On the Hadamard lemma and the Lipschitz condition

A. I. Perov

Chair of Nonlinear Oscillations, Voronez State University, Voronezh, Russia

Abstract: We prove that the difference of values of a nonlinear Lipschitz continuous differential operator is always representable as the difference of values of some linear Lipschitz continuous differential operator with the same Lipschitz constant. The proof is based on the Hadamard lemma, provided that, in addition to the above requirements, the nonlinearity is continuously differentiable in spatial variables. In general case the proof is based on various criteria of the weak compactness and on various approximating statements obtained by the Steklov averaging technique.

Keywords: ordinary differential equations, Hadamard lemma, Lipschitz condition, Steklov averaging.

UDC: 517.956

Received: 27.09.2007

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:12, 30–40

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