tariboon, J: Nonlocal fractional-order boundary value problems with generalized Riemann-Liouville integral boundary conditions . J. Comput. Anal. Appl . 23 , 1281 - 1296 ( 2017 )
7. Mei , DZ , Peng, JG , Gao, JH: General fractional differential equations of order α ∈ (1, 2) and type β ∈ [0, 1] in Banach spaces . Semigroup Forum 94 , 712 - 737 ( 2017 )
8. El-Borai , MM: Some probability densities and fundamental solutions of fractional evolution equations . Chaos Solitons Fractals 14 , 433 - 440 ( 2002 )
9. Zhou , Y , Jiao, F: Nonlocal Cauchy problem for fractional evolution equations . Nonlinear Anal., Real World Appl . 11 , 4465 - 4475 ( 2010 )
10. Wang , J , Zhou , Y: A class of fractional evolution equations and optimal controls . Nonlinear Anal., Real World Appl . 12 , 262 - 272 ( 2011 )
11. Chen , P , Zhang, PX , Li, Y: Study on fractional non-autonomous evolution equations with delay . Comput. Math. Appl . 73 , 794 - 803 ( 2017 )
12. Chen , P , Li, Y , Li , Q: Existence of mild solutions for fractional evolution equations with nonlocal initial conditions . Ann. Pol. Math. 110 , 13 - 24 ( 2014 )
13. Chen , P , Li, Y , Zhang, X: On the initial value problem of fractional stochastic evolution equations in Hilbert spaces . Commun. Pure Appl. Anal. 14 , 1817 - 1840 ( 2015 )
14. Wang , R , Ma, Q: Some new results for multi-valued fractional evolution equations . Appl. Math. Comput . 257 , 285 - 294 ( 2015 )
15. Zhao , J , Wang , R: Mixed monotone iterative technique for fractional impulsive evolution equations . Miskolc Math. Notes 17 , 683 - 696 ( 2016 )
16. Jabeena , T , Lupulescu, V: Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions . J. Nonlinear Sci. Appl . 10 , 141 - 153 ( 2017 )
17. Chen , P , Li, Y , Zhang, X: Existence and uniqueness of positive mild solutions for nonlocal evolution equations . Positivity 19 , 927 - 939 ( 2015 )
18. Deimling , K : Nonlinear Functional Analysis. Springer, New York ( 1985 )
19. Guo , D , Lakshmikantham , V : Nonlinear Problems in Abstract Cone. Academic Press, Orlando ( 1988 )
20. Pazy , A : Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York ( 1983 )
21. Engel , K , Nagel, R : One-Parameter Semigroups for Linear Evolution Equations . Springer, New York ( 1995 )
22. Mainardi , F , Paradisi, P , Gorenflo, R: Probability distributions generated by fractional diffusion equations . In: Kertesz, J , Kondor , I (eds.) Econophysics: An Emerging Science . Kluwer Academic, Dordrecht ( 2000 )
23. Hernandez , E , Sakthivel, R , Tanaka, AS: Existence results for impulsive evolution differential equations with state-dependent delay . Electron. J. Differ. Equ . 2008 , Article ID 28 ( 2008 )