S.S. Mirzoev. Baku State University 23 Z. Khalilov Str., Baku AZ1148, Azerbaijan A.R. Aliev

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1 Jurnal f Mathematical Physics, Analysis, Gemetry 03, vl. 9, N., pp. 076 On the Bundary Value Prblem with the Operatr in Bundary Cnditins fr the Operatr-Dierential Equatin f Secnd Order with Discntinuus Cecients S.S. Mirzev Baku State University 3 Z. Khalilv Str., Baku AZ48, Azerbaijan mirzyevsabir@mail.ru A.R. Aliev Baku State University 3 Z. Khalilv Str., Baku AZ48, Azerbaijan Institute f Mathematics and Mechanics f NAS f Azerbaijan 9 F. Agayev Str., Baku AZ4, Azerbaijan alievaraz@yah.cm L.A. Rustamva Institute f Applied Mathematics, Baku State University 3 Z. Khalilv Str., Baku AZ48, Azerbaijan lamia_rus@rambler.ru Received January 5, 0 Sucient cnditins fr regular slvability f the bundary value prblem fr an elliptic peratr-dierential equatin f secnd rder cnsidered n the psitive semi-axis are btained. Nte that the principal part f the equatin cntains a discntinuus cecient, and the bundary cnditin invlves a linear peratr. Key wrds: Hilbert space, peratr-dierential equatin, discntinuus cecient, regular slvability, Furier transfrmatin, intermediate derivatives. Mathematics Subject Classicatin 00: 34B40, 35J5, 47D03. This wrk was supprted by the Science Develpment Fundatin under the President f the Republic f Azerbaijan - Grant N. EIF-0-(3-8/8/. c S.S. Mirzev, A.R. Aliev, and L.A. Rustamva, 03

2 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva Dedicated t the 80th birthday f Prfessr F.S. Rfe-Beketv. Intrductin Let A be a self-adjint psitive denite peratr in a separable Hilbert space H. It is knwn that the dmain f the peratr A θ (θ 0 becmes a Hilbert space H θ with respect t the scalar prduct (x, y θ = (A θ x, A θ y, x, y Dm(A θ. Fr θ = 0, we cnsider that H 0 = H. We dente by L ((a, b; H, a < b +, the Hilbert space f all vectr functins dened n (a, b, with the values in H, which have the nite nrm f L ((a,b;h = b a f(t H dt (see []. Further, we dente by L(X, Y a set f the linear bunded peratrs acting frm the Hilbert space X t anther Hilbert space Y. Fr Y = X, we cnsider L(X, X = L(X. We als dente the spectrum f the peratr ( by σ(. We intrduce the linear space W ((a, b; H = { u(t : A u(t L ((a, b; H, u (t L ((a, b; H } with the nrm u W ((a,b;h = ( A u L ((a,b;h + u L ((a,b;h The lineal becmes a Hilbert space []. Here and further the derivatives are understd in the sense f distributin thery. Fr a =, b = +, we will assume that L ((, + ; H L (R; H, W ((, + ; H W (R; H, R = (, +. Fr a = 0, b = +, we will suppse that L ((0, + ; H L (R + ; H, W ((0, + ; H W (R + ; H, R + = (0, +. Besides the spaces intrduced, we will use the fllwing subspaces: W (R + ; H = { u(t : u(t W (R + ; H, u(0 = u (0 = 0 }, { W,T (R + ; H = u(t : u(t W (R + ; H, u(0 = T u (0, T L(H. }, H Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

3 On the Bundary Value Prblem with the Operatr in Bundary Cnditins Nw we pass t the statement f the bundary value prblem studied. We cnsider the peratr dierential equatin f the frm u (t + ρ(ta u(t + A u (t = f(t, t R +, ( satisfying the bundary cnditins at zer u(0 = T u (0, ( where f(t, u(t are the functins with values in H, T L(H, H 3, A is a linear unbunded peratr, A is a self-adjint psitive denite peratr in H, ρ(t = α if t (0,, and ρ(t = β if t (, +, and α, β are psitive unequal numbers. Fr deniteness, we suppse that α β. Denitin. Prblem (, ( is called regularly slvable if fr every functin f(t L (R + ; H there exists a functin u(t W (R +; H satisfying equatin ( almst everywhere in R +, bundary cnditin ( hlds in the sense f cnvergence f the space H 3, i.e., and the estimate takes place. lim t 0 u(t T u (t H 3 = 0, u W (R + ;H cnst f L (R + ;H A review article f A.A. Shkalikv [3] cntains a detailed analysis f the results f bth the authr himself and ther authrs btained n the prblems f slvability and spectral prblems, mainly the bundary value prblems fr peratr dierential equatins when the cecients in the bundary cnditins are nly cmplex numbers. Amng the results, we especially mark ut the papers f M.G. Gasymv [46] and his fllwers. Nte that these bundary value prblems d nt lse their actuality (see, fr example, the recent papers f S.S. Mirzev and M.Yu. Salimv [7], A.R. Aliev and S.S. Mirzev [8], A.R. Aliev [9]. Nevertheless, there are rather few wrks n slvability and the studying f the spectrum, the cmpleteness f rt vectrs and elementary slutins f bundary value prblems fr peratr-dierential equatins when the cecients f the bundary cnditins are peratrs. The rst wrks n this subject were the papers f F.S. Rfe-Beketv [0], V.A. Ilyin and A.F. Filippv [], M.L.Grbachuk [], S.Y. Yakubv and B.A. Aliev [3]. Later in this directin there appeared an interesting paper f M.G. Gasymv and S.S. Mirzev [4], in which bth the prblems f slvability and sme spectral aspects f the bundary value prblems fr elliptic type peratr dierential equatins f the secnd rder cnsidered n the semiaxis were Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 09

4 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva studied by using riginal methds. This wrk fund its prper develpment in the papers f S.S. Mirzev and his fllwers (see [59]. The present paper aims t btain apprpriate slvability results f the paper by M.G. Gasymv and S.S. Mirzev [4] fr the case when the principal part f the equatin cntains discntinuus (piecewise cnstant cecient. Such prblems are f interest nt nly because they cntain apprpriate bundary value prblems, in the bundary cnditins in which the cecients are cmplex numbers, but als because they can be applied t a wider range f the prblems fr partial dierential equatins and a number f prblems in mechanics, in particular, nn-standard prblems in the thery f elasticity f multilayered bdies. Fr simplicity, a pint f discntinuity is taken. Here we nte that a regular slvability f the bundary value prblems fr peratr dierential equatins f the secnd rder with discntinuus cecients, when the cecients in the bundary cnditin are nly cmplex numbers, is studied in paper [0] and develped in [].. Main Results We begin with cnsidering the prblem u (t + ρ(ta u(t = f(t, t R +, (3 u(0 = T u (0, (4 where f(t L (R + ; H, u(t W (R +; H. As can be seen, equatin (3 is btained frm ( at A = 0. The fllwing therem is true. Therem. Let the peratr T α,β = E + αt A + β α α + β ( αt A E e αa have a bunded inverse peratr in H 3, where E is an identity peratr in H and e ta is a semigrup f bunded linear peratrs generated by the peratr A. Then the peratr P 0, acting frm the space W,T (R +; H t the space L (R + ; H, P 0 u(t u (t + ρ(ta u(t, u(t induces an ismrphism between these spaces. W,T (R + ; H, P r f. We will shw that the equatin P 0 u(t = 0 has nly the trivial slutin in the space W,T (R +; H. Indeed, the general slutin f the equatin 0 Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

5 On the Bundary Value Prblem with the Operatr in Bundary Cnditins P 0 u(t = 0 frm the space W (R +; H has the frm u 0 (t = { where the vectrs ϕ, ϕ, ϕ 3 H 3 have v (t = e αta ϕ + e α( ta ϕ, t (0,, v (t = e β(t A ϕ 3, t (, +,. Frm the cnditin u 0 (t v (0 = T v (0, v ( = v (, v ( = v (. Thus fr the vectrs ϕ, ϕ, ϕ 3 we get the system f equatins ϕ + e αa ϕ = T ( αaϕ + αae αa ϕ e αa ϕ + ϕ = ϕ 3, αae αa ϕ + αaϕ = βaϕ 3. Frm this system, in turn, it implies that ϕ + Cnsequently, ϕ 3 = α αa α β e ϕ β ϕ, ϕ = α β α + β e αa ϕ, W,T (R +; H we α β e αa ϕ + α β αt Aϕ αt Ae αa ϕ = 0. α + β α + β T α,β ϕ ( E + β α ( αt A + αt A E e αa ϕ = 0. α + β Since by the cnditin f the therem the peratr T α,β has a bunded inverse peratr in H 3, then frm the last equatin it fllws that ϕ = 0. Cnsequently, ϕ = ϕ 3 = 0, i.e., u 0 (t = 0. Frm the cnditin f the therem it is clear that at any f(t L (R + ; H the equatin P 0 u(t = f(t has a slutin frm the space W,T (R +; H and this slutin has the representatin where u(t = { u (t, t (0,, u (t, t (, +, u (t = e α t s A A f(sds + T α,β α ( αt A Ee α(t+a 0, Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

6 α β α ( α + β S.S. Mirzev, A.R. Aliev, and L.A. Rustamva 0 e α( sa A f(sds + α 0 e α(s A A f(sds + + e β(s A A f(sds + T α + β α,β (E + αt Ae α( ta α β α ( α + β +T 0 + e α( sa A f(sds + e β(s A A f(sds α + β α,β ( αt A Ee α( ta α β α ( α + β u (t = β +T e αsa A f(sds, e β t s A A f(sds + T α,β (E + αt A e β(t A α + β β α e α( sa A f(sds + β + e β(s A A f(sds α,β ( αt A Ee β(t A e αa e α(s A A f(sds α + β + β + e β(s A A f(sds. We nte that the fulllment f bundary cnditin (4 is veried directly. In additin, fr any u(t W,T (R +; H we have 0 P 0 u L (R + ;H = u + ρa u L (R + ;H ( u L (R + ;H + max(α ; β A u L (R + ;H max(; α ; β u W (R +;H, i.e., the peratr P 0 : W,T (R +; H L (R + ; H is bunded. Then the assertin f the therem fllws frm the Banach therem n the inverse peratr. The therem is prved. Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

7 On the Bundary Value Prblem with the Operatr in Bundary Cnditins Frm Therem it implies that the nrms P 0 u L (R + ;H and u W (R + ;H W,T (R +; H. are equivalent in the space The fllwing therem is true. Therem. Suppse that the peratr T α,β has a bunded inverse peratr in H 3, and the peratr B = A A is bunded in H, mrever, B < N T, where N T = sup 0 u(t W,T (R +;H Then prblem (, ( is regularly slvable. Au L (R + ;H P 0u L (R + ;H. P r f. Denting by P the peratr acting frm the space t the space L (R + ; H in the fllwing way: P u(t A u (t, u(t W,T (R + ; H, we can rewrite prblem (, ( in the frm f the peratr equatin P 0 u(t + P u(t = f(t, W,T (R +; H where f(t L (R + ; H, u(t W,T (R +; H. Since by Therem the peratr P 0 has a bunded inverse P0 acting frm the space L (R + ; H t the space W,T (R +; H, then after substitutin u(t = P0 v(t we btain the equatin (E + P P 0 v(t = f(t in the space L (R + ; H. And due t the fact that fr any v(t L (R + ; H P P 0 v L (R + ;H = P u L (R + ;H B Au L (R + ;H N T B P 0 u L (R + ;H = N T B v L (R + ;H, then fr N T B < the peratr E +P P0 is invertible in the space L (R + ; H and thus u(t = P 0 (E + P P 0 f(t, u W (R + ;H cnst f L (R + ;H. The therem is prved. We nte that the prblem f estimating the number N T arises here. Fr this purpse we will use the idea f the prcedure ered in []. Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 3

8 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva First we prve the fllwing statement. Lemma. Fr any u(t W,T (R +; H there takes place the inequality P 0 u L (R + ;H α β ( u L (R + ;H + αβ A u L (R + ;H + β Au L (R + ;H + βre(at x, x where x = u (0 H. P r f. Multiplying bth sides f equatin (3 by ρ (t, we get ρ ρ f = P0 u ρ = u + ρ A u L (R + ;H L (R + ;H, L (R + ;H = ρ u ρ + A u Re ( u, A u L (R + ;H L (R + ;H L (R + ;H. Nw, integrating by parts, we have Re ( u, A u L (R + ;H = Re ( A u (0, A 3 u(0 + Au L (R + ;H Then ρ P0 u On the ther hand, fr u(t = Re (AT x, x + Au L (R + ;H. ρ = u ρ + A u L (R + ;H L (R + ;H + Au L (R + ;H + Re (AT x, x W,T (R +; H we have P 0 u ρ L (R + ;H α P0 u Taking int accunt equality (5 in (6, we btain ( ρ P 0 u L (R + ;H α u ρ + A u L (R + ;H L (R + ;H. (5 L (R + ;H. (6 L (R + ;H + Au L (R + ;H + Re(AT x, x ( α u β L (R + ;H + α A u L (R + ;H + Au L (R + ;H + Re(AT x, x = α β S(u, 4 Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

9 On the Bundary Value Prblem with the Operatr in Bundary Cnditins where S(u = u L (R + ;H + αβ A u L (R + ;H + β Au L (R + ;H + βre(at x, x. (7 Lemma is prved. Fr further peratins we factrize the cnsidered in the space H 4 plynmial peratr pencil f the frm P (λ; γ; A = λ 4 E + αβa 4 βλ A + γλ A. Lemma. Fr γ [0, β (α + β the peratr pencil P (λ; γ; A is invertible n an imaginary axis and is represented in the frm where P (λ; γ; A = F (λ; γ; AF ( λ; γ; A, F (λ; γ; A = (λe ω (γa(λe ω (γa λ E + a (γλa + a (γa, (8 and Reω k (γ < 0, k =,, a (γ = β (α + β γ, a (γ = α β. P r f. Let γ [0, β (α + β. Then fr σ σ(a and λ = iξ, ξ R, we have Since P (iξ; γ; σ = ξ 4 + αβσ 4 + βξ σ γξ σ ( ( ξ = σ 4 4 ξ + αβ + β σ4 σ γ ξ ξ σ = σ 4 4 ξ + αβ + β σ4 σ ( ξ ( σ ξ γ σ 4 4 ξ + αβ + β ξ 4 + αβ + β ξ σ4 σ σ 4 σ sup ξ σ 0 γ sup ξ σ 0 ξ σ ξ 4 σ 4 + αβ + β ξ σ = ξ σ ξ 4 σ 4 + αβ + β ξ σ. β (α + β, it fllws that P (iξ; γ; σ > 0 fr γ [0, β (α + β. Cnsequently, the plynmial P (iξ; γ; σ has n rts n the imaginary axis fr γ [0, β (α + β. Therefre, this plynmial has tw rts ω (γσ and ω (γσ frm the left Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 5

10 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva half-plane and tw rts ω (γσ and ω (γσ frm the right half-plane, i.e., Reω k (γ < 0, k =,, and ω (γ = ω (γ. Thus, where Since Reω k (γ < 0, k =,, then P (λ; γ; σ = F (λ; γ; σf ( λ; γ; σ, (9 F (λ; γ; σ = (λ ω (γσ(λ ω (γσ = λ + a (γλσ + a (γσ. a (γ = (ω (γ + ω (γ = (ω (γ + ω (γ = Reω (γ > 0. And as a (γ = αβ and a (γ = ω (γω (γ = ω (γω (γ = ω (γ > 0, we btain that a (γ = α β. Further, frm equality (9 it fllws that a (γ a (γ = γ + β, i.e., a (γ = a (γ + β γ = α β + β γ > 0. Cnsequently, a (γ = β (α + β γ. Nw, using the spectral decmpsitin f the peratr A, frm equality (9 we btain the assertin f the lemma. Lemma is prved. Nw we prve the lemma which plays an imprtant rle in ur investigatin. Lemma 3. Fr any u(t fllwing identity hlds: S(u γ Au L (R + ;H = R(γx, x + F where F (λ; γ; A is dened frm equality (8, and W,T (R +; H and γ [0, β (α R(γx, x = β (α + β Re(AT x, x + ( d dt ; γ; A u L (R + ;H β (α + β γ + β the, (0 ( x H + α β AT x H. ( P r f. First we dente by D T (R + ; H the lineal f all innitely dierentiable functins with values in H, having cmpact supprts n R + and satisfying bundary cnditin ( at zer. By density and trace therems [], this lineal is dense everywhere in W,T (R +; H. Cnsequently, it is enugh t prve (0 fr the functins frm D T (R + ; H. It is bvius that fr u(t D T (R + ; H and γ [0, β (α + β the fllwing equality hlds: ( d F dt ; γ; A u L (R + ;H = u + a (γau + a (γa u L (R + ;H = u L (R + ;H 6 Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

11 On the Bundary Value Prblem with the Operatr in Bundary Cnditins +a (γ Au L (R + ;H + a (γ A u L (R + ;H + a (γre ( u, Au L (R + ;H +a (γre ( u, A u L (R + ;H + a (γa (γre ( Au, A u L (R + ;H. ( On the ther hand, applying integratin by parts, we btain the equalities Re(u, A u L (R + ;H = Au L (R + ;H Re(AT x, x, Re(u, Au L (R + ;H = x H, Re(Au, A u L (R + ;H = AT x H. Taking int accunt equalities frm ( and the values a (γ and a (γ frm Lemma, we have ( d F dt ; γ; A u L (R + ;H = u L (R + ;H + αβ A u L (R + ;H +(β (α +β γ Au L (R + ;H α β Au L (R + ;H α β Re(AT x, x β (α + β γ x H α β β (α + β γ AT x H = S(u γ Au L (R + R(γx, x, (3 ;H where S(u and R(γx, x are dened frm (7 and (, respectively. The validity f (0 fllws frm (3. Lemma 3 is prved. Obviusly that S(u is the nrm in the space W (R +; H which is equivalent t the initial nrm u W (R + ;H. The studies abve allw us t assert that the fllwing therem is true. Therem 3. The number S 0, dened as S 0 = sup 0 u W (R +;H is nite and S 0 =. β (α +β Au L (R + ;H S (u, P r f. Clearly, fr u(t W (R +; H, fr any γ [0, β (α + β frm equality (0 we btain S(u γ Au L (R + ;H = F ( d dt ; γ; A u L (R + ;H. (4 Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 7

12 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva Passing in (4 t the limit as γ β (α + β, we have i.e., S(u β (α + β Au L (R + ;H, Au L (R + ;H S (u. β (α + β Cnsequently, the number S 0. We shw that here the equality β (α +β. Fr this purpse, fr any ε > 0 it is sucient t hlds, i.e., S 0 = β (α +β cnstruct a vectr functin u ε (t W (R +; H such that E(u ε S(u ε (β (α + β + ε Au L (R + ;H < 0. Let ψ H 4, ψ =, and g(t be a scalar functin having a twice cntinuus derivative in R, and g(t, g (t L (R. Then, using Plancherel's therem n the Furier transfrmatin, we btain = where + E(g(tψ S(g(tψ (β (α + β + ε Ag (tψ L (R;H ([ξ 4 E + αβa 4 + βξ A (β (α + β + εξ A ]ψ, ψ ĝ(ξ dξ = + q ε (ξ, ψ ĝ(ξ dξ ( ψ =, (5 q ε (ξ, ψ = ξ 4 + αβ A ψ (α β + εξ Aψ. ( α β +ε Aψ, the It is bvius that fr xed ψ, at the pints ξ = ± functin q ε (ξ, ψ takes its minimum value which is equal t h ε (ψ = αβ A ψ (α ε β + Aψ 4. Since ψ H 4 is an arbitrary vectr ( ψ =, it can be chsen. Namely, if the peratr A has an eigenvectr, then we may chse this vectr as ψ. Indeed, in this ( case Aψ = µψ, and h ε (ψ = αβµ 4 α β + ε µ 4 < 0. If µ is a cntinuus spectrum, then it is pssible t nd a vectr ψ H 4 such that A k ψ = µ k ψ + (δ 8 Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

13 On the Bundary Value Prblem with the Operatr in Bundary Cnditins ( at δ 0, k =,, 3, 4. Then h ε (ψ = αβµ 4 α β + ε µ 4 + (δ < 0 fr suciently small δ > 0. Thus, it is always pssible t nd a vectr ψ H 4 q ε (ξ, ψ < 0. After chsing the vectr ψ, by using the ( ψ = such that min ξ cntinuity f the functin q ε (ξ, ψ at ξ, there exists an interval (η (ε, η (ε in which q ε (ξ, ψ < 0. Nw we cnstruct the functin g(t. Let ĝ(ξ be an arbitrary twice cntinuusly dierentiable functin in R whse supprt is cntained in the interval (η (ε, η (ε. Then frm equality (5 we get E(g(tψ = η (ε q ε (ξ, ψ ĝ(ξ dξ < 0, η (ε and g(t = η(ε ĝ(ξe iξt dξ. π η (ε The therem is prved. R e m a r k. We nte that, in general, S(u is nt a psitive number fr all u(t W,T (R +; H. Indeed, let u(t = e λ0t ϕ 0, and ϕ 0 be sme eigenvectr f the peratr A crrespnding t λ 0, ϕ 0 =. Then, S(u = u L (R + ;H + αβ A u L (R + ;H βre ( u, A u L (R + ;H = λ 0e λ0t λ ϕ 0 + αβ 0 e λ0t ϕ 0 L (R + ;H βre (λ 0e λ0t ϕ 0, λ 0e λ0t ϕ 0 L (R + ;H L (R + ;H = ( + αβλ 4 0 e λ0t ϕ 0 L (R + ;H βλ4 0 e λ0t ϕ 0 L (R + ;H = ( + αβ βλ 4 0 e λ0t ϕ 0 = ( + αβ L (R + ;H βλ3 0. Obviusly, we can require T = A, i.e., the cnditin T u (0 = u(0 t be satised. Then it is clear that fr β αβ > the expressin S(u is negative, and fr β αβ = S(u = 0, u(t W,T (R +; H. Thus, fr S(u in the space W,T (R +; H t be equivalent t the initial nrm u W (R + ;H, additinal cnditins shuld be impsed n the peratr T. Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 9

14 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva Lemma 4. Let x H. If min R(0x, x > 0, then x H = fr any u(t W,T (R +; H. S(u cnst u W (R +;H P r f. Frm (0, fr γ = 0 we get Since frm the cnditin f the lemma that fr u W (R + ;H = the inequality hlds. Since S(u = u W (R +;H S ( S(u R(0x, x. min R(0x, x = c > 0, it is bvius x H = S(u c u u W (R + ;H, then fr all u(t W,T (R +; H is valid. The lemma is prved. Frm Lemma 4 it fllws that c S T = S(u c u W (R +;H sup 0 u W,T (R +;H sup 0 u(t W,T (R +;H Au L (R + ;H S (u Au L (R + ;H u W (R +;H = d, and d < by the therem n intermediate derivatives []. Cnsequently, S T <. Since W,T (R +; H W (R +; H, then S T S 0 =. β (α +β Therem 4. Let the cnditins f Lemma 4 be satised and ReAT 0 in H. Then S T =. β (α + β 0 Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

15 On the Bundary Value Prblem with the Operatr in Bundary Cnditins P r f. If the cnditins f the therem are satised, then R(γx, x > 0 fr any γ [0, β (α + β. Cnsequently, by Lemma 3 it fllws that fr any u(t W,T (R +; H and γ [0, β (α + β the inequality S(u γ Au L (R + ;H is true. Passing in the last inequality t the limit as γ β (α + β, we have S T. β (α + β Thereby, S T =. β (α + β The therem is prved. Basing n the btained results, estimating the nrm f the intermediate derivative peratr A d dt : W,T (R +; H L (R + ; H with respect t the nrm P 0 u L (R + ;H and taking int accunt Lemma, we give the exact frmulatin f the cnditinal Therem in the fllwing frm. Therem 5. Suppse that the cnditins f Lemma 4 are satised, ReAT 0 in H and the peratr B = A A is bunded in H, mrever, ( B < α + α. Then prblem (, ( is regularly slvable. β R e m a r k. In Therem 5, the cnditin ReAT 0 in H. invertibility f the peratr T α,β in the space H 3 prvides Nw we will specify the value f S T under the cnditin min x H = Re(AT x, x The fllwing therem hlds. Therem 6. Suppse that Lemma 4 are satised. Then S T = min x H = Re(AT x, x β (α + β β (α + β min x H = <0. < 0 and the cnditins f Re(AT x, x + α β AT x H. Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

16 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva P r f. First we nte that since the pencil F (λ; γ; A fr γ [0, β (α + β is represented in the frm F (λ; γ; A = (λe ω (γa(λe ω (γa, where Reω k (γ < 0, k =,, then the Cauchy prblem ( d F dt ; γ; A u = 0, (6 u(0 = T x, u (0 = x, x H, (7 has the unique slutin u(t; γ; x W (R +; H represented in the frm u(t; γ; x [ ] = e ω(γta (ω (γt x A x + e ω(γta (A x ω (γt x. ω (γ ω (γ It fllws easily that u(t; γ; x W (R + ;H d (γ x H. Further, using the uniqueness f the slutin f Cauchy prblem (6, (7 and taking int accunt the Banach therem n the inverse peratr, we have u(t; γ; x W (R + ;H d (γ x H, d (γ > 0. (8 Nw we nte that frm Therem 4 it fllws that S T >. β (α + β Cnsequently, S T (0, β (α + β. Then, in equality (0, taking fr u(t the slutin u(t; γ; x f Cauchy prblem (6, (7, fr x H =, we btain R(γx, x = S(u(t; γ; x γ Au (t; γ; x L (R + ;H = S(u(t; γ; x( γs T >0 (9 fr γ [0, S. Taking int accunt Lemma 4 and inequality (8, we btain By that, T R(γx, x cd (γ( γs T > 0. min R(γx, x > 0 fr γ [0, S T x H = fr the case ω (γ = ω (γ.. The same argument can be used Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

17 On the Bundary Value Prblem with the Operatr in Bundary Cnditins Cntinuing, by denitin S T, fr all γ (S T, β (α + β there exists a functin v(t; γ W,T (R +; H such that Cnsequently, fr γ (S T i.e., S(v(t; γ < γ Av (t; γ L (R + ;H. ( R(γx γ, x γ + d F dt ; γ; A v(t; γ, β (α + β frm equality (0 it fllws that L (R + ;H min R(γx, x R(γx γ, x γ < 0. x H = < 0, x γ = v (0; γ, Thus, since min R(γx, x is a cntinuus functin and it changes its sign at x H = the pint γ = S T, then min R(S T x, x = 0. x H = T cmplete the prf, we cnsider the functinal f the frm Q(γ; x = β (α + β γ +β Re(AT x, x (α +β, x + α β AT x H =. H Since then Q(γ; x = R(γx, x + α β AT x H R(γx, x, min x H = Q(S T ; x min x H = R(S T x, x = 0. On the ther hand, and, cnsequently, ( R(S T x, x = Q(S T ; x + α β AT x H ( Q(S T ; x + α β AT H H ( min x H = Q(S T ; x + α β AT H H min x H = R(S T x, x = 0. Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 3

18 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva Thus we have min x H = Q(S T ; x = 0. S, β (α + β S T = β Re(AT x, x (α + β min. x H = + α β AT x H Then, taking int accunt min x H = Re(AT x, x < 0, frm the abve we get S T = β (α + β β (α + β min x H = Re(AT x, x + α β AT x. H The therem is prved. In the case when min x H = Re(AT x, x < 0, Therem is frmulated as fllws. Therem 7. Suppse that the cnditins f Therem 6 are satised, the peratr T α,β is bunded invertible in the space H 3 and the peratr B = A A is bunded in H, mrever ( B < α + α β β (α + β Then prblem (, ( is regularly slvable. min x H = Re(AT x, x + α β AT x. H Despite the fact that equatin ( was presented in a mre general frm in the paper f A.R.Aliev [8], in this paper nly the case when ReAT 0 is studied. Mrever, ur results imprve the results btained in [8]. References [] E. Hille and R.S. Fillips, Functinal Analysis and Semi-Grups. Amer. Math. Sc. Cll. Publ. 3, Amer. Math. Sc., Prvidence, 957; Izdat. Instran. Lit., Mscw, 96. [] J.L. Lins and E. Magenes, Nn-Hmgeneus Bundary Value Prblems and Applicatins. Dund, Paris, 968; Mir, Mscw, 97; Springer-Verlag, Berlin, Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.

19 On the Bundary Value Prblem with the Operatr in Bundary Cnditins [3] A.A. Shkalikv, Elliptic Equatins in Hilbert Space and Assciated Spectral Prblems. Trudy Sem. Petrvsk. (989, N. 4, 404. (Russian (Transl.: J. Sviet Math. 5 (990, N. 4, [4] M.G. Gasymv, On the Thery f Plynmial Operatr Pencils. Dkl. Akad. Nauk SSSR 99 (97, N. 4, (Russian [5] M.G. Gasymv, The Multiple Cmpleteness f Part f the Eigen- and Assciated Vectrs f Plynmial Operatr Bundles. Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 6 (97, N. -3, 347. (Russian [6] M.G. Gasymv, The Slubility f Bundary Value Prblems fr a Class f Operatr Dierential Equatins. Dkl. Akad. Nauk SSSR 35 (977, N. 3, (Russian [7] S.S. Mirzev and M.Yu. Salimv, Cmpleteness f Elementary Slutins t a Class f Secnd Order Operatr Dierential Equatins. Sib. Mat. Zh. 5 (00, N. 4, (Russian (Transl.: Sib. Math. J. 5 (00, N. 4, [8] A.R. Aliev and S.S. Mirzev, On Bundary Value Prblem Slvability Thery fr a Class f High-Order Operatr Dierential Equatins. Funkts. Anal. Prilzhen. 44 (00, N. 3, (Russian (Transl.: Funct. Anal. Appl. 44 (00, N. 3, 09. [9] A.R. Aliev, On the Slvability f Initial Bundary Value Prblems fr a Class f Operatr Dierential Equatins f Third Order. Mat. Zam. 90 (0, N. 3, (Russian (Transl.: Math. Ntes 90 (0, N. 3, [0] F.S. Rfe-Beketv, Expansin in Eigenfunctins f Innite Systems f Dierential Equatins in the Nnself-Adjint and Self-Adjint Cases. Mat. Sb. 5(93 (960, N. 3, (Russian [] V.A. Ilyin and A.F. Filippv, The Nature f the Spectrum f a Self-Adjint Extensin f the Laplace Operatr in a Bunded Regin. Dkl. Akad. Nauk SSSR 9 (970, N., (Russian (Transl.: Sviet Math. Dkl. (970, [] M.L. Grbachuk, Cmpleteness f the System f Eigenfunctins and Assciated Functins f a Nnself-Adjint Bundary Value Prblem fr a Dierential-Operatr Equatin f Secnd Order. Funkts. Anal. Prilzhen. 7 (973, N., (Russian (Transl.: Funct. Anal. Appl. 7 (973, N., [3] S.Y. Yakubv and B.A. Aliev, Fredhlm Prperty f a Bundary Value Prblem with an Operatr in Bundary Cnditins fr an Elliptic Type Operatr Dierential Equatin f the Secnd Order. Dkl. Akad. Nauk SSSR 57 (98, N. 5, (Russian [4] M.G. Gasymv and S.S. Mirzev, On the Slvability f the Bundary Value Prblems fr the Operatr Dierential Equatins f Elliptic Type f the Secnd Order. Di. Uravn. 8 (99, N. 4, (Russian (Transl.: Di. Eq. 8 (99, N. 4, Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N. 5

20 S.S. Mirzev, A.R. Aliev, and L.A. Rustamva [5] S.S. Mirzev and S.G. Veliev, On the Estimatin f the Nrms f Intermediate Derivatives in Sme Abstract Spaces. J. Math. Phys., Anal., Gem. 6 (00, N., [6] S.S. Mirzev and S.G. Veliev, On Slutins f ne Class f Secnd-Order Operatr Dierential Equatins in the Class f Hlmrphic Vectr Functins. Ukr. Mat. Zh. 6 (00, N. 6, (Russian (Transl.: Ukr. Math. J. 6 (00, N. 6, [7] S.S. Mirzev and R.F. Safarv, On Hlmrphic Slutins f Sme Bundary Value Prblems fr Secnd-Order Elliptic Operatr Dierential Equatins. Ukr. Mat. Zh. 63 (0, N. 3, (Russian (Transl.: Ukr. Math. J. 63 (0, N. 3, [8] A.R. Aliyev, T the Thery f Slvability f the Secnd Order Operatr Dierential Equatins with Discntinuus Cecients. Transactins f Natinal Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 4 (004, N., [9] A.R. Aliev and S.F. Babayeva, On the Bundary Value Prblem with the Operatr in Bundary Cnditins fr the Operatr Dierential Equatin f the Third Order. J. Math. Phys., Anal., Gem. 6 (00, N. 4, [0] S.S. Mirzev and A.R. Aliev, On a Bundary Value Prblem fr Secnd-Order Operatr Dierential Equatins with a Discntinuus Cecient. Prc. Inst. Math. Mech. Acad. Sci. Azerb. 6(4 (997, 7. [] S.S. Mirzev and L.A. Rustamva, On Slvability f One Bundary Value Prblem fr Secnd Order Operatr Dierential Equatins with Discntinuus Cecient. Int. J. Appl. Cmput. Math. 5 (006, N., 900. [] S.S. Mirzev, Cnditins fr the Well-Dened Slvability f Bundary Value Prblems fr Operatr Dierential Equatins. Dkl. Akad. Nauk SSSR 73 (983, N., 995. (Russian (Transl.: Sviet Math. Dkl., 8 (983, Jurnal f Mathematical Physics, Analysis, Gemetry, 03, vl. 9, N.