By Cora Sadosky

ISBN-10: 0824783026

ISBN-13: 9780824783020

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Extra resources for Analysis and partial differential equations. Dedicated to Mischa Cotlar

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Show that T is a contraction. 10. Let the mapping T : [a, b] → [a, b] satisfy the condition |T x − T y| ≤ k|x − y|, for all x, y ∈ [a, b]. (a) Is T a contraction? (b) If T is continuously differentiable, show that T satisfies a Lipschitz condition. (c) Does the converse of (b) hold? 11. Apply the Banach fixed theorem to prove that the following system of equations has a unique solution: 2ξ1 + ξ2 + ξ3 = 4 ξ1 + 2ξ2 + ξ3 = 4 ξ1 + ξ2 + 2ξ3 = 4 12. Show that x = 3x2/3 , x(0) = 0 has infinitely many solutions, x, given by x(t) = 0 if t < c and x(t) = (t − c)3 if t ≥ c, where c > 0 is any constant.

13 Theorem If a sequence {xn } of points of X converges to a point x ∈ X, then the set of numbers ρ(xn , θ) is bounded for every fixed point θ of the space X. 8. In some spaces the limit of a sequence of elements is directly defined. If we can introduce in this space a metric such that the limit induced by the metric coincides with the initial limit, the given space is called metrizable. 9. It is known that in the Cauchy convergence criterion ensures the existence of the limit. Yet in any metric space the fulfillment of the Cauchy convergence criterion does not ensure the existence of the limit.

This is known as the usual metric in . 4 4 + + (ii) Euclidean space n , unitary space n , and complex plane Let X be the set of all ordered n-tuples of real numbers. If (ξ1 , ξ2 , . . , ξn ) and y = (η1 , η2 , . . , ηn ) then we set n (ξi − ηi )2 ρ(x, y) = i=1 It is easily seen that ρ(x, y) ≥ 0. Furthermore, ρ(x, y) = ρ(y, x). 1) Preliminaries 13 n Let, z = (ζ1 , ζ2 , . . , ζn ). 3] n 1/2 n (ξi − ηi )(ηi − ζi ) ≤ i=1 (ξi − ηi ) 1/2 n 2 2 (ηi − ζi ) i=1 i=1 ≤ ρ(x, y)ρ(y, z) Thus, ρ(x, z) ≤ ρ(x, y) + ρ(y, z).

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Analysis and partial differential equations. Dedicated to Mischa Cotlar by Cora Sadosky


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