ON THE STABILITY OF STATIONARY SOLUTIONS OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION

In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation ϵx″(t)


Introduction
Let (B II" ]]) be a complex Banach space, 0 the zero element in B, and (B) the Banach space of bounded linear operators on B with the operator norm, denoted also by the symbol I1" I1" For a B-valued function, continuity and differentiability refer to continuity and differentiability in the B-norm.For an (B)-valued function, con- tinuity is the continuity in the operator norm.For operator A, the sets (r(A) and p(A) are its spectrum and resolvent set, respectively.140 A. YA.DOROGOVTSEV and O. YU.TROFIMCHUK In the following, we will consider random element son the same complete probability space (fl,,P).The uniqueness of a random process that satisfies an equation, is its uniqueness up to stochastic equivalence.We consider only B-valued random functions which are continuous with a probability of one.All equalities with random elements in this article are always equalities with a probability one.For a given equation, we consider only solutions which are measurable with respect to the right-hand side random process.
It is well known that the stationary solutions of difference and differential equations are steady with respect to various perturbations of the right-hand side and perturbation of coefficients.For example, see [5].In the present work, it is shown that stability has a place with respect to perturbations such as degeneracy of the equation.
In the first part of this paper, we consider the following equation ex"(t) + x'(t) Ax(t) + / E(t-s)x(s)ds + (t), t e R (1) containing a parameter in B.Here A E (B) is fixed operator, is a stationary process in B and E E C([0, + cx), (B)) is a function satisfying the condition We suppose that the following condition o'(A) g iR O holds.Under condition (2) the function

t<O; t>O
Here P_ and P+ are Riesz spectral projectors corresponding to the spectral sets o'(A) N {z Rez < 0} and o'(A) V {z Rez > 0}, respectively.Let S be the class of all stationary B-valued processes {(t): t R} which possess continuous derivatives of all orders on R with a probability one and such that, for some numbers L L > 0, C C > 0, 5 > 0, the following inequalities Vn>_O:E{ sup lib(n)(s) ll)_LCn O<s<5 hold.The notations S(L, C, ) and 5' will be used.Then we have the follow- ing result.
On the Stability of Stationary Solutions 141 Theorem 1: Let A (B) be an operator satisfying (2).Suppose that S and ab < 1.Then there exists e o > 0 such that for every e with el < eo, the equation (1)   has a stationary solution x e S, which for every bounded subset J of R, satisfies where Yo is a unique stationary solution of the equation x'(t) Ax(t) + / E(t-s)x(s)ds + (t), t R. --00 (3) The process x is a unique solution of (1) in the class of all stationary connected processes in S.
This theorem is proved in Section 2. The method of proof uses a modification of the proof of Theorem 1 in [7] about the stability of stationary solutions for equation (1) with E 0. Remark 1: The asymptotic expansion for a stationary solution of (1) is obtained.Remark 2: The assumption (2) is equivalent to the existence of a unique stationary solution {x(t) lt R} with E I I x(0)[I < + c of the equation x'(t) Ax(t) + (t), t R for every stationary process {(t)[t E R) with Eli (0)I] < +oc, see [3, pp. 201- Remark 3: The general approach to the analysis of the Cauchy problem for deter- ministic differential equations containing a small parameter leads to the appearance of boundary layer summands in the asymptotic expansion of solution [10].These summands are absent in the asymptotic expansion of the stationary solution in the considered problem.
Remark 4: The problem of the existence of stationary solutions for difference and differential stochastic equations has been investigated by many authors.See, for example, monograph [1], surveys [2,4] and article [6].
Corollary 1: Let A e (B) be an operator satisfying (1).Suppose that e S.
Then there exists o > 0 such that for every e with < Co, the equation tx"(t) + x'(t) Ax(t) + (t), t e R (4) has a unique stationary solution xe G S, which, for every bounded subset J of satisfies I I o( )II Fo, where x o is a unique stationary solution of the equation x'(t) Ax(t) + (t), t R.
The second part of this paper deals with the asymptotic expansion of the station- ary with respect to time solution of a boundary value problem containing a small parameter.The following definition is necessary.Definition 1: A B-valued random function u defined on Q: R x [0, 7r] is time- stationary if Vt E RVn NV{(tl, x 1),.. ., (in, xn)} C QV{D,..., On} C %(B): where %(B) is the Borel r-algebra of B.
Theorem 2: Let A L(B) be an operator satisfying the following condition {k 2 + ia k e N, a R} C p(a). (5) Suppose that g C3o and E S with a number 8>0 and ab < l.Then there exists e 0 > 0 such that for every e with [e < eo, the boundary value problem ,'t(t, ; ) + u(t, ; ) '(, ; ) Au(t, z; e) + g(x)(t), t R, z [0, ,(t, o; ) ,(t, ; ) o ,t n has a unique time-stationary solution u(., .; e) with (6) E ( sup O<s<,o<< which, for every t R, satisfies E ( sup <_s< +8,0<x < r where v is the unique time-stationary solution of the following boundary value problem for a heat equation This theorem is proved in Section 3. Remark 5: Condition (5) is a necessary and sufficient condition of the existence of a time-stationary solution for boundary value problem (7) [8].
Remark 6: Note that, if e > 0, problem (6) is a boundary value problem for a hy- perbolic equation and that, if 0, we have a boundary value problem for a para- bolic equation.
Pemark 7: The study of the asymptotic behavior of a solution u(., .;e) of the telegraph equation from (6) as ---,0 + has also physical sense [9].In order to prove Theorem 1, a few lemmas will be needed.
Lemma 1: Let A E (B) be an operator satisfying (2).Suppose that S.
the equation x'(t)-Ax(t) + (t), t R has a unique stationary solution x S, which can be presented in the form (i) A stationary process x S is a unique stationary solution of the equation ( 3).(ii) A stationary process x S is a unique stationary solution of the equation 8 t e (S) Proof: The result is a consequence of Lemma 1.
Lemma 3: Let A (B) be an operator satisfying (2) and ab < 1. is a stationary process in B, which, for some 5 > O, satisfies Suppose that o_<t_< Then the equation ( 8) has a unique stationary solution x, which satisfies E( sup I[x(t) ll' < +cx:" (9) o_<t_<, Proof: Let S O be the class of all stationary connected B-valued processes x which are stationary connected with and, for given 5 > 0, satisfy (9).Let us introduce the operator (Tx)(t)" / G(t s) / E(s u)x(u)duds / / G(t s)(s)ds, t R.
Then Tx S O and E ( sup o_<t_< I I (Tx)(t)-(Ty)(t) I I -abE sup ] \ o<t< therefore T is a continuous operator on S 0. Set o(t) / a(t- tER, then x 0S0and su.
x(t)" XO(t -}-[Xl(t x0(t)] +...-}-[Xn(t Xn-l(t)] +"" converges with a probability one for every t R and this convergence is uniform over the bounded subset of R with a probability one.By continuity of T we have x Tx.
The solution x of (8) is unique.Lamina 4: Let A G (B) be an operator satisfying (2) and ab < 1. Suppose that is a stationary process in B, which, for some 5 > O, satisfies E [ sup \ 0_<t_< J Then equation (3) has a unique stationary solution x, which satisfies (9).
Proof: The result is an immediate consequence of Lemma 2 and Lemma 3.
Proof: We return to the proof of Lemma 3 where the stationary solution x for equation (3) was given.From the inclusion S(L, C, 5) and representation for every k k 0 and x o S(bL, C, 5).For the process x I --:gO' we have X l(t) Xo(t /G(it)/ E(v)xo(t-it-v)ditdv, t e i. (xn xn 1) E S(b(ab)nL, C, 5), n > 1.
Lemma 5 is proved.
Proof of Theorem 1: Let S(L,C, 6).We shall construct the asymptotic expansion for a solution of (1) in the following way.From Lemma 5, equation ( 3) has a unique stationary solution Yo S(bcL, C, 5).Note that y' S(bcLC2, C, 5).
Let Yl be a unique stationary solution for equation yi(t) AYl(t + / E(t-s)Yl(S)ds-yg(t), t e R.This solution exists from Lemma 5 and Yl S(b2c2LC2, C, 5).
If the processes Yo, Yl,'",Yn-1 for n > 1 are already constructed we will define process Yn as a unique stationary solution of the equation which satisfies y'n(t) Ayn(t + j E(t-s)yn(s)ds y'_ (t), t e R, Yn S( bn + lcn + 1LC2n, C, 5).
It is clear that the processes Yn, n > 0 are stationary connected [3].
for every t e R and 1 < 0: (1 -ab)/(2bC2), the series for y converges uniformly on bounded subsets of R with a probability one.This shows that y is continuous on R with a probability one stationary process.
By exactly the same arguments as those used above, we claim that the series for Ye are also absolutely and uniform convergent on bounded subsets of R with probability one and we have y(t) -t-Ye(t) E (n -t-ly,r(t _ _ Aye(t + f E(t-s)ye(s)ds-AYo(t f E(I-s)Yo(s)ds + y'o(t) Aye(t)+ i E(t-s)ye(s)ds + (t), t E R.
To complete the proof of Theorem 1 we need show only the uniqueness.It is sufficient to prove the following fact.If z is stationary connected with the process ac solution of (1), which satisfies E ( suo IIz(t) ll< +oo, E ( su.IIz'(t) ll< + t,o<_t<_ ] \o<_t<_ ] then z-x e.We apply Lemma 4 in the following way.The difference u: x -z is a stationary process which satisfies the equation and eu"(t) + u'(t) Ax(t) + i E(t-s)u(s)ds, t e R (ii) has a unique stationary solution vk(.c) such that (, sup I I t)II where v k is a unique stationary solution of the equation V'k(t + k2vk(t) Avk(t + gk(t), t e R, and J is a bounded subset of R.Moreover, for every t E R, we have and E (t<_s<_t-l-hsup I I Vk(8;C)]1)<--21gk iil,k E ( sup t<s<t+8 I I Vk(;)-Vk() I I ) 21g LL,C= I, ( if I1 _< k, where Ll,k: / I I Gk(S) I I ds < -q-cx3 R and G k is Green's function for operator A-k2I; k > 1.It follows from the properties of Gk that L L1, k -<k 2 k20 k > ko, where a number L can be chosen to be independent of k.
Now we shall remark, that by virtue of boundedness of an operator A, the numbers ok, k > 1 are identifiable and not depending on k.Really, let k 0 be the least natural number such that a spectrum of an operator A-(c2c-k)I resides in the left half-plane.Then the spectrum of an operator A-(a2c-k2)I, k > k 0 also resides in the left half-plane and it is possible to put %: min{Q,c2,...,%0} > 0. Thus, for every c, c < Co, all equations (13) have a unique stationary solution.R and [c _< c 0. This implies that the series (17) converges absolutely and uniformly on [t,t + 6] x [0, Tr] with the probability one and the random function u(.,.;c) is a continuous, time-stationary with respect of time variable, random functions.In addition, sup I I u(, ; )I1% < + .
Hence, the random function u(., .;c)for c with I 1< is a time-stationary solution of (6).
This solution is unique.To see this, we observe that for any t R, the elements {vk(t; e)} are Fourier coefficients of u(t, .; ) e C2([0, 7r], B) which determine u(t, .; ) uniquely with the probability one.See, for example [3] for details.By Corollary 1, the solutions of (13) are also determined uniquely with a probability one.
Similarly, by repeating the above arguments, we conclude that random function v(t,x)" E vk(t)sinkx' (t,x) e Q k=l is a unique, stationary with respect to time variable, solution of (7) and E ( sup t<s<t+5,0<x<Tr for every t R. Note that the random functions u(., .;c),cl _< c 0 and v are time- stationary connected.

P
{w: u(w; t k + t, Xk) e Dk} P {w" u(w; tk, xk) e Dk} k=l k=l On the Stability of Stationary Solutions 143 2. Asymptotic Expansion of the Stationary Solution of Equation (1) every k > O, we have R On the Stability of Stationary Solutions 145 and (x 1 -x0) E S(ab2L, C, 6).By induction, we find This is the corollary of Theorem 1 in[3, pp.201-202].Lemma 2: Let A (B) be an operator satisfying (2).Suppose that