TAILIEUCHUNG - On Hochstadt–Lieberman theorem for impulsive Sturm–Liouville problems with boundary conditions polynomially dependent on the spectral parameter

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: Turkish Journal of Mathematics 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: • Accepted/Published Online: • Final Version: 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 () for x ∈ [a0 , b0 ], with the boundary conditions { a(λ)y ′ (a0 , λ) − b(λ)y(a0 , λ) = 0, () 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, () 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

• Toán học
• Sturm–Liouville problem
• Interior discontinuities
• Hochstadt–Lieberman theorem
• Boundary conditions
• Polynomially dependent on the spectral parameter