By George Boole

ISBN-10: 0828401284

ISBN-13: 9780828401289

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**Extra info for A treatise on differential equations**

**Sample text**

10) for some constant a > 0. 10). So these two results overlap, but somehow they are complementary. t). 11) For a given closed set G, we denote E = {+ E G : max V ( z t ( + ) ( s )=) max V ( + ( s ) for ) all t 2 0}, -r~~
~~

A process on X is a mapping u : R x X x R+ --+ X satisfying the following properties: (i) u is continuous; (ii) U(a,O) = I , the identity; (iii) U ( o s, t)U(a,s) = U ( a ,s t ) . A process u is said to be a p-periodic process, p > 0, if U ( u + p , t ) = U ( a ,t ) for u E R, t E R+. Suppose f : R x C --+ R" is completely continuous, and let x ( u , $ ) denote the solution of the RDDE(f), + + qt)= f ( t , . t ) , = 4. 1) We assume that z is uniquely defined for t 2 a. 3 implies that z ( a ,$ ) ( t )is continuous in u,4, t for a E R, 4 E C , and t 2 u.

If $ E M , then q($)E M c El which implies that 0 = q ( $ ) ( O ) = x ( $ ) ( t ) , t E R; in particular, 0 = x ( t ) = $ ( t ) for t E [-r,O]. Hence, M = {0}, proving the corollary. 4. Suppose D i s stable, u(s),v(s),w(s): R+ 4 R+ are continuous and nondecreasing, u(s),v(s) > 0 f o r s > 0, u(0) = v(0) = w(0) = 0. T h e following statements are true: (i) If there is a V : R x C + R s u c h that t h e n x = 0 of NDDE(D, f ) is uniformly stable. (ii) If, in addition t o (i), lim,,+, u ( s ) = $00, t h e n solutions of the NDDE(D,f) are uniformly bounded.

### A treatise on differential equations by George Boole

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