We also prove a version of the Poincar´e Recurrence Theorem

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December 2017

LINEAR HAMILTONIAN SYSTEM IN SCALE OF HILBERT SPACES AND THE POINCAR ´E RECURRENCE THEOREM

Oleg Zubelevich

Abstract. We consider an initial value problem for linear Hamiltonian system in the scale of Hilbert spaces and prove an existence and uniqueness theorem. We also prove a version of the Poincar´e Recurrence Theorem.

1. Statement of the problem

Linear differential equations in Banach spaces and the semigroup theory have become a classical topic of PDE since the fifties of the last century.

One of the central results of this theory is the Hille–Yosida theorem [10], [5]. This theorem provides necessary and sufficient conditions that a densely defined operator A:E→Eof a Banach spaceEgenerates a strongly continuous semirgoup{e^tA}or, in other words, the initial value problem

˙

x=Ax, x(0) = ˆx has a good enough solutionx(t) =e^tAx.ˆ

This theorem is formulated in terms of spectrum of the operatorA. The point is that the spectrum of an operator is not always simple to comprehend.

In this short paper we consider a Hamiltonian system in the scale of Hilbert spaces.

Apparently there are no direct ways to solve such systems by means of the standard spectral methods.

To motivate the statement of our main problem we first simulate the whole construction by means of the easiest example. The matrixJ (see below) in this example is very simple therefore the corresponding result follows from the Hille–Yosida theorem. But, in general case, our main theorem is not deduced from the Hille–Yosida theorem.

Let Ω ⊂ R^m be a bounded domain with smooth enough boundary ∂Ω. It is well known that the Laplace operator ∆ has the countable system of eigenfunctions

2010 Mathematics Subject Classification: 35A01, 37K05, 35A02

Keywords and phrases: Linear PDE; infinite dimensional Hamiltonian system.

259

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ej(x)∈C^∞(Ω),

−∆ej =λ²_jej, ej(∂Ω) = 0, j∈N. These functions form orthogonal basis ofL²(Ω) and

0< λ1≤λ2≤. . . , λj→ ∞ as j→ ∞.We also assume thatke_jk_L2(Ω)= 1.

Introduce a space Y_s=n

u(x) =

∞

X

k=1

u_ke_k | kuk²_s=

∞

X

k=1

u²_kλ^2s_k <∞o

, s∈R.

One may treat elements ofYsas distributions:

φ=

∞

X

k=1

φkek∈ D(Ω), (u, φ) =

∞

X

k=1

ukφk. It is not hard to show that the last sum is convergent.

The spacesYs are Hilbert spaces (all the Hilbert spaces we use are over the field R) with the inner products

(u, v)s=

∞

X

k=1

λ^2s_k ukvk, v(x) =

∞

X

k=1

vkek ∈Ys.

The spaces Ys are referred to as fractional power spaces associated to the Laplace operator. Observe thatY0=L²(Ω), Y1=H₀¹(Ω).

The power of the Laplace operator is defined in the natural way (−∆)^µu=

∞

X

k=1

λ^2µ_k ukek,

and (−∆)^µ:Y_s→Y_s−2µ, µ∈R, is an isometric isomorphism.

Consider the wave equation

utt= ∆u, u(t, x)|_x∈∂Ω= 0, u(0, x) = ˆu(x), ut(0, x) = ˆv(x). (2) Introduce a vector-valued function f(t, x) = (u(t, x), w(t, x))^T and rewrite our problem as follows

f_t=J Bf, J =

0 1

−1 0

, B=

(−∆)^1/2 0 0 (−∆)^1/2

. The initial conditions take the form

w(0, x) = (−∆)^−1/2v(x).ˆ

It is convenient to solve problem (2) inY_s, u(t,·), w(t,·)∈Y_swith ˆ

u∈Ys, vˆ∈Y_s−1.

Observe that the Schr¨odinger equation iψt = ∆ψ, ψ = ψ1 +iψ2 can also be written in the form ξt = J Bξ, ξ = (ψ1, ψ2)^T where B is a positively defined self- adjoint operator andJ is a skew-symmetric operator.

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1.1 Abstract construction

In this section we describe the above situation axiomatically. All the inessential positive constants we denote by the lettersc, c₁, c₂, . . .or bycwith another subscript.

Let

Xs+δ ⊂Xs, δ >0, s∈R (3)

be a scale of real Hilbert spaces with inner products (·,·)_s and the corresponding normsk · ks, k · ks≤ k · ks+δ. The spaceXs+δ is dense inXs.In the above example Xs=Ys×Ys.

Let

J :X_s→X_s, (J x, y)_s=−(J y, x)s

stand for a bounded skew-symmetric operator. This means that the operator J is defined onS

s∈RXs andJ(Xs)⊂Xs. The same holds for all other operators, yet we do not stress on this.

If it is not specified anything else, the parameter t belongs to the intervalIT = [0, T], T >0.

Introduce a bounded operatorB:X_s→X_s−1. This operator enjoys the following properties.

1. the operatorB is self-adjoint and non-negative

(Bx, y)_s= (x, By)_s, (Bx, x)_s≥0, x, y∈X_s+1; 2. all the powersB^σ :Xs+σ→Xsare bounded and coercive

(B^σx, x)s≥ckxk²_s+σ/2, σ≥0, x∈Xs+σ; 3. the operator−B^σ generates aC₀−semigroup [10]

e^−tB^σ :Xs→Xs

such that for anyx∈Xs+σ we have

1 t

e^−tB^σ−I

x+B^σx _s→0, as t→0+.

Moreover, e^−tB^σ(Xs) ⊂ Xs+δ, t ∈ (0, T] and for any x ∈ Xs and for any δ∈(0, cδ) the estimate

ke^−tB^σxk_s+δ ≤ c1

t^δ/σkxk_s, c₁=c₁(s, δ, σ) (4) is fulfilled.

The main object of our study is the following Hamiltonian initial value problem

˙

x=J B^ˆ^σx, x(0) = ˆx∈Xˆs (5) with some fixed constants ˆσ >0, ˆs∈R.

1.2 Several remarks on accepted hypotheses

In the above example all the hypotheses can be checked by direct calculation.

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Observe that hypotheses on the operatorJ can be considerably relaxed. Actually, one can put J:Xs→Xs−µ, µ >0 to be a bounded and skew-symmetric operator:

(x, J y)s=−(J x, y)s, x, y∈Xs+µ.

This generalization does not bring any essential difficulties in the proofs of the theorems but makes the formulas more complicated than we have now.

Formula (4) is very standard in the parabolic semigroup theory [9], [1], [11].

In the sequel we do not use all the Hilbert spaces contained in the scale {X_s}.

Actually we use only the spaces with indexes ˆ

s, sˆ±γ, sˆ−γ−σ,ˆ sˆ−γ−3ˆσ/2, sˆ−γ−ˆ2σ, ˆs−σ/2,ˆ ˆs+ ˆσ.

But the scales arise in the applications. Furthermore we believe that the scale{Xs} is more acceptable object than several different spaces which look like a cumbersome and artificial construction.

2. Main theorems

LetC_w(I_T, X_s) stand for the spaceC(I_T,X˜_s) and ˜X_sis the spaceX_sendowed with the weak topology.

Theorem 2.1. 1. Problem (5)has a weak solution

x(t)∈L^∞(IT, Xˆs)∩Cw(IT, Xˆs−γ) with arbitraryT >0 and arbitrary γ >σ. That isˆ

(a) for any ψ∈X_ˆ_s−γ it follows that ψ, x(t)

s−γˆ → ψ, x(t0)

ˆs−γ

ast→t0∈IT;

(b) for anyu(t)∈C¹(IT, Xs+γˆ ), u(T) =u(0) = 0 it follows that Z T

0

˙

u(t), x(t)

ˆ sdt=

Z T

0

B^ˆ^σJ u(t), x(t)

ˆ sdt.

For almost allt∈I this solution enjoys the inequality

kx(t)k_ˆ_s≤ckxkˆ _ˆ_s. (6) The constantcdoes not depend on T.

2. Assume in addition that embeddings (3)are compact. Then the solution x(t)∈C(IT, Xˆs−γ)∩C¹(IT, Xs−γ−ˆˆ σ) (7) is unique. This is a classical solution in the spaceXs−γˆ .

Indeed, under the conditions of the second part of the theorem system (5) possesses the Hamiltonian

H(x) = 1

2 B^ˆ^σx, x

s−γ−2ˆˆ σ=1

2 B^ˆ^σ/2x, B^σ/2^ˆ x

s−γ−2ˆˆ σ, x=x(t) (8)

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which is the first integral to this system:

H˙ = B^σ^ˆx,x˙

s−γ−2ˆˆ σ= B^σ^ˆx, J B^σ^ˆx

ˆs−γ−2ˆσ= 0.

Actually system (5) possesses the continuum set of Hamiltonians:

Hs(x) =1

2 B^ˆ^σx, x

s, s≤sˆ−γ−2ˆσ.

Using formula (8) we see that any solution of the kind (7) satisfies the estimate c17kx(t)k²_s−γ−3ˆ_ˆ _σ/2≤H(x(t)) =H(ˆx)≤c18kˆxk²_s−γ−3ˆ_ˆ _σ/2. (9) This implies the uniqueness in Theorem 2.1.

Let us formulate a version of the Poincar´e Recurrence Theorem.

Theorem2.2. Assume that both parts of Theorem 2.1 are fulfilled. Then there is an increasing sequencet_k → ∞, such that

kx(tk)−xkˆ _s−γ−3ˆ_ˆ _σ/2→0.

Unlike classical Poincar´e Recurrence Theorem this assertion deals with a dynam- ical system on non compact infinite dimensional phase space. Another version of the recurrence theorem for an infinite dimensional system on non compact space has been proved for a hydrodynamic problem in [6].

3. Proofs of the theorems 3.1 Auxiliary lemmas

Lemma 3.1. Let µ stand for the standard Lebesgue measure in R+ = (0,∞). Let τ ⊂R+ be a full measure set µ(R+\τ) = 0. Then there is a number a >0 such that {ak}k∈N⊂τ.

Proof. Assume the converse: for anya >0 there existsk∈Nsuch thatak /∈τ. This implies that

R+= [

k∈N

Mk, Mk={a >0|ak /∈τ}=R+\(τ /k).

It follows thatµ(Mk) = 0. This gives a contradiction.

Lemma 3.2 (the Banach-Steinhaus Theorem [3], [8]). Let Aν:X →Y, ν ∈[ν1, ν2]

be a set of bounded operators of a Banach spaceX to the Banach space Y. Assume that for eachx∈X we have

kAνx−AxkY →0 (10) asν →ν0∈[ν1, ν2]. Then the operatorA:X →Y is also bounded and the convergence (10)is uniform on any compact set of variable x.

Moreover, for any continuous function f : [t1, t2]→X it follows thatkAνf(t)− Af(t₀)kY →0 asν→ν₀ andt→t₀∈[t₁, t₂].

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Proof. Observe only that F ={f(t)|t∈[t1, t2]} is a compact set inX as an image of the compact set [t1, t2] under the continuous mappingf. So that by the previous propositions A_νxconverges toAxuniformly inx∈F. Lemma 3.3. Let x∈X_s−s₀, s0>0. Then the mapping

t7→e^−tB^γx is a continuous mapping of(0, T] toXs.

Proof. Lett→˜t−, ˜t >0. We have e^−tB^γ−e⁻^˜^tB^γ

x=e^−tB^γ^/2 I−e^−(˜^t−t)B^γ

e^−tB^γ^/2x.

One yields the estimate

e^−tB^γ^/2 I−e^−(˜^t−t)B^γ

e^−tB^γ^/2x s

≤ c t^s⁰^/γ

I−e^−(˜^t−t)B^γ

e^−tB^γ^/2x s−s0

. The mapping

t7→e^−tB^γ^/2x

is a continuous mapping of the small neighbourhood of the point ˜t to X_s−s₀. From Lemma 3.2 it follows that

I−e^−(˜^t−t)B^γ

e^−tB^γ^/2x s−s0

→0.

Ift→˜t+ then the argument is trivial e^−tB^γ−e⁻^˜^tB^γ

x= e^−(t−^˜^t)B^γ−I

e⁻^˜^tB^γx, e⁻^˜^tB^γx∈X_s.

Lemma 3.4. Assume that a function u∈C([0, t_∗))satisfies the inequality 0≤u(t)≤A+B

Z t

0

u(ξ)

(t−ξ)^αdξ, t >0 (11) here A, B, αare positive constants, α <1. Thensup_t∈[0,t_∗₎u(t)<∞.

Proof. Let us take numbersp, qsuch that 1

p+1

q = 1, 1< p < 1 α. Then the right-hand side of (11) is not greater than

A+BZ t 0

dξ (t−ξ)^αp

¹_pZ t

0

u^qdξ¹_q . So that

u(t)≤A+CZ t 0

u^qdξ¹_q

(12) with some positive constantC.

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Define a function

w(t) =

((u(t)−A)^q ifu(t)≥A, 0 ifu(t)< A.

The function w is non-negative and continuous in [0, t∗). Observe also that u ≤ A+w^1/q.

Inequality (12) implies w(t)≤C^q

Z t

0

Φ w(ξ)

dξ, Φ(η) = A+η^1/q^q

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Recall the following fact.

Proposition 3.5 ( [2]). Let a non negative function v∈C([0, t1]) be such that v(t)≤K

Z t

0

Φ v(ξ)

dξ, K= const>0.

Assume the functionΦ∈C[0,∞)to be positive and non decreasing.

Assume also that

Ψ(η) = Z η

0

dξ Φ(ξ) → ∞ asη→ ∞. Then

v(t)≤Ψ⁻¹ Kt , hereΨ⁻¹ is the inverse function.

By this Proposition and due to formula (13) the functionwis bounded in [0, t∗).

Thusuis also bounded.

3.2 Proof of the first part of Theorem 2.1

We use a version of the classical parabolic regularization method [7].

We will approximate problem (5) with the following sequence of “paraboli” problems

˙

xn =J B^σ^ˆxn−anB^γxn, xn(0) = ˆxn =e^−aⁿ^Bx,ˆ (14) herean= 1/n, n∈N.Note that ˆxn∈Xs, s∈R.

Consider corresponding integral equation xn(t) =e^−aⁿ^tB^γˆxn+

Z t

0

e^−aⁿ^(t−ξ)B^γJ B^σ^ˆxn(ξ)dξ (15) and the operator

Fn[x(·)] =e^−aⁿ^tB^γxˆn+ Z t

0

e^−aⁿ^(t−ξ)B^γJ B^σ^ˆx(ξ)dξ.

Lemma 3.6. The operator Fn takes the space C(Iτ, Xs)to itself. Here τ >0, s∈R are arbitrary constants.

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Proof. First note that ift→0+ thene^−aⁿ^tB^γxˆ→xˆ inXsand kFn[x(·)](t)ks≤

Z t

0

e^−aⁿ^(t−ξ)B^γJ B^σ^ˆx(ξ) _sdξ

≤ Z t

0

c₁

a_n(t−ξ)σ/γˆ kJ B^σ^ˆx(ξ)k_s−ˆ_σdξ

≤ Z t

0

c14

an(t−ξ)^σ/γ^ˆ kx(ξ)ksdξ→0.

ThusFn[x(·)](t) is a continuous function int= 0.

For 0≤t⁰⁰≤t⁰ ≤τ one has the identity

Fn[x(·)](t⁰)− Fn[x(·)](t⁰⁰) =e^−aⁿ^t⁰⁰^B^γ(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)ˆxn

+ Z t⁰

t⁰⁰

e^−aⁿ^(t⁰^−ξ)B^γJ B^σ^ˆx(ξ)dξ +

Z τ

0

χ_[0,t00](ξ)(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)e^−aⁿ^(t⁰⁰^−ξ)B^γJ B^σ^ˆx(ξ)dξ, (16) here χis the indicator function.

Assume thatt⁰⁰>0.The integral in the middle of the right-hand side of formula (16) is not greater than

Z t⁰

t⁰⁰

ke^−aⁿ^(t⁰^−ξ)B^γJ B^ˆ^σx(ξ)ksdξ

≤ Z t⁰

t⁰⁰

c₂

an(t⁰−ξ)σ/γˆ kJ B^σ^ˆx(ξ)ks−ˆσdξ

≤c₃ Z t⁰

t⁰⁰

1

a_n(t⁰−ξ)ˆσ/γkx(ξ)k_sdξ

≤c3max

ξ∈Iτ

kx(ξ)ks

Z t⁰

t⁰⁰

1

a_n(t⁰−ξ)^σ/γ^ˆ dξ→0 (17) as t⁰→t⁰⁰ ort⁰⁰→t⁰.

The expression under the last integral in the formula (16) is majorated byL¹-summable function ofξ∈Iτ:

k(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)e^−aⁿ^(t⁰⁰^−ξ)B^γJ B^ˆ^σx(ξ)k_s≤ c₅

a_n(t⁰⁰−ξ)ˆσ/γkx(ξ)k_s. Fixt⁰ and fixξ < t⁰ and letξ < a < t⁰. Let a constanta∈(ξ, t⁰). By Lemma 3.3 the mapping

t⁰⁰7→e^−aⁿ^(t⁰⁰^−ξ)B^γJ B^ˆ^σx(ξ) is a continuous mapping of [a, t⁰] to Xs. By Lemma 3.2

(Aν =e^−aⁿ^νB^γ−I, ν =t⁰−t⁰⁰, X =Y =Xs)

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it follows that

k(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)e^−aⁿ^(t⁰⁰^−ξ)B^γJ B^ˆ^σx(ξ)ks→0 ast⁰⁰→t⁰.

Whent⁰ →t⁰⁰, the asymptotics

k(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)e^−aⁿ^(t⁰⁰^−ξ)B^γJ B^ˆ^σx(ξ)k_s→0, ξ < t⁰⁰ is obtained directly. Thus by the dominated convergence theorem we have

Z τ

0

χ_[0,t⁰⁰_](ξ)(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)e^−aⁿ^(t⁰⁰^−ξ)B^γJ B^σ^ˆx(ξ)dξ _s→0 ast⁰→t⁰⁰ ort⁰⁰→t⁰.

Analogously one has

k(e^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I)e^−aⁿ^t⁰⁰^B^γxˆ_nks→0

ast⁰→t⁰⁰ ort⁰⁰→t⁰.

Lemma 3.7. For any n∈Nand for anys∈Requation (15)has a unique solution xn(t)∈C(IT, Xs).

Proof. After already prepared estimates it is clear that for sufficiently smallτ >0 the mappingFn is a contraction ofC(Iτ, Xs) and thus there is a fixed pointFn[xn] =xn. This is true for alls∈Rbut τ depends ons.

The solution xn can actually be extended to the whole interval IT. Assume the converse: there is a positive constant t_∗ < T such that the solution xn(t) ∈ C([0, t_∗), Xs) but the limit lim_t→t_∗₋xn(t) does not exist inXs. If this limit would exist we could take x(t_∗) as a new initial condition and use again the contraction mapping principle to prolong the solution forward overt_∗.

Observe that e^−tB^σx

_s≤cs,σkxks, t≥0, t∈IT

see for example [10]. Thus taking in estimate (17)t⁰=tandt⁰⁰= 0 and replacingx withx_n, from equation (15) we get

kxn(t)ks≤c4kˆxnks+c3

Z t

0

kxn(ξ)ks

a_n(t−ξ)^σ/γ^ˆ dξ.

Consequently, by Lemma 3.4 the solutionxn(t) is bounded on [t, t∗).

To get the contradiction it is sufficient to observe that ift⁰, t⁰⁰→t_∗−then kxn(t⁰)−xn(t⁰⁰)ks→0.

Sincexn is a fixed point ofFn, the proof repeats the argument of Lemma 3.6.

Lemma 3.8. The function x_n(t) ∈ C¹(I_T, X_s), s ∈ R solves initial value problem (14).

Proof. In this lemma we already know that the function x(ξ) belongs to C(I_T, X_s)

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for alls∈R. One must check that fors∈Rit follows that

h→0lim

x_n(t+h)−x_n(t)

h + −J B^σ^ˆ+anB^γ

Fn[xn](t)

_s= 0, fort >0 and

h→0+lim

xn(h)−xˆn

h −J B^ˆ^σxˆn+anB^γxˆn

_s= 0.

This follows from the same argument which is employed in Lemma 3.2. One must use formula (16) taking oncet⁰ =t+h, t⁰⁰=t >0, h >0 and after this checking the caset⁰⁰=t−h, t⁰ =t >0.

The operator

1 h

e^−aⁿ^hB^γ−I

appears instead of the operatore^−aⁿ^(t⁰^−t⁰⁰^)B^γ−I. To pass to the corresponding limits one must use again the dominated convergence theorem and Lemma 3.2.

For example, let us show that A_h= 1

h

Z t+h

t

e^−aⁿ^(t+h−ξ)B^γJ B^σ^ˆx(ξ)dξ−J B^σ^ˆx(t) _s→0 as h→0+. Observe that sincex(t)∈C(IT, Xs+ˆσ) the function

ξ→

e^−aⁿ^(t+h−ξ)B^γJ B^ˆ^σx(ξ)dξ−J B^σ^ˆx(t) _s

is continuous and therefore by the Mean Value Theorem for integrals we have Ah≤ 1

h Z t+h

t

e^−aⁿ^(t+h−ξ)B^γJ B^σ^ˆx(ξ)−J B^ˆ^σx(t) _sdξ

=

e^−aⁿ^(t+h−η)B^γJ B^σ^ˆx(η)−J B^σ^ˆx(t)

_s, η∈[t, t+h].

We obtain x(η)→x(t) inXs+ˆσ asη →t; thus

J B^ˆ^σx(η)→J B^ˆ^σx(t), e^−aⁿ^(t+h−η)B^γJ B^σ^ˆx(η)→J B^σ^ˆx(t)

in Xs. The last asymptotic goes from Lemma 3.2.

Lemma 3.9. The following inequality holds sup_n∈Nmaxt∈ITkxn(t)kˆs≤c7kxkˆ ˆs. Proof. Due to equation (14) and sinceJ is a skew symmetric operator we obtain

d

dt B^ˆ^σx_n(t), x_n(t)

ˆs−ˆσ/2=−2an B^ˆ^σ+γx_n(t), x_n(t)

ˆs−ˆσ/2≤0.

Hence,

c₈kx_n(t)k_s_ˆ≤ B^ˆ^σx_n(t), x_n(t)

ˆs−ˆσ/2≤ B^ˆ^σx_n(0), x_n(0)

ˆs−ˆσ/2

= B^ˆ^σe^−aⁿ^Bx, eˆ ^−aⁿ^Bxˆ

s−ˆˆ σ/2= B^σ/2^ˆ e^−aⁿ^Bˆx, B^ˆ^σ/2e^−aⁿ^Bxˆ

ˆs−ˆσ/2

≤c9kˆxksˆ.

Corollary 3.10. The following inequality holds

sup

n∈N

maxt∈IT

kx˙n(t)ks−γˆ ≤c10kˆxksˆ.

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Proof. This directly follows from Lemma 3.9 and equation (14).

From Lemma 3.9 it follows that the sequence {x_n(t)}is *-weakly relatively compact inL^∞(IT, Xsˆ). Recall that for any normed spaceX, the strongly closed ball of X⁰ is aσ(X⁰, X)-compact set [10]. Note also thatL^∞(I_T, X_ˆ_s) = L¹(I_T, X_ˆ_s)0

[4].

Lemma 3.9, its Corollary 3.10 and the third Ascoli theorem [8] imply that the sequence{xn(t)}is relatively compact inC_w(I_T, X_ˆ_s−γ). Note also that each bounded inX_ˆ_s−γ set is weakly relatively compact.

Furthermore since the closed ball of Xs−γˆ is weakly compact, it is also weakly complete.

Hence the sequence{xn(t)}contains a subnet{xn_α}with some directed setA 3α such that

limA xn_α =x

inC_w(I_T, X_ˆ_s−γ) and inL^∞(I_T, X_s_ˆ) equipped with *-weak topology; andkxkL^∞(I_T,X_ˆ_s)≤ c₇kˆxk_s_ˆ.

By virtue of equation (14) it remains to pass to the limit in

− Z T

0

( ˙u(t), xn_α(t)

ˆ sdt=

Z T

0

(−B^σ^ˆJ−an_αB^γ)u(t), xn_α(t)

ˆ sdt.

3.3 Proof of the second part of Theorem 2.1

From Lemma 3.9 and Corollary 3.10 by the Third Ascoli theorem [8] it follows that the sequence {xn(t)} is relatively compact in C(IT, Xs−γˆ ). Here we use the hypothesis on compactness of embeddingXsˆ⊂Xˆs−γ.

Thus there is a subnet{xn_α} ⊂ {xn} such that in addition to enumerated above properties it is convergent tox(t) inC(IT, Xˆs−γ) asα∈ A.

In the spaceC(IT, Xˆs−γ−ˆσ) we pass to the limit in (15) and obtain that x(t) is a solution to integral equation

x(t) = ˆx+ Z t

0

J B^σ^ˆx(ξ)dξ. (18)

Differentiating equation (18) int in the spaceXs−γ−ˆˆ σ we obtain

˙

x=J B^σ^ˆx, x˙ ∈C(IT, Xˆs−γ−ˆσ).

Theorem 2.1 is proved.

3.4 Proof of Theorem 2.2

We employ Lemma 3.1. Let τ stand for the set of valuest >0 such thatx(t)∈Xsˆ

and inequality (6) holds. Then putt⁰_i=ai.Observe thatt⁰_i−t⁰_j ∈τ providedi > j.

Lete^{tJ B}^σ^ˆxˆstand for the solution with initial condition ˆx∈Xˆs. Sincee^(t⁰ⁱ^−t⁰^j^{)J B}^σ^ˆxˆ∈ Xsˆwe can write

e^t⁰ⁱ^{J B}^σ^ˆxˆ−e^t⁰^j^{J B}^σ^ˆxˆ=e^t⁰^j^{J B}^ˆ^σ (e^(t⁰ⁱ^−t⁰^j^{)J B}^σ^ˆ−I)ˆx .

(12)

The sequence{x(t⁰_i)}is bounded inXsˆ. Thus it contains a subsequence{x(˜ti)}, {t˜i} ⊂ {t⁰_i} that is convergent inXˆs−γ−3ˆσ/2,

kx(˜ti)−x(˜tj)ks−γ−3ˆˆ σ/2→0 as i, j→ ∞.

By the same argument as in formula (9) we can write c17

e^(˜^tⁱ⁻^˜^t^j^{)J B}^ˆ^σxˆ−xˆ

2

s−γ−3ˆˆ σ/2≤H

e^(˜^tⁱ⁻^˜^t^j^{)J B}^σ^ˆ−I ˆ x

=H

e^˜^t^j^{J B}^σ^ˆ e^(˜^tⁱ⁻^˜^t^j^{)J B}^ˆ^σ−I ˆ x

≤c18kx(˜ti)−x(˜tj)k²_ˆ_s−γ−3ˆ_σ/2. (19) It remains to choose a sequencej =j(i) such thatti= ˜ti−t˜_j(i)→ ∞andj(i)→ ∞ as i→ ∞. Indeed, from inequality (19) it follows that

ke^tⁱ^{J B}^σ^ˆxˆ−xˆ

_ˆ_s−γ−3ˆ_σ/2→0.

Theorem 2.2 is proved.

Acknowledgement. The author wishes to thank Professor D. V. Treschev for useful discussions.

Partially supported by grants RFBR 12-01-00441., Science Sch.-2964.2014.1 References

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(received 05.12.2016; in revised form 07.01.2017; available online 05.04.2017)

Deptartment of Theoretical mechanics, Mechanics and Mathematics Faculty, M. V.

Lomonosov Moscow State University, Russia, 119899, Moscow, Vorob’evy gory, MGU E-mail: [email protected]