Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter

In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of [Disp. Item 1. lu:=−u′′+[q(x)+2ρp(x)]u=ρ2u,x∈(0,π) ...]–[Disp. Item 3. V(u):=u′(π,ρ)+(H1ρ+H0)u(π,ρ)=0, ...] except for arbitrary one eigenvalue and the Borg type theorem of [Disp. Item 1. lu:=−u′′+[q(x)+2ρp(x)]u=ρ2u,x∈(0,π) ...]–[Disp. Item 3. V(u):=u′(π,ρ)+(H1ρ+H0)u(π,ρ)=0, ...] except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.