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]

In lines 8–11 of Lu (2009) [18, p. 2977], we wrote: “For integer m⩾3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)–(L3), then the functional Lτ is C2-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 Lτ is C2-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].