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

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_tu(t) =Au(t) +f(·, t, u), u(0) =u0, (1.1) where ^CD_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_tu(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_tf(t) =J_t^m−αf^(m)(t), (2.1) whereJ_t^α, α≥0, is a fractional integral for functionf(t) given by

J_t^αf(t) = 1 Γ(α)

Z t

0

(t−τ)^α−1f(τ)dτ, withJ⁰=I,I is identity operator.

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

RLD_t^αf(t) = d^m

dt^mJ_t^m−αf(t). (2.2)

Recall that ^CD^α_t is a left inverse of J_t^α, i.e. ^CD_t^αJ_t^αf = 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^αCD^α_tf(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+αf(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_tf(·, t) = lim

η→0⁺ C

ηD^α_tf(·, t) in L(E), (2.4)

where

C

ηD^α_tf(·, t) = 1 Γ(m−α)

Z t

η

f^(m)(·, τ)

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

k^CD_t^αf(·, t)−^C_ηD^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+1dτ

≤ 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+1dτ

≤ 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_aD^α_tf ∈C(a, b], the following holds

f(t) =f(a) + 1

Γ(1 +α)(^C_aD^α_tf)(ξ)(t−a)^α, a≤ξ≤t, t∈(a, b], where^C_aD^α_tf 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

zⁿ

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)/αexp(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/αt), 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)^α1, follows that there exists a constantC₁>0 such that

E_α,β(ωt^α)≤C₁(ωt^α)^(1−β)/αexp (ωt^α)^1/α

=C1ω^(1−β)/αt^1−βexp(ω^1/αt)

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

SinceE_α,βis a continuous function, for allt∈[0,∞), we have that, fort∈[0,(^T_ω)^α1], there exists a constantC₂ such that

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

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^αA)u₀+ Z t

0

E_α((t−τ)^αA)^RLD^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^dtmdm_t^(Sα−m^α)ε(t)k_L(E)=C <+∞.

(iii) For everyT >0 there existN ∈N,M >0 andε₀∈(0,1) such that sup

t∈[0,T)

k^CD^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 < ε₀. 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^dtmdm_t^(Nα−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^CD_t^γ(N_α)_ε(t)k_L(E)≤M ε^a, ε < ε₀, 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^CD^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))⁰k_L(E)

(t−τ)^α dτ

≤ 1

Γ(1−α) lim

η→0⁺

Z t

η

k((S_α)_ε(τ))⁰k_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))⁰k_L(E)

(t−τ)^α dτ

≤ lim

η→0⁺

(t−η)^1−α Γ(2−α) M1ε^−N

≤ T^1−α

Γ(2−α)M₁ε^−N.

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

sup

t∈(0,T)

k^CD^α_t((Sα)ε(t)(Tα)ε(t))k_L(E)≤M ε^−N. It remains to prove the moderate bound for k^CD^α_t((Sα)ε(t)(Tα)ε(t))

_t=0k. 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

=^CD^α_t((S_α)_ε(t))

_t=0(T_α)_ε(0) + (S_α)_ε(0)^CD_t^α((T_α)_ε(t)) _t=0.

Estimating in norm, we obtain a moderate bound fork^CD_t^α((Sα)ε(t)(Tα)ε(t)) _t=0k.

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)_εis 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_εk_L(E)≤M ε^a, ε < ε₀.

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^dtmdm_tα−m^Gε(t,·)kH^m =C <+∞.

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

t∈[0,T)

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

t∈(0,T)

kd^m

dt^mGε(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^dtmdm_tα−m^Gε(t,·)kH^m =C <+∞.

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

t∈[0,T)

k^CD_t^γG_ε(t,·)kH^m ≤M ε^a, ε < ε₀, γ∈ {0, . . . , m−1, α}, sup

t∈(0,T)

kd^m

dt^mG_ε(t,·)k_Hm ≤M ε^a, ε < ε₀.

(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,∞) : H¹(R)) orGα([0,∞) :H²(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_tu(t) =Au(t), t >0; u(0) =x, (4.1) where ^CD^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= (^CD^α_tSα)(t)x

_t=0,

sinceJ_t^αCD^α_tSα(t)x=Sα(t)x−xand for all functionsv∈C(R+;E) holds limt↓0

J_t^αv(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^αA) =

∞

X

n=0

t^nαAⁿ

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^ω1t, t≥0.

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

Z ∞

0

e^−λtSα(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−1R(λ^α, A)x= Z ∞

0

e^−λtTα(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^αA), t≥0.

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

(iii) For everyt≥0,

CD^α_tS_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^αA) and therefore by Theorem 4.6, S_α(t) =E_α(t^αA). 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)^RLD_τ^1−αf(τ)dτ = Z t

0

(t−τ)^α−1E_α,α((t−τ)^αA)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)^RLD_τ^1−αf(τ)dτ = Z t

0

∞

X

n=0

1

Γ(1 +nα)(t−τ)^nαA^{n RL}D^1−α_τ f(τ)dτ

=

∞

X

n=0

Z t

0

1

Γ(1 +nα)(t−τ)^nαA^{n RL}D^1−α_τ f(τ)dτ

=

∞

X

n=0

J_t^nα+1A^{n RL}D^1−α_t f(t) =

∞

X

n=0

J_t^nα+αAⁿf(t)

=

∞

X

n=0

1 Γ(nα+α)

Z t

0

(t−τ)^nα+α−1Aⁿf(τ)dτ

= Z t

0

(t−τ)^α−1

∞

X

n=0

1

Γ(nα+α)(t−τ)^nαAⁿf(τ)dτ

= Z t

0

(t−τ)^α−1E_α,α((t−τ)^αA)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)^RLD^1−α_τ f(τ)dτ = Z t

0

(t−τ)^α−1E_α,α((t−τ)^αA)∂_τf dτ +t^α−1Eα,α(t^αA)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)^RLD^1−α_τ f(τ)dτ =

∞

X

n=0

d

dtJ_t^nα+αAⁿf(t).

Further, since d

dtJ_t^αf(t) =^RLD^1−α_t f(t) =^CD_t^1−αf(t) +f(0)t^α−1

Γ(α) =J_t^αd

dtf(t) +f(0)t^α−1 Γ(α) , we have

∞

X

n=0

d

dtJ_t^nα+αAⁿf(t) =

∞

X

n=0

J_t^nα+αAⁿd dtf(t) +

∞

X

n=0

J_t^nαt^α−1Aⁿf(0) Γ(α)

=

∞

X

n=0

J_t^nα+αAⁿd dtf(t) +

∞

X

n=0

t^nα+α−1

Γ(α+nα)Aⁿf(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)^RLD^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^CD^α_tS_α(t) =AS_α(t), for a continuous function its fractional integral J_t^αis a continuous function too and ^CD_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^αu(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= (^CD^α_t(Sα)1ε)(t)

_t=0−(^CD_t^α(Sα)2ε)(t) _t=0

=^CD^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^CD_t^α((Sα)1ε−(Sα)2ε)(t)

_t=0k_L(E)≤M ε^a.

It implies that for everya∈Rthere existsM >0 such thatkAe_1ε−Ae_2εk ≤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)^RLD^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−1Eα,α((t−τ)^αAeε)Neε(Tα)ε(τ)xdτ

= Z t

0

(t−τ)^α−1

∞

X

n=0

(t−τ)^nαAeⁿ_ε

Γ(α+nα) Ne_ε(T_α)_ε(τ)xdτ

= Z t

0

∞

X

n=0

(t−τ)^(n+1)α−1Aeⁿ_ε

Γ((n+ 1)α) Neε(Tα)ε(τ)xdτ

=

∞

X

n=1

Z t

0

(t−τ)^nα−1Aeⁿ⁻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−1kAeⁿ⁻¹_ε Ne_ε(T_α)_ε(τ)kdτ

≤ kNeεk sup

t∈[0,T)

k(Tα)ε(t)k

∞

X

n=1

kAeεkⁿ⁻¹ 1 Γ(nα)

T^nα nα

≤ kNe_εk sup

t∈[0,T)

k(Tα)_ε(t)kT^α α

∞

X

n=0

T^nαkAe_εkⁿ Γ(α+nα)

=kNeεk sup

t∈[0,T)

k(Tα)ε(t)kT^α

α Eα,α(T^αkAeεk), 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^CD^α_t((Sα)ε−(Tα)ε)(t)k ≤ kNeεk sup

t∈[0,T)

k(Tα)ε(t)k

∞

X

n=0

kAeεkⁿ Γ(nα)· T^nα

nα

=kNeεk sup

t∈[0,T)

k(Tα)ε(t)k ·Eα(T^αkAeεk).

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−1Eα,α((t−τ)^αAeε)Neε

d

dτ(Tα)ε(τ)xdτ+t^α−1Eα,α(t^αAeε)Neεx.

Then, for everyT₁>0 andt∈[T₁, 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−1kEα,α((t−τ)^αAeε)Neε

d

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

≤ lim

η→0⁺ sup

τ∈[η,T)

E_α,α((T−τ)^αkAe_εk)kNe_εk sup

τ∈[η,T)

k d

dτ(T_α)_ε(τ)k(T−η)^α α +T₁^α−1E_α,α(T^αkAe_εk)kNe_εk.

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

sup

t∈[0,T)

k^CD_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,∞) :H¹(R)) is said to be ofh_ε-type if it has a representative G_εsuch that

kGεk_H1 =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^αAeε) =

∞

X

n=0

t^nαAeⁿ_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_εk^1/α).

Sinceh_ε≤C(log 1/ε)^α, we have sup

t∈[0,T)

k(Sα)ε(t)k ≤M ε^{−T C1/α},

forεsmall enough. Also, since^CD^α_t(Sα)ε(t) =Aeε(Sα)ε(t), for everyt≥0, we have for everyεsmall enough

k^CD^α_t(S_α)_ε(t)k ≤ kAe_εkk(S_α)_ε(t)k ≤C(log1

ε)^αM ε^{−T C1/α} ≤CM ε^{−α−T C1/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)Aeⁿ⁺¹_ε

=t^α−1Aeε

∞

X

n=0

t^nα Γ(α+nα)Aeⁿ_ε

=t^α−1Ae_εE_α,α(t^αAe_ε).

Then, for everyT₁>0 andt∈[T₁, T), the estimate in norm is kd

dt(S_α)_ε(t)k ≤T₁^α−1kAe_εkE_α,α(t^αkAe_εk)

≤T₁^α−1kAeεkCα(1 +kAeεk^(1−α)/α)·exp(kAeεk^1/αT)(1 +T^1−α)

≤T₁^α−1Cα(kAeεk+kAeεk^1/α)·exp(kAeεk^1/αT)(1 +T^1−α)

≤T₁^α−1C_α((log1

ε)^α+ log1

ε)·exp(C^1/αTlog1

ε)(1 +T^1−α)

≤2T₁^α−1Cα(1 +T^1−α)ε^{−1−C1/αT}.

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 =L²(R). Instead of the Cauchy problem (4.1) with closed and densely defined operatorAonL²(R) with domainD(A) =H¹(R), we will consider fractional Cauchy problem given by

CD^α_tu(t) =Au(t), t >e 0;u(0) =x,

where Ae is a generalized linear bounded operator L²-associated with A, i.e., for everyu∈H¹(R), the following holds

k(A−Ae_ε)uk_L2 →0, ε→0.

Theorem 6.1. Let 0 < α < 1. Suppose that u0 ∈ Gα(H¹(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) :=

∂_tf(x, t, u)satisfy the same conditions asf(x, t, u).

Let the operator Ae ∈ SG(H¹(R)) be of h_ε-type, with h_ε =o (log(log 1/ε)^α)^α , such that kAeεuεkL² ≤hεkuεkL², for uε∈H¹(R).

Then for every 0 < α < 1 there exists a unique generalized solution u ∈ Gα([0,∞) :H¹(R))to the Cauchy problem

CD^α_tu(t) =Au(t) +e f(·, t, u), u(0) =u₀. (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)^RLD_τ^1−αf(·, τ, u_ε)dτ, (6.2) whereSα∈ SGα([0,∞) :L(H¹(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(H¹(R))) given bySα(t) = E_α(t^αA) (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,∞) : H¹(R)). First, we show that the solution satisfies

lim

t→0⁺k

d dtu_ε(t,·)

t^α−1 k_H1=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−1E_α,α((t−τ)^αAe_ε)∂_τf(·, τ, u_ε(τ))dτ +t^α−1E_α,α(t^αAe_ε)f(·,0, u_0ε),

(6.4)

and by to the notation g₁(x, t, u) = ∂_uf(x, t, u) and g₂(x, t, u) = ∂_tf(x, t, u), we have

kd

dtuε(t,·)kL²≤ kd

dt(S_α)ε(t)u0εkL²

+ Z t

0

(t−τ)^α−1Eα,α((t−τ)^αkAeεk)k∂τf(·, τ, uε(τ))k_L2dτ +t^α−1Eα,α(t^αkAeεk)kf(·,0, u0ε)k_L2

≤ kd

dt(Sα)ε(t)u0εkL²

+ Z t

0

(t−τ)^α−1Eα,α((t−τ)^αkAeεk)kg1kL^∞k∂τuε(τ)k_L2dτ +

Z t

0

(t−τ)^α−1Eα,α((t−τ)^αkAeεk)kg2kL^∞kuε(τ)kL²dτ +t^α−1Eα,α(t^αkAeεk)kf(·,0, u0ε)k_L2.

(6.5)

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

t→0⁺k

d dtu_ε(t,·)

t^α−1 kL² =C <+∞.

Further, differentiation of (6.4) with respect tox, we have k∂_xd

dtu_ε(t,·)k_L2

≤ kd

dt(S_α)ε(t)∂xu0εk_L2

+ Z t

0

(t−τ)^α−1E_α,α((t−τ)^αkAe_εk)k∂_x∂_τf(·, τ, u_ε(τ))k_L2dτ +t^α−1E_α,α(t^αkAe_εk)k∂xf(·,0, u_0ε)kL²

≤ kd

dt(S_α)ε(t)∂xu0εk_L2

+ Z t

0

(t−τ)^α−1E_α,α((t−τ)^αkAe_εk)k∂xg₁kL^∞k∂τu_ε(τ)kL²dτ +

Z t

0

(t−τ)^α−1Eα,α((t−τ)^αkAeεk)kg1kL^∞k∂x∂τuε(τ)k_L2dτ +

Z t

0

(t−τ)^α−1Eα,α((t−τ)^αkAeεk)k∂xg2kL^∞kuε(τ)kL²dτ +

Z t

0

(t−τ)^α−1E_α,α((t−τ)^αkAe_εk)kg2kL^∞k∂xu_ε(τ)kL²dτ +t^α−1Eα,α(t^αkAeεk)[k∂uf(·,0, u0ε)kL^∞k∂xu0εkL²

+k∂xf(·,0, u0ε)kL^∞ku0εk_L2].

(6.6)

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

t→0⁺k∂_xdt^du_ε(t,·)

t^α−1 k_L2 =C <+∞, and finally we have that property (6.3) is satisfied.

Further, we prove that one has the moderate bound for k^CD_t^γuε(t,·)k_H1, γ ∈ {0, α}, andk_dt^du_ε(t,·)k_H1. First, we prove the moderate bound fork^CD_t^γu_ε(t,·)k_H1, and consider the cases:

Case 1: γ= 0. From the representation (6.2) and Proposition 4.8 we obtain kuε(t)k_L2 ≤ k(Sα)ε(t)u0εk_L2+

Z t

0

(t−τ)^α−1kEα,α((t−τ)^αAeε)k kf(·, τ, uε)k_L2dτ.

Next, using the estimate forE_α,α one gets kEα,α(t^αAeε)k ≤

∞

X

n=0

t^nαkAe_εkⁿ

Γ(α+nα)=Eα,α(t^αkAeεk)

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

Denote

Mf_T := sup

t∈[0,T)

kEα,α(t^αAe_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]).