Existence of solutions for first-order Hamiltonian stochastic impulsive differential equations with Dirichlet boundary conditions

The paper sets up the dual action functional χ and claims existence of a critical point via the mountain pass lemma, but key steps in the C1/PS analysis rely on identities that only hold at critical points (e.g., substituting Jv = ∇H*(t, v̇) for arbitrary v) and on unproven bounds, so the proof as written has gaps. See the definition of χ and its derivative (2.4)–(2.5) and Theorem 2.3, followed by the PS and geometry arguments where Jv is replaced by ∇H*(t, v̇) and where v̇ is identified with ∇H(Jv) along a general sequence; both substitutions are invalid outside critical points (e.g., the inequality chain in Theorem 3.2/3.3 and the sequential compactness step) . The model’s solution, while correctly deriving the formal Euler–Lagrange/jump conditions, incorrectly assumes χ is convex and treats the symplectic term ½∫(Jv̇,v) as a continuous linear functional, then invokes direct minimization on Lp(Ω; W0^{1,p}); this mischaracterizes the indefinite quadratic term and does not establish weak lower semicontinuity/coercivity needed for a minimizer. Thus, both arguments are incomplete.