Stability and Boundedness of Solutions to Some Nonautonomous Multidimensional Nonlinear Systems

The paper’s Theorem 1 proves Lyapunov stability of the origin by combining (i) a pivot/comparison inequality that bounds the state via a scalar auxiliary ODE, x(t;x0) ≤ V(z(t;z0)) with z0 = V(x0) (equation (2.9)), and (ii) stability of the scalar linear system (3.3), which implies a uniform multiplicative bound sup_{t≥0} z(t;1) < ∞. The proof then constructs a sublevel set S_r = {V ≤ r} whose closure is contained in Ω2 and shows forward invariance via contradiction, yielding stability and the global-in-time comparison for small initial data; if Ω2 = ℝ^n, the comparison holds for all initial conditions. These are exactly the steps in the candidate solution: Step 1 establishes the uniform bound S from scalar stability; Steps 2–3 build an invariant sublevel set and extend the comparison to all t; Step 4 translates V-bounds into Lyapunov stability using r_ε := min_{||x||=ε} V(x); Step 5 notes the global case when Ω2 = ℝ^n. The paper’s derivation of (3.6) and its proof strategy (choose r so that the sublevel set is strictly inside Ω2 and argue by contradiction) align point-for-point with the candidate’s argument. Therefore, both are correct and essentially the same proof. See the paper’s statement of the pivot inequality (2.9) and auxiliary equation (2.8) , and Theorem 1 together with its proof sketch that builds the invariant sublevel set and obtains (3.6) from the scalar bound .