Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967–3034]

Abstract In lines 8–11 of Lu (2009) [18, p. 2977] we wrote: “For integer m ⩾ 3 , if M is C m -smooth and C m − 1 -smooth L : R × T M → R satisfies the assumptions (L1)–(L3), then the functional L τ is C 2 -smooth, bounded below, satisfies the Palais–Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4] )”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that L τ is C 2 -smooth is true if and only if for every ( t , q ) the function v ↦ L ( t , q , v ) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20] .