By Earl A. Coddington
"Written in an admirably cleancut and cost-effective style." — Mathematical Review. an intensive, systematic first path in easy differential equations for undergraduates in arithmetic and technology, requiring purely simple calculus for a heritage, and together with many workouts designed to strengthen students' strategy in fixing equations. With difficulties and solutions. Index.
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Additional info for An Introduction to Ordinary Differential Equations (Dover Books on Mathematics)
Xm+1 − xm | = |f (xn−1 ) − f (xn−2 )| + . . + |f (xm ) − f (xm−1 )| ≤ θ(|xn−1 − xn−2 | + . . + |xm − xm−1 |) n−1 ≤ θν |x1 − x0 | ν=m ≤ θm 1 |x1 − x0 | (taking into consideration that θ < 1). 1−θ As θ < 1 we deduce, ∀ε > 0 ∃ N ∈ N ∀ n, m ≥ N : |xn − xm | < ε. This shows that (xn )n∈N is a Cauchy sequence. 44 4. The Banach Fixed Point Theorem. The Concept of Banach Space As R is complete, the sequence (xn ) converges to an ξ ∈ R and as I is closed, ξ ∈ I. Furthermore, as f is continuous on account of (1), we have: f (ξ) = lim f (xn ) = lim xn+1 = lim xm = ξ.
4 A metric on a space X is a mapping d:X ×X →R such that (i) (ii) (iii) ∀ x, y ∈ X : d(x, y) ≥ 0 and d(x, y) = 0 only then when x = y (positive deﬁniteness) ∀ x, y ∈ X : d(x, y) = d(y, x) (symmetry) ∀ x, y, z ∈ X : d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) A sequence (xn )n∈N ⊂ X is said to converge (relative to d) to x ∈ X if lim d(xn , x) = 0. 1. We now deﬁne next the concept of Cauchy sequence in a normed vector space: 46 4. The Banach Fixed Point Theorem. t. · ) if · . A sequence ∀ ε > 0 ∃ N ∈ N ∀ n, m ≥ N : vn − vm < ε .
8 Let f : D → R be a diﬀerentiable function. We say that f is twice diﬀerentiable at x ∈ D if the derivative f of f is diﬀerentiable at x. We write f (x) := (f ) (x) (the derivative of f at x), 2 2 d and also ddxf2 (x) in place of f (x). ) Inductively, we say that a k-times diﬀerentiable function f : D → R is (k + 1)-times diﬀerentiable at x ∈ D if the kth derivative dk f (k) = dx k f is diﬀerentiable at x. It is (k+1)-times continuously diﬀerentiable in D if the (k + 1)st derivative f (k+1) exists and is continuous in D.
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