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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
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is Lipschitz continuous, then it satisfies the Lipschitz condition. There exists , such that is continuous, so we can suppose is continous in a rectangular region , . Since is a closed region, can reach its maximum value in this ...
The existence theorem for nonlinear differential equations is discussed.