Normalized Eigenfunctions of Discontinuous Sturm-Liouville Type Problem with Transmission Conditions

Transcription

1 Applied Mathematical Sciences, Vol., 007, no. 5, Normalized Eigenfunctions of Discontinuous Sturm-Liouville Type Problem with Transmission Conditions Z. Adoğan Gaziosmanpaşa University, Faculty of Science-Art Department of Mathematics, 6040 Toat, Turey M. Demirci Gaziosmanpaşa University, Faculty of Science-Art, Department of Mathematics, 6040 Toat, Turey O. Sh. Muhtarov Gaziosmanpaşa University, Faculty of Science-Art, Department of Mathematics, 6040 Toat, Turey Abstract The goal of this study is to investigate some spectral properties of one discontinuous Sturm-Liouville problem, where the eigenparameter appears not only in the differential equation, but also in the boundary and transmissions conditions. We introduce an operator-theoretic interpretation and obtain asymptotic approximate formulae for normalized eigenfunction. Mathematics Subject Classification: 34L0 Keywords: Discontinuous Sturm-Liouville Problems, Eigenvalues, Eigenfunctions, Transmission Conditions, Eigenparameter-Dependent Boundary Conditions, normalized eigenfunction Introduction It is well-nown that many topics in mathematical physics require the investigation of the eigenvalues and eigenfunctions of Sturm-Liouville type boundary-

2 574 Z. Adoğan, M. Demirci and O. Sh. Muhtarov value problems. Therefore, the Sturmian theory is one of the most actual and extensively developing fields of theoretical and applied mathematics. Particularly, in recent years, highly important results in this field have been obtained for the case when the eigenparameter appears not only in the differential equation but also in the boundary conditions. The literature on such results is voluminous and we refer to,3,,8 and corresponding bibliography cited therein. In particular, 3,5,,3,5 and 6 contains many references to problems in physics and mechanics. It should be mentioned that in the series of S. and Y. Yaubov s wors published in recent years they have constructed an abstract theory of boundary-value problems with an eigenparameter in the boundary conditions see 7,8,9 and corresponding bibliography. While the general theory and methods of boundary-value problems with continuous coefficients are highly developed, very little is nown about a general character of similar problems with discontinuities. The main aim of this study is to extend some results of the standard Sturm- Liouville problems to the discontinuous case. Namely, we shall investigate the Sturm-Liouville equation τu := pxu + qxu = λu, to hold in finite interval a, b except one inner point c a, b, subject to the eigenparameter-dependent boundary conditions L u : = λ α ua α u a α ua α u a = 0 L u : = λ β ub β u b + β ub β u b = 0 3 and the eigenparameter-dependent transmission conditions L 3 u : = uc +0 uc 0 = 0 4 L 4 u : = γ u c +0 γ u c 0 + λδ + δ uc = 0 5 where px = for x a, c and px = for x c, b ;λ is complex p p eigenparameter; qx is real-valued and continuous in both a, c and c, b, and has finite limits qc ± 0 := lim qx ;p i,α i,α x c±0 i,β i,β i and δ i, i =, are real numbers. Consequently, we are interested in two types of generalizations of classical Sturm-Liouville problems: First, we wish to consider more general differential equations, for which the coefficient at the highest derivative may have the discontinuity at one inner point of the considered interval. Naturally, in this point of discontinuity two supplementary transmission conditions are given. A second generalization allows boundary and transmission conditions depending on an eigenparameter.

3 Sturm-Liouville type problem 575 It must be noted that some special cases of the considered problem -5 arise after an application of the method of separation of variables to the varied assortment of physical problems. For example, some boundary-value problems with transmission conditions arise in heat and mass transfer problems see, for example, 5, in vibrating string problems when the string loaded additionally with point masses see, for example,3 and in diffraction problems see,for example,6. Properties such as isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel bases of a system of root functions, distributions of eigenvalues of some discontinuous boundary value problems with transmission conditions and its applications to the corresponding initial-boundary-value problems for parabolic equations have been investigated in 6-9,5 and 8. Also, some problems with transmission conditions which arise in mechanics thermal conduction problem for a thin laminated plate were studied in the article 5. Similar problems for differential equations with discontinuous coefficients were investigated by Rasulov, M.L. in monographs 0,. But, the considered discontinuous problems in these wors do not contain transmission conditions. An Operator Formulation in the Adequate Hilbert Space In this section we shall introduce the special inner product in the Hilbert space H := L a, b C 3 and a symmetric linear operator A defined on this Hilbert Space such a way that the problem -5 can be considered as the eigenvalue problem of this operator. Throughout this paper we shall assume that the coefficients γ,γ and δ have the same signwithout losing the generality we shall assume that γ,γ and δ are positive, ρ := α α α α > 0 and ρ := β β β β > 0. 6 Let us introduce a new equivalent inner product on H = L a, b C 3 by c b F, G := p γ fxgxdx + p γ fxgxdx + γ f g + γ f g + f 3 g 3 ρ ρ δ a c 7 for F := fx,f,f,f 3,G:= gx,g,g,g 3 H 8 which is connected with coefficients of our problem. For a short exposition we shall use the following notations: B a u : = α ua α u a,b au :=α ua α u a,

4 576 Z. Adoğan, M. Demirci and O. Sh. Muhtarov B b u : = β ub β u b,b b u :=β ub β u b, T c u : = γ u c +0 γ u c 0 + δ uc, and T c u := δ uc. In the Hilbert space H we define a linear operator A with domain of definition DA := {F =fx,f,f,f 3 :fis absolutely continuous in a, b, f is absolutely continuous on both a, c and c, b, and has a finite limitsf c ± 0 = lim f x,τf L a, b, x c±0 f = B af,f = B bf,f 3 = T cf} and action law 9 AF =τf,b a f, B b f,t c f. 0 Thus, the problem -5 can be written in the form AF = λf where F =fx,b af,b bf,t cf DA. Naturally, by eigenvalues and eigenfunctions of the problem -5 we mean eigenvalues and first components of corresponding eigenelements of the operator A, respectively. Theorem. 9 Theorem The linear operator A is symmetric. Corollary. 9 Corollary All eigenvalues of the problem -5 are real. We can now assume that all eigenfunctions of the problem -5 are realvalued. 3 Construction and Asymptotic Approximation of Fundamental Solutions Let φ x, λ be the solution of the Cauchy problem p u + qxu = λu, x a, c ua = α + λα 3 u a = α + λα 4 It is nown that φ x, λ is an entire function of λ C for each fixed x a, csee 3. Now, we shall investigate the differential equation on c, b

5 Sturm-Liouville type problem 577 together with spacial type initial conditions uc = φ c 0,λ 5 u c = γ φ λδ + δ c 0,λ φ c 0,λ 6 γ Prove that this initial-value problem has an unique solution u = φ λ x φ x, λ, which also is an entire function of parameter λ C for each fixed x c, b. Let the sequence u n x, λ be defined by the recurrence formulas u 0 x, λ =0, u n x, λ = φ c 0,λ+ + x c γ φ c 0,λ γ γ λδ + δ φ c 0,λ x c qy λ u n y, λx ydy 7 for n =,, 3,...where u 0 x, λ =0. Since φ c 0,λ and φ c 0,λ are entire functions of λ, each term of sequence u n x, λ, n =0,,,...,so are for every x c, b. Let γ Q := max qx, Kλ := φ x c,b c 0,λ γ + λδ + δ φ c 0,λ γ b c 8 and K R = max Kλ 9 λ R where R>0 arbitrary real number. From 7 it follows that x u n+ x, λ u n x, λ Q + R c γ u n y, λ u n y, λ x ydy 0 in the closed sphere λ R for n =, 3, 4,.... From this equality it can be obtained by induction that u n+ x, λ u n x, λ K R Q + R n+ x c n+ n+! In view of this inequality the series u n x, λ u n x, λ n=

6 578 Z. Adoğan, M. Demirci and O. Sh. Muhtarov converges, uniformly with respect to λ if λ R, and with respect to x over c<x b, since the series K R Q + R n+ x c n+ n= n+! is covergent uniformly. Further from 7 we have x u n x, λ u n x, λ = qy λu n y, λ u n y, λ dy, n From which it follows that c 3 4 u n x, λ u n x, λ =qx λu n x, λ u n x, λ, 5 By virtue of each of the series x qy λu n y, λ u n y, λ dy 6 n= c and qx λu n x, λ u n x, λ 7 n= convergent, uniformly with respect to λ over λ R, and with respect to x over c, b. Maing use 4 and 5 we see that the first and second differentiated series n= u n x, λ u n x, λ and n= u n x, λ u n x, λ 8 also converge uniformly with respect to x over c, b. Denote φ x, λ := u n x, λ u n x, λ 9 n= i.e. φ x, λ = lim u n x, λ. 30 n Now, taing into account 9 and 5, we have φ x, λ = u nx, λ u n x, λ n= = qx λ u n x, λ u n x, λ n= = qx λ φ x, λ,

7 Sturm-Liouville type problem 579 so φ x, λ satisfies the differential equation on c, b. It also satisfies the initial conditions 5 and 6. Consequently maing use, 3, 4, 5 and 6 we see that, the function φx, λ is defined by φx, λ = { φ x, λ for x a, c φ x, λ for x c, b 3 and satisfies the differential equation on whole a, c c, b, one of the boundary conditions namely, boundary condition and both transmission conditions 4 and 5. Again, in view of the 3 the differential equation on c, b has an unique solution χ λ x χ x, λ satisfying the initial conditions and χ b, λ =β + λβ 3 χ b, λ =β + λβ 33 which is an entire function of λ for fixed x. The function χ λ x χ x, λ we will define in terms of χ x, λ by initial value problem p u + qxu = λu, x a, c 34 uc = χ c +0,λ 35 u c = γ χ λδ + δ c +0,λ+ χ c +0,λ. 36 γ By applying the same technique as in definition of φ x, λ we can prove that the initial-value problem has an unique solution u = χ λ x χ x, λ which is an entire function of λ for each fixed x a, c. By virtue of 3-36 the solution χx, λ is defined by γ χx, λ = { χ x, λ for x a, c χ x, λ for x c, b 37 and satisfies the differential equation on whole a, c c, b, the other boundary conditions 3 and both transmission conditions 4 and 5 it follows from the differential equation that each of the Wronsian s w λ :=W λ φ,χ ; x =φ x, λχ x, λ φ x, λχ x, λ 38 and w λ :=W λ φ,χ ; x =φ x, λχ x, λ φ x, λχ x, λ 39

8 580 Z. Adoğan, M. Demirci and O. Sh. Muhtarov is independent of x for x a, c and x c, b respectively. Therefore putting x = c 0 and x = c + 0 respectively and using 5, 6,35 and 36 it can be shown that γ w λ =γ w λ. 40 We shall denote both side of this equation by wλ; In 9 we proved the next theorem wλ :=γ w λ =γ w λ 4 Theorem 3. 9 Theorem 5 The eigenvalues of the problem -5 are consist of the zeros of the functions wλ and Δλ := γ γ wλ+λδ + δ χ λ cφ λ c. 4 Lemma 3. Let λ = s. Then the following integral and integro-differential equations hold for =0an =: dx φ λx = α + s α d dx cos p s x a p s α + s α d dx sin p s x a 43 + p s x a dx sin p s x y qyφ λ ydy dx φ λx = φ λ c d dx cos p s x c 44 + γ φ λ p s γ c s δ + δ d φ λ c γ dx sin p s x c + p s x c dx sin p s x y qyφ λ ydy Proof. For proving it is enough substitute s φ λ y +pyφ λy and s φ λ y+pyφ λ y instead of qyφ λy and qyφ λ y in the integral terms of the 43 and 44 respectively and integrate by parts twice. Lemma Lemma 7Let λ = s, Ims = t. Then if α 0 dx φ λx = s α dx cos p s x a + O s + e t p x a δ 45 dx φ λx = s 3 α cos p s c a d p γ dx sin p s x c +O s + e t p x c+p c a 46

9 Sturm-Liouville type problem 58 as λ, while if α =0 dx φ λx = sα p dx φ λx = s α δ sin p s c a p p γ dx sin p s x a + O s e t p x a 47 dx sin p s x c +O s + e t p x c+p c a 48 as λ =0,. Each of these asymptotic equalities holds uniformly for x. Similarly one can establish the following Lemma. Lemma 3.4 Let λ = s, Ims = t. Then if β 0 dx χ λx = s3 β δ cos p s c b d p γ dx sin p s x c 49 +O s + e t p b c+p c x dx χ λx = s β dx cos p s x b + O s + e t p b x 50 as λ, while if β =0 dx χ λx = s β δ sin p s c b d p p γ dx sin p s x c 5 +O s + e t p b c+p c x dx χ λx = sβ p dx sin p s x b + O s + e t p b x 5 as λ =0,. Each of these asymptotic equalities holds uniformly for x. By substituting the obtained asymptotic formulas for φ i x, λ and χ i x, λ i =, in the definitions of wλ and Δλ we can establish the following Theorem. Theorem 3.5 Let λ = s, Ims = t. Then the characteristic functions wλ and Δλ has the following asymptotic representations: Case :If β 0and α 0, then Δλ = s 6 δ α β cos p sc a cos p sb c 53 +O s 5 e t p b c+p c a wλ = s 6 δ γ α γ β cos p sc a cos p sb c 54 +O s 5 e t p b c+p c a

10 58 Z. Adoğan, M. Demirci and O. Sh. Muhtarov Case :If β 0and α =0, then Δλ = s 5 δ α p β sin p sc a cos p sb c 55 +O s 4 e t p b c+p c a wλ = s 5 α γ δ β sin p sc a cos p sb c 56 γ p +O s 4 e t p b c+p c a Case 3:If β =0and α 0, then Δλ = s 5 α p β δ cos p sc a sin p sb c 57 +O s 4 e t p b c+p c a wλ = s 5 α γ δ β cos p sc a sin p sb c 58 γ p +O s 4 e t p b c+p c a Case 4:If β =0and α =0, then Δλ = s 4 α p p β δ sin p sc a sin p sb c 59 +O s 3 e t p b c+p c a wλ = s 4 α β γ δ sin p sc a sin p sb c 60 γ p p +O s 3 e t p b c+p c a. Corollary 3.6 The eigenvalues of the problem -5 is bounded below. Proof. Putting s = it t > 0 in the above formulas it follows that Δ t and w t as t. Consequently, Δλ 0 and wλ 0 for λ negative and sufficiently large in module. 4 Asymptotic Formulae for Eigenvalues and Eigenfunctions We are now ready to find the asymptotic approximation formulas for the eigenvalues of the considered problem -5.

11 Sturm-Liouville type problem 583 Since the eigenvalues are coincided with the zeros of the entire functions Δλ and wλ, it follows that they have no finite limits. Moreover, all eigenvalues are real and bounded below by the Corollaries. and 3.6. Therefore, we may renumber them as λ 0 λ λ which counted according to their multiplicity. Below we shall denote λ n = s n. Theorem 4. 9 Theorem 5 The boundary-value-transmission problem -5 has an precisely denumerable many real eigenvalues, whose behaviour may be expressed by two sequence { } { λ n and λ n} with following asymptotics as n : Case :If β 0and α 0, then s n = n 5 + O, s n p b c n = n + + O 6 p c a n Case :If β 0and α =0, then s n = p b c n + + O n, s n = p c a n + O n 6 Case 3:If β =0and α 0, then s n = p b c n + O n, s n = p c a n + + O n 63 Case 4:If β =0and α =0, then s n = p b c n + O, s n = n p c a n + O. 64 n Asymptotic approximations formulae for eigenfunctions are given as follows. Let β 0and α 0Case. By putting 6in the 45and 46we get φ λ n x = α n 5 p b c φ λ n = α n + p c a and φ λ n x = α δ p γ p b c +O n φ λ n x = α δ p γ p c a cos n 5 cos n + n 5 3 cos n 5 p x a + O n p b c x a + O n c a p c a sin n 5 p b c n + 3 cos n + sin n + x c b c p x c + O n. p c a

12 584 Z. Adoğan, M. Demirci and O. Sh. Muhtarov Hence, the eigenfunctions φx, λ n and φx, λ n have the following asymptotic representations for Case β 0and α 0: α n p b c 5 cos n 5 px a p b c + O n, for x a, c φx, λ n= p α δ n 3 γ p b c 5 cos n 5 pc a p b c sin n 5 x c b c +O n, for x c, b 65 p c a n + cos n + n + x a c a + O n, for x a, c + O n, for x c, b φx, λ n = α A n β p c a cos n + px b p c a 66 Similar formulae in the other cases are as follows: Case :If β 0 and α = 0, then p α n p b c + sin n + px a p b c + O, for x a, c φx, λ n = p α δ n 3 γ p b c + sin n + pc a p b c sin n + x c b c +O n, for x c, b 67 φx, λ p α p c a n sin n x a c a + O, for x a, c n= A n β p c a n cos n px b p c a + O n, for x c, b 68 Case 3:If β = 0 and α 0, then α p b c n cos n px a p b c + O n, for x a, c φx, λ 3 n= p α δ γ p b c n cos n pc a p b c sin n x c b c +O n, for x c, b 69 φx, λ α n p c a + cos n + x a c a + O n, for x a, c n= A nβ n p p c a + sin n + px b p c a + O, for x c, b 70 Case 4:If β = 0 and α = 0, then α p p b c n sin n px a p b c + O, for x a, c φx, λ n = α δ p p γ p b c n sin n pc a p b c sin n x c b c +O n, for x c, b 7

13 Sturm-Liouville type problem 585 α n φx, λ n = p p c a sin n x a c a + O, for x a, c A nβ 7 n p p c a sin n px b p c a + O, for x c, b where A n is bounded by O n, and φλ x are linearly dependent in the case when λ 0 is an eigenvalue. Therefore, χ λ x is used instead of φ λ x in the interval x c, b for the functions of φx, λ n. All this asymptotic approximations are hold uniformly for x. Remar : We can show that the solutions φx, λ 0 and χx, λ 0 are linearly dependent in the case when λ 0 is an eigenvalue. Therefore, the asymptotic approximations of χx, λ n and χx, λ n do not require individual consideration. 5 Asymptotic Formulas for Normalized Eigenfuctions It is evident that the four-component vectors Φ n := φ λ n x,b a φ λ n,b b φ λ n,t c φ λ, n n =0,,, are the eigenelements of the operator A corresponding to the eigenvalues λ n. For n m, Φ n, Φ m =0 n, m =0,,, since A is symmetric. Denoting ψ λ n x := φ λ x n Φ n 75 it is easily seen that the eigenelements. Ψ n := ψ λ n x,b aψ λ n,b bψ λ n,t cψ λ n, n =0,,, are orthonormal that is where δ j i is the ronecer delta. AΨ n = λ n Ψ n and Ψ n, Ψ m = δ j i 77 Lemma 5. The following asymptotic equalities holds: Case :If α 0, then B aφ λ n =O, B bφ λ n =O n and T cφ λ n =O n 78 Case :If α =0, then B aφ λ n =O, B bφ λ n =O n and T cφ λ n =O n. 79

14 586 Z. Adoğan, M. Demirci and O. Sh. Muhtarov Proof. From the formulas, 3 and 5 we get B a u = λ B au, B b u = λ B bu and T c u = λ T cu. 80 then B a φ λ n = B a φ λ n = λ B aφ λ n B bφ λ n = B bφ λ n = λ B bφ λ n T c φ λ = n λ T cφ λ n 8 from Lemma 7 in 9 we have for α 0 B a φ λ n = α φ λ n a α φ λ a n = O s and for α =0 B b φ λ n = β φ λ n b β φ λ b n = O s 4 T c φ λ n = γ φ λ c +0 γ φ n λ c 0 + δ uc n = O s 4 B a φ λ n =O,B b φ λ n =O s 3 and T c φ λ n =O s 3. 8 Now applying Theorem in 9 we have for α 0 for α =0 B a φ λ n =O n,b b φ λ n =O n 4 and T c φ λ n =O n 4 83 B a φ λ n =O,B b φ λ n =O n 3 and T c φ λ n =O n Substituting 83 and 84 into 80, and taing into account Theorem in 9 the proof is complete.

15 Sturm-Liouville type problem 587 Theorem 5. Let Φ n be defined as in 73. Then the following asymptotic formulas hold for Case :If β 0and α 0, then α Φ n = δ n 5 3 γ p b c cos n 5 p c a b c + O n 85 p b c Case :If β 0and α =0, then γ α Φ n = δ n + 3 γ p b c sin n + p c a b c + O n 86 p b c Case 3:If β =0and α 0, then Φ n = α 3 γ δ n γ p b c cos n p c a b c + O n 87 p b c Case 4:If β =0and α =0, then γ α Φ n = δ n p γ p b c sin n p c a b c + O n 88 p b c Proof. Let β 0 and α 0. In this case, using Asymptotic approximation formula of the eigenfunction φx, λ n we have c p γ c φλ x n dx = p γ α n 5 p b c a a cos n 5 p x a + O n dx p b c c = p γ α n 5 4 p b c a cos n 5 p x a dx + O n p b c = p γ α n 5 4 c a+o 89 n 3 p b c

16 588 Z. Adoğan, M. Demirci and O. Sh. Muhtarov similarly, we have p γ b c φλ n dx = δ α n 5 6 γ p b c cos n 5 p c a b c+o n p b c Using 89, 90 and Lemma 5. we get consequently, Φ n = δ α γ cos n 5 6 p b c n 5 p c a p b c b c+o n 5 Φ n = δ α n 5 3 γ p b c cos n 5 p c a b c+o n p b c which proves the formula 85. The proofs of the other formulas are similar. Theorem 5.3 The first components of the normalized eigenelement 76 have the following asymptotic representation as n : Case :If β 0and α 0, then ψ λ n x = γ δ p b c cos n 5 p x a p b c n 5 cos n 5 p c a p b c b c +O n, for x a, c 9 p b c sin n 5 x c b c + O n, for x c, b ψ λ n x = α cosn+ x a c a γ p α c a+γ p β A nb c β A n cosn+ p x b p c a γ p α c a+γ p β A nb c + O n for x a, c + O 9 n, for x c, b

17 Sturm-Liouville type problem 589 Case :If β 0and α =0, then γ γ p δ sin n+ p x a p b c p ψ λ n x = b c n+ sin n+ p b c c a p b c +O n, for x a, c 3 γ p b c sin n + x c b c + O n, for x c, b 93 α sinn x a c a ψ λ x = p β An b c + O n, for x a, c γ n p c a n Anβ n cos p x b p c a p β An γ b c + O n, for x c, b 94 Case 3:If β =0and α 0, then γ γ δ cos n p x a p b c n cos n p b c c a p ψ λ n x = b c p b c +O n, for x a, c p b c sin n x c b c + O n, for x c, b 95 α cosn+ x a c a ψ λ x = p α + O γ c a n, for x a, c n Anβ n+ sin p x b p c a p p α γ c a p c a n+ + O 96 n, for x c, b Case 4:If β =0and α =0, then sin n p x a p b c γ γ δ n sin n p b c c a p ψ λ n x = b c p b c +O n, for x a, c γ p b c sin n x c b c + O n, for x c, b 97 α sinn x a c a + O ψ λ x = p γ α c a+γ β A n b c n, for x a, c n Anβ n sin p x b p c a + O 98 p γ α c a+γ β A n b c n, for x c, b Each of this asymptotic equalities hold uniformly for x.

18 590 Z. Adoğan, M. Demirci and O. Sh. Muhtarov Proof. Let β 0 and α 0. In this case, from 85 it follows that Φ n = γ α δ 3 n 5 p b c cos n 5 p c a p b c b c +O n 99 putting asymptotic approximation formula for the eigenfunction φx, λ n, 99 into 75 we find the required asymptotic formula 9. Similarly, we can drive the other formulas. References Adoğan, Z. Demirci, M. and Muhtarov, O. Sh., Discontinuous Sturm- Liouville Problem with Eigenparameter-Dependent Boundary and Transmission Conditions, Acta Applicandae Mathematicae, , Binding, P.A., Browne, P. J., and Watson, B.A., Strum-Liouville problems with boundary conditions rationally dependent on the eigenparameter II, Journal of Computational and Applied Mathematics, 48 00, Fulton, C.T., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc.Roy.Soc.Edin., 77A 977, Kerimov, N.B. and Memedov, Kh.K., On a boundary value problem with a spectral parameter in the boundary conditions, Sibirs. Math. Zh., ,no ,. English translation: Siberian Math. J. 40, no, 8-90, Liov, A. V. and Mihailov, Yu. A., The Theory of Heat and Mass Transfer, Qosenergaizdat, 963 Russian. 6 Muhtarov, O. Sh., Discontinuous boundary-value problem with spectral parameter in boundary conditions, Turish Journal of Mathematics, 8994, Muhtarov, O. Sh. and Demir, H., Coerciveness of the discontinuous initial-boundary value problem for parabolic equations, Israel Journal of Mathematics, 4, 999, Muhtarov, O. Sh., Kandemir, M. and Kuruoğlu, N., Distribution of Eigenvalues for the discontinunous boundary value problem with functional manypoint conditions, Israel Journal of Mathematics, 9 00,

19 Sturm-Liouville type problem 59 9 Muhtarov, O. Sh. and Yaubov, S., Problems for Ordinary Differential Equations with Transmission Conditions, Applicable Analysis, 8 00, Rasulov,M.L., Metods of Contour Integratian, North-Holland, Amsterdam,967. Rasulov,M.L., Application of the Contour Integral Method, Naua, Moscow,975 Russian. Shaliov, A. A., Boundary Value Problems For Ordinary Differential Equations with a Parameter in Boundary Conditions, Trydy Sem. Imeny I. G. Petrowsgo, 9 983, 90-9 Russian. 3 Tihonov, A.N. and Samarsii, A.A., Equations of Mathematical Physics, Oxford and New Yor, Pergamon, Titchmarsh, E.C., Eigenfunction expensions associated with second order differential equations I, nd edn London: Oxford Univ. Press., Titeux, I. and Yaubov, Ya., Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients, Math. Models Methods Appl. Sc., , Voitovich, N.N., Katsenelbaum, B.Z. and Sivov, A.N., Generalized method of eigen-vibration in the theory of Diffraction, Naua, Mosow, 997 Russian 7 Yaubov,S., Completeness of Root Functions of Regular Differential Operators, Logman, Scientific and Technical, New Yor, Yaubov, S. and Yaubov, Y., Abel basis of root functions of regular boundary value problems, Math. Nachr., , Yaubov,S. and Yaubov,Y., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton London New Yor Washington, D.C, 000. Received: April 7, 007