Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 199, pp. 1–28.

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

DETERMINATION OF THE ORDER OF FRACTIONAL DERIVATIVE AND A KERNEL IN AN INVERSE PROBLEM

FOR A GENERALIZED TIME FRACTIONAL DIFFUSION EQUATION

JAAN JANNO

Abstract. A generalized time fractional diffusion equation containing a lower order term of a convolutional form is considered. Inverse problem to determine the order of a fractional derivative and a kernel of the lower order term from measurements of states over the time is posed. Existence, uniqueness and stability of the solution of the inverse problem are proved.

1. Introduction

Subdiffusion processes in porous, fractal, biological etc. media are described by differential equations containing fractional time (time and space) derivatives [1, 2, 13, 14, 27].

In many practical situations parameters of media or model are unknown or scarcely known. They can be determined solving inverse problems for governing differential equations.

Analytical and numerical study of inverse problems for fractional diffusion equa- tions is undergoing an intensive development during the present decade. Series of papers are devoted to problems to determine unknown source terms [4, 19, 22, 25, 28], boundary conditions [8], initial conditions [12], coefficients [3, 15, 11], orders of derivatives [3, 7, 15, 18] and nonlinear terms [9, 20, 21, 23].

Fractional time derivatives in diffusion models result from postulating the power
law waiting time density of a stochastic processes going on in micro-level. However,
there are no convincing arguments that the waiting time density has to be exactly
of the power law. In the present paper we consider a more general model that is
governed by an equation that involves “almost” fractional time derivative. Namely,
we replace the power function t^{β−1} occurring in the fractional derivative by the
sum oft^{β−1}and a convolution oft^{β−1} with an arbitrary kernelm.

We pose an inverse problem to reconstruct β and mfrom measurements of the states over the time. We prove the existence and uniqueness of the solution of the inverse problem and establish a stability estimate form with respect to the data.

Results are global in time. Moreover, we deduce an explicit formula for β and present a numerical example. The analysis is implemented in the Fourier domain.

2010Mathematics Subject Classification. 35R30, 80A23.

Key words and phrases. Inverse problem; fractional diffusion; fractional parabolic equation.

c

2016 Texas State University.

Submitted February 15, 2016. Published July 25, 2016.

1

2. Formulation of direct and inverse problems

Continuous time random walk models of subdiffusion with power law waiting time densities yield in macro-level differential equations that contain fractional derivatives of order between 0 and 1. The simplest equation of such kind is [1, 2, 14, 27]

Ut(x, t) =D^{1−β}Uxx(x, t),

whereU is the state variable,xis the space variable,tis the time and
D^{1−β}U(x, t) = d

dt Z t

0

(t−τ)^{β−1}

Γ(β) ^{U}(x, τ)dτ

is the Riemann-Liouville fractional derivative of the order 1−β with 0< β <1.

An equation that corresponds to general waiting time densities is [2, Eq. (10)]

Ut= d dt

Z t

0

M(t−τ)Uxx(x, τ)dτ, (2.1) where M is an arbitrary function. Because of the physical background,M is pos- itive, decreasing and has a weak singularity at t = 0. Let us suppose that the functionM has the form

M(t) = t^{β−1}

Γ(β)+t^{β−1}

Γ(β)∗m(t) (2.2)

with some kernelm, where∗denotes the time convolution; i.e., v1∗v2(t) =

Z t

0

v1(t−τ)v2(τ)dτ.

Then the equation (2.1) readsUt=D^{1−β}(Uxx+m∗Uxx).

Our aim is to pose and study an inverse problem to determine the order of the fractional derivativeβ and the kernelm in this equation. But before we proceed, we generalize this equation a bit:

Ut=D^{1−β}(Uxx+m∗Uxx+m^{0}∗Uxx) +G. (2.3)
The functionGis a source term. The inclusion of the addend withm^{0}has a math-
ematical reason. Namely, the study of stability in Section 7 requires a previously
proved existence result for an inverse problem that contains the additional term
with m^{0}. Therefore, it makes sense to incorporate this term already from the
beginning. On the other hand, m^{0} can be interpreted as an initial guess for an
unknown kernel of the formm^{0}+m. In this case, the perturbation part mof the
kernel is to be determined in the inverse problem.

Next we transform the equation under consideration to a more common in the
mathematical literature form. To this end we introduce the operator of fractional
integrationI^{α}defined by the formula

I^{α}v(t) = t^{α−1}

Γ(α)∗v(t) = Z t

0

(t−τ)^{α−1}
Γ(α) v(τ)dτ.

Applying the operatorI^{1−β}to the equation (2.3), we reach the equivalent equation
t^{−β}

Γ(1−β)∗[Ut−G] =Uxx+m∗Uxx+m^{0}∗Uxx.

We mention that the left hand side of (2.7) contains the Caputo derivative of the
orderβ ofU, i.e. _{Γ(1−β)}^{t}^{−β} ∗Ut.

Let us formulate the following initial-boundary value problem for this equation in a bounded domain (x1, x2)×(0, T):

t^{−β}

Γ(1−β)∗[Ut(x, t)−G(x, t)] =Uxx(x, t) +m∗Uxx(x, t) +m^{0}∗Uxx(x, t),
(x, t)∈(x_{1}, x_{2})×(0, T),

U(x,0) =U0(x), x∈(x1, x2),

B1U(·, t) =b1(t), B2U(·, t) =b2(t), t∈(0, T),

(2.4)

whereB_{1}andB_{2} are boundary operators atx=x_{1}andx=x_{2}, respectively. More
precisely,

for anyj∈ {1; 2} eitherBjv=v(xj) or

Bjv=v^{0}(xj) +θjv(xj) withθj∈R, (−1)^{j}θj≥0. (2.5)
Here and in the sequel we use for x- and t-dependent functionsv(x, t) the al-
ternative notation v(·, t) that means a function of t with values as functions of
x.

To formulate an inverse problem, let us introduce an observation functional Φ that maps functions defined on the interval [x1, x2] ontoR. For instance, Φ can be defined as follows:

Φ[v] =v(x_{0}) or Φ[v] =v^{0}(x_{0}) +ϑv(x_{0}) or Φ[v] =
Z x2

x_{1}

κ(x)v(x)dx,
wherex_{0}∈[x_{1}, x_{2}],ϑ∈R,κ: (x_{1}, x_{2})→Rare given. It is natural to assume that
Φ does not coincide with any of the boundary operators, i.e. Φ6=B1 and Φ6=B2.

Now we are in a situation to formulate theinverse problem. GivenG, m^{0},U0, b_{1}
and b_{2}, find the pair (β, m) such that the solutionU of the (direct) problem (2.4)
satisfies the additional condition

Φ[U(·, t)] =H(t), t∈(0, T), (2.6) whereHis a prescribed function (observation of the physical stateU).

It is more convenient to deal with a problem with homogeneous boundary con-
ditions. Then it is possible to interpret the second order space derivative in the
equation (2.4) as a linear operator in some functional space. Let b^{U} be a func-
tion satisfying the nonhomogeneous boundary conditions, i.e. B_{1}b^{U}(·, t) =b_{1}(t) and
B_{2}b^{U}(·, t) =b_{2}(t) for t ∈(0, T). Performing the change of variablesU =b^{U}+u, we
obtain the following equation and conditions foru:

t^{−β}

Γ(1−β)∗[ut(x, t)−g(x, t)]

=u_{xx}(x, t) +f(x, t) +m∗[u_{xx}(x, t) +ψ(x, t)] +m^{0}∗u_{xx}(x, t),
(x, t)∈(x1, x2)×(0, T),

u(x,0) =ϕ(x), x∈(x1, x2),
B_{1}u(·, t) = 0, B_{2}u(·, t) = 0, t∈(0, T),

(2.7)

and

Φ[u(·, t)] =h(t), t∈(0, T), (2.8)
whereg=G−b^{U}t,ψ=b^{U}xx,f =b^{U}xx+m^{0}∗b^{U}xx,ϕ=U0−b^{U}(·,0) andh=H−Φ[b^{U}].

The relations (2.7) form a direct problem foru. Theinverse problemconsists in determining (β, m) such that the solutionuof (2.7) satisfies the additional condition (2.8).

In this article we prove well-posedness results for the inverse problem with the
componentm in spacesL_{p}(0, T), wherep∈[1,∞). This covers as particular cases
functions M of the form M(t) = ^{t}_{Γ(β)}^{β−1} +Pn

i=1cit^{s}^{i}^{−1}, where si > β (for with
such M, see [7]). Then m(t) = Pn

i=1
c_{i}Γ(s_{i})

Γ(s_{i}−β)t^{s}^{i}^{−β−1}. Another example of m is
the exponentially decreasing flux relaxation (memory) kernelm(t) =Pn

i=1c_{i}e^{−α}^{i}^{t},
whereα_{i}>0 [24].

3. Abstraction and reformulation in Fourier domain

Let X be a Hilbert space and A : D(A) → X be a linear operator with the
domainD(A)⊆X. Moreover, letg, f, ψ: (0, T)→X,m^{0}, h: (0, T)→R, be given
functions,ϕ∈X a given element and Φ :D(A)→Ra given linear functional.

In the abstract inverse problem we seek for a number β and a function m : (0, T)→Rsuch that a solutionu: [0, T]→X of the (forward) problem

t^{−β}

Γ(1−β)∗[u^{0}(t)−g(t)]

=Au(t) +f(t) +m∗[Au(t) +ψ(t)] +m^{0}∗Au(t), t∈(0, T),
u(0) =ϕ

(3.1)

satisfies the additional condition

Φ[u(t)] =h(t), t∈(0, T). (3.2)

Firstly, let us formulate a theorem that gives sufficient conditions for the well- posedness of the abstract direct problem (3.1).

Theorem 3.1. Assume thatAis closed and densely defined in X and satisfies the following property:

ρ(A)⊃Σ(βπ/2), ∃M >0 :k(λ−A)^{−1}k ≤ M

|λ| ∀λ∈Σ(βπ/2), (3.3)
where ρ(A) is the resolvent set ofA and Σ(θ) ={λ∈C:|argλ|< θ}. Let XA be
the domain of A endowed with the graph norm kzkX_{A} = kzk+kAzk. Moreover,
assume ϕ ∈ XA, f, ψ, g ∈ W_{1}^{1}((0, T);X) and m, m^{0} ∈ L^{1}(0, T). Then (3.1) has
a unique solution in the space C([0, T];X_{A}) and _{Γ(1−β)}^{t}^{−β} ∗u^{0} ∈ C([0, T];X). The
solution continuously depends on ϕ, f, ψ, g, m and m^{0} in norms of the mentioned
spaces.

The above theorem follows from [17, Theorem 2.3 and Proposition 1.2].

Remark 3.2. DefineX =L2(x1, x2). Then the operatorA=_{dx}^{d}^{2}2 with the domain
D(A) ={w:z∈W_{2}^{2}(x_{1}, x_{2}),B1w= 0, B2w= 0} (3.4)
satisfies the assumptions of Theorem 3.1 (see [10, Theorem 3.1.3]). Consequently,
Theorem 3.1 applies to the problem (2.7).

Our next step is to reformulate the abstract inverse problem (3.1), (3.2) in the Fourier domain. Let us further assume that

the spectrum of A is discrete, the eigenvalues λi, i = 1,2, . . . of the operator −A are nonnegative, ordered in the usual manner, i.e. 0 ≤ λ1 ≤ λ2 ≤ . . . and the corresponding eigenvectors vi, i= 1,2, . . ., form an orthonormal basis inX.

(3.5)

Remark 3.3. It is well-known that the operatorA = _{dx}^{d}^{2}2 with the domain (3.4)
satisfies the property (3.5).

We expand the functions involved in (3.1), (3.2) as follows:

u(t) =

∞

X

i=1

ui(t)vi, g(t) =

∞

X

i=1

gi(t)vi, f(t) =

∞

X

i=1

fi(t)vi,

ψ(t) =

∞

X

i=1

ψi(t)vi, ϕ=

∞

X

i=1

ϕivi,

(3.6)

where ui : [0, T] → R, gi, fi, ψi : (0, T) →R, ϕi ∈R are the Fourier coefficients.

Moreover, let us denote

γi= Φ[vi], i= 1,2, . . . .

Taking the inner product of the equalities (3.1) with the elements vi, i= 1,2, . . ., and inserting the series ofuinto (3.2), we obtain

t^{−β}

Γ(1−β)∗[u^{0}_{i}(t)−gi(t)] +λiui(t)

=fi(t) +m∗[ψi(t)−λiui(t)]−m^{0}∗λiui(t), t∈(0, T), ui(0) =ϕi,
(3.7)
wherei= 1,2, . . .,

∞

X

i=1

γ_{i}u_{i}(t) =h(t), t∈(0, T). (3.8)
The relations (3.7) represent the direct problem, reformulated in the Fourier do-
main. The correspondinginverse problemis stated as follows.

Inverse Problem (IP). Giveng_{i}, f_{i}, ψ_{i}, ϕ_{i},i= 1,2, . . .andm^{0}, h, findβ andmsuch
that solutionsu_{i} of (3.7) satisfy the condition (3.8).

4. Notation and preliminaries Let us introduce the Bessel potential spaces

H_{p}^{s}(0, T) =n

v|_{[0,T]}:v∈H_{p}^{s}(R) ={w:F^{−1}((1 +|ω|^{2})^{s}^{2}Fw)∈L_{p}(R)}o
for 1< p <∞,s >0 and their subspaces

0H_{p}^{s}(0, T) ={v|[0,T]:v∈H_{p}^{s}(R),suppv⊆[0,∞)}.

Here F is the Fourier transform and the symbolv|_{[0,T]} stands for the restriction
onto [0, T] of a function defined onR.

In casen∈Nthe spaceH_{p}^{n}(0, T) coincides with the Sobolev space
W_{p}^{n}(0, T) ={w:w^{(j)}∈Lp(0, T), j= 0, . . . , n}.

Remark 4.1. When s ∈ (0,1), p ∈ (1,1/s) it holds 0H_{p}^{s}[0, T] = H_{p}^{s}(0, T). On
the other hand, whens∈(0,1),p∈(^{1}_{s},∞) the spaceH_{p}^{s}(0, T) is embedded in the
space of continuous on [0, T] functionsC[0, T] andw∈H_{p}^{s}(0, T)⇔w=w(0) +w,
w(0)∈R,w∈0H_{p}^{s}(0, T) (see [26, p. 27-28]).

We use of the following abbreviation for the norms in Lebesgue spacesLp(0, T):

kwkp:=kwk_{L}_{p}_{(0,T)}.

Let us formulate a lemma that describes the functions ^{t}_{Γ(s)}^{s−1}∗mwherem∈Lp(0, T).

Lemma 4.2. . Lets∈(0,1),p∈(1,∞). The operator of fractional integration of
the order s, given byI^{s}z = ^{t}_{Γ(s)}^{s−1} ∗z, is a bijection from Lp(0, T)onto 0H_{p}^{s}(0, T),
the inverse of I^{s}is the Riemann-Liouville fractional derivativeD^{s}=_{dt}^{d}I^{1−s} and

kwks,p:=kD^{s}wkp

is a norm in0H_{p}^{s}(0, T).

The above lemma follows from [26, Corollary 2.8.1].

In our analysis we will use the Mittag-Leffler functions E_{β} and E_{β,β} in case
β∈(0,1). The functions E_{β} andE_{β,γ} are defined by the following power series:

Eβ(t) =

∞

X

k=0

t^{k}

Γ(βk+ 1), Eβ,γ(t) =

∞

X

k=0

t^{k}
Γ(βk+γ).

Note thatEβ is a generalization of the exponential function. Indeed, in caseβ = 1
it holds Eβ(t) = e^{t}. Like the exponential function, Eβ and Eβ,γ are also entire
functions. Moreover,Eβ(−t) andEβ,γ(−t) are completely monotonic fort∈[0,∞)
and

Eβ(0) = 1, Eβ,β(0) = 1

Γ(β), E_{β}^{0} = 1

βEβ,β (4.1)

(see [5]).

Next we prove a lemma that will be applied in a treatment of the direct problem (3.7).

Lemma 4.3. Let z ∈ H_{r}^{1−β}(0, T) with some β ∈ (0,1), r ∈ (1,_{1−β}^{1} ) and y ∈
L1(0, T),λ, w0∈R. Then the Cauchy problem

t^{−β}

Γ(1−β)∗w^{0}(t)+ t^{−β}

Γ(1−β)∗y∗w^{0}(t)+λw(t) =z(t), t∈(0, T), w(0) =w0 (4.2)
has a unique solutionwin the space W_{r}^{1}(0, T). This solution has in casey= 0 the
representation

w(t) =w0Eβ −λt^{β}
+

Z t

0

(t−τ)^{β−1}Eβ,β

−λ(t−τ)^{β}

z(τ)dτ. (4.3)
Proof. By Lemma 4.2, Remark 4.1 and the relationw=I^{1}w^{0}+w_{0}, (4.2) is equiv-
alent to

w^{0}(t) +y∗w^{0}(t) +λD^{1−β}(I^{1}w^{0}(t) +w_{0}) =D^{1−β}z(t), t∈(0, T), w(0) =w_{0}.

Since D^{1−β}I^{1} = D^{1−β}I^{1−β}I^{β} = I^{β}, the obtained equation for w^{0} is the Volterra
equation of the second kind

w^{0}(t) +
Z t

0

hy(t−τ) +λ(t−τ)^{β−1}
Γ(β)

iw^{0}(τ)dτ

=D^{1−β}z(t)−λw_{0}t^{β−1}

Γ(β), t∈(0, T).

(4.4)

The right-hand sideD^{1−β}z−λw0t^{β−1}

Γ(β) belongs toLr(0, T). By well-known results
for the Volterra equations of the second kind [6], the equation (4.4) has a a unique
solutionw^{0} ∈Lr(0, T). This proves the existence and uniqueness assertions of the
lemma.

It remains to prove the formula (4.3). From [5, p. 172-173], it follows that the second addend in (4.3), i.e.

ω(t) :=

Z t

0

(t−τ)^{β−1}Eβ,β

−λ(t−τ)^{β}
z(τ)dτ

solves the equation D^{β}ω+λω = z. Since ω(0) = 0 we have D^{β}ω = _{Γ(1−β)}^{t}^{−β} ∗ω^{0}.
Consequently, we obtain the relation

t^{−β}

Γ(1−β)∗ω^{0}(t) +λω(t) =z(t), t∈(0, T), ω(0) = 0. (4.5)
Further, by [5, (4.10.16)], the functionφ(t) :=Eβ −λt^{β}) solves the equation

D^{β}φ+λφ= t^{−β}
Γ(1−β).

This yields _{Γ(1−β)}^{t}^{−β} ∗φ^{0}(t) +λφ(t) = 0. Moreover,φ(0) = 1. Therefore, for the first
addend in (4.3), i.e. χ(t) :=w0Eβ −λt^{β}

the relations
t^{−β}

Γ(1−β)∗χ^{0}(t) +λχ(t) = 0, t∈(0, T), χ(0) =w0 (4.6)
are valid. The summa w=ω+χsolves (4.2) with y= 0. Summing the formulas

ofω andχwe obtain (4.3).

Let us introduce further auxiliary material. We use the following family of
weighted norms in the spaces_{0}H_{p}^{s}(0, T) andL_{p}(0, T):

kwks,p;σ=ke^{−σt}D^{s}wkL_{p}(0,T), and kwkp;σ=ke^{−σt}wkL_{p}(0,T),
whereσ≥0. Evidently, the equivalence relations

e^{−σT}kwk_{s,p} ≤ kwk_{s,p;σ}≤ kwk_{s,p}, e^{−σT}kwk_{p}≤ kwk_{p;σ}≤ kwk_{p} (4.7)
are valid. Moreover, by the dominated convergence theorem, in casep <∞,

kwks,p;σ →0 and kwkp;σ→0 asσ→ ∞. (4.8)

Lemma 4.4. Let β∈(0,1). Then the functions
Eeβ,i(t) =t^{β−1}Eβ,β

−λit^{β}

(4.9) satisfy the following estimates:

kλ_{i}Ee_{β,i}k_{1;σ}≤1 (4.10)

kλ^{1−}_{i} Ee_{β,i}k_{1;σ}≤cβ,

σ^{β}, 0< ≤1, (4.11)
fori= 1,2, . . ., wherecβ,is a constant independent ofσandi. The symbolk · k1;σ

denotes the norm k · kp;σ in case p= 1.

Proof. Using the positivity ofE_{β,β}(−t) fort≥0 and (4.1) we deduce
kλiEeβ,ik1;σ=

Z T

0

e^{−σt}λit^{β−1}Eβ,β(−λit^{β})dt

≤ Z T

0

λit^{β−1}Eβ,β(−λit^{β})dt

=− Z T

0

d

dtE_{β}(−λit^{β})dt=E_{β}(0)−E_{β}(−λiT^{β}).

SinceEβ(−t) is positive fort≥0 andEβ(0) = 1 we reach (4.10). Further, taking
the asymptotical relationEβ,β(−t) =O(t^{−2}) as t→ ∞ (see [16, Thm. 1.2.1]) into
account, we have λit^{β}δ

Eβ,β(−λit^{β}) ≤c^{1}_{β,δ} fort ≥ 0 and 0≤δ ≤2 with some
constantc^{1}_{β,δ}. Thus, for 0< ≤1 we deduce

kλ^{1−}_{i} Eeβ,ik1;σ=
Z T

0

e^{−σt} λit^{β}^{1−}

t^{β−1}Eβ,β(−λit^{β})dt

≤c^{1}_{β,1−}

Z T

0

e^{−σt}t^{β−1}dt

= c^{1}_{β,1−}

σ^{β}
Z σT

0

e^{−s}s^{β−1}ds

< c^{1}_{β,1−}

σ^{β}
Z ∞

0

e^{−s}s^{β−1}ds.

This implies (4.11).

Finally, we point out the Young’s theorem for convolutions that will be an im- portant tool in our computations:

kw1∗w2kp3≤ kw1kp1kw2kp2, where 1 p1

+ 1 p2

= 1 + 1 p3

. (4.12)

5. Results for direct problem in Fourier domain In this section, we prove two propositions for the direct problem (3.7).

Proposition 5.1. Let β ∈ (0,1), m, m^{0} ∈ L_{1}(0, T) and f_{i}, ψ_{i} ∈ H_{r}^{1−β}(0, T),
g_{i}∈L_{r}(0, T)with somer∈(1,_{1−β}^{1} ). Then the problem (3.7)has a unique solution
u_{i}∈W_{r}^{1}(0, T). Moreover, the following assertions are valid:

(i) if

kmk1;σ+km^{0}k1;σ ≤1

2 (5.1)

then the estimate
ku^{0}_{i}kr;σ+λikuik1−β,r;σ≤C0

λi|ϕi|+kfik1−β,r;σ+kψik1−β,r;σ+kgikr;σ

(5.2)
holds, whereC_{0} is a constant independent ofσandi;

(ii) if (5.1) is satisfied and fi, ψi ∈ L∞(0, T), I^{1−β}gi ∈ L∞(0, T) then the
estimate

λ_{i}kuik∞;σ≤C_{1}

λ_{i}|ϕi|+kfik∞;σ+kψik∞;σ+kI^{1−β}g_{i}k∞;σ

(5.3)
holds, whereC_{1} is a constant independent ofσandi.

Proof. Sincem, m^{0}∈L_{1}(0, T), the Volterra equation of the second kind
y(t) + (m+m^{0})∗y(t) +m(t) +m^{0}(t) = 0, t∈(0, T),

has a unique solutiony ∈L1(0, T) (see [6, Theorem 3.1]). From this equation we obtain the operator relations

(I+y∗) I+ (m+m^{0})∗

= I+ (m+m^{0})∗

(I+y∗) =I,

where I is the unity operator. Applying the operator I+y∗ to the equation in (3.7) we obtain the problem

t^{−β}

Γ(1−β)∗[u^{0}_{i}(t) +y∗u^{0}_{i}(t)] +λ_{i}u_{i}(t) =fe_{i}(t), t∈(0, T),
ui(0) =ϕi,

(5.4)

where fei(t) = fi(t) +y∗fi(t) +I^{1−β}gi(t) +y∗I^{1−β}gi(t) + (m+y∗m)∗ψi(t).

Conversely, applying the operatorI+ (m+m^{0})∗to the equation in (5.4), we reach
(3.7). Therefore, problems (3.7) and (5.4) are equivalent. From the assumptions of
the proposition, Lemma 4.2 and Remark 4.1 we have

fei(t) =fi(t) +y∗ t^{−β}

Γ(1−β)∗D^{1−β}fi(t) +I^{1−β}gi(t)
+y∗ t^{−β}

Γ(1−β)∗g_{i}(t) + (m+y∗m)∗ t^{−β}

Γ(1−β)∗D^{1−β}ψ_{i}(t)

=f_{i}(t) +I^{1−β}g_{i}(t) + t^{−β}
Γ(1−β)∗

y∗D^{1−β}f_{i}(t) +y∗g_{i}(t)
+ (m+y∗m)∗D^{1−β}ψi(t)

,

where y∗D^{1−β}fi+y∗gi+ (m+y∗m)∗D^{1−β}ψi ∈Lr(0, T). This impliesfei ∈
H_{r}^{1−β}(0, T). In view of Lemma 4.3, the problem (5.4) has a unique solution in
W_{r}^{1}(0, T). This proves the existence and uniqueness assertion of the proposition.

Further, let us prove (i). For this purpose, we represent the solution of (3.7) by means of the formula (4.3). Using the abbreviation (4.9) we have

ui(t) =ϕiEβ −λit^{β}
+

Z t

0

Eeβ,i(t−τ)h

fi(τ) + τ^{−β}

Γ(1−β)∗gi(τ) +ψi∗m(τ)i

dτ− Z t

0

Eeβ,i(t−τ)λiui∗[m(τ) +m^{0}(τ)]dτ.

(5.5)

In view of the relation I =I^{1−β}D^{1−β} = _{Γ(1−β)}^{t}^{−β} ∗D^{1−β} that holds inH_{r}^{1−β}(0, T)
we obtain

ui(t) =ϕiEβ −λit^{β}
+

Z t

0

Eeβ,i(t−τ)

×h τ^{−β}

Γ(1−β)∗ D^{1−β}f_{i}(τ) +g_{i}(τ)

+ τ^{−β}

Γ(1−β)∗D^{1−β}ψ_{i}∗m(τ)i
dτ

− Z t

0

Ee_{β,i}(t−τ)λ_{i} τ^{−β}

Γ(1−β)∗D^{1−β}u_{i}∗[m(τ) +m^{0}(τ)]dτ.

(5.6)

Applying the operatorD^{1−β} =_{dt}^{d} ^{t}_{Γ(β)}^{β−1}∗ and taking the relations ^{t}_{Γ(β)}^{β−1} ∗ _{Γ(1−β)}^{t}^{−β} = 1
and

d

dtE_{β} −λ_{i}t^{β}

=−λ_{i}Ee_{β,i}(t), (5.7)

following from (4.1) and (4.9), we reach the expression
D^{1−β}ui(t) =−λiϕi

Z t

0

Eeβ,i(t−τ)τ^{β−1}

Γ(β)dτ+ϕi

t^{β−1}
Γ(β)
+

Z t

0

Eeβ,i(t−τ)h

D^{1−β}fi(τ) +gi(τ) +D^{1−β}ψi∗m(τ)i
dτ

− Z t

0

Eeβ,i(t−τ)λiD^{1−β}ui∗[m(τ) +m^{0}(τ)]dτ.

Next we multiply this equality byλie^{−σt}, bring the factore^{−σt}inside the integrals
and use the relation

e^{−σt}[w_{1}(t)∗w_{2}(t)] =e^{−σt}w_{1}(t)∗e^{−σt}w_{2}(t).

Thereupon we estimate the obtained expression in the normk · krand apply (4.12).

As a result we obtain
λikD^{1−β}uikr;σ≤λi|ϕi|

kλiEeβ,ik1;σ+ 1

t^{β−1}
Γ(β)
_{r;σ}

+kλiEeβ,ik1;σ

kD^{1−β}fi+gikr;σ+kD^{1−β}ψikr;σkmk1;σ

+kλiEe_{β,i}k1;σλ_{i}kD^{1−β}u_{i}kr;σ[kmk1;σ+km^{0}k1;σ].

(5.8)

Using (4.10) we obtain

λikuik1−β,r;σ≤2ˆcβ,rλi|ϕi|+kfik1−β,r;σ+kgikr;σ+kψik1−β,r;σkmk1;σ

+ [kmk1;σ+km^{0}k1;σ]·λikuik_{1−β,r;σ},

where ˆc_{β,r} = k^{t}_{Γ(β)}^{β−1}k_{r}. In case (5.1) is valid, we estimate kmk_{1;σ} and kmk_{1;σ}+
km^{0}k1;σ by ^{1}_{2}, bring the term ^{1}_{2}λikuik_{1−β,r;σ} to the left-hand side and multiply the
obtained inequality by 2. This results in

λ_{i}ku_{i}k_{1−β,r;σ}≤C_{4}

λ_{i}|ϕ_{i}|+kf_{i}k_{1−β,r;σ}+kψ_{i}k_{1−β,r;σ}+kg_{i}k_{r;σ}

, (5.9)

whereC4 is a constant.

Further, applyingD^{1−β} to (3.7) we deduce

u^{0}_{i}=−λiD^{1−β}u_{i}+D^{1−β}f_{i}+g_{i}+ D^{1−β}ψ_{i}−λiD^{1−β}u_{i}

∗m−λiD^{1−β}u_{i}∗m^{0}. (5.10)

Here we used that
D^{1−β}(f∗m) = d

dtI^{β}(f∗m) = d
dt

t^{β−1}

Γ(β)∗f∗m

= d dt

t^{β−1}
Γ(β)∗f

∗m+ t^{β−1}
Γ(β)∗f

(t)|t=0·m

=D^{1−β}f∗m+ t^{β−1}

Γ(β)∗ t^{−β}

Γ(1−β)∗D^{1−β}f

(t)|t=0·m

=D^{1−β}f∗m+ I^{1}D^{1−β}f

(t)|t=0·m=D^{1−β}f∗m

is valid for any f ∈ H_{r}^{1−β}(0, T). Assuming (5.1) and using (5.9) from (5.10) we
obtain

ku^{0}_{i}kr;σ≤λikuik1−β,r;σ+kfik1−β,r;σ+kgikr;σ

+ kψik1−β,r;σ+λikuik1−β,r;σ

kmk1;σ+λikuik1−β,r;σkm^{0}k1;σ

≤C_{5}

λ_{i}|ϕi|+kfik1−β,r;σ+kψik1−β,r;σ+kgikr;σ

(5.11)

with a constantC5. Adding (5.9) and (5.11) we reach (5.2).

Finally, let us prove (ii). To this end, let us return to the equality (5.5). Multi-
plying (5.5) byλie^{−σt}and estimating the result we obtain

λikuik_{∞;σ} ≤λi|ϕi|+kλiEeβ,ik1;σ

kfi+I^{1−β}gik_{∞;σ}+kψik_{∞;σ}kmk1;σ

+kλiEeβ,ik1;σλikuik_{∞;σ}[kmk1;σ+km^{0}k1;σ].

Observing (4.10) and (5.1) we deduce (5.3).

Proposition 5.2. Let β ∈ (0,1), m, m^{0} ∈ Lp(0, T) with some p ∈ (1,∞) and
fi ∈W_{p}^{1}(0, T),ψi∈W_{1}^{1}(0, T), gi∈0H_{p}^{β}(0, T). Thenu^{0}_{i}+qβ,i∈0H_{p}^{β}(0, T), where
ui is the solution of (3.7)and

q_{β,i}(t) = λ_{i}ϕ_{i}−f_{i}(0)

E_{β,β} −λit^{β}

t^{β−1}= λ_{i}ϕ_{i}−f_{i}(0)

Ee_{β,i}(t). (5.12)
Moreover, in the case

T^{p−1}^{p} kmkp;σ+km^{0}kp;σ

≤ 1

2 (5.13)

the estimates

λiku^{0}_{i}+qβ,ikp;σ≤C2kλiEeβ,ik1;σ

λi|ϕi|+|fi(0)|

+ (|ψi(0)|+kψ^{0}_{i}k1;σ)kmkp;σ+kf_{i}^{0}kp;σ+kgikβ,p;σ

,

(5.14)
ku^{0}_{i}+q_{β,i}k_{β,p;σ}≤C_{3}

λ_{i}|ϕ_{i}|+|f_{i}(0)|

+ (|ψ_{i}(0)|+kψ_{i}^{0}k_{1;σ})kmk_{p;σ}+kf_{i}^{0}k_{p;σ}+kg_{i}k_{β,p;σ}
,

(5.15)
hold, whereC_{2} andC_{3} are constants independent of σandi.

Before proving Proposition 5.2, we prove a lemma concerning the functionqβ,i. Lemma 5.3. The function qβ,i satisfies the equations

t^{−β}

Γ(1−β)∗qβ,i+λiI^{1}qβ,i=λiϕi−fi(0), (5.16)
D^{β}q_{β,i}+λ_{i}q_{β,i}= 0. (5.17)

Proof. In caseλ_{i}= 0 we haveq_{β,i}(t) =−f_{i}(0)^{t}_{Γ(β)}^{β−1} and since _{Γ(1−β)}^{t}^{−β} ∗^{t}_{Γ(β)}^{β−1} = 1, the
relations (5.16) and (5.17) are immediate. Letλi >0. By Lemma 4.3, the function

¯

qi(t) =Eβ −λit^{β}

is a solution of the equation _{Γ(1−β)}^{t}^{−β} ∗q¯i0+λiq¯i= 0. Multiplying
this equation by _{λ}^{1}

i fi(0)−λiϕi

we obtain
t^{−β}

Γ(1−β)∗ 1 λi

f_{i}(0)−λ_{i}ϕ_{i}

¯

q_{i}^{0}+ f_{i}(0)−λ_{i}ϕ_{i}

¯

q_{i}= 0. (5.18)
On the other hand, in view of (4.1) and the definitions of ¯qi, qβ,i it holds the
formula_{λ}^{1}

i f_{i}(0)−λiϕ_{i}

¯

q^{0}_{i}=q_{β,i}. Integrating, multiplying byλ_{i}and observing that

¯

q_{i}(0) = 1 we have another formula f_{i}(0)−λ_{i}ϕ_{i}

¯

q_{i}(t) =λ_{i}I^{1}q_{β,i}(t) +f_{i}(0)−λ_{i}ϕ_{i}.
Using these relations in (5.18) we arrive at (5.16). Finally, differentiating (5.16) we

come to (5.17).

Proof of Proposition 5.2. Since W_{1}^{1}(0, T)⊂H_{r}^{1−β}(0, T) and 0H_{p}^{β}(0, T)⊂Lr(0, T)
for r ∈ (0,_{1−β}^{1} ), r ≤ p, by Proposition 5.1, problem (3.7) has a unique solution
u_{i}∈W_{r}^{1}(0, T). Differentiating (5.5) and observing (5.7), (5.12) we obtain

u^{0}_{i}(t) +q_{β,i}(t)

= Z t

0

Eeβ,i(t−τ)h

f_{i}^{0}(τ) +D^{β}gi(τ) +ψi(0)m(τ) +ψ^{0}_{i}∗m(τ)i
dτ

− Z t

0

Ee_{β,i}(t−τ)λ_{i}h

ϕ_{i} m(τ) +m^{0}(τ)

+u^{0}_{i}∗ m(τ) +m^{0}(τ)i
dτ.

(5.19)

From u^{0}_{i} ∈ L_{1}(0, T) and the assumptions of the proposition, the right-hand side
of this relation belongs toL_{p}(0, T). Therefore, u^{0}_{i}+q_{β,i} ∈L_{p}(0, T). Multiplying
(5.19) by λ_{i}e^{−σt}, representing u^{0}_{i} as u^{0}_{i} =−q_{β,i}+u^{0}_{i}+q_{β,i} at the right-hand side
and using (4.12) as well as the relationkmk1;σ≤T^{p−1}^{p} kmkp;σ we obtain

λiku^{0}_{i}+qβ,ikp;σ≤ kλiEeβ,ik1;σ kf_{i}^{0}kp;σ+kgikβ,p;σ

+|ψi(0)|kmkp;σ+kψ_{i}^{0}k1;σkmkp;σ

+kλ_{i}Ee_{β,i}k_{1;σ} λ_{i}|ϕ_{i}|+λ_{i}kq_{β,i}k_{1,σ}

kmk_{p;σ}+km^{0}k_{p;σ}
+kλiEeβ,ik1;σT^{p−1}^{p} λiku^{0}_{i}+qβ,ikp;σ kmkp;σ+km^{0}kp;σ

. Note that

λ_{i}kq_{β,i}k_{1,σ} ≤λ_{i}|ϕ_{i}|+|f_{i}(0)| (5.20)
by (4.10) and (5.12). Thus, using (4.10) we deduce

λiku^{0}_{i}+qβ,ikp;σ≤ kλiEeβ,ik1;σ

hkf_{i}^{0}kp;σ+ (|ψi(0)|+kψ^{0}_{i}k1;σ)kmkp;σ

+kgikβ,p;σ+ 2λi|ϕi|+|fi(0)|

kmkp;σ+km^{0}kp;σi
+T^{p−1}^{p} kmkp;σ+km^{0}kp;σ

·λiku^{0}_{i}+qβ,ikp;σ.
In the case (5.13), from this relation we obtain (5.14).

Further, differentiating (3.7) we have

D^{β}u^{0}_{i}+λ_{i}u^{0}_{i}=f_{i}^{0}+D^{β}g_{i}+ ψ_{i}(0)−λ_{i}ϕ_{i}

m+ (ψ^{0}_{i}−λ_{i}u^{0}_{i})∗m

−λ_{i}ϕ_{i}m^{0}−λ_{i}u^{0}_{i}∗m^{0}. (5.21)

Adding (5.21) and (5.17) we obtain

D^{β}(u^{0}_{i}+qβ,i) =−λi(u^{0}_{i}+qβ,i) +f_{i}^{0}+D^{β}gi+ ψi(0)−λiϕi
m

+ (ψ_{i}^{0}−λiu^{0}_{i})∗m−λiϕim^{0}−λiu^{0}_{i}∗m^{0}. (5.22)
By the assumptions of the proposition and the relationsu^{0}_{i}∈L_{1}(0, T),u^{0}_{i}+q_{β,i}∈
Lp(0, T), the right-hand side of (5.22) belongs to Lp(0, T). Therefore, D^{β}(u^{0}_{i}+
q_{β,i}) ∈ L_{p}(0, T). This implies the assertion u^{0}_{i} +q_{β,i} ∈ 0H_{p}^{β}(0, T). Estimating
(5.22) we have

ku^{0}_{i}+q_{β,i}kβ,p;σ

≤λiku^{0}_{i}+qβ,ikp;σ+kf_{i}^{0}kp;σ+kgikβ,p;σ+h

|ψi(0)|+kψ_{i}^{0}k1;σ

ikmkp;σ

+h

λi|ϕi|+λikqβ,ik1;σ+T^{p−1}^{p} λiku^{0}_{i}+qβ,ikp;σ

i kmkp;σ+km^{0}kp;σ

.
Using (5.14) for λiku^{0}_{i}+qikp;σ, (4.10) for kλiEeβ,ik1;σ, (5.20) for λikqβ,ik1;σ, and
estimatingkmkp;σ+km^{0}kp;σ by ^{1}_{2}T^{p−1}^{p} we obtain (5.15).

6. Uniqueness

In the sequel we use the notationsui[β, m] andui[m] to indicate the dependence of the solution of (3.7) on the pair (β, m) andm.

Theorem 6.1. Letfi, ψi∈H_{r}^{1−s}^{1}(0, T)∩C[0, T],gi ∈Lr(0, T),I^{1−s}^{2}gi∈L_{∞}[0, T],
i= 1,2, . . .with some s1∈[0,1),s2∈(0,1],r∈(1,∞) andm^{0}∈L1(0, T). More-
over, assume

∞

X

i=1

|γi|λi|ϕi|<∞,

∞

X

i=1

|γi|kfik1−s1,r <∞,

∞

X

i=1

|γi|kfik∞<∞,

∞

X

i=1

|γ_{i}|kψ_{i}k_{1−s}_{1}_{,r}<∞,

∞

X

i=1

|γ_{i}|kψ_{i}k_{∞}<∞,

∞

X

i=1

|γi|kgikr<∞,

∞

X

i=1

|γi|kI^{1−s}^{2}gik_{∞}<∞

(6.1)

and ∞

X

i=1

γi(λiϕi−fi(0))6= 0,

∞

X

i=1

γi(λiϕi−ψi(0))6= 0. (6.2)
If(β_{j}, m_{j})∈(s_{1}, s_{2})×L_{1}(0, T),j= 1,2, are solutions of the inverse problem, then
β_{1}=β_{2} andm_{1}=m_{2}.

Proof. Without loss of generality we may assume r <min{_{1−β}^{1}

1;_{1−β}^{1}

2}. In view
of Proposition 5.1, the problems (3.7) with the data (βj, mj)∈(s1, s2)×L1(0, T)
have unique solutionsu_{j,i}:=u_{i}[β_{j}, m_{j}]∈W_{r}^{1}(0, T)⊂C[0, T], i= 1,2, . . .,j= 1,2.

Due to (4.8) there existsσ >0 such thatkmjk1;σ+km^{0}k1;σ≤ ^{1}_{2},j= 1,2. In view
of the estimates (5.2), (5.3), the assumptions (6.1) and the equivalence relations of
weighted norms (4.7) we have

∞

X

i=1

|γ_{i}|λ_{i}ku_{j,i}k_{∞}<∞,

∞

X

i=1

|γ_{i}|ku^{0}_{j,i}k_{r}<∞. (6.3)

This implies P∞

i=1γiλiuj,i ∈ C[0, T], P∞

i=1γiλiuj,i

_{t=0}= P∞

i=1γiλiϕi and P∞

i=1γiu^{0}_{j,i}∈Lr(0, T).

Moreover, from (3.8) we obtain h^{0} = P∞

i=1γiu^{0}_{j,i}, j = 1,2. In view of this
relation, from (3.7) we deduce the expressions

t^{−β}^{j}

Γ(1−βj)∗(h^{0}−

∞

X

i=1

γigi)

=

∞

X

i=1

γ_{i}(f_{i}−λ_{i}u_{j,i}) +

∞

X

i=1

γ_{i}(ψ_{i}−λ_{i}u_{j,i})∗m_{j}−

∞

X

i=1

γ_{i}λ_{i}u_{j,i}∗m^{0},

(6.4)

forj= 1,2. From the relationsfi, ψi∈C[0, T] and the third and fifth inequality in (6.1) we haveP∞

i=1γifi ∈C[0, T] and P∞

i=1γiψi ∈C[0, T]. Therefore, the right-
hand side of (6.4) belongs toC[0, T]. We obtain_{Γ(1−β}^{t}^{−}^{βj}

j)∗(h^{0}−P∞

i=1γigi)∈C[0, T],
j= 1,2. Taking the limitt→0^{+} in (6.4), we have

lim

t→0^{+}

t^{−β}^{j}

Γ(1−βj)∗(h^{0}−

∞

X

i=1

γigi) =

∞

X

i=1

γi(fi(0)−λiϕi), j= 1,2. (6.5)
Suppose thatβ_{1}< β_{2}. Then

t^{−β}^{1}

Γ(1−β_{1})∗(h^{0}−

∞

X

i=1

γ_{i}g_{i}) = t^{β}^{2}^{−β}^{1}^{−1}

Γ(β_{2}−β_{1})∗ζ(t), ζ(t) = t^{−β}^{2}

Γ(1−β_{2})∗(h^{0}−

∞

X

i=1

γ_{i}g_{i}).

Sinceζ∈C[0, T] it holds lim_{t→0}+ t^{β}^{2}^{−β}^{1}^{−1}

Γ(β_{2}−β1)∗ζ(t) = 0. Thus, lim_{t→0}+ t^{−β}^{1}

Γ(1−β1)∗(h^{0}−
P∞

i=1γ_{i}g_{i}) = 0. But this with (6.5) contradicts to the assumption (6.2). Similarly
we reach the contradiction in caseβ_{1}> β_{2}. Consequently, β_{1}=β_{2}.

Denoteβ :=β1=β2 and subtract the equalities (6.4) withj = 2 andj= 1:

∞

X

i=1

γi(ψi−λiu1,i)∗(m1−m2)−

∞

X

i=1

γiλi(u1,i−u2,i)∗(m2+m^{0})

−

∞

X

i=1

γ_{i}λ_{i}(u_{1,i}−u_{2,i}) = 0.

(6.6)

The differencesvi=u1,i−u2,i,i= 1,2, . . ., solve the problems
t^{−β}

Γ(1−β)∗v^{0}_{i}+λ_{i}v_{i}=−λ_{i}v_{i}∗(m_{2}+m^{0}) + (ψ_{i}−λ_{i}u_{1,i})∗(m_{1}−m_{2}),
v_{i}(0) = 0.

(6.7) Let us consider the problems

t^{−β}

Γ(1−β)∗w^{0}_{i}+λiwi =−λiwi∗(m2+m^{0}) +ψi−λiu1,i,
wi(0) = 0

(6.8)
for i = 1,2, . . .. By Proposition 5.1, these problems have the unique solutions
w_{i}∈W_{r}^{1}(0, T)⊂C[0, T],i= 1,2, . . .andλ_{i}kwik∞;σ≤C_{1}(kfik∞;σ+λ_{i}ku1,ik∞;σ).

( (3.7) takes the form of (6.8), if we replace the data vector (f_{i}, g_{i}, ψ_{i}, m, m^{0}, ϕ) by
(ψ_{i}−λ_{i}u_{1,i},0,0,0, m_{2}+m^{0},0).) The properties of w_{i} with (6.1) and (6.3) yield
the relationsP∞

i=1γ_{i}λ_{i}w_{i}∈C[0, T] andP∞

i=1γ_{i}λ_{i}w_{i}

_{t=0}= 0. One can immediately
check that v_{i} = w_{i}∗(m_{1}−m_{2}) solves (6.7). By the uniqueness of the solution

of (6.7), it holdsu1,i−u2,i =wi∗(m1−m2), i= 1,2, . . .. Consequently, we can transform (6.6) as follows:

∞

X

i=1

γi

n

ψi−λiu1,i−λiwi∗(m2+m^{0})−λiwi

o∗(m1−m2)(t) = 0, (6.9)
fort∈(0, T). By Titchmarsh convolution theorem, there existT1≥0 andT2≥0
such thatT_{1}+T_{2}=T and

∞

X

i=1

γi

n

ψi−λiu1,i−λiwi∗(m2+m^{0})−λiwi

o

(t) = 0 (6.10)
a.e. t∈(0, T1), and (m^{1}−m^{2})(t) = 0 a.e. t∈(0, T2). But since the function at the
left-hand side of (6.10) is continuous and possesses the limitP∞

i=1(ψi(0)−λiϕi)6= 0
as t → 0^{+}, the equality T1 = 0 is valid. Consequently, (m1−m2)(t) = 0 a.e.

t∈(0, T). This completes the proof.

7. Existence Let us introduce the function

Qβ,ϕ,f(t) =

∞

X

i=1

γiqβ,i(t)t^{1−β}=

∞

X

i=1

γi λiϕi−fi(0)

Eβ,β −λit^{β}

. (7.1)

Firstly, we prove a proposition that gives a necessary consistency condition for
h^{0}+Qβ,ϕ,f(t)t^{β−1}.

Proposition 7.1. Let (β, m)∈ (0,1)×L_{p}(0, T) with some p∈ (1,∞) solve IP.

Assume that fi ∈ W_{p}^{1}(0, T), ψi ∈ W_{1}^{1}(0, T), gi ∈ 0H_{p}^{β}(0, T), i = 1,2, . . ., m^{0} ∈
Lp(0, T)and

∞

X

i=1

|γi|λi|ϕi|<∞,

∞

X

i=1

|γi|kfikW_{p}^{1}(0,T)<∞,

∞

X

i=1

|γ_{i}|kψ_{i}k_{W}1

1(0,T)<∞,

∞

X

i=1

|γ_{i}|kg_{i}k_{β,p}<∞.

(7.2)

Thenh^{0}+Q_{β,ϕ,f}(t)t^{β−1}∈0H_{p}^{β}(0, T).

Proof. Since0H_{p}^{β}(0, T),→Lr(0, T) and W_{1}^{1}(0, T),→H_{r}^{1−β}(0, T) for r∈(1,_{1−β}^{1} ),
r≤p, Proposition 5.1 yieldsui∈W_{r}^{1}(0, T). Moreover, (7.2) implies the inequalities
P∞

i=1|γi|kfik1−β,r <∞,P∞

i=1|γi|kψik1−β,r <∞andP∞

i=1|γi|kgikr<∞. There exist σ such that (5.13) (hence also (5.1)) is valid. Applying (5.2) we obtain the relation P∞

i=1|γi|ku^{0}_{i}kr;σ <∞. Thus, h^{0} =P∞

i=1γ_{i}u^{0}_{i} ∈L_{r}(0, T). Further, (5.15)
with (7.2) implies P∞

i=1|γ_{i}|ku^{0}_{i}+q_{β,i}k_{β,p;σ} < ∞. Since h^{0}(t) +Q_{β,ϕ,f}(t)t^{β−1} =
P∞

i=1γi(u^{0}_{i}+qβ,i)(t), we deducekh^{0}+Qβ,ϕ,f(t)t^{β−1}kβ,p;σ<∞. This with Lemma

4.2 implies the assertion of the proposition.

For the statement and proof of an existence theorem, we define the following balls in the spaceLp(0, T):

B%,σ={w∈Lp(0, T) :kwkp;σ≤%}

and introduce the notation d=

ϕi|i=1,...,∞, fi|i=1,...,∞, ψi|i=1,...,∞, gi|i=1,...,∞, m^{0}, h