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In the present paper, we consider an inverse problem for the Sturm–Liouville operator with a finite number of discontinuities at interior points and boundary conditions polynomially dependent on the spectral parameter on an arbitrary finite interval, and prove the Hochstadt–Lieberman-type theorem for this problem. | Turk J Math (2018) 42: 3002 – 3009 © TÜBİTAK doi:10.3906/mat-1807-77 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article On Hochstadt–Lieberman theorem for impulsive Sturm–Liouville problems with boundary conditions polynomially dependent on the spectral parameter 1 Seyfollah MOSAZADEH1,∗,, Aliasghar Jodayree AKBARFAM2 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran 2 Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Received: 10.07.2018 • Accepted/Published Online: 25.09.2018 • Final Version: 27.11.2018 Abstract: In the present paper, we consider an inverse problem for the Sturm–Liouville operator with a finite number of discontinuities at interior points and boundary conditions polynomially dependent on the spectral parameter on an arbitrary finite interval, and prove the Hochstadt–Lieberman-type theorem for this problem. Key words: Sturm–Liouville problem, interior discontinuities, Hochstadt–Lieberman theorem, boundary conditions polynomially dependent on the spectral parameter 1. Introduction We consider the boundary value problem $ generated by the second-order differential equation of Sturm– Liouville (S-L) type y ′′ + (λ − q(x))y = 0 (1.1) for x ∈ [a0 , b0 ], with the boundary conditions { a(λ)y ′ (a0 , λ) − b(λ)y(a0 , λ) = 0, (1.2) c(λ)y ′ (b0 , λ) − d(λ)y(b0 , λ) = 0, and the transmission (discontinuous) conditions { y(xp + 0) = αp y(xp − 0), p = 1, 2, 3, ., ℓ, (1.3) y ′ (xp + 0) = αp−1 y ′ (xp − 0), p = 1, 2, 3, ., ℓ, where λ is the spectral parameter, q is a real-valued function in L2 (a0 , b0 ), αp ∈ R and αp ̸= 0 for p = 1, 2, 3, ., ℓ , a0 0 , and some sequences Rk → ∞ as k → ∞. (2) lim|x|→∞ |f (ix)| = 0 . Then f ≡ 0. Proof 2 See Proposition B.6 of [5]. 3. Main results In this section, first we prove a Hochstadt–Lieberman-type theorem for S-L problems with one interior discontinuity and boundary conditions polynomially .