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By S. B. Norkin

Booklet through Norkin, S. B.

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Let B be a Banach space over the field R of real numbers and let C be a nonempty compact subset of B. Also, on a bounded interval Z, let F be an infinite and equicontinuous set of C-valued functions. Show that F contains an infinite sequence which is uniformly convergent on Z. I-11. Let B be a Banach space over the field R of real numbers and let C be a nonempty compact and convex subset of B. Assume that a C-valued function f (t, x) is continuous on a region R={(t,x)ERxB:It-rl <-a,llx-t1l <-b}, where r E R, t E B, a > 0, b > 0, and 11 11 is the norm of B.

Assumption 1. For the initial-value problem (P), assume that (i) f(t, y", e) is continuous in (t, y, e) in an open set Do in the (t, y, e)-space, (ii) ¢'(t) = f (t, 6(t), co) and (t, ¢(t), eo) E Do on an interval Zo = {t : tt < t < t2}, where eo is a fixed value of the variable e, (iii) y ' = ¢(t) is unique in the sense that i f 1r1 = f (t, tl'(t), co) and co) E Do on a subinterval I of the interval Zo and if y(r) = fi(r) at a point r of the interval Z, then L,(t) = ¢(t) identically on the interval Z.

Let d be the distance between two compact sets F1 and F2. Since &(c,µ) is continuous in p, there exists, for each k, a real number µk such that IµkI < c-r and d. & (c, µk), F1) = Since the family ilk) : k = 1, 2, ... } is bounded and equicontinuous on the interval Io, there exists a subsequence j = 1,2.... +oo i--+00 urn 1Pk, (t, µk1) = 0(t) exists uniformly on To. (A) but * (c) 4 F1 U F2. This is a contradiction (cf. Figure 10). III. =c t=c FIGURE 10. FIGURE 9. Case 2 (general case). Fl) and A2 = A n R({c} x F2), where {c} x F; = {(c, yl : y" E Fj } (j = 1, 2).

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