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Let the problem $$y' = f(x,y) , y(x_0) = y_0$$ Let $f$ an continuous function, bounded on $R = \{(x,y) \in \mathbb{R}^2, |x-x_0| \leq a , |y-y_0| \leq b\}$ such that $f$ is Lipschitz compared with $y.$

How we can use the fixed point method to prouve that this Cachy problem admits a unique solution?

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See the proof sketch in the Wikipedia: http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem. Complete proof in http://www.math.byu.edu/~grant/courses/m634/f99/lec4.pdf.

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