Reconstruction of the Volterra-type integro-differential operator from nodal points

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1 Keski Boudary Value Problems 18 18:47 R E S E A R C H Ope Access Recostructio of the Volterra-type itegro-differetial operator from odal poits Baki Keski * * Correspodece: bkeski@cumhuriyet.edu.tr Departmet of Mathematics, Faculty of Sciece, Cumhuriyet Uiversity, Sivas, Turkey Abstract I this work, the Sturm Liouville problem perturbated by a Volterra-type itegro-differetial operator is studied. We give a uiqueess theorem ad a algorithm to recostruct the potetial of the problem from odal poits zeros of eigefuctios. MSC: 34A55; 34B7; 34B4; 34B37; 34K9; 34K1; 47G Keywords: Sturm Liouville equatio; Iverse odal problem; Itegro-differetial equatio 1 Itroductio We cosider the boudary value problem L geerated by the covolutio-type Sturm Liouville itegro-differetial operator y qxy x with boudary coditios Mx, ty t dt = y, x, π 1 y hy =, y πhyπ= 3 ad with the discotiuity coditios { y π =αy π, y π =α 1 y π, 4 where is the spectral parameter; α is a positive real costat; qx admx, t are realvalued fuctios from the class L, π adw 1, π, respectively. Without loss of geerality, we assume that π qxmx, x dx =. The first result of the iverse odal Sturm Liouville problem was give by McLaughli i [1]. I this work, she proved that the potetial of the cosidered problem ca be uiquely determied by a give dese subset of the zeros of the eigefuctios called The Authors 18. This article is distributed uder the terms of the Creative Commos Attributio 4. Iteratioal Licese which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial authors ad the source, provide a lik to the Creative Commos licese, ad idicate if chages were made.

2 KeskiBoudary Value Problems 18 18:47 Page of 8 odal poits. I 1989, Hald ad McLaughli studied more geeral boudary coditios ad gave some umerical schemes for the recostructio of the potetial from a give dese subset of odal poits []. Yag provided a algorithm to determie the coefficiets of the Sturm Liouville problem by usig the give odal poits i [3]. Iverse odal problems for differet types of operators have bee extesively well studied i several papers see [4 14]ad[15]. Iverse problems for itegro-differetial operators ad the other classes of olocal operators are more difficult to ivestigate. The classical methods are ofte ot applicable for such problems. I preset, the studies cocerig the perturbatio of a differetial operator by a Volterra-type itegral operator, amely the itegro-differetial operator, cotiue to be performed ad are begiig to have a sigificat place i the literature see [16 1], ad []. Iverse odal problem for this type of operator was first discussed by [3]. It is show i this study that the potetial fuctio ca be determied by usig odal poits, while the coefficiet of the itegral operator is kow. The iverse odal problem for Dirac-type itegro-differetial operators was first ivestigated by [4]. I this work, it is show that the coefficiets of the differetial part of the operator ca be determied by usig odal poits, ad odal poits also give partial iformatio about the itegral part. I the preset paper we ivestigate the iverse odal problem for Volterra-type itegrodifferetial operator. This type of operator has previously bee addressed i [5]ad[6]. Prelimiaries Let ϕx, be the solutioof 1 with the iitial coditios ϕ, =1, ϕ, =h 5 ad the ump coditios 4. We have the followig itegral equatios of the solutio of 1: for x < π, for x > π, ϕx, = cos x h si x x x si x t t si x t qtϕt, dt Mt, τϕ τ, dτ dt; 6 ϕx, =α cos x α cos π x h [ α si x si π x ] π/ [ π/ t x π/ x π/ α si x t α si π x t [ α si x t α si π x t si x t qtϕt, dt t ] qtϕt, dt ] Mt, τϕ τ, dτ dt si x t Mt, τϕ τ, dτ dt, 7

3 KeskiBoudary Value Problems 18 18:47 Page 3 of 8 where α ± = 1 α ± 1. By virtue of the above equatios, we have the followig asymptotic α relatios for sufficietly large :forx < π, ϕx, = cos x si x x x h qt dt Mt, t dt 1 o exp τ x ; 8 for x > π, ϕx, =α cos x α cos π x α si x 1 I 1xα I xo exp τ x, 9 where I 1 x =h x qt dt x Mt, t dt, I x = π/ qt Mt, t dt x π/ qt Mt, t dt,adτ = Im. Defie a fuctio as follows: :=ϕ π, Hϕπ,. 1 This etire fuctio is called a characteristic fuctio of the problem L ad the zeros of it areeigevalues ofthe problem L.Forsufficietlylarge, byvirtueof8ad9, we have the followig asymptotic formula: where { = α δ 1 si π cos π δ } o exp τ π, 11 δ 1 =h α H [ δ = α H α π π qtmt, t dt, π/ ] qtmt, t dt qtmt, t dt. It ca be easily show that the sequece { } satisfies the followig asymptotic relatio for : = μ 1 π o 1 ad 1 = 1 μ 1 3 π o, 3 where μ = δ 1 1 δ. Lemma 1 The eigefuctio ϕx, correspodig to the eigevalue has exactly zeros {x : 1, =, 1}, amely odal poits, i, π, such that <x < x1 < < x 1 < π

4 KeskiBoudary Value Problems 18 18:47 Page 4 of 8 ad the umbers {x } have the followig asymptotic formulae for sufficietly large : for x, π, x = 1/π 1/π 1/π 1/π δ 1 δ I 1x π o 1, =k, k Z, δ 1 δ I 1x o 1, =k 1,k Z; π 13 ad for x π, π, x = 1/π 1/π 1/π 1/π δ 1 δ α δ 1 δ α I 1 x α I x π o 1, =k, k Z, ρ δ 1 δ α δ 1 δ α I 1 x α I x o 1, =k 1,k Z, π ρ 14 where ρ = α 1 α. Proof By virtue of 8ad9, we get the followig asymptotic formula for eigefuctio ϕx, : ϕx, =cos x si x e τ x I 1 xo for x < π, ϕx, =α cos x α cos π x α si x I 1 x α si π x e τ x I xo for x < π for sufficietly large, uiformlyix. Sice the zeros of eigefuctios are odal poits, from ϕx, =,weget α cos x α cos π x α si x I 1 x α si π cos x I x α si x cos π I x e τ x o =, which implies that cot x α α cos π α si π I x = α si π α I 1 x α cos π I x e τ x o, cot x = α si π α I 1 x α cos π I x o e τ x α 1 α α cos π α si π I x α = 1 α si π α ρ I 1 x α cos π I x e τ x o,

5 KeskiBoudary Value Problems 18 18:47 Page 5 of 8 which is equivalet to π ta x = 1 α si π α ρ I 1 x α cos π I x e τ x o for x > π. Taylor s expasios formula for the arctaget yields x = 1/π 1 α si π α ρ I 1 x α cos π I x e τ x o. If we divide both sides of this equality by ad take accout of the asymptotic formula of,weget x = 1/π 1/π μ π 1 α μ α I 1 x 1 α I x ρ 1 o. The proofof 14is completed.equatio13 ca be proved similarly. Let = 1 be the set of zeros of eigefuctio, i.e., = {x : =k, k Z}, 1 = {x : =k 1, k Z}.Foreachfixedx, π, there exists a sequece x m m =,1,whichcovergestox. Therefore, from Lemma 1, we ca show that the followig fiite limits exist: where lim x 1 π lim x 1 π = f m x forx < π, 15 = g m x forx > π, 16 f m x= μ mx π I 1x forx < π, 17 g m x= μ mx π I 1xσ m for x > π, 18 where σ m = 1m α μ m I 1 π h ρ m.put F m x= { f m x forx < π, g m x forx > π. 19

6 KeskiBoudary Value Problems 18 18:47 Page 6 of 8 The followig theorem shows that if oe of qxormx, x is give, the the other oe ca be determied uiquely by usig a dese subset of the give odal set. Theorem 1 Thegivedesesubsetoftheodalset or 1 uiquely determies qx Mx, x, a.e. o, π ad the coefficiets h, H, ad α of the boudary ad discotiuity coditios. qxmx, x ad the costats h, H, ad α cabecostructedbythefollowig formulae: 1. Foreachfixedx, π, choose a sequece x, i.e., lim x = x;. Fid F m x from equatio 19 ad calculate h = F, μ = F πf F π F π, qxmx, x=f x μ π, F F π α = F F π, π π π I 1 = F πf F F, H = α μ hα I 1 π/ F α α. Example 1 Cosider the followig BVP: y qxy x Mx, ty t dt = y, x, π, y hy =, L : y πhyπ=, y π =αy π, y π =α 1 y π, where qxadmx, t are real-valued fuctios from the class L, π adw 1, π, respectively, ad h, H, α are ukow coefficiets we cofirmed o the assumptios of the problem L.Let{x } be the zeros of the eigefuctio of the cosidered problem i, π with the followig asymptotics: If x, π, x = 1/π 1/π 5 π si 1/ π 1 o. If x π, π, x = 1/π 1/π 5 π 1/ si π 6/5 1 o, the we ca calculate that F x= { 4x 5π si x for x < π, 4x 5π si x 4 5 for x > π.

7 KeskiBoudary Value Problems 18 18:47 Page 7 of 8 Accordig to Theorem 1, h = F =1, π π μ = F πf F F = 4 5, qxmx, x=f xμ = cos x, π F F π α = F F π =, H = α μ hα I 1 π/ F α α =. 3 Coclusio I this paper we have ivestigated the discotiuous iverse odal problem for Volterra type itegro-differetial operator. We showed that if oe of qx ormx, x isgive, the the other oe ca be determied uiquely by usig oly the give odal poits. Fudig Not applicable. Abbreviatios Not applicable. Availability of data ad materials Not applicable. Ethics approval ad coset to participate Not applicable. Competig iterests The author declares that he has o competig iterests. Authors cotributios The author read ad approved the fial mauscript. Publisher s Note Spriger Nature remais eutral with regard to urisdictioal claims i published maps ad istitutioal affiliatios. Received: 18 December 17 Accepted: 6 March 18 Refereces 1. McLaughli, J.R.: Iverse spectral theory usig odal poits as data a uiqueess result. J. Differ. Equ. 73, Hald, O.H., McLaughli, J.R.: Solutios of iverse odal problems. Iverse Probl. 5, Yag, X.-F.: A solutio of the odal problem. Iverse Probl. 13, Browe, P.J., Sleema, B.D.: Iverse odal problem for Sturm Liouville equatio with eigeparameter deped boudary coditios. Iverse Probl. 1, Buteri, S.A., Shieh, C.T.: Iverse odal problem for differetial pecils. Appl. Math. Lett., Buteri, S.A., Shieh, C.T.: Icomplete iverse spectral ad odal problems for differetial pecil. Results Math. 6, Cheg, Y.H., Law, C.-K., Tsay, J.: Remarks o a ew iverse odal problem. J. Math. Aal. Appl. 48, Guo, Y., Wei, G.: Iverse problems: dese odal subset o a iterior subiterval. J. Differ. Equ. 55, Law, C.K., She, C.L., Yag, C.F.: The iverse odal problem o the smoothess of the potetial fuctio. Iverse Probl. 151, Erratum: Iverse Probl. 17, Ozka, A.S., Keski, B.: Iverse odal problems for Sturm Liouville equatio with eigeparameter depedet boudary ad ump coditios. Iverse Probl. Sci. Eg. 38, Shieh, C.-T., Yurko, V.A.: Iverse odal ad iverse spectral problems for discotiuous boudary value problems. J. Math. Aal. Appl. 347, Yag, X.-F.: A ew iverse odal problem. J. Differ. Equ. 169,

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