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Extra info for A (Terse) Introduction to Linear Algebra (Student Mathematical Library, Vol. 44)
29), requiring 24 I. BASIC CONCEPTS only continuity of the right-hand side of this equation (cf. 131). The situation is considerably more complex for the existence of a solution to the basic initial value problem in the singular case for an equation of neutral type. We consider, for example, the equation with a single deviating argument, K(t) = f (t,x(t) ,x(t-T(t)) ,k(t-r(t))) (33) and assume that the singular case occurs - the set reduces to a single point t0' Then, if the solution is defined only in a sufficiently small right neighborhood of the point to, the set Et reduces to 0 a single point (since T(tO)=O and r(t) increases in this neighborhood more slowly than (t) and, consequently, the initial condition has the form x(to) = x o .
R(t,x )-ti does not change. D. 41). If in Eq. r(t,x) is continuous in some neighborhood of the point (to,#(to)1 , 4 (to+-T(t I ~ o ) ) ) is ) continuous, the functions T and f satisfy Lipschitz conditions in all arguments except the first, the initial function $(t) is defined on the half-axis (--,to] and satisfies a Lipschitz condition on each finite half-interval, then there exists a unique, continuous solution x,(t) of the basic initial value problem for Eq. (47) for tO F u r t h e r , l e t x ( t a ) be a r e a l r o o t of Eq. (34) and t h e f u n c t i o n f c o n t i n u o u s i n a neighborhood of t h e p o i n t -( t0 xL( t o ,)x (t,), ? ( t o) )w i t h r e s p e c t t o a l l i t s arguments and s a t i s f i e s a L i p s c h i t z c o n d i t i o n w i t h r e s p e c t t o a l l arguments b e g i n n i n g w i t h t h e second, w i t h c o n s t a n t s L2,L3,L1. I f t h e n on some i n t e r v a l t o t h e r i g h t of t h e i n i t i a l p o i n t t o t h e r e e x i s t s a unique c o n t i n u o u s l y d i f f e r e n t i a b l e s o l u t i o n of Eq.
F u r t h e r , l e t x ( t a ) be a r e a l r o o t of Eq. (34) and t h e f u n c t i o n f c o n t i n u o u s i n a neighborhood of t h e p o i n t -( t0 xL( t o ,)x (t,), ? ( t o) )w i t h r e s p e c t t o a l l i t s arguments and s a t i s f i e s a L i p s c h i t z c o n d i t i o n w i t h r e s p e c t t o a l l arguments b e g i n n i n g w i t h t h e second, w i t h c o n s t a n t s L2,L3,L1. I f t h e n on some i n t e r v a l t o t h e r i g h t of t h e i n i t i a l p o i n t t o t h e r e e x i s t s a unique c o n t i n u o u s l y d i f f e r e n t i a b l e s o l u t i o n of Eq.