Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 293, pp. 1–24.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

GENERALIZED UNIFORMLY CONTINUOUS SOLUTION OPERATORS AND INHOMOGENEOUS FRACTIONAL EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS

MILOˇS JAPUNDˇZI ´C, DANIJELA RAJTER- ´CIRI ´C Communicated by Mokhtar Kirane

Abstract. We consider Cauchy problem for inhomogeneous fractional evo- lution equations with Caputo fractional derivatives of order 0< α <1 and variable coefficients depending onx. In order to solve this problem we in- troduce generalized uniformly continuous solution operators and use them to obtain the unique solution on a certain Colombeau space. In our solving proce- dure, instead of the original problem we solve a certain approximate problem, but therefore we also prove that the solutions of these two problems are asso- ciated. At the end, we illustrate the applications of the developed theory by giving some appropriate examples.

1. Introduction

Fractional evolution equations have been studied very often in the previous decades because of their numerous applications. Many well known problems are, in fact special cases of fractional evolution equations. For instance, both time frac- tional diffusion problem and time fractional reaction-advection-diffusion problem are of that type. In literature, authors have mainly considered several cases: homo- geneous case, case whenf is linear or case with constant coefficients. In this paper, we want to study semilinear problem which also includes space variable coefficients, i.e. the equation of the type

CD^{α}_{t}u(t) =Au(t) +f(·, t, u), u(0) =u0, (1.1)
where ^{C}D_{t}^{α} is the Caputo’s fractional derivative of order 0 < α < 1 and A is a
linear, closed operator densely defined on some Banach space.

Semilinear fractional Cauchy problem with variable coefficients in general case has been solved approximately, usually applying different numerical methods. One of the reasons why we have considered fractional equations in the framework of the Colombeau theory of generalized functions is the intention that these equations be treated using operator’s approach, that is, applying the solution operators as generalization of semigroup of operators.

2010Mathematics Subject Classification. 35R11, 46F30, 26A33.

Key words and phrases. Fractional evolution equation; fractional Duhamel principle;

generalized Colombeau solution operator; fractional derivative; Mittag-Leffler type function.

c

2017 Texas State University.

Submitted October 21, 2016. Published November 27, 2017.

1

In our solving procedure, instead of the original problem (1.1) we consider the approximate problem

CD^{α}_{t}u(t) =Au(t) +e f(·, t, u), u(0) =u0, (1.2)
where Aeis a generalized linear bounded operator associated (in certain sense) to
the original operator A. Therefore we will pay special attention to comparison
analysis of these two problems. To solve the approximate problem, we introduce a
notion of generalized uniformly continuous solution operator generated byA. Thise
generalized solutions operator is, in fact, a generalization of generalized uniformly
continuous semigroup of operators. (For α= 1 a generalized solution operator is
defined as a generalized semigroup of operators). Generalized uniformly continuous
semigroups were introduced in [18] and the theory has been developed later in [19] in
order to use the theory of semigroups in solving some partial differential equations
with singularities in some generalized function spaces.

Solution operators as a generalization of C0 semigroups and cosine families of operators are introduced by Bazhlekova in [5]. In [4] and [5] the corresponding solution operator theory was developed for solving some homogeneous fractional evolution problems. We remark that in some literature the solution operator is also called fractional resolvent family or fractional resolvent operator function (see e.g.

[7, 16]).

In this article, we solve problem (1.2) in the framework of the Colombeau the- ory. The theory of Colombeau generalized functions is developed in order to make possible studying some nonlinear differential equations that can not be treated neither classically (there is no classical solution) nor in distributional sense (non- linear problems include the multiplication and the multiplication of distribution is not well defined). For the Colombeau theory in general we refer, for example, to [6, 8, 17, 20].

In [14] we considered a special case of (1.2) for α = 1 and with Colombeau generalized operator Ae defined by space fractional derivatives. In this paper we make a step further by considering the problem with fractional time derivative of order 0< α <1. We obtain the unique solution to the problem (1.2) in a certain Colombeau space. In case when A is a differential operator (integer of fractional order) the regularization is necessary in order to obtain bounded operators. Our method admits variable coefficients in both ˜AandA.

This article is organized as follows. Fractional derivatives and some useful esti- mates involving them are investigated in Section 2. A part of this section is devoted to the Mittag-Leffler function since it has an important role in defining the solution operator. Colombeau spaces that we use later in the paper are defined in Section 3. In Section 4 we define uniformly continuous solution operators and prove some basic properties. In Section 5 we introduce the Colombeau uniformly continuous solution operators and develop the corresponding theory. After setting the frame- work theory, in Section 6 we investigate the inhomogeneous problem (1.2). We prove that the problem has a unique solution in a certain Colombeau space. Since in the whole paper, instead of the original problem (1.1) we study the corresponding approximate problem (1.2), Section 7 is devoted to a comparison analysis of these two problems. Finally, in the last section we illustrate how one can use our the- ory in solving some fractional evolution problems appearing in applications, such as time and time-space fractional diffusion equation and also time-space fractional

reaction-advection-diffusion equation. In these problems the corresponding differ- ential operators will be in the form of regularized operators, in order to transform unbounded differential operators into (integral) bounded operators.

2. Time fractional derivatives and some useful estimates In this section we recall definitions of fractional derivatives with respect to time variable and give some useful estimates for fractional derivatives and Mittag-Leffler function that we will use later.

2.1. Fractional derivatives with respect to time variable. The Caputo frac- tional derivative of orderα,m−1< α≤m,m∈N, has the form (see, for example, [1, 2, 15, 23, 26])

CD^{α}_{t}f(t) =J_{t}^{m−α}f^{(m)}(t), (2.1)
whereJ_{t}^{α}, α≥0, is a fractional integral for functionf(t) given by

J_{t}^{α}f(t) = 1
Γ(α)

Z t

0

(t−τ)^{α−1}f(τ)dτ,
withJ^{0}=I,I is identity operator.

The Riemann-Liouville fractional derivative of orderα,m−1< α≤m,m∈N, is given by

RLD_{t}^{α}f(t) = d^{m}

dt^{m}J_{t}^{m−α}f(t). (2.2)

Recall that ^{C}D^{α}_{t} is a left inverse of J_{t}^{α}, i.e. ^{C}D_{t}^{α}J_{t}^{α}f = f for a continuous
function f, but in general it is not a right inverse [10, Theorems 3.7, 3.8]. In
general, for an absolutely continuous functionf and 0< α <1, the following holds
J_{t}^{αC}D^{α}_{t}f(t) =f(t)−f(0).

The following holds for a Riemann-Liouville fractional derivative:

Proposition 2.1 ([28, Lemma 2.1]). For all α∈(m−1, m]andβ ≥0, it holds
J_{t}^{β+α}f(t) =J_{t}^{β+m RL}D^{m−α}_{t} f(t). (2.3)
Remark 2.2. Taking into account the properties of Mittag-Leffler function and its
integer order derivatives (especially at zero), the Colombeau space from which we
choose generalized solution operators will be defined using the spaceC^{m−1}([0,∞) :
L(E))∩C^{m}((0,∞) :L(E)) and supposing some additional properties (see Definition
3.1 and Definition 3.2). It is the space of continuously differentiable functions with
respect totand with values in spaceL(E), where (E,k · k) is a Banach space and
L(E) is the space of all linear continuous mappings fromE intoE with the norm

kAk_{L(E)}= sup

x∈E, x6=0

kAxk_{E}
kxkE

.

The following lemma will play an important role in later proofs.

Lemma 2.3. Let (E,k · k) be a Banach space and L(E) the space of all linear
continuous mappings from E into E. Let m−1 < α < m, m ∈ N. Suppose
that (·, t) → f(·, t) ∈ C^{m−1}([0,∞) : L(E))∩C^{m}((0,∞) : L(E)) is such that
lim_{t→0}+

f^{(m)}(·,t)
t^{α−m}

_{L(E)}=C <+∞. Then

CD^{α}_{t}f(·, t) = lim

η→0^{+}
C

ηD^{α}_{t}f(·, t) in L(E), (2.4)

where

C

ηD^{α}_{t}f(·, t) = 1
Γ(m−α)

Z t

η

f^{(m)}(·, τ)

(t−τ)^{α−m+1}dτ. (2.5)
Proof. Fixm∈Nandαsuch that m−1< α < m. Then

k^{C}D_{t}^{α}f(·, t)−^{C}_{η}D^{α}

tf(·, t)k_{L(E)}

≤ 1

Γ(m−α) Z η

0

kf^{(m)}(·, τ)k_{L(E)}
(t−τ)^{α−m+1} dτ

= 1

Γ(m−α) Z η

0

kf^{(m)}(·, τ)k_{L(E)}
τ^{α−m}

τ^{α−m}
(t−τ)^{α−m+1}dτ

≤ 1

Γ(m−α) sup

τ∈[0,η]

kf^{(m)}(·, τ)
τ^{α−m} k_{L(E)}

Z η

0

τ^{α−m}
(t−τ)^{α−m+1}dτ

≤ 1

Γ(m−α) sup

τ∈[0,η]

kf^{(m)}(·, τ)

τ^{α−m} k_{L(E)} η^{α−m+1}

(α−m+ 1)(t−η)^{α−m+1}.

Lettingη→0^{+} one easily gets (2.4).

The following assertion is so-called fractional mean value theorem.

Theorem 2.4 ([21]). Let 0< α <1. Fort→f(t)∈C[a, b] and^{C}_{a}D^{α}_{t}f ∈C(a, b],
the following holds

f(t) =f(a) + 1

Γ(1 +α)(^{C}_{a}D^{α}_{t}f)(ξ)(t−a)^{α}, a≤ξ≤t, t∈(a, b],
where^{C}_{a}D^{α}_{t}f is defined as in (2.5).

2.2. Mittag-Leffler function. The two-parameter Mittag-Leffler functionEα,βis given by

Eα,β(z) =

∞

X

n=0

z^{n}

Γ(β+nα), z∈C, α >0, β∈C.
Whenβ= 1 we shortly writeE_{α,1}(z)≡E_{α}(z).

If 0< α <2 andβ >0 then, for|z| → ∞, Eα,β(z) = 1

αz^{(1−β)/α}exp(z^{1/α}) +εα,β(z), |argz| ≤ απ

2 , (2.6)

where

εα,β(z) =−

N−1

X

n=1

z^{−n}

Γ(β−αn)+O(|z|^{−N}),
for someN ∈N,N 6= 1 (see [9]).

Using the previous asymptotic expansion when |z| → ∞, one can get a very useful estimation for the two-parameter Mittag-Leffler function.

Proposition 2.5. Let 0< α <2 andβ >0. Then

Eα,β(ωt^{α})≤Cα,β(1 +ω^{(1−β)/α})(1 +t^{1−β}) exp(ω^{1/α}t), ω≥0, t≥0. (2.7)
Proof. Forω= 0 and allt≥0, the inequality is trivially satisfied. Fix 0< α <2,
β > 0 and ω > 0. Choose an arbitrarily large T > 0. Then from (2.6), for all
t >(^{T}_{ω})^{α}^{1}, follows that there exists a constantC_{1}>0 such that

E_{α,β}(ωt^{α})≤C_{1}(ωt^{α})^{(1−β)/α}exp (ωt^{α})^{1/α}

=C1ω^{(1−β)/α}t^{1−β}exp(ω^{1/α}t)

≤C1(1 +ω^{(1−β)/α})(1 +t^{1−β}) exp(ω^{1/α}t).

SinceE_{α,β}is a continuous function, for allt∈[0,∞), we have that, fort∈[0,(^{T}_{ω})^{α}^{1}],
there exists a constantC_{2} such that

Eα,β(ωt^{α})≤C2≤C2(1 +ω^{(1−β)/α})(1 +t^{1−β}) exp(ω^{1/α}t).

TakingCα,β= max{C1, C2}we obtain the inequality (2.7).

The linear Cauchy problem (1.1) with Caputo fractional derivatives has been
considered in [28], but in some special spaces ofL^{p} functions whose Fourier trans-
forms are compactly supported in a some domain G, and the following result was
obtained.

Proposition 2.6 (Fractional Duhamel principle [28]). The solution of the Cauchy problem (1.1)is given by

u(t) =E_{α}(t^{α}A)u_{0}+
Z t

0

E_{α}((t−τ)^{α}A)^{RL}D^{1−α}_{τ} f(τ)dτ. (2.8)
3. Colombeau spaces

Let (E,k · k) be a Banach space and L(E) the space of all linear continuous mappings fromE intoE.

Definition 3.1. Letm−1 < α < m, m∈N. SE_{M}^{α,m}([0,∞) :L(E)) is the space
of nets

(Sα)ε: [0,∞)→ L(E), ε∈(0,1), with the following properties:

(i) t→(Sα)ε(t)∈C^{m−1}([0,∞) :L(E))∩C^{m}((0,∞) :L(E)).

(ii) lim_{t→0}+k^{dtm}^{dm}_{t}^{(S}α−m^{α}^{)}^{ε}^{(t)}k_{L(E)}=C <+∞.

(iii) For everyT >0 there existN ∈N,M >0 andε_{0}∈(0,1) such that
sup

t∈[0,T)

k^{C}D^{γ}_{t}(Sα)ε(t)k_{L(E)}≤M ε^{−N}, ε < ε0, γ∈ {0, . . . , m−1, α},
sup

t∈(0,T)

kd^{m}

dt^{m}(S_{α})_{ε}(t)k_{L(E)}≤M ε^{−N}, ε < ε_{0}.
Similarly we define the following space.

Definition 3.2. Letm−1 < α < m, m∈N. SNα,m([0,∞) :L(E)) is the space of nets

(N_{α})_{ε}: [0,∞)→ L(E), ε∈(0,1)
with the following properties:

(i) t→(N_{α})_{ε}(t)∈C^{m−1}([0,∞) :L(E))∩C^{m}((0,∞) :L(E)).

(ii) lim_{t→0}+k^{dtm}^{dm}_{t}^{(N}α−m^{α}^{)}^{ε}^{(t)}k_{L(E)}=C <+∞.

(iii) For everyT >0 anda∈Rthere existM >0 andε0∈(0,1) such that sup

t∈[0,T)

k^{C}D_{t}^{γ}(N_{α})_{ε}(t)k_{L(E)}≤M ε^{a}, ε < ε_{0}, γ∈ {0, . . . , m−1, α},
sup

t∈(0,T)

kd^{m}

dt^{m}(Nα)ε(t)k_{L(E)}≤M ε^{a}, ε < ε0.

(3.1)

Whenm= 1 we denote

SE_{M}^{α,1}([0,∞) :L(E)) =SE_{M}^{α}([0,∞) :L(E)),
SNα,1([0,∞) :L(E)) =SNα([0,∞) :L(E)).

In this article, Caputo’s fractional derivative in the problem of consideration is of order 0 < α <1 and therefore, from now on, we will consider only that case.

Hence, further we investigate spacesSE_{M}^{α}([0,∞) :L(E)) andSNα([0,∞) :L(E)),
although all the assertions we give can be extended for allm∈N.

Proposition 3.3. The space SE_{M}^{α}([0,∞) : L(E)) is an algebra with respect to
composition of operators, and SNα([0,∞) : L(E)) is an ideal of SE_{M}^{α}([0,∞) :
L(E)).

Proof. Fix 0< α <1 and letS_{α} and T_{α} are from the space SE_{M}^{α}([0,∞) :L(E)).

Then, it easily follows that S_{α}(t)T_{α}(t) satisfies the properties (i) and (ii) from
Definition (3.1). The fact thatS_{α}(t)T_{α}(t) satisfies the property (iii) forγ∈ {0,1},
can be proved in the usual way as in the case of Colombeau spaces with integer
order derivatives.

Let us prove that (iii) is satisfied for γ =α, too. Indeed, for t ∈(η, T), where η >0 is arbitrarily small andT >0, using the property (2.4) we have

k^{C}D^{α}_{t}((S_{α})_{ε}(t)(T_{α})_{ε}(t))k_{L(E)}

≤ 1

Γ(1−α) lim

η→0^{+}

Z t

η

k((Sα)ε(τ)(Tα)ε(τ))^{0}k_{L(E)}

(t−τ)^{α} dτ

≤ 1

Γ(1−α) lim

η→0^{+}

Z t

η

k((S_{α})_{ε}(τ))^{0}k_{L(E)}k(T_{α})_{ε}(τ)k_{L(E)}

(t−τ)^{α} dτ

+ 1

Γ(1−α) lim

η→0^{+}

Z t

η

k(Sα)ε(τ)k_{L(E)}k((Tα)ε(τ))^{0}k_{L(E)}

(t−τ)^{α} dτ

≤ lim

η→0^{+}

(t−η)^{1−α}
Γ(2−α) M1ε^{−N}

≤ T^{1−α}

Γ(2−α)M_{1}ε^{−N}.

Thus, we obtain the moderate bound fort∈(0, T), i.e.

sup

t∈(0,T)

k^{C}D^{α}_{t}((Sα)ε(t)(Tα)ε(t))k_{L(E)}≤M ε^{−N}.
It remains to prove the moderate bound for k^{C}D^{α}_{t}((Sα)ε(t)(Tα)ε(t))

_{t=0}k. From
Theorem 2.4 we obtain

CD^{α}_{t}((Sα)ε(t)(Tα)ε(t))
_{t=0}

= 1

Γ(1 +α) lim

t→0^{+}

(S_{α})_{ε}(t)(T_{α})_{ε}(t)−(S_{α})_{ε}(0)(T_{α})_{ε}(0)
t^{α}

= 1

Γ(1 +α) lim

t→0^{+}

((Sα)ε(t)−(Sα)ε(0))(Tα)ε(t)
t^{α}

+ 1

Γ(1 +α) lim

t→0^{+}

(Sα)ε(0))((Tα)ε(t)−(Tα)ε(0))
t^{α}

=^{C}D^{α}_{t}((S_{α})_{ε}(t))

_{t=0}(T_{α})_{ε}(0) + (S_{α})_{ε}(0)^{C}D_{t}^{α}((T_{α})_{ε}(t))
_{t=0}.

Estimating in norm, we obtain a moderate bound fork^{C}D_{t}^{α}((Sα)ε(t)(Tα)ε(t))
_{t=0}k.

Thus, (iii) is satisfied.

Similarly, one can prove that (T_{α})_{ε}(t)(S_{α})_{ε}(t), also satisfies all properties from
Definition (3.1). Thus, the space SE_{M}^{α}([0,∞) : L(E)) is an algebra. One can
similarly prove that the space SNα([0,∞) : L(E)) is an ideal of SE_{M}^{α}([0,∞) :

L(E)).

Now we can define a Colombeau-type space as a factor algebra by
SGα([0,∞) :L(E)) = SE_{M}^{α}([0,∞) :L(E))

SNα([0,∞) :L(E)). (3.2)
For every 0< α <1 elements ofSGα([0,∞) :L(E)) will be denoted byS= [(Sα)ε],
where (S_{α})_{ε}is a representative of the class.

Similarly, one can define the following spaces: SEM(E) is the space of nets of linear continuous mappings

Aε:E→E, ε∈(0,1),

with the property that there exists constantsN ∈N, M >0 and ε0 ∈(0,1) such that

kAεk_{L(E)}≤M ε^{−N}, ε < ε0.

SN(E) is the space of nets of linear continuous mappingsAε:E→E,ε∈(0,1), with the property that for everya∈R, there existM >0 andε0∈(0,1) such that

kA_{ε}k_{L(E)}≤M ε^{a}, ε < ε_{0}.

The Colombeau space of generalized linear operators onE is defined by SG(E) = SEM(E)

SN(E) .

Elements of SG(E) will be denoted byA = [Aε], where Aε is a representative of the class.

Finally, we introduce the Colombeau space within which we will solve (1.2).

We give the definitions for arbitrary m ∈ N. Let m−1 < α < m, m ∈ N.
E_{M}^{α}([0,∞) :H^{m}(R)) is the space of nets

Gε: [0,∞)×R→C, ε∈(0,1), with the following properties:

(i) G_{ε}(·,·)∈C^{m−1}([0,∞) :H^{m}(R))∩C^{m}((0,∞) :H^{m}(R)).

(ii) lim_{t→0}+k^{dtm}^{dm}_{t}α−m^{G}^{ε}^{(t,·)}kH^{m} =C <+∞.

(iii) For everyT >0 there existM >0, N∈Nandε0>0 such that sup

t∈[0,T)

k^{C}D_{t}^{γ}Gε(t,·)kH^{m}≤M ε^{−N}, ε < ε0, γ∈ {0, . . . , m−1, α},
sup

t∈(0,T)

kd^{m}

dt^{m}Gε(t,·)kH^{m}≤M ε^{−N}, ε < ε0.

(3.3)

It is an algebra with respect to multiplication.

Similarly, form−1< α < m, m∈N, N_{α}([0,∞) :H^{m}(R)) is the space of nets
Gε∈ E_{M}^{α}([0,∞) :H^{m}(R)) with the following properties:

(i) Gε(·,·)∈C^{m−1}([0,∞) :H^{m}(R))∩C^{m}((0,∞) :H^{m}(R)).

(ii) lim_{t→0}+k^{dtm}^{dm}_{t}α−m^{G}^{ε}^{(t,·)}kH^{m} =C <+∞.

(iii) For everyT >0 anda∈Rthere existM >0 andε0>0 such that sup

t∈[0,T)

k^{C}D_{t}^{γ}G_{ε}(t,·)kH^{m} ≤M ε^{a}, ε < ε_{0}, γ∈ {0, . . . , m−1, α},
sup

t∈(0,T)

kd^{m}

dt^{m}G_{ε}(t,·)k_{H}m ≤M ε^{a}, ε < ε_{0}.

(3.4)

The spaceNα([0,∞) :H^{m}(R)) is an ideal ofE_{M}^{α}([0,∞) :H^{m}(R)).

The quotient space

G_{α}([0,∞) :H^{m}(R)) = E_{M}^{α}([0,∞) :H^{m}(R))
Nα([0,∞) :H^{m}(R))

is the corresponding Colombeau generalized function space related to the Sobolev
spaceH^{m}. Again, in this paper we will consider only the casem= 1 andm= 2, i.e.

the solution of our fractional evolution problem will be an element of G_{α}([0,∞) :
H^{1}(R)) orGα([0,∞) :H^{2}(R)).

In a similar way, by omitting variable t, one can define spaces E_{M}^{α}(H^{m}(R)),
Nα(H^{m}(R)), andGα(H^{m}(R)).

4. Uniformly continuous solution operators

Consider the Cauchy problem for the fractional evolution equation of order α with 0< α <1,

CD^{α}_{t}u(t) =Au(t), t >0; u(0) =x, (4.1)
where ^{C}D^{α}_{t} is the Caputo fractional derivative of order α, andA is a linear and
bounded operator defined on a Banach spaceE. The more general case whenAis
a closed linear operator densely defined in a Banach spaceE was considered in [5].

As it is pointed out in [5], the problem (4.1) is well-posed if and only if the Volterra integral equation

u(t) =x+ Z t

0

g_{α}(t−τ)Au(τ)dτ (4.2)

is well-possed, wheregα(t) is defined for α >0, by
g_{α}(t) =

(t^{α−1}/Γ(α), t >0,

0, t≤0.

In the general case when A is a closed linear operator densely defined in a Banach space E, strongly continuous solution operator for (4.1) is introduced in [5]. Similarly, when A is linear and bounded, we introduce uniformly continuous solution operator.

Definition 4.1. A familySα(t), t≥0, of linear and bounded operators on Banach spaceEis called a uniformly continuous solution operator for (4.1) if the following conditions are satisfied:

(i) Sα(t) is a uniformly continuous function fort≥0 andSα(0) =I, whereI is identity operator onE.

(ii) ASα(t)x=Sα(t)Ax, for allx∈E,t≥0.

(iii) Sα(t)xis a solution of (4.2) for allx∈E, t≥0.

Definition 4.2. The infinitesimal generatorAof a uniformly continuous solution operatorSα(t),α >0,t≥0, for (4.1) is defined by

Ax= Γ(1 +α) lim

t↓0

S_{α}(t)x−x

t^{α} , (4.3)

for allx∈E.

The generatorAcould also be defined as
Ax= (^{C}D^{α}_{t}Sα)(t)x

_{t=0},

sinceJ_{t}^{α}^{C}D^{α}_{t}Sα(t)x=Sα(t)x−xand for all functionsv∈C(R+;E) holds
limt↓0

J_{t}^{α}v(t)

gα+1(t) =v(0) (see [5]).

Remark 4.3. In the case 0< α≤1, the definition given by (4.3) also follows from Theorem 2.4.

Definition 4.4([5]). The solution operatorSα(t) is called exponentially bounded if there exist constantsM ≥1 andω≥0 such that

kSα(t)k ≤M e^{ωt}, t≥0.

Theorem 4.5 ([5, Theorem 2.5]). Let α > 0. Then exponentially bounded uni- formly continuous solution operator Sα(t) is the solution operator for the Cauchy problem (4.1)if and only if A∈ L(E).

From Definition 4.2 it follows that every solution operator has a unique infini-
tesimal generator. If Sα(t) is a uniformly continuous solution operator satisfying
kSα(t)k ≤ M e^{ωt}, for some M ≥ 1 and ω ≥ 0, its infinitesimal generator is a
bounded linear operator.

On the other hand, every bounded linear operatorAis the infinitesimal generator of a uniformly continuous solution operator given by

Sα(t) =Eα(t^{α}A) =

∞

X

n=0

t^{nα}A^{n}

Γ(1 +nα), α >0, t≥0.

For every 0 < α≤1 this solution operator is unique as asserted in the following theorem.

Theorem 4.6. Let 0 < α ≤ 1 and let Sα(t) and Tα(t) be exponential bounded uniformly continuous solution operators with infinitesimal generators A and B, respectively. IfA=B then Sα(t) =Tα(t), for everyt≥0.

Proof. SinceSα(t) is exponential bounded there exist constantsM ≥1 andω1≥0 such that

kSα(t)k ≤M e^{ω}^{1}^{t}, t≥0.

Then forReλ > ω1 andx∈E we have
λ^{α−1}R(λ^{α}, A)x=

Z ∞

0

e^{−λt}Sα(t)xdt,

where R(λ, A) = (λI−A)^{−1} stands for the resolvent operator ofA. Similarly, for
Tα(t) there exists ω2≥0 such that forReλ > ω2 andx∈E we have

λ^{α−1}R(λ^{α}, A)x=
Z ∞

0

e^{−λt}Tα(t)xdt,

andSα(t) =Tα(t) follows from the uniqueness of the Laplace transform.

Proposition 4.7. LetS_{α}(t),0< α≤1,t≥0, be a uniformly continuous solution
operator satisfyingkS_{α}(t)k ≤M e^{ωt}, for someM ≥1 andω≥0. Then

(i) There exists a unique bounded linear operatorAsuch that
S_{α}(t) =E_{α}(t^{α}A), t≥0.

(ii) The operator A in (i) is the infinitesimal generator of solution operator Sα(t).

(iii) For everyt≥0,

CD^{α}_{t}S_{α}(t) =AS_{α}(t) =S_{α}(t)A.

Proof. Fix 0< α≤1. From Theorem 4.5 we know that the infinitesimal generator
of S_{α}(t) is a bounded linear operator A. Also,A is the infinitesimal generator of
E_{α}(t^{α}A) and therefore by Theorem 4.6, S_{α}(t) =E_{α}(t^{α}A). All others assertions of

the proposition follow from (i).

Integral representation stated in the next proposition will often be used in prov- ing some auxiliary results as well as in proving our main result.

Proposition 4.8. Let0< α <1 and letSα(t)be a solution operator generated by A. Then

Z t

0

S_{α}(t−τ)^{RL}D_{τ}^{1−α}f(τ)dτ =
Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}A)f(τ)dτ. (4.4)
Proof. Fix 0< α <1. Taking into account the relation 2.3 in Proposition 2.1 one
gets

Z t

0

Sα(t−τ)^{RL}D_{τ}^{1−α}f(τ)dτ =
Z t

0

∞

X

n=0

1

Γ(1 +nα)(t−τ)^{nα}A^{n RL}D^{1−α}_{τ} f(τ)dτ

=

∞

X

n=0

Z t

0

1

Γ(1 +nα)(t−τ)^{nα}A^{n RL}D^{1−α}_{τ} f(τ)dτ

=

∞

X

n=0

J_{t}^{nα+1}A^{n RL}D^{1−α}_{t} f(t) =

∞

X

n=0

J_{t}^{nα+α}A^{n}f(t)

=

∞

X

n=0

1 Γ(nα+α)

Z t

0

(t−τ)^{nα+α−1}A^{n}f(τ)dτ

= Z t

0

(t−τ)^{α−1}

∞

X

n=0

1

Γ(nα+α)(t−τ)^{nα}A^{n}f(τ)dτ

= Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}A)f(τ)dτ.

Similarly, the first order derivative of the previously integral representation has the following form.

Proposition 4.9. Let0< α <1 and letSα(t)be a solution operator generated by A. Then

d dt

Z t

0

S_{α}(t−τ)^{RL}D^{1−α}_{τ} f(τ)dτ =
Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}A)∂_{τ}f dτ
+t^{α−1}Eα,α(t^{α}A)f(0).

(4.5)

Proof. Fix 0< α <1. From the proof of Proposition 4.8 it follows that d

dt Z t

0

Sα(t−τ)^{RL}D^{1−α}_{τ} f(τ)dτ =

∞

X

n=0

d

dtJ_{t}^{nα+α}A^{n}f(t).

Further, since d

dtJ_{t}^{α}f(t) =^{RL}D^{1−α}_{t} f(t) =^{C}D_{t}^{1−α}f(t) +f(0)t^{α−1}

Γ(α) =J_{t}^{α}d

dtf(t) +f(0)t^{α−1}
Γ(α) ,
we have

∞

X

n=0

d

dtJ_{t}^{nα+α}A^{n}f(t) =

∞

X

n=0

J_{t}^{nα+α}A^{n}d
dtf(t) +

∞

X

n=0

J_{t}^{nα}t^{α−1}A^{n}f(0)
Γ(α)

=

∞

X

n=0

J_{t}^{nα+α}A^{n}d
dtf(t) +

∞

X

n=0

t^{nα+α−1}

Γ(α+nα)A^{n}f(0),
and similarly to the proof of Proposition 4.8 one finally gets the relation (4.5).

Motivated by Proposition 2.6 we give the fractional Duhamel principle in the case of solution operator.

Proposition 4.10. The solution of the Cauchy problem (1.1) with Caputo frac- tional derivative is given by

u(t) =Sα(t)u0+ Z t

0

Sα(t−τ)^{RL}D^{1−α}_{τ} f(·, τ, u)dτ, (4.6)
whereS_{α}(t)is a solution operator generated byA. The solution above is called mild
solution to the problem (1.1).

Proof. Since^{C}D^{α}_{t}S_{α}(t) =AS_{α}(t), for a continuous function its fractional integral
J_{t}^{α}is a continuous function too and ^{C}D_{t}^{α} is a left inverse of fractional integralJ_{t}^{α}
for allα≥0 and all continuous functions, it can be easily shown thatu(t) given by

(4.6) satisfies the Cauchy problem (1.1).

Remark 4.11. The solution of the Cauchy problem (1.1) can also be represented by Caputo fractional derivative, but in that case one must additionally suppose thatf(·,0, u0) = 0.

5. Generalized uniformly continuous solution operators First, recall that every linear and bounded operator on Banach space E is a closed and densely defined operator in E. Therefore, all results in the previous section continue to be valid in the case of linear and bounded operators on Banach space.

Instead of the Cauchy problem (4.1) with closed and densely defined operator A, let us now consider fractional Cauchy problem given by

CD_{t}^{α}u(t) =Au(t),e t >0;u(0) =x, (5.1)
whereAeis a generalized linear bounded operator.

Definition 5.1. Let 0< α <1. Sα ∈ SGα([0,∞) :L(E)) is called a Colombeau uniformly continuous solution operator for (5.1) if it has a representative (Sα)ε

which is a uniformly continuous solution operator for (5.1) and for every ε small enough.

Proposition 5.2. Let0< α <1 and let(Sα)1εand(Sα)2ε be representatives of a
generalized uniformly continuous solution operatorS_{α}, with infinitesimal generators
Ae_{1ε} andAe_{2ε}, respectively, for εsmall enough. Then

Ae_{1ε}−Ae_{2ε}∈ SN(E).

Proof. Fix 0< α <1. Then we have
Ae1ε−Ae2ε= (^{C}D^{α}_{t}(Sα)1ε)(t)

_{t=0}−(^{C}D_{t}^{α}(Sα)2ε)(t)
_{t=0}

=^{C}D^{α}_{t}((S_{α})_{1ε}−(S_{α})_{2ε})(t)
t=0.
Since

(S_{α})_{1ε}−(S_{α})_{2ε}∈ SNα([0,∞) :L(E)),
we have that, for everya∈R, there existsM >0 such that

k^{C}D_{t}^{α}((Sα)1ε−(Sα)2ε)(t)

_{t=0}k_{L(E)}≤M ε^{a}.

It implies that for everya∈Rthere existsM >0 such thatkAe_{1ε}−Ae_{2ε}k ≤M ε^{a}.

Thus,Ae_{1ε}−Ae_{2ε}∈ SN(E).

Definition 5.3. Ae∈ SG(E) is called the infinitesimal generator of a Colombeau
uniformly continuous solution operatorS_{α}∈ SGα([0,∞) :L(E)), 0< α <1, ifAe_{ε}
is the infinitesimal generator of the representative (S_{α})_{ε}, for everyεsmall enough.

Proposition 5.4. Let 0 < α < 1. Let Ae be the infinitesimal generator of a
Colombeau uniformly continuous solution operatorS_{α}, andBethe infinitesimal gen-
erator of a Colombeau uniformly continuous solution operator T_{α}. If Ae=B, thene
S_{α}=T_{α}.

Proof. Fix 0< α <1 and let Neε =Aeε−Beε ∈ SN(E). Then from the property (iii) in Proposition 4.7 we obtain

CD_{t}^{α}((S_{α})_{ε}−(T_{α})_{ε})(t)x=Ae_{ε}((S_{α})_{ε}−(T_{α})_{ε})(t)x+Ne_{ε}(T_{α})_{ε}(t)x.

By using fractional Duhamel principle (4.6) and since (Sα)ε(0) = (Tα)ε(0) =I, one gets

((Sα)ε−(Tα)ε)(t)x= Z t

0

(Sα)ε(t−τ)^{RL}D^{1−α}_{τ} Neε(Tα)ε(τ)xdτ. (5.2)

Then, from the integral representation given in Proposition 4.8 we have ((Sα)ε−(Tα)ε)(t)x=

Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}Aeε)Neε(Tα)ε(τ)xdτ

= Z t

0

(t−τ)^{α−1}

∞

X

n=0

(t−τ)^{nα}Ae^{n}_{ε}

Γ(α+nα) Ne_{ε}(T_{α})_{ε}(τ)xdτ

= Z t

0

∞

X

n=0

(t−τ)^{(n+1)α−1}Ae^{n}_{ε}

Γ((n+ 1)α) Neε(Tα)ε(τ)xdτ

=

∞

X

n=1

Z t

0

(t−τ)^{nα−1}Ae^{n−1}_{ε}

Γ(nα) Neε(Tα)ε(τ)xdτ.

Fort∈[0, T),T >0, we obtain estimate
k((S_{α})_{ε}−(T_{α})_{ε})(t)k ≤

∞

X

n=1

1 Γ(nα)

Z t

0

(t−τ)^{nα−1}kAe^{n−1}_{ε} Ne_{ε}(T_{α})_{ε}(τ)kdτ

≤ kNeεk sup

t∈[0,T)

k(Tα)ε(t)k

∞

X

n=1

kAeεk^{n−1} 1
Γ(nα)

T^{nα}
nα

≤ kNe_{ε}k sup

t∈[0,T)

k(Tα)_{ε}(t)kT^{α}
α

∞

X

n=0

T^{nα}kAe_{ε}k^{n}
Γ(α+nα)

=kNeεk sup

t∈[0,T)

k(Tα)ε(t)kT^{α}

α Eα,α(T^{α}kAeεk),
and using the estimate (2.7) forEα,α we have

k((Sα)ε−(Tα)ε)(t)k

≤ kNeεk sup

t∈[0,T)

k(Tα)ε(t)kT^{α}

α Cα(1 +kAeεk^{(1−α)/α})(1 +T^{1−α}) exp(TkAeεk^{1/α})

= Cα

α kNeεk sup

t∈[0,T)

k(Tα)ε(t)k(1 +kAeεk^{(1−α)/α})(T+T^{α}) exp(TkAeεk^{1/α}).

Now, we consider the caseγ=α. Fort∈[0, T),T >0, one similarly gets
k^{C}D^{α}_{t}((Sα)ε−(Tα)ε)(t)k ≤ kNeεk sup

t∈[0,T)

k(Tα)ε(t)k

∞

X

n=0

kAeεk^{n}
Γ(nα)· T^{nα}

nα

=kNeεk sup

t∈[0,T)

k(Tα)ε(t)k ·Eα(T^{α}kAeεk).

Differentiation of integral representation (5.2) with respect to t, using integral representation (4.5) one gets

d

dt((Sα)ε−(Tα)ε)(t)x

= Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}Aeε)Neε

d

dτ(Tα)ε(τ)xdτ+t^{α−1}Eα,α(t^{α}Aeε)Neεx.

Then, for everyT_{1}>0 andt∈[T_{1}, T), the estimate in norm is
kd

dt((S_{α})_{ε}−(T_{α})_{ε})(t)k

≤ lim

η→0^{+}

Z t

η

(t−τ)^{α−1}kEα,α((t−τ)^{α}Aeε)Neε

d

dτ(Tα)ε(τ)kdτ
+t^{α−1}Eα,α(t^{α}kAeεk)kNeεk

≤ lim

η→0^{+} sup

τ∈[η,T)

E_{α,α}((T−τ)^{α}kAe_{ε}k)kNe_{ε}k sup

τ∈[η,T)

k d

dτ(T_{α})_{ε}(τ)k(T−η)^{α}
α
+T_{1}^{α−1}E_{α,α}(T^{α}kAe_{ε}k)kNe_{ε}k.

Finally, since Ne_{ε} ∈ SN(E) it follows that for every a∈R there exists M > 0
such that

sup

t∈[0,T)

k^{C}D_{t}^{γ}((S_{α})_{ε}−(T_{α})_{ε})(t)k_{L(E)}≤M ε^{a}, γ∈ {0, α},
sup

t∈(0,T)

kd

dt((Sα)ε−(Tα)ε)(t)k_{L(E)}≤M ε^{a},

i.e. (Sα)ε−(Tα)ε∈ SNα([0,∞) :L(E)).

Definition 5.5. Let hε be a positive net satisfying hε ≤ ε^{−1}. It is said that
Ae∈ SG(E) is ofhε-type if it has a representativeAeεsuch that

kAeεk_{L(E)}=O(hε), ε→0.

An elementG∈ Gα([0,∞) :H^{1}(R)) is said to be ofh_{ε}-type if it has a representative
G_{ε}such that

kGεk_{H}1 =O(hε), ε→0.

The following proposition holds for generalized operators.

Proposition 5.6. Let 0 < α < 1. Every Ae ∈ SG(E) of hε-type, where hε ≤
C(log 1/ε)^{α}, is the infinitesimal generator of some generalized uniformly continuous
solution operatorSα∈ SGα([0,∞) :L(E)).

Proof. Fix 0< α <1. From Theorem 4.5 one knows that every linear and bounded
operator Ae_{ε} is the infinitesimal generator of some uniformly continuous solution
operator (Sα)ε(t) defined by

(Sα)ε(t) =Eα(t^{α}Aeε) =

∞

X

n=0

t^{nα}Ae^{n}_{ε}
Γ(1 +nα).

Let us show that (Sα)ε ∈ SE_{M}^{α}([0,∞) : L(E)). From the inequality for Mittag-
Leffler function it follows that there exists constantM >0 such that

k(Sα)_{ε}(t)k ≤Mexp(tkAe_{ε}k^{1/α}).

Sinceh_{ε}≤C(log 1/ε)^{α}, we have
sup

t∈[0,T)

k(Sα)ε(t)k ≤M ε^{−T C}^{1/α},

forεsmall enough. Also, since^{C}D^{α}_{t}(Sα)ε(t) =Aeε(Sα)ε(t), for everyt≥0, we have
for everyεsmall enough

k^{C}D^{α}_{t}(S_{α})_{ε}(t)k ≤ kAe_{ε}kk(S_{α})_{ε}(t)k ≤C(log1

ε)^{α}M ε^{−T C}^{1/α} ≤CM ε^{−α−T C}^{1/α}.

It remains to prove the moderate bound fork_{dt}^{d}(Sα)ε(t)k. First, we have
d

dt(S_{α})_{ε}(t) =

∞

X

n=0

t^{(n+1)α−1}
Γ(α+nα)Ae^{n+1}_{ε}

=t^{α−1}Aeε

∞

X

n=0

t^{nα}
Γ(α+nα)Ae^{n}_{ε}

=t^{α−1}Ae_{ε}E_{α,α}(t^{α}Ae_{ε}).

Then, for everyT_{1}>0 andt∈[T_{1}, T), the estimate in norm is
kd

dt(S_{α})_{ε}(t)k ≤T_{1}^{α−1}kAe_{ε}kE_{α,α}(t^{α}kAe_{ε}k)

≤T_{1}^{α−1}kAeεkCα(1 +kAeεk^{(1−α)/α})·exp(kAeεk^{1/α}T)(1 +T^{1−α})

≤T_{1}^{α−1}Cα(kAeεk+kAeεk^{1/α})·exp(kAeεk^{1/α}T)(1 +T^{1−α})

≤T_{1}^{α−1}C_{α}((log1

ε)^{α}+ log1

ε)·exp(C^{1/α}Tlog1

ε)(1 +T^{1−α})

≤2T_{1}^{α−1}Cα(1 +T^{1−α})ε^{−1−C}^{1/α}^{T}.

Thus finally we have (Sα)ε∈ SE_{M}^{α}([0,∞) :L(E)).

Note that a Colombeau uniformly continuous solution operator always possess an infinitesimal generator and it is unique. That follows from the fact that its representative is a classical uniformly continuous solution operator for which there exists a unique infinitesimal generator.

6. Existence and uniqueness result

In this section we specify the Banach space, i.e. we takeE =L^{2}(R). Instead
of the Cauchy problem (4.1) with closed and densely defined operatorAonL^{2}(R)
with domainD(A) =H^{1}(R), we will consider fractional Cauchy problem given by

CD^{α}_{t}u(t) =Au(t), t >e 0;u(0) =x,

where Ae is a generalized linear bounded operator L^{2}-associated with A, i.e., for
everyu∈H^{1}(R), the following holds

k(A−Ae_{ε})uk_{L}2 →0, ε→0.

Theorem 6.1. Let 0 < α < 1. Suppose that u0 ∈ Gα(H^{1}(R)) and let the func-
tionf(x, t, u)be continuously differentiable with respect tot, globally Lipschitz with
respect to x and u with bounded second order derivative with respect to u and
f(x, t,0) = 0. Also, suppose that ∂xf(x, t, u) and ∂tf(x, t, u) are globally Lips-
chitz function with respect to u. Let g1(x, t, u) := ∂uf(x, t, u) and g2(x, t, u) :=

∂_{t}f(x, t, u)satisfy the same conditions asf(x, t, u).

Let the operator Ae ∈ SG(H^{1}(R)) be of h_{ε}-type, with h_{ε} =o (log(log 1/ε)^{α})^{α}
,
such that kAeεuεkL^{2} ≤hεkuεkL^{2}, for uε∈H^{1}(R).

Then for every 0 < α < 1 there exists a unique generalized solution u ∈
Gα([0,∞) :H^{1}(R))to the Cauchy problem

CD^{α}_{t}u(t) =Au(t) +e f(·, t, u), u(0) =u_{0}. (6.1)

An equivalent integral equation for the solution (i.e. mild solution) is given by
u_{ε}(t) = (S_{α})_{ε}(t)u_{0ε}+

Z t

0

(S_{α})_{ε}(t−τ)^{RL}D_{τ}^{1−α}f(·, τ, u_{ε})dτ, (6.2)
whereSα∈ SGα([0,∞) :L(H^{1}(R))) is a Colombeau uniformly continuous solution
operator generated by A.e

Remark 6.2. The existence of a solution for integral equation (6.2) can be proved using a Banach principle of a fixed point.

Proof of Theorem 6.1. Fix 0 < α < 1. Since the operator Ae is of hε-type, with
hε=o((log log 1/ε)^{α}), it is obvious that the operatorAeis the infinitesimal generator
of a Colombeau solution operatorSα ∈ SGα([0,∞) :L(H^{1}(R))) given bySα(t) =
E_{α}(t^{α}A) (see Proposition 5.6). Also, from (4.6) we know that (6.2) represents ae
solution to (6.1).

Let us show that this solution is an element of Gα([0,∞) : H^{1}(R)). First, we
show that the solution satisfies

lim

t→0^{+}k

d
dtu_{ε}(t,·)

t^{α−1} k_{H}1=C <+∞. (6.3)

Indeed, after differentiation of (6.2) with respect tot, using the first order derivative of integral representation (4.5) one gets

d dtuε(t,·)

= d

dt(S_{α})_{ε}(t)u_{0ε}+
Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}Ae_{ε})∂_{τ}f(·, τ, u_{ε}(τ))dτ
+t^{α−1}E_{α,α}(t^{α}Ae_{ε})f(·,0, u_{0ε}),

(6.4)

and by to the notation g_{1}(x, t, u) = ∂_{u}f(x, t, u) and g_{2}(x, t, u) = ∂_{t}f(x, t, u), we
have

kd

dtuε(t,·)kL^{2}≤ kd

dt(S_{α})ε(t)u0εkL^{2}

+ Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}kAeεk)k∂τf(·, τ, uε(τ))k_{L}2dτ
+t^{α−1}Eα,α(t^{α}kAeεk)kf(·,0, u0ε)k_{L}2

≤ kd

dt(Sα)ε(t)u0εkL^{2}

+ Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}kAeεk)kg1kL^{∞}k∂τuε(τ)k_{L}2dτ
+

Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}kAeεk)kg2kL^{∞}kuε(τ)kL^{2}dτ
+t^{α−1}Eα,α(t^{α}kAeεk)kf(·,0, u0ε)k_{L}2.

(6.5)

After applying the Gronwall’s inequality to (6.5) one gets lim

t→0^{+}k

d
dtu_{ε}(t,·)

t^{α−1} kL^{2} =C <+∞.

Further, differentiation of (6.4) with respect tox, we have
k∂_{x}d

dtu_{ε}(t,·)k_{L}2

≤ kd

dt(S_{α})ε(t)∂xu0εk_{L}2

+ Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}kAe_{ε}k)k∂_{x}∂_{τ}f(·, τ, u_{ε}(τ))k_{L}2dτ
+t^{α−1}E_{α,α}(t^{α}kAe_{ε}k)k∂xf(·,0, u_{0ε})kL^{2}

≤ kd

dt(S_{α})ε(t)∂xu0εk_{L}2

+ Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}kAe_{ε}k)k∂xg_{1}kL^{∞}k∂τu_{ε}(τ)kL^{2}dτ
+

Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}kAeεk)kg1kL^{∞}k∂x∂τuε(τ)k_{L}2dτ
+

Z t

0

(t−τ)^{α−1}Eα,α((t−τ)^{α}kAeεk)k∂xg2kL^{∞}kuε(τ)kL^{2}dτ
+

Z t

0

(t−τ)^{α−1}E_{α,α}((t−τ)^{α}kAe_{ε}k)kg2kL^{∞}k∂xu_{ε}(τ)kL^{2}dτ
+t^{α−1}Eα,α(t^{α}kAeεk)[k∂uf(·,0, u0ε)kL^{∞}k∂xu0εkL^{2}

+k∂xf(·,0, u0ε)kL^{∞}ku0εk_{L}2].

(6.6)

Again, after applying the Gronwall’s inequality one gets lim

t→0^{+}k∂_{x}_{dt}^{d}u_{ε}(t,·)

t^{α−1} k_{L}2 =C <+∞,
and finally we have that property (6.3) is satisfied.

Further, we prove that one has the moderate bound for k^{C}D_{t}^{γ}uε(t,·)k_{H}1, γ ∈
{0, α}, andk_{dt}^{d}u_{ε}(t,·)k_{H}1. First, we prove the moderate bound fork^{C}D_{t}^{γ}u_{ε}(t,·)k_{H}1,
and consider the cases:

Case 1: γ= 0. From the representation (6.2) and Proposition 4.8 we obtain
kuε(t)k_{L}2 ≤ k(Sα)ε(t)u0εk_{L}2+

Z t

0

(t−τ)^{α−1}kEα,α((t−τ)^{α}Aeε)k kf(·, τ, uε)k_{L}2dτ.

Next, using the estimate forE_{α,α} one gets
kEα,α(t^{α}Aeε)k ≤

∞

X

n=0

t^{nα}kAe_{ε}k^{n}

Γ(α+nα)=Eα,α(t^{α}kAeεk)

≤Cα(1 +kAeεk^{(1−α)/α})(1 +t^{1−α}) exp(tkAeεk^{1/α}).

Denote

Mf_{T} := sup

t∈[0,T)

kEα,α(t^{α}Ae_{ε})k. (6.7)
Note that forα= 1 it followsMfT := sup_{t∈[0,T)}kS(t)k, whereS(t) is a generalized
uniformly continuous semigroup of operators generated by the operatorAe(see [14]).