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By Elliott Ward Cheney

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If the components of L depend on t 6 /, the norm of i_(i) on / is denned as 26 Note that in this case the interval over which the supremum is taken is indicated by the index /. In a similar way we introduce the norm of a vector field A(£,L) with (t,i_) e / x £1 as When using these norms we have to establish whether the suprema indeed exist. 2) indeed exists. 1) we associate the (Euclidean) matrix norm with Aj the eigenvalues of Z3Tzi. Now consider the initial value problem (IVP) Anticipating the existence of the solution, which will be considered in §2, we investigate the uniqueness of solutions.

See Fig. 2. , and the solution exists uniquely on their union. If / and fi are open and bounded, the 7Qi will become smaller and smaller, because x(t) will approach the boundary of / x fi. If / and Q are closed and bounded the procedure will terminate for certain i — n, because either tn is at the boundary of I, or x(tn) is at the boundary of fi. We conclude that in all cases a maximal existence interval can be found. It may be very hard to estimate the size of this interval in advance without solving the problem explicitly.

An essential ingredient is Picard iteration, a procedure of successive substitutions. 1) is automatically differentiable, so the proof of the existence theorem can be formulated in terms of continuous functions. , of continuous functions: This so-called Picard mapping P maps continuous functions on continuous functions. By construction, the x*(£) are even differentiable. 2) that all x* satisfy the initial condition X*(£Q) = XQ. The proof to be given below shows that the iterates xz converge to a unique fixed point of the Picard mapping provided that f is Lipschitz continuous.

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Introduction to Approximation Theory by Elliott Ward Cheney


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