A REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS STURM-LIOUVILLE OPERATOR WITH DELAYED ARGUMENT

Electronic Journal of Differential Equations, Vol. 217 217, No. 14, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu A REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS STURM-LIOUVILLE OPERATOR WITH DELAYED ARGUMENT MEHMET BAYRAMOGLU, AZAD BAYRAMOV, ERDOĞAN ŞEN Abstract. In this study, we obtain a formula for the regularized sums of eigenvalues for a Sturm-Liouville problem with delayed argument at two points of discontinuity. 1. Introduction We consider the boundary value problem for the differential equation y x + qxyx x + 2 yx =, 1.1 on, h 1 h 1, h 2 h 2,, with boundary conditions and transmission conditions y cos α + y sin α =, 1.2 y cos β + y sin β = 1.3 yh 1 = δyh 1 +, 1.4 y h 1 = δy h 1 +, 1.5 yh 2 = γyh 2 +, 1.6 y h 2 = γy h 2 + 1.7 where the real-valued function qx is continuous in, h 1 h 1, h 2 h 2, and has finite limits qh 1 ± = lim qx, qh 2 ± = lim qx, x h 1± x h 2± the real-valued function x is continuous in, h 1 h 1, h 2 h 2, has finite limits h 1 ± = lim x, h 2 ± = lim x, x h 1± x h 2± if x, h 1, then x x ; if x h 1, h 2, then x x h 1 ; if x h 2,, then x x h 2 ; where is a spectral parameter; h 1, h 2, α, β, δ, γ are arbitrary real numbers such that < h 1 < h 2 < and. The goal of this article is to calculate the regularized trace for the problem 1.1 1.7. 21 Mathematics Subject Classification. 47A1, 47A55. Key words and phrases. Delay differential equation; transmission conditions; regularized trace. c 217 Texas State University. Submitted June 16, 216. Published April 18, 217. 1

2 M. BAYRAMOGLU, A. BAYRAMOV, E. ŞEN EJDE-217/14 The theory of regularized trace of ordinary differential operators has long history. Gelfand and Levitan 4 first obtained the trace formula for the Sturm-Liouville differential equation. After this study several mathematicians were interested in developing trace formulas for different differential operators. The current situation of this subject and studies related to it are presented in the comprehensive survey paper 1. The regularized trace formulas for differential equations with the operator coefficients are found in 1, 2, 5, 6, 7, 12. However, there are a small number of works on the regularized trace for the differential equations with retarded argument; see 9, 13. Pikula 9 investigated the problem of second order: y + qxyx τ = y, y hy = y + Hy =, yx τ = yϕx τ, x τ, ϕ = 1 and obtained trace formula of first order: if τ, then n τ n 2 = qϕτ h2 + H 2 and if τ, then n=1 n=1 n τ n 2 2 cos nτ h + H + 2 = qϕτ h2 + H 2 + h + H 2 2 τ b 1 2 2τ + 4 8 qtdt. τ τ qtdt 2 qτ + q qtdt + 4 τ qτ Here the definition of b can be found in 9. Yang 13 obtained formulas of the first regularized trace for a discontinuous boundary value problem with retarded argument and with transmission conditions at the one point of discontinuity. As in this problem, if the retardation function in 1.1 and δ = 1, γ = 1 we have the formula of the first regularized trace for the classical Sturm-Liouville operator see 4 which is called Gelfand-Levitan formula 1. Note that, the trace formulas can be used in the inverse problems of spectral analysis of differential equations 1 and have applications in the approximate calculation of the eigenvalues of the related operator 3, 1. 2. A formula for the regularized trace First we give the asymptotic behavior of solutions for large values of the spectral parameter. Let ω 1 x, be a solution of 1.1 on, h 1, satisfying the initial conditions ω 1, = sin α and ω 1, = cos α. 2.1 These conditions define a unique solution of 1.1 on, h 1 8, p. 12. After defining the above solution, then we will define the solution ω 2 x, of 1.1 on h 1, h 2 by means of the solution ω 1 x, using the initial conditions ω 2 h 1, = δ 1 ω 1 h 1, and ω 2h 1, = δ 1 ω 1h 1,. 2.2 These conditions define a unique solution of 1.1 on h 1, h 2.

EJDE-217/14 TRACE FORMULA FOR A DELAY DIFFERENTIAL OPERATOR 3 After describing the above solution, then we will give the solution ω 3 x, of 1.1 on h 2, by means of the solution ω 2 x, using the initial conditions ω 3 h 2, = γ 1 ω 2 h 2, and ω 3h 2, = γ 1 ω 2h 2,. 2.3 These conditions define a unique solution of 1.1 on h 2,. Consequently, the function ωx, is defined on, h 1 h 1, h 2 h 2, by ω 1 x,, x, h 1, ωx, = ω 2 x,, x h 1, h 2, ω 3 x,, x h 2, 2.4 is a solution of 1.1 on, h 1 h 1, h 2 h 2,, which satisfies one of the boundary conditions and the transmission conditions 1.4 1.7 Then the following integral equations hold: ω 1 x, = sin α cos x cos α sin x 1 qτ sin x τω 1 τ τ, dτ, ω 2 x, = 1 δ ω 1h 1, cos x h 1 + ω 1h 1, sin x h 1 δ 1 h 1 qτ sin x τω 2 τ τ, dτ, ω 3 x, = 1 γ ω 2h 2, cos x h 2 + ω 2h 2, sin x h 2 γ 1 h 2 qτ sin x τω 3 τ τ, dτ. 2.5 2.6 2.7 Differentiating 2.5 and 2.6 with respect to x, we obtain ω 1x, = sin x cos α cos x Moreover qτ cos x τω 1 τ τ, dτ, ω 2x, = δ ω 1h 1, sin x h 1 + ω 1h 1, cos x h 1 δ h 1 qτ cos x τω 2 τ τ, dτ, ω 3x, = γ ω 2h 2, sin x h 2 + ω 2h 2, cos x h 2 γ qτ cos x τω 3 τ τ, dτ, h 2 ω 1 x, < const., x, h 1, 2.8 2.9 2.1

4 M. BAYRAMOGLU, A. BAYRAMOV, E. ŞEN EJDE-217/14 see 11. Then from 2.5 we have ω 1 x, = sin α cos x cos α sin α sin x qτ sin x τ cos τ τdτ + cos α 2 qτ sin x τ sin τ τdτ + O 1 3 = sin α cos x cos α sin α sin x 2 2 + cos α 2 2 Therefore, and ω 2 x, = sin α δ 2δ + cos α 2 2 δ ω 3 x, = sin α 2 + cos α 2 2 qτ sin x 2τ + τdτ cos α 2 2 qτ sin x τdτ qτ cos x τdτ qτ cos x 2τ + τdτ + O 1 3. 2.11 cos x cos α δ sin α sin x 2δ qτ sin x 2τ + τdτ cos α 2 2 δ qτ sin x τdτ qτ cos x τdτ qτ cos x 2τ + τdτ + O 1 3. 2.12 cos α cos x sin x sin α 2 qτ sin x 2τ + τdτ cos α 2 2 qτ sin x τdτ qτ cos x τdτ qτ cos x 2τ + τdτ + O 1 3. 2.13 The solution ωx, defined above is a nontrivial solution of 1.1 satisfying conditions 1.2 and 1.4 1.7. Putting ωx, in 1.3, we obtain the characteristic equation W ω, cos β + ω, sin β =. 2.14 The set of eigenvalues of boundary value problem 1.1 1.7 coincides with the set of the squares of roots of 2.14, and eigenvalues are simple. From 2.5-2.1 and 2.14 we obtain W { 1 1 γ δ sin α cos h 1 cos α sin h 1 1 qτ sin h 1 τω 1 τ τ, dτ cos h 2 h 1

EJDE-217/14 TRACE FORMULA FOR A DELAY DIFFERENTIAL OPERATOR 5 1 δ + 1 + 1 γ 1 sin α sin h 1 + cos α cos h 1 h2 h 1 qτ cos h 1 τω 1 τ τ, dτ sin h 2 h 1 δ qτ sin h 2 τω 2 τ τ, dτ cos h 2 sin α cos h 1 cos α sin h 1 qτ sin h 1 τω 1 τ τ, dτ 1 δ sin α sin h 1 + cos α cos h 1 + h2 h 1 sin h 2 h 1 qτ cos h 1 τω 1 τ τ, dτ cos h 2 h 1 qτ cos h 2 τω 2 τ τ, dτ sin h 2 1 { + 1 sin α cos h 1 cos α γ δ sin h 1 1 1 δ 1 h 2 qτ sin τω 3 τ τ, dτ qτ sin h 1 τω 1 τ τ, dτ sin α sin h 1 + cos α cos h 1 + h2 } cos β cos h 2 h 1 qτ cos h 1 τω 1 τ τ, dτ sin h 2 h 1 1 qτ sin h 2 τω 2 τ τ, dτ sin h 2 h 1 + 1 sin α cos h 1 cos α γ δ sin h 1 qτ sin h 1 τω 1 τ τ, dτ sin h 2 h 1 1 δ + sin α sin h 1 + cos α cos h 1 h2 h 1 qτ cos h 1 τω 1 τ τ, dτ cos h 2 h 1 qτ cos h 2 τω 2 τ τ, dτ cos h 2 h 2 qτ cos τω 3 τ τ, dτ } sin β,

6 M. BAYRAMOGLU, A. BAYRAMOV, E. ŞEN EJDE-217/14 which implies W = sin + sin β sin β γ sin β cos β cos β γ cos β h2 h 1 sinα β cos qτ cos τω 1 τ τ, dτ qτ cos τω 2 τ τ, dτ h 2 qτ cos τω 3 τ τ, dτ h2 h 1 qτ sin τω 1 τ τ, dτ qτ sin τω 2 τ τ, dτ h 2 qτ sin τω 3 τ τ, dτ. cos α cos β δ sin 2.15 From 2.11-2.13, we obtain ω 1 τ τ, = sin α cos τ τ cos α sin τ τ τ τ qt 1 sin τ τ t 1 dt 1 2 2 τ τ ω 2 τ τ, = sin α δ 2δ ω 3 τ τ, = sin α 2 2 τ τ qt 1 sin τ τ 2t 1 + t 1 dt 1 + O 1 3, 2δ cos τ τ cos α δ τ τ sin τ τ qt 1 sin τ τ t 1 dt 1, 2.16 2.17 qt 1 sin τ τ 2t 1 + t 1 dt 1 + O 1 3, 2.18 cos α cos τ τ τ τ τ τ sin τ τ qt 1 sin τ τ t 1 dt 1 qt 1 sin τ τ 2t 1 + t 1 dt 1 + O 1 3, 2.19

EJDE-217/14 TRACE FORMULA FOR A DELAY DIFFERENTIAL OPERATOR 7 Using 2.15 2.19 we obtain W Denote sinα β cos α cos β sin + cos sin qτcos τ + cos 2τ + τdτ 2 sin α cos β qτsin τ + sin 2τ + τdτ 2 cos α sin β + qτsin τ sin 2τ + τdτ 2 τ τ + qτ qt 1 sin τ t 1 dt 1 dτ 4 + 4 4 4 qτ τ τ qt 1 sin 2τ + τ + t 1 dt 1 dτ τ τ qτ qt 1 sin τ 2t 1 + t 1 dt 1 dτ τ τ qτ qt 1 sin 2τ + τ + 2t 1 t 1 dt 1 dτ + O 1 3. 2.2 A 1 = 1 2 B 1 = 1 4 B 2 = 1 4 C 1 = 1 4 C 2 = 1 4 E 1 = 1 4 E 2 = 1 4 qτ cos τdτ, A 2 = 1 qτ sin τdτ, 2 τ τ qτ qt 1 cos 2τ τ t 1 dt 1 τ τ qτ qt 1 sin 2τ τ t 1 dt 1 τ τ qτ qt 1 cos τ + t 1 dt 1 τ τ qτ qt 1 sin τ + t 1 dt 1 D 1 = 1 2 qτ cos 2τ τdτ, D 2 = 1 qτ sin 2τ τdτ, 2 τ τ qτ qt 1 cos τ + 2t 1 t 1 dt 1 τ τ qτ qt 1 sin τ + 2t 1 t 1 dt 1

8 M. BAYRAMOGLU, A. BAYRAMOV, E. ŞEN EJDE-217/14 H 1 = 1 4 H 2 = 1 4 τ τ qτ qt 1 cos 2τ τ 2t 1 + t 1 dt 1 τ τ qτ qt 1 sin 2τ τ 2t 1 + t 1 dt 1 dτ. Then from 2.2, we obtain Define W sinα β cos α cos β sin + cos sin A 1 cos + A 2 sin + D 1 cos + D 2 sin sin α cos β A 1 sin A 2 cos + D 1 sin D 2 cos cos α sin β + A 1 sin A 2 cos D 1 sin + D 2 cos + C 1 sin C 2 cos B 1 sin B 2 cos + E 1 sin E 2 cos H 1 sin H 2 cos + O 1 3. 2.21 W sin. 2.22 Denote by n, n Z, zeros of the function W, then we have n = n, n Z, and it is simple algebraically except for. Denote by C n the circle of radius, < ε < 1 2, centered at the origin n = n, n Z, and by Γ N the counterclockwise square contours with four vertices A = N + ε + N i, B = N ε + N i, C = N ε N i, D = N + ε N i, where i = 1 and N is a natural number. Obviously, if C n or Γ N, then W M e Im M > by using a similar method in 14. Thus, on

EJDE-217/14 TRACE FORMULA FOR A DELAY DIFFERENTIAL OPERATOR 9 C n or Γ N, from 2.21 and 2.22, we have W W = 1 + cot α cot β + A 1 + D 1 cot + A 2 + D 2 + cot α cot βa 2 cot α + cot βd 2 + C 2 B 2 + E 2 H 2, cot/ 2 + cot α cot β + cot β cot αa 1 + cot α + cot βd 1 2 + C 1 + B 1 E 1 + H 1 2 + O 1 3. Expanding ln W W ln W W by the Maclaurin formula, we find that = cot α cot β + A 1 + D 1 cot + A 2 + D 2 + cot α cot βa 2 cot α + cot βd 2 + C 2 B 2 + E 2 H 2 cot/ 2 + cot α cot β + cot β cot αa 1 + cot α + cot βd 1 2 + C 1 + B 1 E 1 + H 1 2 2.23 2.24 1 cot α cot β + A1 + D 1 2 2 2 cot 2 + A 2 + D 2 2 2 + 2 cot α cot β + A 1 + D 1 A 2 + D 2 2 cot + O 1 3. It is well known see 11 that the spectrum of problem 1.1 1.7 is discrete and n n + O1 as n. 2.25 Next we present the more exact asymptotic distribution of the spectrum. Using the residue theorem we have n n = 1 ln W 2i C n W d = 1 cot α cot β + A 1 + D 1 cot d 2i C n 1 A 2 + D 2 d 2i C n 1 cot α cot βa 2 cot α + cot βd 2 2i C n cot + C 2 B 2 + E 2 H 2 d 2

1 M. BAYRAMOGLU, A. BAYRAMOV, E. ŞEN EJDE-217/14 1 cot α cot β + cot β cot αa 1 + cot α + cot βd 1 2i C n d C 1 + B 1 E 1 + H 1 + 1 2i + 1 2i + 1 2i C n C n + O 1 n 3. 2 cot α cot β + A 1 + D 1 2 cot 2 2 2 d A 2 + D 2 Cn 2 2 2 d cot α cot β + A 1 + D 1 A 2 + D 2 cot 2 d which implies, using residue calculation, that n = n + cot β cot α A 1n D 1 n n + 2D 2 n cot α + A 1 n + D 1 na 2 n + D 2 n C 2 n + B 2 n E 2 n + H 2 n /n 2 cot α cot β + A 1n + D 1 n 2 n 3 + O 1 n 3. Thus we have proven the following theorem. 2.26 Theorem 2.1. The spectrum of problem 1.1 1.7 has the 2.26 asymptotic distribution for sufficiently large n. Finally, we will get regularized trace formula for problem 1.1 1.7. The asymptotic formula 2.25 for the eigenvalues implies that for all sufficiently large N, the numbers n with n N are inside Γ N, and the numbers n with n > N are outside Γ N. It follows that 2 + 2 + N 2 n n 2 n=n = 1 2 ln W 2i Γ n W d = 1 2cot α cot β + A 1 + D 1 cot d 2i Γ n 1 2A 2 + D 2 d 2i Γ n 1 2 2D 2 cot α A 1 + D 1 A 2 + D 2 2i Γ n cot + C 2 B 2 + E 2 H 2 d 1 2i Γ n 2 cot α cot β + cot β cot αa 1 + cot α + cot βd 1

EJDE-217/14 TRACE FORMULA FOR A DELAY DIFFERENTIAL OPERATOR 11 d C 1 + B 1 E 1 + H 1 + 1 2i + 1 2i Γ n cot α cot β + A 1 + D 1 2 cot 2 d A 2 + D 2 Γn 2 d + O 1 N by calculations, which implies 2 + 2 + N N n=n 2 n n 2 = 2 cot α cot β + A 1 n + D 1 n n=n 2 cot α cot β + A 1 + D 1 N n=n 2 2D2 n cot α A 1 n + D 1 na 2 n + D 2 n 1 + C 2 n B 2 n + E 2 n H 2 n n + T 2 cot α cot β + cot β cot αa 1 + cot α + cot βd 1 C 1 + B 1 E 1 + H 1 2.27 cot α cot β + A 1 + D 1 2 + A 2 + D 2 2 + O 1 N, where T = Res = {2 2D 2 cot α + A 1 + D 1 A 2 + D 2 cot } C 2 + B 2 E 2 + H 2. Passing to the limit as N in 32, we have 2 + 2 + n= { 2 n n 2 2 cot β cot α A 1n D 1 n + 2D 2 n cot α + A 1 n + D 1 na 2 n + D 2 n 1 } + C 2 n B 2 n + E 2 n + H 2 n n = 2 cot β cot α A 1 D 1 cot 2 α cot 2 β 2cot α cot βa 1 + D 1 A 1 + D 1 2 + A 2 + D 2 2 2 cot α cot β + cot β cot αa 1 + cot α + cot βd 1 C 1 + B 1 E 1 + H 1 + T. 2.28

12 M. BAYRAMOGLU, A. BAYRAMOV, E. ŞEN EJDE-217/14 The series on the left side of this equality is called the regularized trace of the problem 1.1 1.7. The main result of this article is given by the following theorem. Theorem 2.2. Formula 2.28 holds for the first regularized trace of the problem 1.1 1.7. Acknowledgements. The authors want to thank the anonymous referee for the valuable comments. References 1 M. Bayramoglu; The trace formula for the abstract Sturm-Liouville equation with continuous spectrum, Akad. Nauk Azerb. SSR, Inst. Fiz., Bakü, Preprint, 6 1986, 34. 2 M. Bayramoglu, H. Sahinturk; Higher order regularized trace formula for the regular Sturm- Liouville equation contained spectral parameter in the boundary condition, Appl. Math. Comput., 186 27 1591-1599. 3 L.A. Dikii; Trace formulas for Sturm-Liouville differential equations, Uspekhi Mat. Nauk, 12 3 1958, 111-143. 4 I. M. Gelfand, B. M. Levitan; On a formula for eigenvalues of a differential operator of second order, Doklady Akademii Nauk SSSR, 88 1953, 593 596 Russian. 5 K. Koklu, I. Albayrak, A. Bayramov; A regularized trace formula for second order differential operator equations, Mathematica Scandinavica, 17 21, 123-138. 6 A. Makin; Regularized trace of the Sturm-Liouville operator with irregular boundary conditions, Electronic Journal of Differential Equations EJDE, 29 27 29, 1-8. 7 F.G. Maksudov, M. Bayramoglu, E. E. Adıguzelov; On a regularized traces of the Sturm- Liouville operator on a finite interval with the unbounded operator coefficient, Doklady Akademii Nauk SSSR, 277 4 1984; English translation: Soviet Math. Dokl., 3 1984, 169-173. 8 S. B. Norkin; Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, AMS, Providence, RI, 1972. 9 M. Pikula; Regularized traces of differential operator of Sturm-Liouville type with retarded argument, Differentsialnye Uravneniya, 26 1 199, 13-19 Russian; English translation: Differential Equations 26 1 199, 92-96. 1 V. A. Sadovnichii, V. E. Podolskii; Traces of operators, Uspekhi Mat. Nauk, 61 26, 89-156; English translation: Russian Math. Surveys 61 26, 885-953. 11 E. Şen, J. Seo, S. Araci; Asymptotic behaviour of eigenvalues and eigenfunctions of a Sturm- Liouville problem with retarded argument, Journal of Applied Mathematics, 213 213, 1-8. 12 E. Şen, A. Bayramov, K. Oruçoğlu; Regularized trace formula for higher order differential operators with unbounded coefficients, Electronic Journal of Differential Equations EJDE, 216 31 216, 1-12. 13 C.-F. Yang; Trace and inverse problem of a discontinuous Sturm-Liouville operator with retarded argument, J. Math. Anal. Appl., 395 212, 3-41. 14 V. A. Yurko; Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2. Mehmet Bayramoglu Institute of Mathematics and Mechanics of NAS of Azerbaijan, 1141, Baku, Azerbaijan E-mail address: mamed.bayramoglu@yahoo.com Azad Bayramov Azerbaijan State Pedagogical University, 1, Baku, Azerbaijan E-mail address: azadbay@gmail.com Erdoğan Şen Department of Mathematics, Namık Kemal University, 593, Tekirdağ, Turkey E-mail address: erdogan.math@gmail.com