Studia Scientiarium Mathematicarum Hungarica 32. (1996)

1-2. szám - Slezák B.: Implicit function theorem in uniform spaces

IMPLICIT FUNCTION THEOREM 3 VVeBx 3 we By 3HB)to Vy G W(f{t0; x0)) 3g:H-^V(x0) VteH: f(t,g{t))=y. If T is equicontinuous at xq then a neighbourhood H' x V'(xq) 3 (to,xo) can be chosen so that for every point of this set there exists a </>: H —> V{xo) implicit function passing through this point. (iii) Suppose that there exists an implicit function g which passes through the point (h,s) G H x P(;co) and continuous at h, furthermore T is equicon­tinuous at the point s. Then f is continuous at the point (h,s). (iv) Let us suppose that every element of T is injective. Then the implic­it functions are unique and the implicit function passing through the point (io^o) is continuous at to-Proof, (i) Let V G Bx and U x V{xo) be a neighbourhood of (to,xo)■ We show that F(U x F(xo)) is a neighbourhood of F(to,xq). It is clear that the following equalities hold: F(U x P(xo)) = {(t, f{t, x)) \{t,x)eU x V(x0)} = W = Umx/(U(^(zo)). teu As T is equiopen at xo the set W\ can be chosen so that (2) ViGt/: W1(f(t,x0))Qf(t,.)(V(x0)) holds. Let W2, W E By, W2 ° W ^W\. As f{.,x 0) is continuous at to the neighbourhood H of to can be chosen so that H QU and f(.,xo)(H) ^ Q W2Íf(to,x0)), that is (3) VteH: {f{t,xo)J(to,xo))eW2 holds. It follows that for every t£ H the set W (/(to, %o)) is a subset of the set fit, .)(P(x0)). Indeed, by (3), (f(t,x0),f{to,x0)) € W2 and (/(t0, x0), y) G W imply that (/(t, xq),y) G W2 o W Q W\. Hence (using (2)) Vi G H: W(f{to,x0)) i Wxifit, x0)) Q f(t, .)(V"(*0)). By (1) we get that H x xo)) C F(H x V(x0)) Q F{U x V(x0)). (ii) Let W be as above, y G W(/(to, £0)) be arbitrary. As Hx xW(f(to,x0)) Q F(H x P(a:o)), we have that Vi G H 3g{t) = x G V^rco): F{t, g{t)) = y, which means that (f, /(t, g(t))) = (t, y). Hence f{t, g{t)) = y. If T is equicontinuous at (to,^o) then using Proposition 2 we get that / and F are continuous at (to, zo). So there exists a neighbourhood H' x V'{xo) of the point (io,a:o) that the inclusion F(H' x P'(xo)) C H x W(f(to,xo))

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