Question

pls explain to me step by step. pls write clearly and dont skip any steps. i will rate ur answer immediately. thanks.

A metric space is a set M together with a distance function p(x, y) "distance" between elements a and y of the set M. The distance function must satisfy that represents the (i) f (x,y) 0 and p(z, y)--0 if and only if x y; (ii) ρ(z,y)=ρ(y,x); (iii) ρ(z, y) ρ(z, z) + ρ(z, y) for all x, y, z in M. Let (M. p) be a metric space. A mapping T from M into M is called a contraction if for some constant a with 0sa<1, and all rand y in M (a) Show that the function f(t,y) = V t) V2 is continous and satisfies the Lipschitz condition for all t, y. z. Hint: show that If,(t, v)l1 for all t and y (b) Prove that the initial value problem y(0)0 has a unique solution y(t) defined on R.

Answer 1

Any query in any step then comment below.. I will explain you..

ว่า Jy vou

Now in part b...

y' = f(t,y) ...and we already prove that f(t,y) is Lipschitz function....

So by existence and uniqueness theorem., If function is lipzhitz global on R then solution of initial value problem is unique ... This means that this problem has unique solution...