16: Let f(t,x) be piecewise continuous in rand Lipschitz in x on [t,,t,]x W with a Lipschitz constant L, where WcR"is . Let y(t) and z(t) be solutions of ý = f(t, y), y(t,)= Yo, 2= f(, 2) + g(t,z), z(t,) = Z,, such that y(t), z(t) eW,Vte[t,,t,]. Suppose that g(t, x)|| < µ, V(t,x)E[t,.t,]× W, for some u>0. |yM)– z(1)|| <|y,- z.|exp[L(t,-t,)]+texp[L(t,-t,)-1}, Vtet,t,). (a) an open connected set. (b) an open set. (c) convex set.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
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16: Let f(t,x) be piecewise continuous in tand Lipschitz in x on [t,t,]x W with a
Lipschitz constant L, where Wc R" is . . Let y(t) and z(t)be solutions of
ý = f(t, y), y(t,)= Y,,
ż = f(t, z) + g(t,z), z(t,) = zo,
such that y(t), z(t) e W ,Vte[t,,!,].
Suppose that g(t,x)|| < µ, V(t,x)E[t,,t,]x W, for some u>0.
|ly©) - z(0|<]y.- z.|lexp[L(t,-t,)] +texp[L(t,-t))-1}, Vte[t,t,].
(a) an open connected set.
(b) an open set.
(c) convex set.
Transcribed Image Text:16: Let f(t,x) be piecewise continuous in tand Lipschitz in x on [t,t,]x W with a Lipschitz constant L, where Wc R" is . . Let y(t) and z(t)be solutions of ý = f(t, y), y(t,)= Y,, ż = f(t, z) + g(t,z), z(t,) = zo, such that y(t), z(t) e W ,Vte[t,,!,]. Suppose that g(t,x)|| < µ, V(t,x)E[t,,t,]x W, for some u>0. |ly©) - z(0|<]y.- z.|lexp[L(t,-t,)] +texp[L(t,-t))-1}, Vte[t,t,]. (a) an open connected set. (b) an open set. (c) convex set.
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