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Let g(x,y) be Lipschitz continuous. Let ? (x) = y , and for n > 0 define ? (x) = y +
Prove that ? (x)  ?(x) on [x - , x + ], for some > 0, where ?(x) solves the ODE ?'(x) = g(x, ?(x)), and ?(x ) = y
The existence theorem for nonlinear differential equations is discussed.