An Impulsive Sturm-Liouville Problem with Boundary Conditions Containing Herglotz-Nevanlinna Type Functions

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1 Appl. Math. Inf. Sci. 9, No. 1, ( Applied Mathematics & Information Sciences An International Journal An Impulsive Sturm-Liouville Problem with Boundary Conditions Containing Herglotz-Nevanlinna Type Functions A. Sinan Ozkan Department of Mathematics, Faculty Science, Cumhuriyet University, 5814, Sivas, Turkey Received: 26 Mar. 214, Revised: 26 Jun. 214, Accepted: 27 Jun. 214 Published online: 1 Jan. 215 Abstract: In this article, an impulsive Sturm Liouville boundary value problem with boundary conditions contain Herglotz Nevanlinna type rational functions of the spectral parameter is considered. It is shown that the coefficients of the problem are uniquely determined by either the Weyl function or by the Prüfer angle or by the classical spectral data consist of eigenvalues and norming constants. Keywords: Inverse Problem, Parameter Dependent Boundary Condition, Herglotz-Nevanlinna Function, Transfer Condition. 1 Introduction Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematical physics, mechanics, electronics, geophysics and other branches of natural sciences. The first inverse problem of a regular Sturm-Liouville operator was brought out by Ambarzumyan in 1929 [1]. However, the most important uniqueness theorem for inverse Sturm-Liouville problem was proved by G. Borg in 1945 [2]. Borg s result has been generalized to various versions until today. A large body of literature has built up, over the years, on problems of Sturm-Liouville type but where the boundary conditions depend on parameter. A boundary condition, rationally dependent on the spectral parameter, has the form a(λy(1b(λy (1, where a(λ and b(λ are polynomials. This equality, in the case when dega(λdegb(λ1, is said as affinely (or linearly dependent boundary conditions. Walter [28] and Fulton [15] have extensive bibliographies and we also refer to Fulton for some physical applications. Inverse problems for some classes of differential operators depending linearly on the parameter were studied in various publications (see [4], [5], [9], [11], [18] and [21]. The more general cases of the polynomials a(λ and b(λ are more difficult to investigate. There are several papers about the spectral problems for differential operators with the boundary conditions rationally dependent on the spectral parameter. ([6]-[8], [1], [14], [2], [22], [25]-[27] and [29]. Binding et al investigated direct and inverse spectral theory for Sturm-Liouville a(λ problem, when is a rational function of b(λ Herglotz Nevanlinna type such that f(λaλ b N f k λ g k, in one boundary condition in [6] and [7]. Spectral problems arising in mechanical engineering and having boundary conditions depending on the spectral parameter can be found in the classical textbook [12] of Collatz. Furthermore, these kinds of problems appear among others in connection with accoustic wave propagation in a rectangular duct with a uniform meanflow profile and walls with finite accoustic impedance [19]. Boundary value problems with transfer conditions inside the interval often appear in applied sciences. Spectral problems for differential operator with the transfer conditions have been studied in [3], [13], [17], Corresponding author asozkan58@gmail.com c 215 NSP

2 26 A. Sinan Ozkan: An Impulsive Sturm-Liouville Problem with... [18] and [21]-[24] where further references and links to applications can be found. The aim of this paper is to present various uniqueness theorems for an inverse Sturm Liouville problem with eigenparameter-dependent-boundary conditions and transfer conditions. We are studied in two types of generalizations of classical Sturm-Liouville problems: First, both of the boundary conditions depend on λ by Herglotz Nevanlinna functions. Second, we have two transfer conditions depending linearly on λ. The conditions of the considered problem have an important place in the class of parameter-dependent conditions. In this case, not only eigenvalues and eigenfunctions of the problem are real, but it is also possible to define norming constants which play an important role in inverse spectral theory. 2 Preliminaries Let us consider the boundary value problem L, generated by the regular Sturm Liouville equation 2 ly : y q(xyλ y, x Ω (d i,d i1 (1 i subject to the boundary conditions U 1 (y : y ( f 1 (λy( (2 U 2 (y : y (1 f 2 (λy(1 (3 and two transfer conditions y(d i α i y(d i y (d i αi 1 y (d i (β i λ γ i y(d i, i1,2 where λ is the spectral parameter; q(x is a real valued function in L 2 (,1; α i, β i and γ i are real numbers, α i >, β i >, d <d 1 < d 2 < d 3 1. We assume that f 1 (λ and f 2 (λ are rational functions of Herglotz Nevanlinna type such that f j (λa j λ b j N j f jk λ g jk, j 1,2, where a j, b j, f jk, g jk are real numbers, <, <, >, >, g j1 < g j2 <...<g jnj. It should be noted that, if f j (λ then the condition (2 and/or (3 are interpreted as the Dirichlet conditions y( y(1. Consider the space H L 2 (,1 C 1 C 1 C 2 and an element in H such that Y (y(x,u,v,w, u (u 1,u 2,...,u N1,u N1 1, v(v 1,v 2,...,v N2,v N2 1, w(w 1,w 2. H is a Hilbert space with the inner product defined by (4 Y,Y : y(xz(xdx u k u k u 1u 1 u 1u 1 α 1w 1 w 1 α 2w 2 w 2 v k v k (5 for Y (y(x,u,v,w, Y (z(x,u,v,w in H. Here, s denotes complex conjugate of the component s. Define the operator T on the domain D(T Y H : i y(x and y (x are absolutely continuous in Ω, ly L 2 (,1; ii u N1 1 y(, v N2 1 y(1; iii w i β i y(d i ; iv y(d i α i y(d i, i1,2, such that T(Y(ly(x,Tu,Tv,Tw (6 where Tu(Tu i, Tv(T v i, Tw(Tw i, g 1i u i f 1i y(, i1, Tu i y ( b 1 y( u k, i 1 g 2i v i f 2i y(1, i1, T v i y (1 b 2 y(1 v k, i 1 Tw i y (d i αi 1 y (d i γ i y(d i, i1,2(9 The following theorem can be proven by using same methods in [7] or [24]. Theorem 1.The eigenvalues of the operator T and the problem L coincide. where It is clear that f j (λ can be written as follows: a j (λ (a j λ b j b j (λ N j N j f j (λ a j(λ b j (λ N j N j f jk i1,i k (7 (8 (1 ( λ g jk (11 (λ g ji, ( λ g jk, j 1,2. (12 Let the functions ϕ(x,λ and ψ(x,λ be the solutions of (1 under the initial conditions ϕ(,λ b 1 (λ, ϕ (,λ (λ (13 ψ(1,λ b 2 (λ, ψ (1,λ (λ (14 c 215 NSP

3 Appl. Math. Inf. Sci. 9, No. 1, (215 / 27 and the conditions (4. It can be proven that ϕ(x,λ and ψ(x,λ also satisfy the following equalities: ϕ 1 (x,λ, x<d 1 ϕ(x,λ ϕ 2 (x,λ, d 1 < x<d 2 (15 ϕ 3 (x,λ, x>d 2 ϕ ϕ 1 (x,λ, x<d 1 (x,λ ϕ 2 (x,λ, d 1 < x<d 2 ϕ 3 (x,λ, x>d 2 ϕ 1 (x,λ λ N11/2 sin λ xo ( λ expτx ϕ 2 (x,λ β [ 1 2 λ 1 cos λ x cos ] λ(2d 1 x ( O λ N11/2 expτx (16 ϕ 3 (x,λ β [ 2 λ 3/2 sin λ x sin λ(2d 1 x 4 sin λ(2d 2 x sin ] λ(2d 2 2d 1 x O ( λ N11 expτx ϕ 1(x,λ λ N11 cos ( λ xo λ N11/2 expτx ϕ 2(x,λ β [ 1 2 λ 3/2 sin λ xsin ] λ(2d 1 x O ( λ N11 expτx ϕ 3(x,λ β [ 2 λ 2 cos λ xcos λ(2d 1 x 4 cos λ(2d 2 x cos ] λ(2d 2 2d 1 x ( O λ N13/2 expτx O ( λ N23/2 expτ(1 x, x<d 1 ψ(x,λ O ( λ N21 expτ(1 x, d 1 < x<d 2 (17 O ( λ N21/2 expτ(1 x, x>d 2 O ( λ N22 expτ(1 x, x<d 1 ψ (x,λ O ( λ N23/2 expτ(1 x, d 1 < x<d 2 O ( λ N21 expτ(1 x, x>d 2 (18 where τ Im λ. Consider the function : W(ϕ,ψ (19 (λϕ(1,λ b 2 (λϕ (1,λ b 1 (λψ (,λ (λψ(,λ and the sequence ρ n : ϕ 2 (x,λ n dx (2 f 1(λ n ϕ 2 (,λ n f 2(λ n ϕ 2 (1,λ n α 1 ϕ 2 (d 1,λ n α 2 ϕ 2 (d 2,λ n. is an entire function and its zeros, namely λ n n are eigenvalues of L. Since a j (g jk and b j (g jk for each j 1,2 and k 1,2,...,N j, g jk is an eigenvalue if and only if ϕ( j 1,g jk, i.e., (g jk. Lemm.i The eigenvaluesλ n n are real numbers. ii The equality (λ n ρ n s n is valid for all n, where s n ψ(,λ n b 1 (λ n ψ (,λ n (λ n (21 Proof.i It is sufficient to prove that the eigenvalues of T are real. For Y in D(T, we calculate using integration by part that TY,Y ly(xy(xdx T v k v k T v 1v N2 1 α 1Tw 1 w 1 α 2Tw 2 w 2 Tu k u k Tu 1u N1 1 ( y (x 2 q(x y(x 2 dxb 1 y( 2 b 2 y(1 2 g 1k N1 2Re u k 2 y(u k g 2k v k 2 y(1v k α 1 γ 1 y(d 1 2 α 2 γ 2 y(d 2 2 Therefore, it can be concluded that TY,Y is real for each Y in D(T. This completes the proof of (i. ii Let λ n g ik and ϕ(x,λ n be the eigenfunction which corresponds to the eigenvalue λ n. The equation (1 can be written for ϕ(x,λ n and ψ(x,λ as follows ψ (x,λq(xψ(x,λλ ψ(x,λ, ϕ (x,λ n q(xϕ(x,λ n λ n ϕ(x,λ n If these equations are (i: multiplied by ϕ(x,λ n and ψ(x, λ, respectively; (ii: subtracted from each other and (iii: integrated over the interval [, 1], the following c 215 NSP

4 28 A. Sinan Ozkan: An Impulsive Sturm-Liouville Problem with... equality is obtained: d [ ϕ (x,λ n ψ(x,λ ψ (x,λϕ(x,λ n ] (22 (λ λ n ψ(x,λϕ(x,λ n dx The initial conditions (13, (14 and the transfer conditions (4 are used to get b 1 (λ b 1 (λ n λ λ n ψ(x,λϕ(x,λ n dx [ f 1(λ f 1 (λ n ] ϕ(,λ n ψ(,λ λ λ n [ f 2(λ f 2 (λ n ] ϕ(1,λ n ψ(,λ λ λ n α 1 ψ(d 1,λϕ(d 1,λ n α 2 ψ(d 2,λϕ(d 2,λ n Letting limit as λ λ n it can be obtained that (λ n ρ n s n (23 If g 1k and/or g 2k are the eigenvalues, ϕ(,g 1k and/or ϕ(1,g 2k, so we see validity of (ii. It is concluded from Lemma-1 that all eigenvalues of L are simple zeros of. 3 Uniqueness Theorems Together with L, consider the problem L :. ly : y q(xyλ y, x Ω 2 ( d i, d i1 (24 i Ũ 1 (y : y ( f 1 (λy( (25 Ũ 2 (y : y (1 f 2 (λy(1 (26 y( d i α i y( d i y ( d i α i 1 y ( d i (27 ( βi λ γ i y( d i, i1,2 It is assumed in what follows that if a certain symbol s denotes an object related to L, then the corresponding symbol s denotes the analogous object related to L. We consider three statements of the inverse problem of the reconstruction of the boundary-value problem L; i by the Weyl function; ii by Prüfer s angle; iii by the spectral dataλ n,ρ n n andλ n, µ n n. 3.1 By the Weyl Function Denote m(λ : ψ(,λ (28 The function m(λ is called Weyl function of the boundary value problem L. It is clear that m(λ is a meromorphic function with poles inλ n n. Let s(x,λ and c(x,λ be the solutions of (1 that satisfy the conditions s(,λc (,λ; c(,λs (,λ1 (29 and (4. The following equalities are valid: ϕ(x,λ (λs(x,λb 1 (λc(x,λ (3 ψ(x,λ 1 [s(x,λm(λϕ(x,λ] b 1 (λ (31 Theorem 2.If m(λ m(λ and f 1 (λ f 1 (λ then L L, i.e., q(x q(x, almost everywhere in Ω; f 2 (λ f 2 (λ and(α i,β i,γ i ( α i, β i, γ i, i1,2. Proof.Let us define the functions P 1 (x,λ and P 2 (x,λ as follows, P 1 (x,λ ϕ(x,λ ψ (x,λ P 2 (x,λ ϕ(x,λ ψ(x,λ ϕ (x,λ ψ(x,λ ϕ(x,λ ψ(x,λ (32 (33 Since f 1 (λ f 1 (λ, ã 1, b 1 b 1,, g 1k g 1k, so (λ ã 1 (λ and b 1 (λ b 1 (λ. From the hypothesis m(λ m(λ and the equalities (3-(33, we get P 1 (x,λ s (x,λc(x,λ s(x,λ c (x,λ P 2 (x,λ s(x,λ c(x,λ s(x,λc(x,λ Therefore P 1 (x,λ and P 2 (x,λ are entire functions of λ. Denote G δ λ : λ k 2, k >δ, λ n n,1,2,... and G δ λ : λ k 2, λn k > δ, n,1,2,... where δ is sufficiently small number. One can easily show by using (15-(19, (32 and (33 that the relations P 1 (x,λ C δ, P 2 (x,λ C δ λ 1/2 (34 hold for λ G δ G δ. Thus, P 1 (x,λ is a function, namely A(x, depends only on x and P 2 (x,λ. Use (32 and (33 again to take ϕ(x,λa(x ϕ(x,λ, ψ(x,λ A(x ψ(x,λ (35 c 215 NSP

5 Appl. Math. Inf. Sci. 9, No. 1, (215 / 29 Since [ [ W ϕ(x,λ, ψ(x,λ ] ] 1 and similarly W ϕ(x,λ, ψ(x,λ 1, then A 2 (x 1 and so ϕ 2 (x,λ ϕ 2 (x,λ. From the asymptotic formula (15 we have d 1 d 1 and d 2 d 2. Suppose that A(x 1 for some x. In this case, we contradict using (15 to the assumptions <, > and >. Hence A(x 1 and ϕ(x,λ ϕ(x,λ and ψ (x,λ ψ(x,λ ψ (x,λ ψ(x,λ It can be obtained from (1, (4, (1 and (14 that q(x q(x, a.e. in Ω; f 2 (λ f 2 (λ and (a i,β i,γ i i1,2 (ã i, β i, γ i 3.2 By the Prüfer angle Denote φ(x,λ : i1,2. cot 1 ψ (x,λ ψ(x,λ if ψ(x,λ, tan 1 ψ(x,λ ψ (x,λ if ψ (x,λ, (36 (37 The function φ(,λ is called Prüfer angle. It can be calculated that the function Φ(x,λ : cotφ(x,λ is the solution of the following initial value problem: d dx Φ(x,λΦ2 (x,λq(x λ Φ(1,λ f 2 (λ Theorem 3.If φ(,λ φ(,λ and f 1 (λ f 1 (λ, L L; i.e. the Prüfer angle φ(,λ and the coefficient f 1 (λ together determine uniquely the problem L. Proof.It is obvious from (28 and (37 that the equality m(λ[ (λ b 1 (λcotφ(,λ] 1 holds. Therefore, under the hypothesis of the theorem m(λ m(λ. This completes the proof. 3.3 By the norming constants Now, we turn the notations in the structure of the operator T and make a relation between the norm of an eigenvector on the space H and the sequence ρ n, defined above. For an element Y (y(x,u,v,w in H, the norm of Y is defined by Y 2 : Y,Y. From (5, we get Y 2 y(x 2 dx u k 2 v k 2 (38 u 1 2 u 1 2 α 1 w 1 2 α 2 w 2 2. Lemm.Let λ n be an eigenvalue of T (or the problem L and Y n eigenvector for λ n. Then, the equality Y n 2 ρ n is valid. Proof.Let λ n g jk. The case λ n g jk requires minor modification in the following proof. Using the structure of D(T and the equalities (5-(9, a direct calculation yields 1 1 Y n 2 1 ϕ 2 (x,λ n dx u k 2 N2 v k 2 u 1 2 u 1 2 α 1 w 1 2 α 2 w 2 2 ϕ 2 (x,λ n dx u k 2 ϕ 2 (,λ n ϕ 2 (1,λ n N2 v k 2 α 1 ϕ 2 (d 1,λ n α 2 ϕ 2 (d 2,λ n ϕ 2 (x,λ n dx ϕ 2 (,λ n ϕ 2 (1,λ n [g 2k λ n ] 2 ϕ 2 (,λ n ϕ 2 (1,λ n [g 1k λ n ] 2 α 1 ϕ 2 (d 1,λ n α 2 ϕ 2 (d 2,λ n. 1 ϕ 2 (x,λ n dx ϕ 2 (,λ n ϕ 2 (1,λ n [g 2k λ n ] 2 [g 1k λ n ] 2 α 1 ϕ 2 (d 1,λ n α 2 ϕ 2 (d 2,λ n. 1 ϕ 2 (x,λ n dx f 1 (λ nϕ 2 (,λ n f 2 (λ nϕ 2 (1,λ n α 1 ϕ 2 (d 1,λ n α 2 ϕ 2 (d 2,λ n ρ n. Theorem 4.If λ n,ρ n n λn, ρ n and n f 1 (λ f 1 (λ then L L; i.e. the spectral data λ n,ρ n n and the coefficient f 1 (λ together determine uniquely the problem L. c 215 NSP

6 21 A. Sinan Ozkan: An Impulsive Sturm-Liouville Problem with... Proof.Recall that, (λ ã 1 (λ and b 1 (λ b 1 (λ when f 1 (λ f 1 (λ. Denote Γ n µ : µ ( λ n ε 2, where ε is sufficiently small number. Consider the contour integral F n (λ m(µ Γ n (µ λ dµ, λ intγ n. From (15-(19, it can be calculated that, m(µ C µ N1 1. Therefore, lim F n(λ. According to Lemma1, we obtain n m(µ m(λ Res (µ λ,λ n n n n ψ(,λ n (λ λ n (λ n b 1 (λ n ρ n (λ λ n (39 (4 (41 Consequently, if λ n λ n, ρ n ρ n for all n and f 1 (λ f 1 (λ then m(λ m(λ. Hence, Theorem2 yields L L. 3.4 By two given spectra We consider the boundary value problem L 1 with the condition y(,λ instead of (2 in L. Let ηn 2 n be the eigenvalues of the problem L 1. It is obvious that η n are zeros of 1 (η : ψ(,η. Theorem 5.If λ n,η n n λn, η n and f 1 (λ f 1 (λ then L L. n Proof.The functions and 1 (η which are entire of order 1, can be represented by Hadamard s factorization 2 theorem as follows C (1 λλn (42 n 1 (η C 1 n (1 ηηn, (43 where C and C 1 are constants which depend only on λ n and η n, respectively. Therefore, and 1 (η 1 (η, when λ n λ n and η n η n for all n. Consequently, the equality (28 yields m(λ m(λ. Hence, the proof is completed by Theorem2. 4 Conclusion The aim of this paper is to give uniqueness theorems for an inverse Sturm Liouville problem with eigenparameter-dependent-boundary conditions and transfer conditions. We are studied in two types of generalizations of classical Sturm-Liouville problems: First, both of the boundary conditions depend on λ by Herglotz Nevanlinna type functions; second, we have two transfer conditions depending linearly on λ. We prove that, if the coefficient f 1 (λ in the first boundary condition is known, the other coefficients of the boundary value problem L can be uniquely determined by each of the following: i The Weyl function M(λ; ii The Prüfer angle φ(, λ; iii The sequence λ n,ρ n consists of eigenvalues and norming constants; iv The sequenceλ n,η n consists of two given spectra. References [1] V. A. Ambartsumyan, Über eine Frage der Eigenwerttheorie, Z. Physik, 53, (1929. [2] G. Borg, Eine Umkehrung der Sturm Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math., 78, 1-96 (1946. [3] R.Kh. Amirov, A.S. Ozkan and B. Keskin, Inverse Problems for Impulsive Sturm-Liouville Operator with Spectral Parameter Linearly Contained in Boundary Conditions, Integral Transforms and Special Functions, 2, (29. [4] A. Benedek and R. Panzone, On inverse eigenvalue problems for a second-order differential equations with parameter contained in the boundary conditions, Notas Algebra y Analisis, 1 13 (198. [5] P.A. Binding, P.J. Browne and B.A. Watson, Inverse spectral problems for Sturm Liouville equations with eigenparameter dependent boundary conditions, J. London Math. Soc., 62, (2. [6] P.A. Binding, P.J. Browne, B.A. Watson, Sturm Liouville problems with boundary conditions rationally dependent on the eigenparameter, I, Proc.Edinburgh Math.Soc., 45, (22. [7] P.A. Binding, P.J. Browne, B.A. Watson, Sturm Liouville problems with boundary conditions rationally dependent on the eigenparameter, II, Journal of Computational and Applied Mathematics, 148, (22. [8] P.A. Binding, P.J. Browne and B.A. Watson,. Equivalence of inverse Sturm Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl., 291, (24. [9] P.J. Browne and B.D. Sleeman, A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems, Inverse Problems, 13, (1997. [1] A. Chernozhukova and G. Freiling, A uniqueness theorem for the boundary value problems with nonlinear dependence on the spectral parameter in the boundary conditions, Inverse Problems in Science and Engineering, 17, (29. [11] M.V. Chugunova, Inverse spectral problem for the Sturm Liouville operator with eigenvalue parameter dependent boundary conditions Oper. Theory: Adv. Appl. 123 (Basel: Birkhauser, (21. [12] L. Collatz, Eigenwertaufgaben mit Technischen Anwendungen, Akademische Verlag, Leipzig, (1963. c 215 NSP

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