ON A MEAN VALUE THEOREM OF THE DIFFERENTIAL CALCULUS OF VECTOR VALUED FUNCTIONS, AND UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS IN A LINEAR NORMED SPACE
Abstract
Suppose that: (1) the vector valued function x(t) is defined for all real t such that a t b, where a < b, and that its values are in a linear normed vector space B (the norm in B will be denoted by ); (2) limt ax( T) = x a) and limt bx(t) = x(b) (i.e., for example limt a x(t) - x(a) =0); (3) the derivative x'(t) exists, and is finite, whenever a < t < b (i.e., there is a vector x'(t) in the space B such that lims t (x(s) - x(t))/(s-t) x'(t) = 0). Then there is a number , with a < < b, such that (x(b)-x(a))/(b-a) x'( ) . This mean value theorem for vector valued functions is proved first, and then it is used to derive uniqueness theorems for the (vector) initial value problem dxdt = f(t,x); s(to) = xo. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- May 16, 1961
- Accession Number
- AD0261013
Entities
People
- A.k. Aziz
- J.b. Diaz
Organizations
- Naval Ordnance Laboratory