AMATH751 A1.pdf - Feipeng Song 20602388 AM 751 Assignment 1...

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AM 751 Assignment 1 Fall 2020 Due: Sept. 28 at 6:00pm 1. Show that if ϕ ( t ) is a solution of the IVP x 0 = f ( x ) , x ( t 0 ) = x 0 (1) defined on ( -∞ , ), then ϕ ( t + t 0 ) is a solution of the IVP x 0 = f ( x ) , x (0) = x 0 (2) on ( -∞ , ). Is it still true if f ( x ) is replaced by f ( t, x )? Justify your answer. 2. Show that each of the functions | x | 1 = n i =1 | x i | , | x | 2 = r n i =1 x 2 i and | x | = max | x i | 1 i n defines a norm on R n . 3. Show that if g m C 1 [[ a, b ] , R n ] for all m = 1, 2, ..., and { g m 0 ( t ) } is uniformly bounded on [ a, b ], then { g m ( t ) } is equicontinuous on[a,b]. 4. Determine if the integral equation has a unique solution for some α > 0 x ( t ) = 2 + Z t 0 1 3 + s 2 q 2 + x ( s ) 2 ds, t [ - α, α ] . 5. Determine if the following IVP has a unique solution. ( x 1 0 = x 1 sint - 2 x 1 2 - x 1 x 2 x 2 0 = 2 t 2 x 2 - x 2 2 - x 1 x 2 . x (0) = x 0 R 2 . 6. Let f C [ Q, R n ], where Q R n +1 is an open set. Assume that f ( t, x ) has continuous partial derivatives with respect to x . Show that f ( t, x ) is locally Lipschitz in x in Q , and Lipschitz in x on any compact and convex subset of Q .

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