Dynamical systems method and a homeomorphism theorem

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Let F be a nonlinear map in a real Hilbert space H. Suppose that ᵤ∈ B₍ᵤ₀,ᵣ₎ [F'(u)]⁻¹≤ m(R), where B(u₀,R)={u:u-u₀≤ R}, R>0 is arbitrary, u₀∈ H is an element. If ᵣ>₀R/m(R)=∞, then F is surjective. If [F'(u)]⁻¹≤ au+b, a≥ 0 and b>0 are constants independent of u, then F is a homeomorphism of H onto H. The last result is known as an Hadamard-type theorem, but we give a new simple proof of it based on the DSM (dynamical systems method).