Existence, uniqueness and stability of the solution of the inverse problem are proved

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−1and 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−βUxx(x, t),

whereU is the state variable,xis the space variable,tis the time and D^1−βU(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+m⁰∗Uxx) +G. (2.3) The functionGis a source term. The inclusion of the addend withm⁰has 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 m⁰. Therefore, it makes sense to incorporate this term already from the beginning. On the other hand, m⁰ can be interpreted as an initial guess for an unknown kernel of the formm⁰+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^αdefined by the formula

I^αv(t) = t^α−1

Γ(α)∗v(t) = Z t

0

(t−τ)^α−1 Γ(α) v(τ)dτ.

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

Γ(1−β)∗[Ut−G] =Uxx+m∗Uxx+m⁰∗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) +m⁰∗Uxx(x, t), (x, t)∈(x₁, x₂)×(0, T),

U(x,0) =U0(x), x∈(x1, x2),

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

(2.4)

whereB₁andB₂ are boundary operators atx=x₁andx=x₂, respectively. More precisely,

for anyj∈ {1; 2} eitherBjv=v(xj) or

Bjv=v⁰(xj) +θjv(xj) withθj∈R, (−1)^jθj≥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(x₀) or Φ[v] =v⁰(x₀) +ϑv(x₀) or Φ[v] = Z x2

x₁

κ(x)v(x)dx, wherex₀∈[x₁, x₂],ϑ∈R,κ: (x₁, x₂)→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, m⁰,U0, b₁ and b₂, 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. B₁b^U(·, t) =b₁(t) and B₂b^U(·, t) =b₂(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)] +m⁰∗u_xx(x, t), (x, t)∈(x1, x2)×(0, T),

u(x,0) =ϕ(x), x∈(x1, x2), B₁u(·, t) = 0, B₂u(·, t) = 0, t∈(0, T),

(2.7)

and

Φ[u(·, t)] =h(t), t∈(0, T), (2.8) whereg=G−b^Ut,ψ=b^Uxx,f =b^Uxx+m⁰∗b^Uxx,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^si−1, where si > β (for with such M, see [7]). Then m(t) = Pn

i=1 c_iΓ(s_i)

Γ(s_i−β)t^si−β−1. Another example of m is the exponentially decreasing flux relaxation (memory) kernelm(t) =Pn

i=1c_ie^−αit, 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,m⁰, 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)∗[u⁰(t)−g(t)]

=Au(t) +f(t) +m∗[Au(t) +ψ(t)] +m⁰∗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)⁻¹k ≤ 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 ∈ W₁¹((0, T);X) and m, m⁰ ∈ L¹(0, T). Then (3.1) has a unique solution in the space C([0, T];X_A) and _Γ(1−β)^t−β ∗u⁰ ∈ C([0, T];X). The solution continuously depends on ϕ, f, ψ, g, m and m⁰ 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^d22 with the domain D(A) ={w:z∈W₂²(x₁, x₂),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^d22 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)∗[u⁰_i(t)−gi(t)] +λiui(t)

=fi(t) +m∗[ψi(t)−λiui(t)]−m⁰∗λiui(t), t∈(0, T), ui(0) =ϕi, (3.7) wherei= 1,2, . . .,

∞

X

i=1

γ_iu_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, . . .andm⁰, 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)^s2Fw)∈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_pⁿ(0, T) coincides with the Sobolev space W_pⁿ(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_Lp(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^sz = ^t_Γ(s)^s−1 ∗z, is a bijection from Lp(0, T)onto 0H_p^s(0, T), the inverse of I^sis the Riemann-Liouville fractional derivativeD^s=_dt^dI^1−s and

kwks,p:=kD^swkp

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_β⁰ = 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−β¹ ) and y ∈ L1(0, T),λ, w0∈R. Then the Cauchy problem

t^−β

Γ(1−β)∗w⁰(t)+ t^−β

Γ(1−β)∗y∗w⁰(t)+λw(t) =z(t), t∈(0, T), w(0) =w0 (4.2) has a unique solutionwin the space W_r¹(0, T). This solution has in casey= 0 the representation

w(t) =w0Eβ −λt^β +

Z t

0

(t−τ)^β−1Eβ,β

−λ(t−τ)^β

z(τ)dτ. (4.3) Proof. By Lemma 4.2, Remark 4.1 and the relationw=I¹w⁰+w₀, (4.2) is equiv- alent to

w⁰(t) +y∗w⁰(t) +λD^1−β(I¹w⁰(t) +w₀) =D^1−βz(t), t∈(0, T), w(0) =w₀.

Since D^1−βI¹ = D^1−βI^1−βI^β = I^β, the obtained equation for w⁰ is the Volterra equation of the second kind

w⁰(t) + Z t

0

hy(t−τ) +λ(t−τ)^β−1 Γ(β)

iw⁰(τ)dτ

=D^1−βz(t)−λw₀t^β−1

Γ(β), t∈(0, T).

(4.4)

The right-hand sideD^1−βz−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 solutionw⁰ ∈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−1Eβ,β

−λ(t−τ)^β z(τ)dτ

solves the equation D^βω+λω = z. Since ω(0) = 0 we have D^βω = _Γ(1−β)^t−β ∗ω⁰. Consequently, we obtain the relation

t^−β

Γ(1−β)∗ω⁰(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^βφ+λφ= t^−β Γ(1−β).

This yields _Γ(1−β)^t−β ∗φ⁰(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)∗χ⁰(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 spaces₀H_p^s(0, T) andL_p(0, T):

kwks,p;σ=ke^−σtD^swkL_p(0,T), and kwkp;σ=ke^−σtwkL_p(0,T), whereσ≥0. Evidently, the equivalence relations

e^−σTkwk_s,p ≤ kwk_s,p;σ≤ kwk_s,p, e^−σTkwk_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−1Eβ,β

−λit^β

(4.9) satisfy the following estimates:

kλ_iEe_β,ik_1;σ≤1 (4.10)

kλ¹⁻_i Ee_β,ik_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^−σtλit^β−1Eβ,β(−λit^β)dt

≤ Z T

0

λit^β−1Eβ,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^βδ

Eβ,β(−λit^β) ≤c¹_β,δ fort ≥ 0 and 0≤δ ≤2 with some constantc¹_β,δ. Thus, for 0< ≤1 we deduce

kλ¹⁻_i Eeβ,ik1;σ= Z T

0

e^−σt λit^β1−

t^β−1Eβ,β(−λit^β)dt

≤c¹_β,1−

Z T

0

e^−σtt^β−1dt

= c¹_β,1−

σ^β Z σT

0

e^−ss^β−1ds

< c¹_β,1−

σ^β Z ∞

0

e^−ss^β−1ds.

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, m⁰ ∈ L₁(0, T) and f_i, ψ_i ∈ H_r^1−β(0, T), g_i∈L_r(0, T)with somer∈(1,_1−β¹ ). Then the problem (3.7)has a unique solution u_i∈W_r¹(0, T). Moreover, the following assertions are valid:

(i) if

kmk1;σ+km⁰k1;σ ≤1

2 (5.1)

then the estimate ku⁰_ikr;σ+λikuik1−β,r;σ≤C0

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

(5.2) holds, whereC₀ is a constant independent ofσandi;

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

λ_ikuik∞;σ≤C₁

λ_i|ϕi|+kfik∞;σ+kψik∞;σ+kI^1−βg_ik∞;σ

(5.3) holds, whereC₁ is a constant independent ofσandi.

Proof. Sincem, m⁰∈L₁(0, T), the Volterra equation of the second kind y(t) + (m+m⁰)∗y(t) +m(t) +m⁰(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+m⁰)∗

= I+ (m+m⁰)∗

(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)∗[u⁰_i(t) +y∗u⁰_i(t)] +λ_iu_i(t) =fe_i(t), t∈(0, T), ui(0) =ϕi,

(5.4)

where fei(t) = fi(t) +y∗fi(t) +I^1−βgi(t) +y∗I^1−βgi(t) + (m+y∗m)∗ψi(t).

Conversely, applying the operatorI+ (m+m⁰)∗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−βfi(t) +I^1−βgi(t) +y∗ t^−β

Γ(1−β)∗g_i(t) + (m+y∗m)∗ t^−β

Γ(1−β)∗D^1−βψ_i(t)

=f_i(t) +I^1−βg_i(t) + t^−β Γ(1−β)∗

y∗D^1−βf_i(t) +y∗g_i(t) + (m+y∗m)∗D^1−βψi(t)

,

where y∗D^1−βfi+y∗gi+ (m+y∗m)∗D^1−βψi ∈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_r¹(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) +m⁰(τ)]dτ.

(5.5)

In view of the relation I =I^1−βD^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−βf_i(τ) +g_i(τ)

+ τ^−β

Γ(1−β)∗D^1−βψ_i∗m(τ)i dτ

− Z t

0

Ee_β,i(t−τ)λ_i τ^−β

Γ(1−β)∗D^1−βu_i∗[m(τ) +m⁰(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_it^β

=−λ_iEe_β,i(t), (5.7)

following from (4.1) and (4.9), we reach the expression D^1−βui(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−βfi(τ) +gi(τ) +D^1−βψi∗m(τ)i dτ

− Z t

0

Eeβ,i(t−τ)λiD^1−βui∗[m(τ) +m⁰(τ)]dτ.

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

e^−σt[w₁(t)∗w₂(t)] =e^−σtw₁(t)∗e^−σtw₂(t).

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

As a result we obtain λikD^1−βuikr;σ≤λi|ϕi|

kλiEeβ,ik1;σ+ 1

t^β−1 Γ(β) _r;σ

+kλiEeβ,ik1;σ

kD^1−βfi+gikr;σ+kD^1−βψikr;σkmk1;σ

+kλiEe_β,ik1;σλ_ikD^1−βu_ikr;σ[kmk1;σ+km⁰k1;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+km⁰k1;σ]·λikuik_1−β,r;σ,

where ˆc_β,r = k^t_Γ(β)^β−1k_r. In case (5.1) is valid, we estimate kmk_1;σ and kmk_1;σ+ km⁰k1;σ by ¹₂, bring the term ¹₂λikuik_1−β,r;σ to the left-hand side and multiply the obtained inequality by 2. This results in

λ_iku_ik_1−β,r;σ≤C₄

λ_i|ϕ_i|+kf_ik_1−β,r;σ+kψ_ik_1−β,r;σ+kg_ik_r;σ

, (5.9)

whereC4 is a constant.

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

u⁰_i=−λiD^1−βu_i+D^1−βf_i+g_i+ D^1−βψ_i−λiD^1−βu_i

∗m−λiD^1−βu_i∗m⁰. (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−βf∗m+ t^β−1

Γ(β)∗ t^−β

Γ(1−β)∗D^1−βf

(t)|t=0·m

=D^1−βf∗m+ I¹D^1−βf

(t)|t=0·m=D^1−βf∗m

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

ku⁰_ikr;σ≤λikuik1−β,r;σ+kfik1−β,r;σ+kgikr;σ

+ kψik1−β,r;σ+λikuik1−β,r;σ

kmk1;σ+λikuik1−β,r;σkm⁰k1;σ

≤C₅

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^−σtand estimating the result we obtain

λikuik_∞;σ ≤λi|ϕi|+kλiEeβ,ik1;σ

kfi+I^1−βgik_∞;σ+kψik_∞;σkmk1;σ

+kλiEeβ,ik1;σλikuik_∞;σ[kmk1;σ+km⁰k1;σ].

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

Proposition 5.2. Let β ∈ (0,1), m, m⁰ ∈ Lp(0, T) with some p ∈ (1,∞) and fi ∈W_p¹(0, T),ψi∈W₁¹(0, T), gi∈0H_p^β(0, T). Thenu⁰_i+qβ,i∈0H_p^β(0, T), where ui is the solution of (3.7)and

q_β,i(t) = λ_iϕ_i−f_i(0)

E_β,β −λit^β

t^β−1= λ_iϕ_i−f_i(0)

Ee_β,i(t). (5.12) Moreover, in the case

T^p−1p kmkp;σ+km⁰kp;σ

≤ 1

2 (5.13)

the estimates

λiku⁰_i+qβ,ikp;σ≤C2kλiEeβ,ik1;σ

λi|ϕi|+|fi(0)|

+ (|ψi(0)|+kψ⁰_ik1;σ)kmkp;σ+kf_i⁰kp;σ+kgikβ,p;σ

,

(5.14) ku⁰_i+q_β,ik_β,p;σ≤C₃

λ_i|ϕ_i|+|f_i(0)|

+ (|ψ_i(0)|+kψ_i⁰k_1;σ)kmk_p;σ+kf_i⁰k_p;σ+kg_ik_β,p;σ ,

(5.15) hold, whereC₂ andC₃ 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+λiI¹qβ,i=λiϕi−fi(0), (5.16) D^βq_β,i+λ_iq_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 _λ¹

i fi(0)−λiϕi

we obtain t^−β

Γ(1−β)∗ 1 λi

f_i(0)−λ_iϕ_i

¯

q_i⁰+ f_i(0)−λ_iϕ_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_λ¹

i f_i(0)−λiϕ_i

¯

q⁰_i=q_β,i. Integrating, multiplying byλ_iand observing that

¯

q_i(0) = 1 we have another formula f_i(0)−λ_iϕ_i

¯

q_i(t) =λ_iI¹q_β,i(t) +f_i(0)−λ_iϕ_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 W₁¹(0, T)⊂H_r^1−β(0, T) and 0H_p^β(0, T)⊂Lr(0, T) for r ∈ (0,_1−β¹ ), r ≤ p, by Proposition 5.1, problem (3.7) has a unique solution u_i∈W_r¹(0, T). Differentiating (5.5) and observing (5.7), (5.12) we obtain

u⁰_i(t) +q_β,i(t)

= Z t

0

Eeβ,i(t−τ)h

f_i⁰(τ) +D^βgi(τ) +ψi(0)m(τ) +ψ⁰_i∗m(τ)i dτ

− Z t

0

Ee_β,i(t−τ)λ_ih

ϕ_i m(τ) +m⁰(τ)

+u⁰_i∗ m(τ) +m⁰(τ)i dτ.

(5.19)

From u⁰_i ∈ L₁(0, T) and the assumptions of the proposition, the right-hand side of this relation belongs toL_p(0, T). Therefore, u⁰_i+q_β,i ∈L_p(0, T). Multiplying (5.19) by λ_ie^−σt, representing u⁰_i as u⁰_i =−q_β,i+u⁰_i+q_β,i at the right-hand side and using (4.12) as well as the relationkmk1;σ≤T^p−1p kmkp;σ we obtain

λiku⁰_i+qβ,ikp;σ≤ kλiEeβ,ik1;σ kf_i⁰kp;σ+kgikβ,p;σ

+|ψi(0)|kmkp;σ+kψ_i⁰k1;σkmkp;σ

+kλ_iEe_β,ik_1;σ λ_i|ϕ_i|+λ_ikq_β,ik_1,σ

kmk_p;σ+km⁰k_p;σ +kλiEeβ,ik1;σT^p−1p λiku⁰_i+qβ,ikp;σ kmkp;σ+km⁰kp;σ

. Note that

λ_ikq_β,ik_1,σ ≤λ_i|ϕ_i|+|f_i(0)| (5.20) by (4.10) and (5.12). Thus, using (4.10) we deduce

λiku⁰_i+qβ,ikp;σ≤ kλiEeβ,ik1;σ

hkf_i⁰kp;σ+ (|ψi(0)|+kψ⁰_ik1;σ)kmkp;σ

+kgikβ,p;σ+ 2λi|ϕi|+|fi(0)|

kmkp;σ+km⁰kp;σi +T^p−1p kmkp;σ+km⁰kp;σ

·λiku⁰_i+qβ,ikp;σ. In the case (5.13), from this relation we obtain (5.14).

Further, differentiating (3.7) we have

D^βu⁰_i+λ_iu⁰_i=f_i⁰+D^βg_i+ ψ_i(0)−λ_iϕ_i

m+ (ψ⁰_i−λ_iu⁰_i)∗m

−λ_iϕ_im⁰−λ_iu⁰_i∗m⁰. (5.21)

Adding (5.21) and (5.17) we obtain

D^β(u⁰_i+qβ,i) =−λi(u⁰_i+qβ,i) +f_i⁰+D^βgi+ ψi(0)−λiϕi m

+ (ψ_i⁰−λiu⁰_i)∗m−λiϕim⁰−λiu⁰_i∗m⁰. (5.22) By the assumptions of the proposition and the relationsu⁰_i∈L₁(0, T),u⁰_i+q_β,i∈ Lp(0, T), the right-hand side of (5.22) belongs to Lp(0, T). Therefore, D^β(u⁰_i+ q_β,i) ∈ L_p(0, T). This implies the assertion u⁰_i +q_β,i ∈ 0H_p^β(0, T). Estimating (5.22) we have

ku⁰_i+q_β,ikβ,p;σ

≤λiku⁰_i+qβ,ikp;σ+kf_i⁰kp;σ+kgikβ,p;σ+h

|ψi(0)|+kψ_i⁰k1;σ

ikmkp;σ

+h

λi|ϕi|+λikqβ,ik1;σ+T^p−1p λiku⁰_i+qβ,ikp;σ

i kmkp;σ+km⁰kp;σ

. Using (5.14) for λiku⁰_i+qikp;σ, (4.10) for kλiEeβ,ik1;σ, (5.20) for λikqβ,ik1;σ, and estimatingkmkp;σ+km⁰kp;σ by ¹₂T^p−1p 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−s1(0, T)∩C[0, T],gi ∈Lr(0, T),I^1−s2gi∈L_∞[0, T], i= 1,2, . . .with some s1∈[0,1),s2∈(0,1],r∈(1,∞) andm⁰∈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_ik_1−s1,r<∞,

∞

X

i=1

|γ_i|kψ_ik_∞<∞,

∞

X

i=1

|γi|kgikr<∞,

∞

X

i=1

|γi|kI^1−s2gik_∞<∞

(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)∈(s₁, s₂)×L₁(0, T),j= 1,2, are solutions of the inverse problem, then β₁=β₂ andm₁=m₂.

Proof. Without loss of generality we may assume r <min{_1−β¹

1;_1−β¹

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_r¹(0, T)⊂C[0, T], i= 1,2, . . .,j= 1,2.

Due to (4.8) there existsσ >0 such thatkmjk1;σ+km⁰k1;σ≤ ¹₂,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_iku_j,ik_∞<∞,

∞

X

i=1

|γ_i|ku⁰_j,ik_r<∞. (6.3)

This implies P∞

i=1γiλiuj,i ∈ C[0, T], P∞

i=1γiλiuj,i

_t=0= P∞

i=1γiλiϕi and P∞

i=1γiu⁰_j,i∈Lr(0, T).

Moreover, from (3.8) we obtain h⁰ = P∞

i=1γiu⁰_j,i, j = 1,2. In view of this relation, from (3.7) we deduce the expressions

t^−βj

Γ(1−βj)∗(h⁰−

∞

X

i=1

γigi)

=

∞

X

i=1

γ_i(f_i−λ_iu_j,i) +

∞

X

i=1

γ_i(ψ_i−λ_iu_j,i)∗m_j−

∞

X

i=1

γ_iλ_iu_j,i∗m⁰,

(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)∗(h⁰−P∞

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

lim

t→0⁺

t^−βj

Γ(1−βj)∗(h⁰−

∞

X

i=1

γigi) =

∞

X

i=1

γi(fi(0)−λiϕi), j= 1,2. (6.5) Suppose thatβ₁< β₂. Then

t^−β1

Γ(1−β₁)∗(h⁰−

∞

X

i=1

γ_ig_i) = t^{β2−β1−1}

Γ(β₂−β₁)∗ζ(t), ζ(t) = t^−β2

Γ(1−β₂)∗(h⁰−

∞

X

i=1

γ_ig_i).

Sinceζ∈C[0, T] it holds lim_t→0+ t^{β2−β1−1}

Γ(β₂−β1)∗ζ(t) = 0. Thus, lim_t→0+ t^−β1

Γ(1−β1)∗(h⁰− P∞

i=1γ_ig_i) = 0. But this with (6.5) contradicts to the assumption (6.2). Similarly we reach the contradiction in caseβ₁> β₂. Consequently, β₁=β₂.

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+m⁰)

−

∞

X

i=1

γ_iλ_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)∗v⁰_i+λ_iv_i=−λ_iv_i∗(m₂+m⁰) + (ψ_i−λ_iu_1,i)∗(m₁−m₂), v_i(0) = 0.

(6.7) Let us consider the problems

t^−β

Γ(1−β)∗w⁰_i+λiwi =−λiwi∗(m2+m⁰) +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_r¹(0, T)⊂C[0, T],i= 1,2, . . .andλ_ikwik∞;σ≤C₁(kfik∞;σ+λ_iku1,ik∞;σ).

( (3.7) takes the form of (6.8), if we replace the data vector (f_i, g_i, ψ_i, m, m⁰, ϕ) by (ψ_i−λ_iu_1,i,0,0,0, m₂+m⁰,0).) The properties of w_i with (6.1) and (6.3) yield the relationsP∞

i=1γ_iλ_iw_i∈C[0, T] andP∞

i=1γ_iλ_iw_i

_t=0= 0. One can immediately check that v_i = w_i∗(m₁−m₂) 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+m⁰)−λiwi

o∗(m1−m2)(t) = 0, (6.9) fort∈(0, T). By Titchmarsh convolution theorem, there existT1≥0 andT2≥0 such thatT₁+T₂=T and

∞

X

i=1

γi

n

ψi−λiu1,i−λiwi∗(m2+m⁰)−λiwi

o

(t) = 0 (6.10) a.e. t∈(0, T1), and (m¹−m²)(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 h⁰+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_p¹(0, T), ψi ∈ W₁¹(0, T), gi ∈ 0H_p^β(0, T), i = 1,2, . . ., m⁰ ∈ Lp(0, T)and

∞

X

i=1

|γi|λi|ϕi|<∞,

∞

X

i=1

|γi|kfikW_p¹(0,T)<∞,

∞

X

i=1

|γ_i|kψ_ik_W1

1(0,T)<∞,

∞

X

i=1

|γ_i|kg_ik_β,p<∞.

(7.2)

Thenh⁰+Q_β,ϕ,f(t)t^β−1∈0H_p^β(0, T).

Proof. Since0H_p^β(0, T),→Lr(0, T) and W₁¹(0, T),→H_r^1−β(0, T) for r∈(1,_1−β¹ ), r≤p, Proposition 5.1 yieldsui∈W_r¹(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|ku⁰_ikr;σ <∞. Thus, h⁰ =P∞

i=1γ_iu⁰_i ∈L_r(0, T). Further, (5.15) with (7.2) implies P∞

i=1|γ_i|ku⁰_i+q_β,ik_β,p;σ < ∞. Since h⁰(t) +Q_β,ϕ,f(t)t^β−1 = P∞

i=1γi(u⁰_i+qβ,i)(t), we deducekh⁰+Qβ,ϕ,f(t)t^β−1kβ,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,...,∞, m⁰, h