AN INVERSE PROBLEM FOR STURM-LIOUVILLE TYPE DIFFERENTIAL EQUATION WITH A CONSTANT DELAY. 1. Introduction

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.3 6, No., 07, DOI: /SJM AN INVERSE PROBLEM FOR STURM-LIOUVILLE TYPE DIFFERENTIAL EQUATION WITH A CONSTANT DELAY ELMIR ČATRNJA, MILENKO PIKULA Abstract. The topic of this paper is a direct ad iverse boudary spectral problems defied by the secod order differetial equatio with costat delay y x + qxyx = λyx where [ π, π ad qx L [0, π] ad boudary coditios y 0 hy0 = 0 ad y π + Hyπ = 0. We will establish properties of the spectral characteristics ad research the iverse problem of recoverig operator parameters from their spectra obtaied by varyig right side boudary coditio. We will prove that the potetial, ad boudary coditios are uiquely determied by the two spectra. We will also prove that the delay is recovered from oe spectrum.. Itroductio The iverse problems i the spectral theory of operators, especially differetial operators, have bee studied sice the 930s. Ambarzumja s paper [] is the first paper i this area, while boo [] gives deeper isight i this topic. Itroducig a deviatig argumet ito this problem, opeed space for may mathematicias to further examie a whole ew set of equatios. A separate chapter of these studies deals with the iverse problem related to the boudary problems of the geerated equatios with a costat delay. I this paper we study differetial operators of Sturm-Liouville-type geerated with secod order differetial equatios with a costat [ delay. π, π. We solve the iverse spectral problem for these operators whe The iverse problem of classical Sturm-Liouville operators is fully solved. The solutio ca be foud i [4]. The methods used for solvig the classical problem trasformatio operator method, method of spectral mappigs ad others do ot 00 Mathematics Subject Classificatio. 34B4, 34A55. Key words ad phrases. Differetial operators with delay, iverse problem, Fourier trigoometric coefficiets, Volterra itegral equatio. Copyright c 07 by ANUBIH.

2 98 ELMIR ČATRNJA, MILENKO PIKULA give solutio for the problems with a costat delay. Because of that, the iverse problem of secod order differetial operators with costat delay is still ot solved. I this paper we study the spectral boudary problem defied by the followig equatio y x + qxyx = λyx, ad the boudary coditios y 0 hy0 = 0. y π + Hyπ = 0, 3 We will also tae qx 0, for x [0,. 4 The case of Dirichlet boudary coditios ad costat delay is solved i [9] ad this paper is its atural cotiuatio. The quadruplet qx, h, H, defies the boudary spectral problem defied by 4. It ca be show that the equatio with the boudary coditio is equivalet with the Volterra itegral equatio yx, z = z cos πz + h si πz + qt si zx tyt, zdt, 5 z 0 where z = λ. Havig i mid that π of equatio 5 < π usig the step method we obtai the solutio yx = z cos zx + h si zx + qt si zx t cos zt dt From 6 we easily get + h qt si zx t si zt dt. z y x = z si zx + hz cos zx + z qt cos zx t cos zt dt h qt cos zx t si zt dt. 6 7

3 AN INVERSE PROBLEM FOR STURM-LIOUVILLE TYPE DIFFERENTIAL EQUATION 99 Itroducig the followig fuctios a s x, z = qt si zx t si zt dt, a sc x, z = qt si zx t cos zt dt, a cs x, z = qt cos zx t si zt dt, a c x, z = qt cos zx t cos zt dt, ad isertig 6 ad 7 ito the coditio 3, we obtai the characteristic fuctio F z of the boudary value problem defied by 4 F z = z + hh z si zπ + h + H cos zπ + a c z + h z a csz + H z a scz + hh z a s z. Here we wrote a s z istead of a s π, z, a sc z istead of a sc π, z, etc. Trasformig itegrals a s z, a sc z, a cs z ad a c z we get 8 a s z = I cos zπ + ã c z, a cs z = I si zπ ã s z, a sc z = I si zπ + ã s z, a c z = I cos zπ + ã c z, where ad ã c z = qθ cos zπ θdθ, ã s z = qθ si zπ θdθ, 0, 0 θ < qθ = q θ +, θ π 9 0, π < θ π I = ã c 0 = qθdθ = π qtdt.

4 00 ELMIR ČATRNJA, MILENKO PIKULA Now, characteristic fuctio ca be writte as follows F z = z + hh si πz + h + H cos πz + z I cos zπ + ã c z + h + H z I si zπ + H h z ã s z + hh z I cos zπ + ã c z. 0 Now, we will fid asymptotic behavior of its zeros. Firstly, we will fid fuctios C ad C so that asymptotic of zeros of characteristics fuctio F z ca be writte i the form z = + C + C C + o Usig elemetary trigoometric trasformatios it ca be easily show that the followig lemma holds. Lemma. Whe followig asymptotic behavior holds z si πz = + C π + C π C π + o, C π si πz = o z, cos πz = C π + O, cos z π = cos + C π C si + o ã ã c z = ã + O, si z π = C si + O, where ã s z = + b + O b., ã = qθ cos θ dθ, b = qθ si θ dθ. By usig this lemma together with λ = z, it is easily show that the followig theorem holds

5 AN INVERSE PROBLEM FOR STURM-LIOUVILLE TYPE DIFFERENTIAL EQUATION 0 Theorem. Eigevalues of Sturm-Liouville spectral problem with costat delay qx, h, H, have followig asymptotic behavior λ = +C 0 + I π cos + π ã+ C si +C si +o,, where C 0 = h + H, π C H + h = π I, C = π 4π I From 0 it is easily deduced that the characteristic fuctio F z is a whole fuctio, eve fuctio ad that it has uity growth i respect to variable z, so it ca be writte i the followig form applyig Hadamard factorizatio theorem F z = π λ 0 z λ z which will be used i the followig sectio. =,. Mai results I this sectio we will show that from the two spectra { λ j }, that match N 0 problems qx, h, H j,, j =, respectively, the iverse problem is uiquely solved. From the mai result of previous sectio we ow that eigevalues λ j asymptotic behavior λ j = +C j 0 +I π cos + π ã+ C j si + C si +o Aalogously as i [9] it ca be show that the followig teorem is valid Theorem. If {λ } N0 the where µ = From 3 we have have,. 3 is spectrum of boudary value problem qx, h, H,, = lim arccos µ, 4 λ+ + λ λ + + λ. Because C 0 C 0 = lim λ λ. C j 0 = π h + H j,

6 0 ELMIR ČATRNJA, MILENKO PIKULA we get ad H H = π C 0 C 0 = π lim λ λ. 5 It is possible to fid subsequeces i of sequece for which From 3 we get get I = π lim cos cos 0, cos 0 cos δ > 0,. λ λ cos cos From 3, bearig i mid that we already determied, H H ad I we C j 0 = lim λ l l I π. 6 cos, j =,. 7 Let s otice that C j 0 ca be recovered without previous owledge of I. Namely, we have C j 0 = lim λ cos λ cos cos cos Before we recover h, H ad H we must write characteristic fuctio F z i a somewhat differet form. From 8 we have Solvig for h we get h = si πz Replacig z = + h = lim H H, j =,. F z F z h z H H si πz + H H cos πz + H H I si zπ + o z z. [ F z F z z z cos πz I si zπ H H i 9 ad lettig we obtai F + + F + ] + o After we have recovered h from 0 we recover H ad H by I cos 8. 9 z +. 0 H j = π Cj 0 h j =,.

7 AN INVERSE PROBLEM FOR STURM-LIOUVILLE TYPE DIFFERENTIAL EQUATION 03 Let us defie the followig fuctios From 8 we have 3. Recoverig fuctio qx Az = H F z H F z + z si πz h cos πz, H H Bz = z F z F z h si πz z cos πz. H H Az = [I cos zπ ã s z] + h z [I si zπ ã s z] Bz = [I si zπ ã s z] h z [I cos zπ ã c z] Itegratig by parts we elimiate z from deomiators ad get ã s z z = si zπ I + J ã c z, z ã c z z = cos zπ I + J ã s z. z Now turs ito Az = [ I I cos zπ + ã c z + h z si zπ + I ] z si zπ J ã cz Bz = [ I si zπ + ã s z + h I z cos zπ + I ] z cos zπ J ã sz. Let A z = Az hi si zπ z Notice that B z 0, whe z 0. So we get I cos zπ, B z = Bz I si zπ A z = ã c z hj ã c z, B z = ã s z hj ã s z Taig z = m, m N 0 we obtai system A m = ã c m hj ã c m, B m = ã s m hj ã s m or A m = m B m = m+ qθ cos mθdθ h m qθ si mθdθ h m+ θ qθ dθ cos mθdθ θ qθ dθ si mπ θdθ.

8 04 ELMIR ČATRNJA, MILENKO PIKULA Let us defie sequeces A m ad B m i the followig way A m = m π A m = ã m hj ã m, B m = m+ π B m = b m hj b m. 3 System 3 relates cosie Fourier coefficiets of the fuctios qθ, with A m ad sie Fourier coefficiets of the fuctios qθ, θ qθ dθ θ qθ dθ with B m. Lemma. Sequeces {A m } m N0 ad {B m } m N0 are Fourier coefficiets of some fuctio f L [0, π]. From this lemma we have fx = A m cos mx + B m si mx. m= Fially, we coclude that the system 3 is equivalet to itegral equatio qx = fx + h 0 qθdθ. 4 Equatio 4 is liear Volterra itegral equatio of secod type. Notice that the erel of this equatio is costat h. From [8] it follows that its uique solutio is give i the form qx = fx + h Herewith, we have solved the iverse problem. 0 Refereces e hxt ftdt. [] Ambarzumja V., Über eie Frage der Eigewerttheorie, Zeitshrift für Physi, -Bd.53., -S., [] G. Freilig ad V. Yuro, Iverse Sturm-Liouville problems ad their applicatios, NOVA Sciece Publishers, New Yor, 00 [3] G. Freilig G. ad V.A. Yuro, Iverse problems for Sturm-Liouville differetial operators with a costat delay, Applied Mathematics letters, 0 [4] B.M. Levita ad I.S. Sargsja, Sturm-Liouville ad Dirac Operators, Naua, Moscow,988; Eglish trasliteratio: Kluwer Academic Publishers, Dordrecht, 99 [5] M. Piula, About determiatio of differetial operator with deviatig argumet, Matematica Motisigri, Vol VI, [6] M. Piula ad T. Marjaović, The costructio of the small potetial for a equatio of Sturm-Liouville type with costat delay, Proceedigs of the II Mathematical Coferece i Prištia, 996, 35 4

9 AN INVERSE PROBLEM FOR STURM-LIOUVILLE TYPE DIFFERENTIAL EQUATION 05 [7] M. Piula ad T. Marjaović, The regulatio idepedet of the potetial symmetrical to ceter [, π] for Sturm-Liouville operator with a costat delay, Facta Uiversitatis, Ser. Math. Iform. 4, 999, 9 [8] A. Polyai, A. Mazhirov, Hadboo of itegral equatios, CRC Press LLC, 998. [9] V. Vladičić ad M. Piula, A iverse problems for Sturm-Liouville-type differetial equatio with a costat delay, Sarajevo Joural of Mathematics, Vol., No., 06, [0] A. M. Zveri, G. A. Kamesii, S. B. Nori, ad L. È. Èl sgol c, Differetial equatios with deviatig argumet, Uspehi Mat. Nau 7 96, o. 04, Russia. MR #403 Received: Jue 6, 06 Revised: October 7, 06 Elmir Čatrja Faculty of Educatio Džemal Bijedić Uiversity of Mostar Uiverzitetsi ampus bb, Mostar Bosia ad Herzegovia elmir.catrja@umo.ba Mileo Piula Departmet of Mathematics,Iformatics ad Physics Uiversity East Sarajevo Alese Šatica, East Sarajevo Bosia ad Herzegovia mpiula@paleol.et