On discontinuous Sturm-Liouville problems with transmission conditions

Transcription

1 J. Math. Kyoto Uiv. (JMKYAZ 44-4 (4, O discotiuous Sturm-Liouville problems with trasmissio coditios By O. Sh. Mukhtarov, MahirKadakal ad F. S. Muhtarov Abstract We cosider a discotiuous Sturm-Liouville equatio together with eigeparameter depedet boudary coditios ad two supplemetary trasmissio coditios at the poit of discotiuity. By modifyig some techiques of [], [] ad [4] we exted ad geeralize some approach ad results of classic regular Sturm-Liouville problems to the similar problems with discotiuities. I particular, we itroduce a special Hilbert space formulatio such a way that the cosidered problem ca be iterpreted as a eigevalue problem of suitable self-adjoit operator, the we costruct the Gree fuctio ad resolvet operator ad derive a asymptotic formulas for eigevalues ad ormalized eigefuctios.. Itroductio The Sturmia theory is a importat aid i solvig may problems of mathematical physics. Usually, the eigevalue parameter appear liearly oly i the differetial equatio of the classic Sturm-Liouville problems. However, i mathematical physics are ecoutered such problems, where eigevalue parameter appear i both differetial equatio ad boudary coditios (various physical applicatios ca be foud i []. There is a substatial literature o this type of problems (see, for example, [], [], [3], [8], [9], [4] ad more recetly [5], [6], [7] ad correspodig refereces cited therei. I these works, oly cotiuous problems have bee ivestigated. The purpose of this paper is to exted some classic results of Sturmia theory to the discotiuous case, i which two supplemetary trasmissio coditios added to the boudary coditios. I fact, we ivestigate both cotiuous ad discotiuous cases (the cases δ =adδ i below, respectively i this study. Let us cosider the Sturm-Liouville equatio (. τu := u + q(xu = λu for x [, (, ] 99 Mathematics Subject Classificatio(s. 34L, 34B4, 34B7 Received October, 3 Revised September, 4

2 78 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov (i.e. o [, ] except oe ier poit x =,whereq(x is a real-valued, cotiuous i both [, ad (, ] ad has fiite limites q(± = lim q(x, x ± together with stadard boudary coditio at x = (. L u := α u( + α u ( =, trasmissio coditios at the poit of discotiuity x = (.3 L u := u( δu(+ =, (.4 L 3 u := u ( δu (+ =, ad eigeparameter depedet boudary coditio at x = (.5 L 4 (λu := λ(β u( β u ( + (β u( β u ( =, where λ C is a complex spectral parameter ad all coefficiets of the boudary ad trasmissio coditios are real costats. Naturally, we assume that α + α,δ, β + β ad β + β. Moreover, we shall assume that ρ := β β β β >. Some special cases of this problem arises after a applicatio of the method of separatio of variables to the varied assortmet of physical problems, such as, i heat ad mass trasfer problems (see, for example, [], i vibratig strig problems whe the strig loaded additioally with poit masses (see, for example, [], i thermal coductio problem for a thi lamiated plate (see, for example, []. Note that such properties as isomorphism, coerciveess with respect to the spectral parameter, completeess of root fuctios, distributios of eigevalues of some discotiuous boudary value problems with trasmissio coditios ad its applicatios to the correspodig iitial-boudary value problems for parabolic equatios have bee ivestigated i [5], [6], [7], [].. Operator-theoretic formulatio i suitable Hilbert space I this sectio, we itroduce the special ier product i the Hilbert space (L (, L (, C ad defie a liear operator A i it such a way that the cosidered problem (. (.5 ca be iterpreted as the eigevalue problem of A. So, we defie a ew Hilbert space ier product o H := (L (, L (, C by for F = ( f(x f F, G H = δ f(xg(xdx + δ f(xg(xdx + δ ρ f g (, G = g(x H. For coveiece we shall use the otatios g R (u :=β u( β u (, R (u :=β u( β u (.

3 O discotiuous Sturm-Liouville problems with trasmissio coditios 78 I this Hilbert space, we costruct the operator A : H H with domai (. D(A = ad actio law (. AF = F = ( f(x f f(x,f (xare absolutely cotiuous i [, (, ] ad have fiite oe-had sided limits f(,f (, respectively; τf L (, L (, L f = L f = L 3 f =;f = R (f ( ( τf f(x with F = R (f R (f D(A. Thus, we ca pose the boudary value-trasmissio problem (. (.5 as ( u(x (.3 AU = λu, U := R (u D(A i the Hilbert space H. It is readily verified that the eigevalues of the operator A coicide with those of the problem (. (.5. Theorem.. The operator A is symmetric. Proof. Let F = ( f(x R (f ad G = ( g(x R (g are arbitrary elemet of D(A. By two partial itegratio we get (.4 AF, G H F, AG H = W (f,g; W (f,g; + δ W (f,g; δ δ δ W (f,g;++ δ ρ (R (fr (g R (fr (g, where, as usual, W (f,g; x deotes the Wroskias of the fuctios f ad g, i.e. W (f,g; x :=f(xg (x f (xg(x. Sice F, G D (A, the first compoets of these elemets, i.e. f ad g satisfy the boudary coditio (.. From this fact, we easily have that (.5 W (f,g; =, sice α ad α are real. Further, as f ad g also satisfy both trasmissio coditios we get (.6 W (f,g; = δ W (f,g;+.

4 78 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov Moreover, the direct calculatios gives (.7 R (fr (g R (fr (g = ρw (f,g;. Now, substitutig (.5 (.7 i (.4 gives AF, G H = F, AG H (F, G D(A, so A is symmetric. The proof is complete. Recallig that the eigevalues of the problem (. (.5 are coicide with the eigevalues of A we have the ext corollary. Corollary.. All eigevalues of the problem (. (.5 are real. As all eigevalues are real it is eough to ivestigate oly the real-valued eigefuctios. Takig this ito accout, we ca ow assume that all eigefuctios of the problem (. (.5 are real-valued. 3. Asymptotic represetatios of the basic solutios Let us defie two basic solutios { { φ (x, λ, x [, χ (x, λ, x [, φ(x, λ = ad χ(x, λ = φ (x, λ, x (, ] χ (x, λ, x (, ] of equatio (. by the followig procedure. At first cosider the ext iitial-value problem: (3. u + q(xu = λu, x [, ], (3. u( = α, (3.3 u ( = α. By virtue of [, Theorem.5] this problem has a uique solutio u = φ (x, λ,whichisaetirefuctioofλ C for each fixed x [, ]. Slightly modifyig the method of [, Theorem.5] we ca prove that the iitial-value problem (3.4 u + q(xu = λu, x [, ], (3.5 u( = β λ + β, (3.6 u ( = β λ + β has a uique solutio u = χ (x, λ, which is a etire fuctio of parameter λ for each fixed x [, ]. The other fuctios φ (x, λ adχ (x, λ cabe defied i terms of φ (x, λ adχ (x, λ, respectively. Applyig the method used i the proof of [3, Theorem ] we ca prove that the iitial-value problem (3.7 u + q(xu = λu, x [, ], (3.8 (3.9 u( = δ φ (,λ, u ( = δ φ (,λ

5 O discotiuous Sturm-Liouville problems with trasmissio coditios 783 has a uique solutio u = φ (x, λ,whichisaetirefuctioofλ for each fixed x [, ]. Similarly, the iitial-value problem (3. (3. (3. u + q(xu = λu, x [, ], u( = δχ (,λ, u ( = δχ (,λ also has a uique solutio u = χ (x, λ, which is a etire fuctio of λ for each fixed x [, ]. By virtue of (3. ad (3.3 the solutio φ(x, λ satisfies the first boudary coditio (.. Moreover, by virtue of (3.8 ad (3.9, φ(x, λ also satisfies both trasmissio coditios (.3 ad (.4. Similarly, by virtue of (3.5, (3.6, (3. ad (3. the other solutio χ(x, λ satisfies the secod boudary coditio (.5 ad both trasmissio coditios (.3 ad (.4. It is well-kow, from the ordiary liear differetial equatios theory, that each of the Wroskias ω (λ =W (φ (x, λ,χ (x, λ ad ω (λ =W (φ (x, λ,χ (x, λ are idepedet o x i [, ] ad [, ], respectively. Lemma 3.. The equality ω (λ =δ ω (λ holds for each λ C. Proof. Sice the above Wroskias are idepedet o x, the usig (3.8, (3.9, (3. ad (3. we have (3.3 ω (λ =φ (,λχ (,λ φ (,λχ (,λ =(δφ (,λ (δχ (,λ (δφ (,λ (δχ (,λ = δ ω (λ. Corollary 3.. The zeros of ω (λ ad ω (λ are coicide. Takig the Lemma 3. ito accout we deote both ω (λ adδ ω (λ by ω(λ. Recallig the defiitios of φ i (x, λadχ i (x, λ we coclude the ext corollary. Corollary 3.3. The fuctio ω(λ is a etire fuctio. Theorem 3.4. The eigevalues of the problem (. (.5 are coicide with the zeros of the fuctio ω(λ. Proof. Let ω(λ =. The W (φ (x, λ,χ (x, λ = for all x [, ]. Cosequetly, the fuctios φ (x, λ adχ (x, λ are liearly depedet, i.e. χ (x, λ =k φ (x, λ,x [, ]

6 784 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov for some k. By usig (3. ad (3.3, from this equality we have α χ(,λ +α χ (,λ =α χ (,λ +α χ (,λ = k (α φ (,λ +α φ (,λ = k (α α + α ( α =, so χ(x, λ satisfies the first boudary coditio (.. Recallig that the solutio χ(x, λ satisfies also the other boudary coditio (.5 ad both trasmissio coditios (.3 ad (.4, we coclude that χ(x, λ is a eigefuctio of the problem (. (.5, i.e. λ is a eigevalue. Thus, each zero of ω(λ is a eigevalue. Now let λ be a eigevalue ad u (x be a ay eigefuctio correspodig to this eigevalue. Suppose, if possible, that ω(λ. Whece W (φ (x, λ,χ (x, λ adw (φ (x, λ,χ (x, λ. So, by virtue of well-kow properties of Wroskias, it follows that each of the pairs φ (x, λ, χ (x, λ ad φ (x, λ, χ (x, λ are liearly idepedet. Therefore the solutio u (x of Equatio (. may be represeted i the form { c φ (x, λ +c χ (x, λ,x [,, u (x = c 3 φ (x, λ +c 4 χ (x, λ,x (, ], where at least oe of the costats c,c,c 3 ad c 4 is ot zero. Cosiderig the true equalities L ν (u (x = c L ν (φ (x, λ + c L ν (χ (x, λ (3.4 + c 3 L ν (φ (x, λ + c 4 L ν (χ (x, λ =, ν=,, 3, 4 as the homogeeous system of liear equatios of the variables c,c,c 3 ad c 4, ad takig (3.8, (3.9, (3. ad (3. ito accout it follows that the determiat of this system is equal to ω (λ φ (,λ χ (,λ δφ (,λ δχ (,λ φ (,λ χ (,λ δφ (,λ δχ (,λ = δ ω (λ ω(λ ω (λ = δ ω3 (λ ad therefore it is ot equal to zero by assumptio. Cosequetly, this homogeeous system of liear equatios has the oly trivial solutio (c,c,c 3,c 4 = (,,,. Thus we get cotradictio, which completes the proof. Theorem 3.5. Let λ = s, Im s = t. The, the followig asymptotic equalities hold as λ : ( I the case α φ (k (x, λ =α d k ( (3.5 cos[s(x +]+O e t (x+, dxk s k φ (k (x, λ =α d k ( (3.6 cos[s(x +]+O e t (x+, δ dxk s k

7 O discotiuous Sturm-Liouville problems with trasmissio coditios 785 for k =ad k =. ( I the case α = (3.7 (3.8 φ (k (x, λ = α s φ (k (x, λ = α δs d k ( si[s(x +]+O dxk ( d k si[s(x +]+O dxk e t (x+ s k e t (x+ s k for k =ad k =. Moreover, each of asymptotic equalities hold uiformly for x. Proof. The above asymptotic formulas for φ (x, λ have bee foud i [, Lemma.7]. But the similar formulas for the solutio φ (x, λ eed idividual cosideratios, sice this solutio is defied by the iitial coditios havig special o-stadard forms. The iitial-value problem (3.7, (3.8, (3.9 ca be trasformed ito a equivalet itegral equatio u(x =δ φ (,λcos λx + δ λ φ (,λsi λx, (3.9 + λ x si[ λ(x y]q(yu(ydy. Let α. Substitutig (3.5 i (3.9 we have (3. φ (x, λ = α δ cos λ(x ++ λ + O ( λ e t (x+. x si[ λ(x y]q(yφ (y, λdy Multiplyig by e t (x+ ad lettig F (x, λ =e t (x+ φ (x, λ, we have the ext asymptotic itegral equatio F (x, λ = α e t (x+ cos λ(x + δ + x si[ ( λ(x y]q(ye t (x y F (y, λdy + O λ. λ Lettig M(λ = max F (x, λ from the last equatio we derive that x [,] M(λ M ( α δ + λ

8 786 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov for some M >. Cosequetly, M(λ =O( as λ,so φ (x, λ =O(e t (x+ as λ. Substitutig this asymptotic equality i the itegral term of the (3. gives (3.6 for the case k =. The proof of (3.6 for the case k =cabe obtaied at oce by differetiatig (3.9 ad the followig the same procedure as i the case k =. The proof of (3.8 is similar to that of (3.6 ad hece omitted. Theorem 3.6. Let λ = s, Im s = t. The, the followig asymptotic formulas hold for the eigevalues of the boudary-value-trasmissio problem (. (.5: Case : β, α (3. s = π( + O ( Case : β, α = (3. s = ( π Case 3: β =, α (3.3 s = ( π Case 4: β =, α = + O + O (3.4 s = π + O (, (, (,. Proof. Let us cosider Case oly. Writig ω (λ =φ (x, λχ (x, λ φ (x, λχ (x, λ for x = ad the usig χ (,λ=β λ + β, χ (,λ=β λ + β as give by (3.5 ad (3.6, respectively, we have the followig represetatio for ω (λ: (3.5 ω (λ =(β λ + β φ (,λ (β λ + β φ (,λ. Now writig x = i (3.6 ad the substitutig i (3.5 we derive that (3.6 ω (λ =δ β α s 3 si( λ+o( s e t. By applyig well-kow Rouche s Theorem (which assert that if f(z ad g(z are aalytic iside ad o a closed cotour Γ, ad g(z < f(z o Γ, the f(z adf(z+g(z have the same umber zeros iside Γ, provided that

9 O discotiuous Sturm-Liouville problems with trasmissio coditios 787 each zeros are couted accordig to their multiplicity o a sufficietly large cotour, it follows that ω (λ has the same umber of zeros iside the cotour as the leadig term i (3.6. Hece, if λ <λ <λ <..., are the zeros of ω (λ ads = λ,wehave (3.7 s = π ( + δ, where δ < π 4, for sufficietly large. By substitutig (3.7 ito (3.6 we have ( δ = O, so the proof completes for Case. The proofs for the other cases are similar. The followig asymptotic formulas hold for the eige- Theorem 3.7. fuctios φ λ (x = of the problem (. (.5: Case : β, α ( α cos (3.8 φ λ (x = α δ cos Case : β, α = { φ (x, λ, x [,, φ (x, λ, x (, ] π( (x + ( π( (x + (3.9 φ λ (x ( ( α π( / si π = ( ( α δ π( / si π Case 3: β =, α ( α cos π (3.3 φ λ (x = α δ cos, x [,, x (, ] (x + (x +,x [,, x (, ] ( (x +, x [, ( π ( Case 4: β =, α = ( α π si (3.3 φ λ (x = α δ π si (x +, x (, ] π(x + ( π(x +, x [,, x (, ] + O (, ( + O, + O (, ( + O.

10 788 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov All this asymptotic formulas hold uiformly for x. Proof. Let us cosider Case oly. Substitutig (3.6 ito the itegral term of (3., it is easy to see that (3.3 x si[ λ(x y]q(yφ (y, λdy = O(e t (x+. Substitutig ito (3. we have (3.33 φ (x, λ = α δ cos ( λ(x ++O λ e t (x+. We already kow that all eigevalues are real. Further, puttig λ = R, R> i (3.6 it follows that ω( R as R +, soω( R for sufficietly large R>. Cosequetly, the set of eigevalues is bouded below. Now, writig λ = s i (3.33 we obtai φ (x, λ = α ( δ cos[s (x +]+O sice t =Ims = for sufficietly large. After some routie calculatios we easily obtai that ( ( cos[s (x +]=cos π( (x + + O. Cosequetly, φ (x, λ = α δ cos Similarly we ca fid that Sice φ (x, λ =α cos φ λ (x = s ( π( (x + + O, (. ( ( π( (x + + O. { φ (x, λ,x [,, φ (x, λ,x (, ], the proof for Case is completed. The proofs for the other cases are similar. 4. Asymptotic formulas for ormalized eigefuctios It is evidet that the two-compoet vectors ( φλ (x (4. Φ := R (φ,=,,,... λ

11 O discotiuous Sturm-Liouville problems with trasmissio coditios 789 are the eigeelemets of the operator A correspodig to the eigevalue λ. For m, (4. Φ, Φ m H =,,m=,,,..., sice A is symmetric. Deotig (4.3 ψ := φ λ (x Φ H, it is easily see that the eigeelemets ( ψλ (x (4.4 Ψ := R (ψ,,m=,,,... λ are orthoormal. That is, where δ m is the kroecker delta. x = ad Ψ, Ψ m H = δ m, Lemma 4.. The followig asymptotic equalities hold: ( i case α (4.5 R (φ λ =O (, ( i case x = ( (4.6 R (φ λ =O. Proof. It follows from the equality ω (λ =that (4.7 λ R (φ λ +R (φ λ =. ( Let α. The from the formula (3.6 we get R (φ λ =β φ λ ( β φ λ ( = β O( β O( s. Now applyig Theorem 3.6 we have (4.8 R (φ λ =O(. Substitutig (4.8 ito (4.7, ad takig Theorem 3.6 ito accout, we get R (φ λ = ( R (φ λ =O. λ

12 79 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov Now let α =. By usig Theorem 3.6 we obtai R (φ λ =β φ λ ( β φ λ ( = β O( s β O( = β O = O(. ( β O( Takig ito accout that λ ( π ad usig (4.7 we have R (φ λ = ( R (φ λ =O λ. The proof is complete. Theorem 4.. Let Φ be defied as i (4.. The the followig asymptotic formulas hold for the orms Φ H of the eigeelemets Φ : Case : If β ad α,the (4.9 Φ H = α ( + O, δ Case : If β ad α =,the (4. Φ H = α ( δ π( / + O, Case 3: If β =ad α,the (4. Φ H = α ( + O, δ Case 4: If β =ad α =,the (4. Φ H = α ( δ π + O. Proof. Let β adα. I this case, usig (3.8 we have (4.3 (φ λ (x dx = α = α [ ( cos π( (x + + O cos ( π( (x + dx + O ( = α + O. ( ] dx (

13 O discotiuous Sturm-Liouville problems with trasmissio coditios 79 Similarly, we have (4.4 ( (φ λ (x dx = α δ + O. Usig (4.5, (4.3 ad (4.4 we get (4.5 Φ H = δ (φ λ (x dx + δ ( ( α = δ + O ( = α δ + O. + ( α δ + O (φ λ (x dx + δ ρ (R (φ λ ( + δ ( ρ O Cosequetly, Φ H = α δ + O ( = α + O δ (, which proves the formula (4.9. Now let β adα =. I this case from (3.9 we get (4.6 Φ H = (φ λ (x dx + δ δ { ( = α δ π( / + O { ( + δ α δ π( / + O ( = 4α δ 3. (π( / + O (φ λ (x dx + δ ρ (R (φ λ ( } 3 ( } ( 3 + O 4 From this it follows that Φ H = α ( δ π( / + O, which proves the formula (4.. The proofs for the other cases are similar. Theorem 4.3. The first compoets of the ormalized eigeelemets (4.4 have the followig asymptotic represetatio as : Case : If β ad α,the

14 79 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov (4.7 sg(α ( δ cos ψ (x = ( α sg cos δ δ π( (x + Case : If β ad α =,the (4.8 ψ (x sg( α ( δ si π = ( α sg si δ δ ( π( (x + ( (x + ( π( (x + Case 3: If β =ad α,the ( + O for x [,, ( + O for x (, ], ( + O for x [,, ( + O for x (, ], (4.9 δ f(xg(xdx + δ f(xg(xdx + δ ρ R (fg =, Case 4: If β =ad α,the (4. sg( α ( δ si ψ (x = ( sg α si δ δ π(x + ( π(x + ( + O ( + O for x [,, for x (, ]. Each of this asymptotic equalities hold uiformly for x. (Here, as usual, sg deotes the sig fuctio (4. Proof. Let β adα. I this case, from (4.9 it follows that ( δ = Φ H α + O. Puttig (3.8 ad (4. ito (4.3 we fid the required asymptotic formula (4.7. Similarly, we ca derive the other formulas (4.8 ( Gree fuctio, resolvet operator ad self-adjoitess of the problem Let A : H H be defied by (. ad (., ad let λ ot be a eigevalue of A. For fidig the resolvet operator R(λ, A =(λi A cosider the operator equatio (5. (λi AU = F

15 O discotiuous Sturm-Liouville problems with trasmissio coditios 793 for F = ( f(x f H. This operator equatio is equivalet to the ihomogeeous differetial equatio (5. u +(λ q(xu = f(x forx [, (, ] subject to ihomogeeous boudary coditios (5.3 (5.4 α u( + α u ( =, λ(β u( β u ( + (β u( β u ( = f ad homogeeous trasmissio coditios (5.5 (5.6 u( δu(+ =, u ( δu (+ =. By applyig the same techiques as i our previous paper [4] we ca prove that the problem (5. (5.6 has a uique solutio u(x, λ, which ca be represeted as (5.7 χ (x, λ ω (λ u(x, λ = χ (x, λ ω (λ x φ (y, λf(ydy+ φ (x, λ ω (λ + δ δ φ (y, λf(ydy+ + φ (x, λ ω (λ x χ (y, λf(ydy χ (y, λf(ydy + δ f for x [,, x x φ (y, λf(ydy χ (y, λf(ydy+f for x (, ]. Deotig χ(x, λφ(y, λ δ for y x, ω(λ (5.8 G(x, y, λ = φ(x, λχ(y, λ δ for x y, ω(λ where x ady, the formula (5.7 reduced to (5.9 u(x, λ = δ G(x, y, λf(ydy + δ G(x, y, λf(ydy + δ f φ(x, λ ω(λ. O the other had, by applyig the fuctioal R to the Gree fuctio with respect to the variable y ad recallig that χ(x, λ satisfies the iitial

16 794 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov coditios (3.5 ad (3.6 we have (5. R (G(x, ; λ = β G(x, ; λ β G(x, ; λ y φ(x, λ = δ ω(λ (β χ(,λ β χ (,λ φ(x, λ = δ ω(λ (β (β λ + β β (β λ + β φ(x, λ = δ ρ ω(λ. Substitutig this ito (5.9 gives (5. u(x, λ = δ G(x, y, λf(ydy + δ + δ ρ R (G(x,,λf. G(x, y, λf(ydy Now itroducig ( G(x,,λ (5. G x,λ = R (G(x,,λ which we call the Gree elemet of the problem (5. (5.6, the last formula (5. takes the form (5.3 u(x, λ = G x,λ, F, where by F we mea ( f(x F =. Now we ca fid the resolvet operator of A i terms of Gree elemet G x,λ. As the fuctio u(x, λ defied by (5. is the solutio of the ihomogeeous boudary-trasmissio problem (5. (5.6 which is equivalet to the operator equatio (5. we have (5.4 R(λ, AF = for arbitrary F H. f ( ( u(x, λ Gx,λ, F R (u(,λ = R G,λ, F Theorem 5.. The operator A is self-adjoit o the Hilbert space H.

17 O discotiuous Sturm-Liouville problems with trasmissio coditios 795 Proof. First, we prove that A is desely defied o H. For this suppose ( g(x G = H g is orthogoal to D(A, i.e. (5.5 δ f(xg(xdx + δ f(xg(xdx + δ ρ R (fg = for all F ( f(x R D(A. Let C (f ([, (, ] be a set of ifiitely differetiable fuctios o [, (, ], each elemet of which vaishes o some eighborhood of the poits x =, x =adx =. It is clear from the defiitio of D(A thatc ([, (, ] {} D(A. By writig (5.5 for all F C ([, (, ] we ca see that g(x is orthogoal to C ([, (, ] i L (, with respect to the followig ier product δ f(xg(xdx + δ f(xg(xdx =forallf C ([, (, ]. Cosequetly, g(x vaishes, sice L (, is complete with respect to the above ier product. The, substitutig g(x = ito (5.5 yields (5.6 R (fg =, for all f L (, such that ( f(x R (f D(A. Choosig F = ( f (x R (f D(A such that R (f =, we have from (5.6 that g =. Cosequetly,G =, so D(A isdeseih. Further, sice A is symmetric it is eough to prove that D(A =D(A, where A is adjoit of A. LetF D(A. We must show that F D(A. By defiitio of A (5.7 AG, F H = G, A F H for all G D(A. From this it follows that (5.8 (ii AG, F = G, ( ii A F. We already kow that (see (5.4 λ = i is regular poit of A ad therefore we ca let (5.9 U = R( i, A( if A F,

18 796 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov that is (5. ( ii AU = if A F. Substitutig this ito (5.8 ad takig ito accout that A is symmetric ad U D(A wehave Cosequetly, (ii AG, F H = G, ( ii AU H = G, iu H G, AU H = ig, U H AG, U H = (ii AG, U H. (5. (ii AG, F U H =forallg H. Sice λ = i is regular poit of A we ca choose G = R(i, A(F U. Substitutig this ito (5. we get F U H =, so F = U ad therefore F D(A. The proof is complete. Departmet of Mathematics Faculty of Arts ad Scieces Gaziosmapasa Uiversity Tokat, Turkey omukhtarov@yahoo.com Departmet of Mathematics Faculty of Arts ad Scieces Odokuz Mayıs Uiversity 5539 Kurupelit-Samsu, Turkey mkadakal@yahoo.com Mathematic ad Mechaic Istitute Azerbaija Sciece Academy Baku, Azerbaija muhtarov@gop.edu.tr Refereces [] G.D.Birkhoff,O the asymptotic character of the solutio of the certai liear differetial equatios cotaiig parameter, Tras. Amer. Soc. 9 (98, 9 3.

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20 798 O. Sh. Mukhtarov, Mahir Kadakal ad F. S. Muhtarov [6] S. Yakubov ad Y. Yakubov, Abel basis of root fuctios of regular boudary value problems, Math.Nachr.97 (999, [7], Differetial-Operator Equatios, Ordiary ad Partial Differetial Equatios, Chapma ad Hall/CRC, Boca Rato,, p. 568.