optimization—theory and applications || the implicit function theorem and the elementary closure...

CHAPTER 8

The Implicit Function Theorem and the Elementary Closure Theorem

8.1 Remarks on Semicontinuous Functionals

As in Section 2.15 let us consider briefly an abstract space S of elements x, and let us assume that a concept a of convergence of sequences X k of elements of S has been defined, satisfying the two main axioms: (a) If [xk] converges to x in S, then any subsequence [XkJ also converges to x; (b) Any sequence of repetitions [x, x, ... ,x, ... ] must converge to x, where x is any element of S. Any such space is called a a-limit space. In Section 2.15 we introduced the concepts of a-lower and a-upper semicontinuity of a functional F: S -+ reals. A functional which is both upper and lower semicontinuous is said to be continuous. Let us show here that, already at this level of generality, quite relevant theorems can be proved. To this effect, let us carryover the usual concepts. Thus, we say that a subset A of S is a-closed if all elements of accumulation of A in S belong to A; that is, if Xo E S is the a-limit of elements Xk of A, then Xo E A. We say that a subset A of S is relatively sequentially a-compact if every sequence [xk] of elements of A possesses a subsequence [XkJ which is a-convergent to an element x of S.

S.Li. If S is a a-limit space, if F:S -+ R is lower semicontinuous on S, then for every real number a, the set Ma = [x E SIF(x) ~ a] is closed. If F is upper semicontinuous, then the sets M~ = [x E SIF(x) 2:: a] are closed.

Indeed, if Xo is a point of accumulation of M a' then there is a sequence Xk of elements Xk E M a with F(xk) ~ a, Xk -+ Xo, and then F(xo) ~ lim infk F(xk) ~ a. The same proof works for upper semicontinuity. In the usual terminology this theorem can be reworded by saying that lower and upper semicontinuous functionals are B-measurable. The same statement (S.l.i) holds for

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272 Chapter 8 The Implicit Function Theorem and the Elementary Closure Theorem

functionals F defined on a O'-closed subset A of S where the M a are the corresponding subsets of A.

S.1.li. Let S be a O'-limit space, let A be a nonempty O'-closed and relatively sequentially O'-compact subset of S, and assume that F is lower semicontinuous at every Xo E A with respect to O'-convergence. Then F is bounded below in A and has an absolute minimum in A.

Analogously, if F is upper semicontinuous, then F is bounded above and has an absolute maximum in A.

Proof. Let m = infA F(x), - 00 ::s;; m < + 00, and take any sequence [xk] of elements of A such that F(Xk) - m as k - 00. We may well assume that F(Xk) ::s;; m + 11k if m is finite, and F(Xk)::S;; - k if m = - 00. First, A is 0'

compact; hence there is a subsequence [XkJ which is O'-convergent to an element Xo of S. Since A is O'-closed, then Xo belongs to A, so F(xo) is defined and is a real number. Then, by lower semicontinuity we have - 00 < F(xo) ::s;; lim infk-+ 00 F(Xk) = m < + 00. Thus, F(xo) is finite, and so is m. Since Xo E A, also F(xo) ~ m. By comparison, we have F(xo) = m, and the existence of the minimum for F on A is proved. An analogous proof holds for upper semicontinuous functionals and maxima. D

As a consequence of (8.l.ii) we derive that any continuous functional on a O'-closed and relatively sequentially O'-compact set A has both an absolute minimum and an absolute maximum.

Statement (8.l.ii) holds even under weaker hypotheses. Indeed, we could assume that F may take on A the value + 00, with F(x) not identically + 00

on A, and F(x) > - 00 for all x E A. Moreover, we could assume that A is nonempty and O'-closed, and only that the sets Ma = [x E AIF(x)::s;; a], if not empty, are relatively sequentially O'-compact. Some authors call such sets A "inf-compact", but we shall not need this terminology. Analogous remarks hold for upper semicontinuous functionals.

The following particularization of the above concepts and statements is important.

S.l.iii. Let S be a real reflexive Banach space of elements x with norm Ilxli, and take in S for O'-convergence the weak convergence in S. Let A be a nonempty closed convex subset of S. Let F:A - R be a functional which is lower semicontinuous in A with respect to weak convergence, and such that F(x)+ 00 as Iixli- + 00, x E A. Then, F is bounded below in A and has an absolute minimum in A.

Proof. First we note that the convex set A is closed in the weak as well as in the strong topology in S. Indeed, by the Banach-Saks-Mazur theorem (cf. Section 10.1), weak and strong closures of a convex set in any Banach space coincide. Let i = inf[ F(x) I x E A], - 00 ::s;; i < + 00, and let N be any real number N > i. Then, the set AN = [x E AIF(x)::s;; N] is nonempty and

8.1 Remarks on Semicontinuous Functionals 273

bounded, since F(x) ~ + 00 as Ilxll ~ + 00 in A. From functional analysis we know that AN, as a bounded subset of a reflexive Banach space, is sequentially compact with respect to weak convergence. Thus, for any minimizing sequence xk, that is, F(xk) ~ i as k ~ + 00, Xk E A, we certainly have Xk E AN C A for all k sufficiently large, and we can choose a subsequence, say still [k], such that Xk ~ Xo E A in the weak convergence of S. By the lower semi continuity in A we have - 00 < F(xo) ~ lim infk F(xd = i < N. Hence, Xo E AN' As in the proof of (S.l.ii), we also have F(xo) ~ i, and finally F(xo) = i.

A functional F:A ~ R on a convex set A of a linear space S is said to be convex in A provided Xb X2 E A, 0 ~ rt. ~ 1, implies F((1 - rt.)Xl + rt.x2) ~ (1 - rt.)F(x l ) + rt.F(x2). The same functional is said to be strictly convex in A provided F is convex in A and strict inequality holds above for all 0 < rt. < 1.

8.l.iv. Under the conditions of (S.l.iii), if F: A ~ reals is strictly convex on A, then the element x E A at which F(x) = i (equivalently, F(x) ~ F(y) for all y'E A) is unique.

Proof. If there were two such elements Xl' x 2 with F(xd = F(X2) = i, Xl' x 2 E A, then F(2- l(Xl + x 2)) < 2- 1F(xd + 2- 1F(X2) = i, a contradiction.

o

Again, let A be a convex subset of the real Banach space S. A functional F:A ~ R is said to have a Gateau derivative F~h at a point X E A provided the limit

lim rt.-l[F(x + rt.h) - F(x)] = F~h (l-O+

exists for every h such that X + h E A. The same functional F is said to be differentiable in A if F~h exists for all x, X + hE A (and F~(h) = - P( - h) at every X in the interior of A).

A stronger concept is often used. Let S* denote the dual space of S, that is, the space of all linear continuous operators z: S ~ R on S. Then the same functional F:A ~ R above is said to have a Frechet derivative at a point x E A provided there is an element F~ of S*, and for every e > 0 a number £5 = £5(e, x) > 0 such thatiF(x + h) - F(x) - F~hl ~ ellhll for allllhil ~ £5, x + hEA.

8.1.v. If F:A ~ R is a convex differentiable functional on a convex subset A of a Banach space S, and if x E A is any point of A where F(x) ~ F(y) for all YEA, then we have also

(S.1.1) F~(y - x) ~ 0 for all YEA.

Conversely, if this relation holds at some x E A, then F(x) ~ F(y) for all YEA.


Proof. If x E A is such that F(x) ::s;; F(y) for all YEA, then for y = x + h E A, the entire segment x + rx.h, 0 ::s;; rx. ::s;; 1, lies in the convex set A, and F(x) ::s;; F(x + rx.h). For 0 < rx. ::s;; 1, we also have

rx.-l(F(x + rx.h) - F(x» ~ 0,

and by taking the limit as rx. -+ 0+ we derive F~h ~ 0, or F~(y - x) ~ 0 for all YEA. Conversely, if x E A and (8.1.1) holds for all YEA, then by convexity

F( (1 - rx.)x + rx.y) ::s;; (1 - rx.)F(x) + rx.F(y), or

F(y) - F(x) ~ rx.-l(F(x + rx.(y - x» - F(x», O<rx.<l.

Since the limit F~(y - x) exists by hypothesis and is nonnegative, we derive, as rx. -+ 0+, that

F(y) - F(x) ~ F~(y - x) ~ 0,

and this holds for all YEA. o

The relation (8.1.1) is called a "variational inequality", and, as we have proved under the assumed hypotheses, (8.1.1) is a characteristic property of the elements x E A at which F has its minimum value. If A = S, then we can take in (8.1.1) y = x ± h, hE S, and (8.1.1) yields F~h = 0 for all hE S. This is the abstract form of Theorem (2.3.ii). Some authors refer to the relation F~h = 0 for all h as the "Euler equation" for F.

As a further particularization of the above considerations, let us assume that n(x, y) is a given symmetric continuous bilinear form on S, that is, n:S x S -+ R, n(x, y) = n(y,x), and n(x, y) is linear and continuous in x for every y, and in y for every x. Let L:S -+ R be a given linear continuous functional on S.

If n(x, y) is coercive, that is, n(x, x) ~ cllxl1 2 for all XES and a constant c > 0, then the functional

F(x) = n(x, x) - 2L(x), XES,

has all the properties we have requested in (8.1.iii) and (8.l.iv). Indeed, IL(x)1 ::s;; Mllxll for some constant M, and F(x) ~ cllxl1 2 - 2Mllxll. Then, F(x) -+ + 00 as Ilxll-+ + 00. Moreover, F is convex and strictly convex in S. Indeed, for 0 ::s;; rx. ::s;; 1 and Xl i= X2, we have

F((1-rx.)Xl + rx.X2)=F(Xl +rx.(X2 -xd)

=n(xi +rx.(X2-Xl), Xl +rx.(X2-Xl»-2L(Xl +rx.(X2- Xl»

=n(xI,xI)+2rx.n(xI, X2-xI)+rx.2n(x2-xI, X2- XI)

-2L(XI)- 2rx.L(X2 -xI)=m+nrx.+ prx.2 =P(rx.).

Here P(O) = F(Xl), P(l) = F(X2), and since p = n(x2 - Xl, X2 - Xl) > 0, the polynomial P is strictly convex in rx. for 0 ::s;; rx. ::s;; 1, so that

F((1- rx.)XI + rx.X2) = P(rx.) < (1 - rx.)P(O) + rx.P(l) = (1 - rx.)F(Xl) + rx.F(X2)

8.2 The Implicit Function Theorem 275

for all 0 < IX < 1. We have proved that F is convex and strictly convex in S. It is easily seen that the Gateau derivative of F is F~h = 2[ n(x, h) - L(h)]. As a corollary of (8.l.iii-v), we have

S.l.vi. For n bilinear, symmetric, continuous, and coercive, and L linear and continuous, then F(x) = n(x, x) - 2L(x) has an absolute minimum in every convex closed subset A of S. The unique point x E A for which F(x) ~ F(y) for all YEA is characterized by the inequality n(x, y - x) ~ L(y - x) for all YEA.

If A = S, then the equality F~h = 0 reduces to n(x, y - x) = L(y - x) for all YES.

8.2 The Implicit Function Theorem

A. An Abstract Form of the Implicit Function Theorem

Given any two metric spaces X, Yand any single valued functionf: X ~ Y, we denote by Yo the image of f, or Yo = [y E Yly = f(x), x E X], and for every subset F of Y we denote by f- 1 F the set f- 1 F = [x E X I f(x) E F]. If f is continuous, then for every closed set F in Y, f- 1 F is also closed; if G is open in Y, then f- 1G is also open. Thus, if f is continuous, then any point y of Yo has a counterimage f-1 y which is a closed subset of X.

Thus, f- 1 is a set valued function, and we shall see in Section 8.5 that f- 1 is upper semicontinuous if f is continuous.

A single valued function cp: Yo ~ X such that cp(y) E f-1 y for every y E Yo is called a partial inverse of f, and for any such cp we have f[cp(y)] = y for all y E Yo, that is, fcp is the identity on Yo. (In the terminology of Section 8.3 cp is a "selection" of the set valued functionf- 1). Here we discuss the question whether, for any continuous single valuedf:X ~ Y, there is a B-measurable single valued partial inverse cp: Yo ~ X. Here we show that the answer is affirmative under some assumptions on X.

S.2.i (A PARTIAL INVERSE THEOREM). Given any two metric spaces X, Y, where X is the countable union of compact subspaces of X, let f:X ~ Y be any continuous mapping, and let Yo = f(X) be the image of X in Y. Then there is a B-measurable map cp: Yo ~ X such that f cp is the identity map on Yo, that is, f[ cp(y)] = y for every y E Yo.

Proof·

(a) Let us suppose that X is replaced by a closed set L c [0, + (0). In this situation, let T(y) = inf f-1(y) for y E Yo. Here f-1(y) is a nonempty set of real nonnegative numbers, and the operator inf applies. Actually, f is a


continuous map, hence f-l(y) is a closed nonempty subset of [0, + 00), and hence f-l(y) has a minimum T(y) = min f- 1(y), whence T(y) E f- 1(y), and f[T(y)] = y for every y E Yo. Let us prove that T: Yo --+ L is a lower semicontinuous (real single valued) function. Suppose this is not the case. Then, there is a point Yo E Yo, a number 8 > 0, and a sequence [Yk] such that Yk E Yo, Yk --+ Yo, T(Yk) ~ T(yo) - 8, or ° ~ Xk ~ Xo - 8, with Xk = T(Yk), f(xk) = Yk> X o = T(yo), f(x o) = Yo. Here [Xk] is a bounded sequence of real numbers; hence there is a subsequence, say [XkJ, with Xks --+ x and ° ~ x ~ X o - 8. Here Xks = T(YkJ E L, and thus x E L, since L is closed. Also, f(xkJ = f(T(YkJ) = Yks --+ Yo, Xks --+ x, hence f(x) = Yo, since f is continuous on L. Thus, x E f- 1(yo) ~ Xo - 8, a contradiction, since Xo = min f- 1(yo). We have proved that T: Yo --+ L is lower semicontinuous in Yo, and hence Bmeasurable because of (8.1.i), and f(T(y)) = Y for every Y E Yo. Theorem (8.2.i) is proved for X replaced by any closed subset L of [0, + 00).

(b) Let us consider now the general case. By general topology (Kelley [1, Theorem 3.28]) we know that any compact metric space X is the continuous image A: K --+ X of some closed subset K of [0, 1], AK = X. (Actually, under some restriction on the compact set X-namely connectedness and local connectedness-we even know that X is the continuous image of an interval, say [0,1], or any interval, and if X is, say, a square or a cube, we say that A is the Peano curve filling a square or a cube. The restrictions on the compact set X are only that X must be connected and locally connected. But we shall not need these particularizations.) In our case X = UX~, where X~, rx = 1,2, ... , is a sequence of compact subsets of X, and we can think of each X ~ as being the continuous image I~: L~ --+ X~, rx = 1, 2, ... , of some closed subset of [0, 1], say L~ C [8, 1 - 8] for some 8 > 0. Let us denote by L the set which coincides with L~ = rx + L~ in [rx, rx + 1], that is, the displacement L~ of L~ in [rx, rx + 1]. Then L is a closed subset of [0, + 00), and we shall denote by I:L --+ X the map which coincides with I~ on L~. Then I is a continuous map of L onto X. We have the situation shown in the picture, and, by (a), there is a B-measurable map T: Yo --+ L such that (j1)(T(y)) = Y for every Y E Yo, or (j1T)(y) = y, since j1 is a continuous map, and Yo =

f(X) = f(l(L)) = (j1)(L). If we take cp = IT, cp: Yo --+ X, we have f(cp(y)) = y

T

~ L.2....X~Y


for every Y E Yo. Obviously, qJ is a B-measurable map, as it is the superposition of a continuous map I on the B-measurable map T. Theorem (8.2.i) is now completely proved. Note that T is a lower semicontinuous function. D

We just mention here the general concept of measure space (X, .91, m), that is, a space X, a a-ring .91 of subsets A of X, and a real valued function m:d ~ R with the following properties: (a) UA = X, and m is a measure, that is, (b) m(0) = 0, where 0 is the empty set; (c) m(A) ~ 0 for all A E .91; (d) E; E .91, i = 1,2, ... , E; n Ej = 0 for all i"l= j, implies m(Uf; 1 E;) = If; 1 m(E;). Then, a real valued function f(x), x E X, is said to be measurable with respect to the measure space (X,d,m) if for every real a the set [x E Xl f(x) < a] is in the a-ring d. For a vector valued function f(x) = (flo ... ,In) we say that f is measurable with respect to (X, .91, m) if each component /; of f is measurable. The most common example of a measure space is that X = R, m is the Lebesgue measure, and .91 is the collection of all Lebesgue measurable subsets of R.

S.2.ii (AN ABSTRACT FORM OF mE IMPLICIT FUNCTION THEOREM). Let S be a measure space, let X and Y be metric spaces where X is the countable union of compact subspaces, let f: X ~ Y be a continuous mapping, let Yo = f(X), and let a:S ~ Y be a measurable map such that a(S) c f(X) = Yo. Then there is a measurable map t/!:S ~ X such that f(t/!(t)) = a(t) for all t E s. Proof. Indeed, by (8.2.i) there is a B-measurable map qJ: Yo ~ X such that f[qJ(Y)] = Y for every Y E Yo. Then, the map t/! = qJa:S ~ X is measurable, ft/!:S ~ Y, and ft/! = f(qJa) = (fqJ)a = a on S. D

Remark. The same theorem (8.2.ii) holds also for any topological space X which is the countable union of compact metrizable subsets, and any Hausdorff space Y. The theorem was proved by E. J. McShane and R. B. Warfield [1 J.

B. Orient or Fields and the Implicit Function Theorem

We assume that a set A is given in the tx-space R1+", x = (xl, . .. ,x"), and that for every (t, x) E A a nonempty set Q(t, x) of points z = (Zl, ... ,z") of


the z-space R" is assigned, or Q(t, x) c R", and this set may depend on (t, x). As mentioned in Section 1.2, we refer to the relation

(8.2.1) dx/dt E Q(t, x)

as an orientor field (or a differential equation with multivalued second member, or a contingent equation, or a differential inclusion). A solution X(t).tl ~ t ~ t2, of(8.2.1) is any vector valued function x(t) = (Xl, ... , xn) such that (a) x(t) absolutely continuous (AC) in [t1> ti]; (b) (t, x(t)) E A for all t E [tl' t2]; (c) dx/dt E Q(t, x(t)) a.e. in [tl' t2]. Thus, for almost all t E [t 1,t2] the direction dx/dt = (x'I, ... ,xln) of the curve x = x(t) at (t, x(t)) is one of the "allowable directions" z E Q(t, x(t)) assigned at (t, x(t)).

An orientor field will be said to be autonomous if Q(t, x) depends on x only and not on t. Nevertheless, every orientor field can be written as an autonomous one by a change of coordinates. Indeed, if we add the vector variable Xo satisfying the differential equation dxo /dt = 1 and initial conditionxO(t1) = t1, and if we then use the(n + I)-vector x = (XO, Xl, ... ,xn) and direction set Q(x) = [z = (ZO, Zl, ... , zn) = (ZO,z), z E Q(XO,x), ZO = 1], then the system (8.2.1) becomes dx/dt E Q(x). We may use this remark in proofs in order to simplify notations.

We return now to the notation of Section 1.12, where we have seen that, if an AC vector function x(t) = (Xl, ... ,xn), tl ~ t ~ t2, is a solution of the differential system x'(t) = f(t,x(t), u(t)), tl ~ t ~ t2, for some u(t) measurable, u(t) E U(t, x(t)), then it can always be written in the form of an AC solution of the orient or field x'(t) E Q(t,x(t)) where Q(t,x) = f(t,x, U(t, x)). We are in a position to show that the converse is also true.

S.2.iii (AN IMPLICIT FUNCTION THEOREM FOR ORIENTOR FIELDS). If A is a closed subset of the tx-space Rl +n, ifU(t, x) is a subset of Rm for every (t, x) E A; if the set M of all (t, x, u) E R1+n+m with (t, x) E A, u E U(t, x), is closed; if f(t, x, u) = (f1> ... ,fn) is continuous on M and Q(t, x) denotes the set Q(t, x) = f(t, x, U(t, x)) in Rn; and if x(t), tl ~ t ~ t2, is an AC vector function such that (t,x(t)) E A for all t E [tl' t2] and x'(t) E Q(t,x(t)) for almost all t E [t1,t2], then there is a measurable function u(t), tl ~ t ~ t2, such that u(t) E U(t,x(t)) and x'(t) = f(t, x(t), u(t)) for almost all t E [tl' t2].

Note that A is the projection of M on the tx-space, and that for every (t, x) E A the set U(t, x) is the projection on the u-space of the intersection of M with the subspace [t = t, x = x] in Rl +n+m. Thus, the assumption that M is closed certainly implies that U(t, x) is a closed subset of Rm. Thus, for every (t, x) E A, the set U(t, x) is necessarily closed. Moreover, if Ao is the projection of the sets A and M on the t-axis, then A o => [tl' t2].

Proof of (8.2.iii). As usual we denote by M the set of all (t, x, u) with (t, x) E A and u E U(t,x), hence Me R1+n+m. Also, we denote by N the set of all (t,x,z) with (t,x) E A, z = f(t,x, u), u E U(t, x); hence N c Rl +2n. Let F:M -4 N denote the continuous map defined by (t, x, u) -4 (t, x, z) with z = f(t, x, u). Here


M and N are metric spaces since MeR 1 + n + m, NcR 1 + 2n; also, M is closed by hypothesis, and M is the countable union of compact subsets, say Ma = [(t,x,u) E Mlltl + Ixl + lui ~ a], a = 1,2, .... Finally, N = F(M). By (S.l.ii) there is a B-measurable map cp: N -+ M such that F cp is the identity map, that is, Fcp(t, x, z) = (t, x, z) for every (t, x, z) E N. Now let us consider the map a: 1 -+ N on 1 = [t1,t2], defined by t -+ (t, x(t), x'(t)), where x(t), t1 ~ t ~ t2, is an AC solution of the orient or field x' E Q(t, x) = f(t, x, U(t, x)). Then ljJ = cpa:1 -+ M, and ljJ maps t into some (t, x(t), u(t)) E M, with F(t, x(t), u(t)) = (t, x(t), x'(t)), or x'(t) = f(t, x(t), u(t)). Actually, a is defined not on all of I, but in the subset 10 of 1 where x'(t) exists, and meas 10 = meas 1 = t2 - t1>

and thus the concluding relation holds in 10 , that is, x'(t) = f(t, x(t), u(t)) a.e. in [t1' t2]. On the other hand, x'(t) is measurable, that is, a is measurable, cp is B-measurable, and hence u(t) is measurable. The implicit function theorem (S.2.iii) is thereby proved. 0

C. Exercises

The following statements are often used. In this book we shall apply them to the case v = 1, G = (a, b) c R, x AC in [a,b], and ~(t) = x'(t), t E [a,b] (a.e.), r = n. 1. Prove the following more general form of (8.2.iii): Let A be a closed subset of the

tx-space Rv+n; for every (t, x) E A let U(t, x) be any subset of Rrn, assume that the set M = [(t,x,u)l(t,x) E A, u E U(t, x)] be closed in w+n+rn, let f(t, x, u) = (ft> ... ,j,.) be a continuous function on M, and for every (t, x) E A let Q(t, x) = f(t, x, U(t, x» = [z 1 z = f(t, x, u), U E U(t, x)] c R'. Let G be a measurable subset of W, and x(t) = (Xl, ... , Xn), ~(t) = (~1, ... , ~'), t E G, be measurable functions on G such that (t, x(t» E A, ~(t) E Q(t, x(t», t E G (a.e.). Then there is a measurable function u(t) = (u l , ... , urn), t E G, such that u(t) E U(t, x(t», ~(t) = f(t, x(t), u(t», t E G (a. e.).

2. Let A be a closed subset of the tx-space W+ n• For every (t, x) E A let U(t, x) be any subset of Rrn. Assume that the set M = Crt, x, u) 1 (t, x) E A, u E U(t, x)] is closed in w+ n+m. Let fort, x, u) and f(t, x, u) = (fl, ... ,f..) be continuous functions on M, and for every (t, x) E A, let (2(t, x) = [(ZO, z) Izo ~fo(t, x, u), z = f(t, x, u), u E U(t, x)] C R'+ 1. Let G be a measurable subset of the t-space W, and let rJ(t), ~(t) = (~l' ... , ~,), x(t) =

(Xl, ... , Xn), t E G, be measurable functions such that (t, x(t» E A, (rJ(t), ~(t» E (2(t, x(t», t E G (a.e.). Then there is a measurable function u(t) = (u l , ... , urn), t E G, such that u(t) E U(t, x(t», rJ(t) ~ fort, x(t), u(t», ~(t) = f(t, x(t), u(t», t E G (a.e.).

3. Let M be a compact subset of the xu-space w+rn, let fo(x, u), f(x, u) = (fl, ... ,f..) be continuous functions on M, and for every (x, z) E Rn +, let

T(x, z) = inf[ ZO 1 ZO ~ fo(x, u), z = f(x, u), (x, u) EM], -00 s T(x,z) s +00.


Let R(x, z) = [u I z = f(x, u), (x, u) E M]. Then (a) T(x, z) = + 00 if R(x, z) is empty; (b) T(x, z) = minE Zo I Zo ~ fo(x, u), (x, u) E R(x, z)] if R(x, z) is not empty; (c) T(x, z) is lower semicontinuous in Rn+,. The same statements hold even iff is continuous and fo is only lower semicontinuous on M.

A further extension is as follows: Let M be a compact subset of the xu-space Rn +m, let fOl(X, u), ... ,foix, u) be lower semicontinuous functions on M, let fl(X, u), ... ,J,(x, u) be continuous functions on M, and for every (x, z) E Rn+, and i = 1, ... ,IX, let T;(x, z) = inf[3i l(31, ••• , 3", Zl, ••• ,z'), 3i ~ foix, u), j = 1, ... , IX,

z' = f.(x, u), s = 1, ... ,r, (x, u) E M]. Then the extended functions Ti(x, z), i = 1, ... , IX,

are lower semicontinuous in Rn+,. 4. Let M be a closed subset of the xu-space Rn+m, and let fo(x, u),f(x, u) = (flo ... ,J,)

and T(x, z) be as in Exercise 3. Then T(x, z) is B-measurable in Rn+,. 5. The same as Exercise 1, except that now G is a measurable subset of RV with finite

measure, and on A, M, f we make the following assumption: For every e > 0 there is a compact subsetK ofG such thatmeas(G - K) < e, the set AK = [(t,x) E Alt E K] is closed, the set M K = [(t, x, u) E Mit E K] is closed, and the function f is continuous on M K • The conclusion is the same. Hint: For every e = k-l, k = 1,2, ... , there is a compact set Kk as above, and by Exercise 1 there is a measurable function Uk(t), t E Kko with Uk(t) E U(t, x(t», ,(t) = f(t, x(t), uk(t» for t E Kk (a. e.). Now take u(t) = Uk(t) for t E Kk - (Kl U ... u Kk- 1), k = 1,2, ....

6. The same as Exercise 2, except that now G is a measurable subset of RV with finite measure, and on A, M, fo, f we make only the following assumption: For every e > 0 there is a compact subset K of G such that meas(G - K) < e, the set AK = [(t,x)EAltEK] is closed, the set M K= [(t,x,u)EMltEK] is closed, and the functions fo, f are continuous on M K' The conclusion is the same.

7. The same as Exercises 2 and 6, except that now G is a measurable subset of RV with finite measure, and we have defined T(t,x,z), (t,x,z) E w+ n+m, by taking

T(t, x, z) = inf[ ZO I ZO ~ fo(t, x, u), z = f(t, x, u), u E U(t, x)]

= inf[zOI(zO,z) E Q(t, x)].

Let ,(t) = W, ... , c;'), x(t) = (xl, . .. ,x"), t E G, be measurable functions such that (t, x(t» E A, T(t, x(t), ,(t» E Q(t, x(t», t E G (a.e.). Prove that (a) T(t, x(t), ,(t» is measurable in G; and (b) there is a measurable function u(t) = (u 1, ••• ,~), t E G, such that u(t) E U(t,x(t», W) = f(t,x(t),u(t», T(t,x(t),W» = fo(t,x(t),u(t», t E G (a.e.).

8.3 Selection Theorems

A. A General Selection Theorem

The question we treat here is most relevant and, as we shall see, will allow a different interpretation and a generalization of what we have discussed in Section 8.2.

Let X, Y be two arbitrary sets or spaces. Let us consider a set valued map F, or x -+ F(x), x E X, F(x) c Y, mapping any element x of X into a subset F(x) of Y. Sometimes F is also called a multifunction from X to Y. Alternatively, we may think of F(x) as a variable subset of Y depending on the point, or parameter, x in X.

8.3 Selection Theorems 281

We say that a single valued map f:X ..... Y, or x ..... f(x), is a selector of F provided f(x) E F(x) for every x E X. Under some assumptions on F, X and Y, we will be able to prove the existence of selectors f having relevant properties.

Now let X be any given set, Y be a metric space, and S be a countably additive family of subsets of X, that is, such that, if A. E S for n = 1, 2, ... , then U:,= 1 A. E S. Let p(p, q) denote the distance function in Y.

8.3.i (LEMMA). If f.:X ..... Y, n = 1,2, ... , is a sequence of maps converging uniformly to a map f: X ..... Y, and such that f;; l( G) E S, n = 1, 2, ... , for every open subset G of Y, then f-l( G) E S for every open subset G of Y.

Proof. For any open set G of Y let us consider the closed set Y - G, and the closed set (Y - G). ofall y E Y at a distance::::; lin from Y - G. For every n let m. be an integer such that ml < m2 < ... , m ...... 00, and p(fm.(x),!(x»::::; lin for all x E X. Let us prove the formula

00

(8.3.1) rl(G) = U f';;n1(y - (Y - G).). n=l

Indeed, if x E r l(G), then y = f(x) E G and there is b > 0 such that N a(Y) c G, that is, the entire open neighborhood N a(Y) of y is contained in G, since G is open. Now for any n with lin < b/2, or n > 2/b, we have

p(fmn(x),y)::::; p(fmn(x),!(x» + p(f(x),y)::::; n- 1 + 0 = n- 1 < b/2,

while all points of(Y - G). are ata distance fromywhich is ;:::b - lin> 21n - lin = lin. Thus, x E f';;n1(y - (Y - G).) for all n > 2/b, and we have proved that in (8.3.1) the relation c certainly holds. Conversely, let x be a point in the second member of(8.3.1). Then x E f';;n1(y - (Y - G).) for some n; hence fmn(x) E Y - (Y - G)., that is, fmn(x) is at a distance> lin from Y - G; and therefore f(x) is at a distance >0 from Y - G, that is,f(x) E G, or x E f-l(G). We have proved that in (8.3.1) also the relation ::l holds. Thus, (8.3.1) is proved. Now Y - G is closed, (Y - G). is also closed, Y - (Y - G). is open. Hence, f';;n1(y - (Y - G).) is in the class S, and so is f-l(G), since S is countably additive. This proves (8.3.i). 0

Let L be a field of subsets of X, that is, L is a collection of subsets of X with the property that A, BEL implies that A u B, A n B, X - A also belong to L. Let S denote the countably additive family induced by L, that is, briefly, the family of all countable unions of elements of L.

8.3.ii (THEOREM: KURATOWSKI AND RYLL-NARDZEWSKI [1]). Let Ybe a complete metric space which is countably separable, that is, there is a countable subset of Y which is everywhere dense in Y. Let X be any set, or space, with afield L of subsets inducing a countably additive family S of subsets of X. Let x ..... F(x), x E X, be a set valued function such that (a) F(x) is a closed subset of Y for every x E X; (b) for every open subset G of Y the set [x E XIF(x) n G #- 0] E S. Then there exists a selector f:X ..... Y such that f-1(G) E S for every open subset G of Y.

Proof· Let R = [ri, i = 1, 2, ... ] be a countable subset of distinct points of Y, everywhere dense in Y. Thus we understand that ri #- r. for i #- s. By modifying the metric space if needed, we may assume that Y has diameter < 1. We shall obtain f as the limit of


maps I.: X - Y with the following properties:

(8.3.2) f;; 1( G) E S for every open subset G of Y;

(8.3.3) p(l.(x), F(x» < 1/2" for every x EX;

(8.3.4) p(I.(X),I._l(X» < 1/2"-1 for all x E X and n = 1,2, ....

Let us proceed by induction. First, take fo = r1 for all x E X, so that (8.3.2-4) are trivially satisfied. Assume that 1.-1 satisfying (8.3.2-4) has been found, and let us determine I.. To do this we take

(8.3.5)

(8.3.6)

(8.3.7)

Let us prove that

q = [xlp(rj,F(x» < 1/2"],

Di = [xlp(rj,J"_I(X»:$; 1/2"-1],

00

X = U Ai for every n. i== 1

Indeed, for every point x E X we have p(f"-I(x),F(x» < 1/2"-1; hence there is some Y E F(x) with P(y,l.-l(X» < 1/2"-1. Since Y = c1 R, there is some i such that

p(r;. y) :$; min[I/2", 1/2"-1 - P(y,l.-l(X»]. Then,

p(r;. F(x» :$; p(rj, y) :$; 1/2",

p(rj,l._l(X»:$; p(rj,Y) + P(y,l.-l(X»:$; 1/2"-1;

hence x E A~. Thus, X = Uj Ai· Denote by Bi the open ball [y E Y I p(y, rj) < 1/2"]. Then (8.3.5) and (8.3.6) become

(8.3.8) q = [x I F(x) II Bi #- 0],

(8.3.9)

From (b) we have q E S, and from (8.3.2) and induction at n - 1 we have Di E S; hence, Ai E S. Consequently, each set Ai = Ui Eii is the countable union of subsets Etl of the field L, and X = UiUi Eij. We may well arrange the double sequence Eu, i, j = 1, 2, ... , into a simple sequence E~ .... , s = 1, 2, ... , so that if E; = Eksm.' then

X=E'1uE~u···uE;u···.

We shall now define I.:X -R as follows: I.(x) =. rks if xEE; - (E'1 U·· ·UE;-I). It remains to show that I. satisfies the relations (8.3.2-4). By definition f;;l(rkJ = E;(E'1 U ... U E; _ d. Since L is a field, it follows that f;; l(rk) E L, and since f;; 1 (rj) =

Uk.=J;; l(rk), we also have f;; l(rj) E S for each i, and finally f;; I(Z) E S for every subset Z of R, since R is countable, and S is countably additive. We have proved (8.3.2).

Now, for a given x E X, let x E E; - (E'1 U ... U E;_ 1), let ks = i, and note that x E E; implies x .E E; c: Ai = q II Di, and (8.3.3), (8.3.4) follow from (8.3.5) and (8.3.6), since I.(x) = rj. We have proved that the sequence I., n = 1, 2, ... , is completely defined and satisfies (8.3.2-4). By (8.3.4) the sequence I. is Cauchy, and since Y is complete, I. converges uniformly to a function f: X - Y. By Lemma (8.3.i), f satisfies the condition (8.3.2). Finally, f(x) E F(x) by (8.3.3). Theorem (8.3.ii) is thereby proved. D


8.3.iii (COROLLARY). Theorem (8.3.ii) still holds even if(b) is replaced by (c) :for every closed subset K of Y the set {x E XIF(x) n K oF- 0} E L.

Proof. Since Y is a metric space, every open set G c Y is an Fs-set, that is, G = Kl U

K2 U' .. is the countable union of closed sets Ks. Then

00

{xIF(x)nGoF-0} = U {xIF(x)nKs oF-0}. o 05= 1

Remark 1. If S = L the proof of (8.3.ii) can be slightly simplified: namely, the decomposition of Ai into sets Eii is not needed, and we can simply define 1. by taking f.(x) = ri

for x E Ai - (A'i u ... u Ai- d.

B. L-measurable Set Valued Functions and L-measurable Selectors

We shall now consider the case where X is an interval [a, b] and L is the field of all Lebesgue measurable subsets of [a, b ].

For Y = R, we know that a real valued map f: X -+ Y is said to be L-measurable if for every real 0( the set [x E Xlf(x) < O(] is L-measurable, that is, belongs to L. Then, by taking complements, intersections, countable unions, and countable intersections, we immediately show that all sets [xlf(x) ~ O(], [xlf(x):$; O(], [xlf(x) > O(], [xlO( < f(x) < P], and finally [xlf(x) E G] where G is any open subset of R, are measurable. Note that here S = L.

For Yany topological space it is now natural to say that a single valued map f: X -+ Y is L-measurable if for every open subset G of Y the set [x E X I f(x) E G] belongs to L. Finally, again for Yany topological space, we shall say that a set valued map x -+ F(x), x E X, F(x) c Y, is L-measurable if for every open subset G of Y the set [x E X I F(x) n

G oF- 0] belongs to L.

8.3.iv THEOREM (KURATOWSKI AND RYLL-NARDZEWSKI [1]). Let Y be a separable complete metric space, and let x -+ F(x) c Y, be a set valued map whose values F(x) are closed subsets of Y.lf F is L-measurable, then there is an L-measurable selector f:X -+ Y with f(x) E F(x) for all x E X.

This is an immediate consequence of (8.3.ii). We know that any measurable single valued function f:X -+ Y, X = [a,b], has the

Lusin property, that is, given Il > 0, there is a compact subset K of X = [a, b] such that meas(X - K) < Il, and f restricted to K is continuous. It can be proved (see, e.g., Castaing and Valadier [I]) that any L-measurable set valued map x -+ F(x), x E X = [a, b], whose values F(x) are closed subsets of Y, has an analogous Lusin type property, namely, given Il > 0, there is a compact subset K of X such that meas(X - K) < Il,

and for every open subset G of Y the set [x E K I F(x) n G oF- 0] is open relative to K. We shall not need in the sequel the full strength of the theory of measurable set

valued functions.


C. Caratheodory's Functions

Let f(x, y) be a given real valued function defined in a product space G x Rn, where G is a given L-measurable subset of some space RS , s ;;:: 1, n ;;:: 1. We say that f is a Caratheodory function if f is continuous in Y for almost all x E G, and is measurable in x for every Y ERn.

8.3.v (SCORZA-DRAGONI [1 ]). If G is any measurable subset of RS and if f(x, y) is defined in G x Rn, is continuous in Y for almost all x E G, and is measurable in x for all Y ERn, then for any 1'/ > ° there is a closed subset K of G such that meas(G - K) < 1'/ and f is continuous on K x Rn.

Proof. For the sake of simplicity we take s = 1, n = 1, G = [0,1], B = [0,1], f defined on G x B; and if Eo is some subset of measure zero on G, we assume f continuous in Y for every x E G - Eo, and measurable in x for every Y E B.

For any given e > ° and integer m let E,rn be the set of all x E [0,1] such that Yl> Y2 E [0,1], IYl - Y21 ::s; m- l implies If(x, Yl) - f(x, Y2)1 ::s; r Ie. Let us prove that Eern is measurable. It is enough to prove that [0,1] - Eern is measurable, and sincemeas Eo =0, it is enough to prove that D = [0,1] - Eern - Eo is measurable. Now if xED, then there is a pair ofreal numbers Yl, Y2 with IYl - Y21 ::s; m- l and If(x, Yl) - f(x, Y2)1 > 3 -Ie. Since x ¢ Eo, f(x, y) is continuous in y, and therefore we may well assume that Yl and Y2 are rational. In other words, D is the union of sets of the form [xllf(x, Yl) -f(x, Y2)1 > r Ie] where Yl> Y2 are rational numbers with IYl - Y21 ::s; m 1. This is a countable family of measurable sets, and thus, D is measurable, and E,rn is measurable.

Here E,rn c E,.rn+ 1, and if E, is the union of all the sets E,rn, m = 1,2, ... , then E, is measurable, and we take F = [0,1] - Eo. Note that for every x E F, f(x, y) is continuous in Y on the compact set B, and hence uniformly continuous in y. In other words, every x E F belongs to some E,rn, or [0,1] - Eo = Ee = Urn Eern, meas Ee = 1. We conclude that there is some mo such that meas Eerno > 1 - 2 -1 el'/, and for all x E Eerno and all Yl> Y2 E [0,1], IYl - Y21 ::s; mo \ we also have If(x, Yl) - f(x, Y2)1 ::s; r Ie.

Let 0= Uo < Ul < ... < uq = 1 be q + 1 equidistant points in [0,1] with q ;;:: mo, so that Uj - Uj-l = q-l ::s; mol. For every j = 1, ... , q, the function f(x,u) is measurable in x. Hence, by Lusin's theorem, there is some closed set Fj C [0,1] with meas Fj > 1 - (2q)-lel'/ and f(x,u) is continuous in x on Fj-in fact uniformly continuous. If V = n1=1 Fj, then meas V> 1- rlet/ and all q functions of x, f(x,u), j = 1, ... , q, are continuous on V-in fact uniformly continuous. Thus, there is some bl = bl(e) such that Xl' X2 E V, IXI - x21::s; bl(e) implies If(Xl,Uj) - f(X2,U)I::s; 3- le, j = 1, ... ,q. Finally, for E~ = V n E,rno and b(e) = min [ ma \ bl (e)], we see that for any two points (Xl> Yl), (X2' Y2) with Xl, X2 E E~, IXI - x21 ::s; b(e), Yl, Y2 E [0,1], IYl - hi ::s; b(e), there is some j with IYl - Ujl, IY2 - Ujl ::s; ma 1 and

If(Xl' Yl) - f(x 2, h)1 ::s; If(Xl' Yl) - f(x l , u)1 + If(xl, u) - f(X2, u)1

+ If(X2' u) - f(X2' Y2)1 ::s; e/3 + e/3 + e/3 = e.

The set E~ depends also on 1'/, but we shall keep 1'/ fixed. Now we take e = ei = r i -1, E; = E~i' i = 1,2, ... , and E* = n;x;l E;. Then meas E* > 1 - 1'/ I;x;l r i- l > 1 - 1'/. Note that f is continuous on E* x B. Indeed, given y > 0, take i = i(y), so that ei < y. If (Xl> Yl), (X2' Y2) are in E* x B, and their distance is < b(e,) = b(y). then certainly Xl> X2 E E;, and If(Xl' Yl) - f(X2, Y2)1 ::s; ei < y. Now E* is measurable, meas E* > 1 - 1'/. Thus, there is some closed subset K of E* with meas K > 1 - 1'/, and f is continuous


on K x B. Statement (8.3.v) is thereby proved under the mentioned restrictions. We leave to the exercises to prove it under the stated general hypotheses. 0

Remark 2. The statement (8.3.5) will have a role in what follows. Here we mention only an immediate application. If f(x, y) satisfies the conditions of (8.3.v), and y(x), x E G, is measurable with values in R", then f(x, y(x)) is measurable in G. Indeed, for every '1 > 0, there is K compact, KeG, with meas(G - K) < '1, such that f(x, y) is continuous in K x R", and there is K' compact, K' c G, with meas(G - K') < '1, such that y(x) is continuous on K'. Then f(x, y(x)) is continuous in K n K', and since meas( G - K n K') < 2'1 with '1 arbitrary, we conclude that f(x, y(x)) is measurable in G.

D. Another Form of the Implicit Function Theorem

As in Section 8.2B, let A = [a, b] x R" (a closed subset ofthe tx-space R"+ 1), let U be a fixed closed subset of the u-space Rm, and let f(t, x, u) = (fh ... , I.) be a Caratheodory function defined on [a, b] x R" x U, that is,! is measurable in t for every (x, u) E R" x U, and is continuous in (x, u) for almost all t E [a, b]. As usual, for every (t, x) E [a, b] x R", let Q(t, x) denote the subset of all z = (zl, ... , z") E R" such that z = f(t, x, u), U E U, that is, Q(t, x) = f(t, x, U). Let x(t) = (xl, ... , x"), a ::;; t::;; b, be any AC function, such that

x'(t) E Q(t, x(t)), tE [a,b] (a.e.).

In other words, x(t) is an AC solution for the orientor field dx/dt E Q(t,x).

8.3.vi (AN IMPLICIT FUNCTION THEoREM). Under the above assumptions there is an Lmeasurable u(t), a ::;; t ::;; b, with u(t) E U, x'(t) = f(t, x(t), u(t)), t E [a, b] (a.e.).

Proof. Two proofs are given here of this statement. I. By (8.3.v), for '1 = k -1, k = 1, 2, ... , there is a closed subset F k of [a, b] such that

f(t,x,u) is continuous in Fk x R" x U and meas Fk > b - a - k- 1• If F = Uk'=l Fk, then F c [a, b], meas F = b - a. From the implicit function theorem (S.2.iii) there is a measurable function Uk(t), t E Fb such that Uk(t) E U for t E Fk, x'(t) = f(t, x(t), uk(t)), t E Fk (a.e.). Then, if we take u(t) = Uk(t) for t E Fk - (F 1 U ... u Fk- 1), then u(t) is defined a.e. in [a, b], it is measurable in [a,b], and u(t) E U, x'(t) = f(t, x(t), u(t)), tE [a,b] (a.e.).

II. Let F 0 denote the set of all t E [a, b] for which either x'(t) does not exist, or it is not finite, or x'(t) E Q(t, x(t)) does not hold, or f(t, x, u) is not a continuous function of (x, u). Then meas F 0 = O. Again, by (8.3.v), for '1 = k-l, k = 1, 2, ... , there is a closed subset F k of [a, b] such that f(t, x, u) is continuous in F k X R" x U, and meas F k > b -a - k- 1• We may well assume that Fo n Fk = 0, that is, Fo c [a,b] - Fk for all k. Let F = Uk'=l Fk, so that meas F = b - a, Fo c [a,b] - F. For t E F let U(t) denote the subset of all u E U such that f(t, x(t), u) = x'(t). We know that U(t) is not empty, since x'(t) E Q(t, x(t)) = f(t, x(t), U) for t E F. Let us prove that the set-valued map t --+ U(t), t E F, U(t) cUe Rm, is L-measurable. Let B., s = 1,2, ... , denote the closed ball with center the origin and radius s in Rm. Let G be any given open subset of Rm. Since f(t, x(t), u) is a continuous function on the compact set Fk x U n B., and x'(t) is continuous on Fb the set Fk.=[tEFklf(t,x(t),u)=x'(t) for some uEUnB.] is closed, and the setF;" = [t E Fklf(t,x(t),u) = x'(t) for some u E GnU n B.] is open in F ks, that is, open relative to Fk •. Hence, the set F~ = U. F;" = [t E Fkl f(t, x(t), u) = x'(t)


for some u E GnU] is B-measurable, and the set F" = Uk Fk is also B-measurable. Finally, F" differs from the set V = [t E [a, b] I f(t, x(t), u) = x'(t) for some u E GnU] by at most a set of measure zero (c: [a, b] - F). Thus, V is measurable, and t ..... U(t) is a set valued L-measurable map. By (S.3.iv) there is an L-measurable u(t), t E [a, b], such that u(t) E U(t), that is, u(t) E U, x'(t) = f(t, x(t), u(t», t E [a, b] (a.e.). 0

Remark 3. Statement (S.3.vi) is a particular case of Section S.2.C, Exercise 5. Indeed, under the conditions of(S.3.vi), by (S.3.v), for every e > 0 there is a compact subset K of [a, b] with meas([ a, b] - K) < e such that f(t, x, u) is continuous on the closed set M K = K x R" x U, and AK = K x R" is also closed. The statement in Section S.2C, Exercise 5 is more general, since there we allow (t,x) to cover an arbitrary set A, U is an arbitrary set U(t, x) which may depend on t and x, and f is an arbitrary function on M = [(t, x, u) I (t, x) E A, u E U(t, x)], with the sole restriction that for e > 0 there is a compact subset K of [a,b] such that meas([a,b] - K) < e, AK = [(t,x) E Alt E K] is closed, MK = [(t, x, u) E Mit E K] is closed, and f(t, x, u) is continuous on M K.

8.4 Convexity, Caratheodory's Theorem, Extreme Points

A. Convexity

Let X be any linear space over the reals (cf. Dunford and Schwartz, [I, p. 49]). A subset K of X is said to be convex provided Xl> X 2 E K,O::5: IX ::5: 1, implies that x = (1 - IX)X1 + IXX2 is also a point of K; that is, the entire segment s = X I X2 between two points of K is contained in K.

Given any m points, Xl' •.• , Xm in X, any point x = AIX I + ... + AmXm with As ~ 0, s = 1, ... , m, Al + ... + Am = 1, is said to be a convex combination of Xl' ... , xm•

Given any subset A of X, we shall denote by co A the smallest convex set in X containing A. This set co A is said to be the convex hull of A. We also denote by A the set of all convex combinations of arbitrarily many points of A. Finally, for every fixed integer m ~ 2, we denote by Am the set of all combinations of at most m points of A. Thus, Am c: Am+ I c: A, A = UAm, where U ranges over all m = 2, 3, ....

8.4.i. A subset A of a linear space X is convex if and only if every convex combination x = D'= I AiXi of points of A belongs to A.

The sufficiency is trivial, since the requirement for m = 2 reduces to the definition of convex set. The necessity is a consequence of the statement:

8.4.0. For any subset A of a linear space X we have Um Am = A = co A.

Proof· (a) Let us prove that A c: co A. It is enough to prove that Am c: co A for every m.

That A2 c: co A is a consequence of the definition of convex set. Let us assume that we have proved that Am- l c: co A, and let us prove that Am c: co A. Indeed, if x = AIXI + ... + AmXm, O:s: As:S: 1, s = 1, ... ,m, Al + ... + As = 1, either all As are zero

8.4 Convexity, Caratheodory's Theorem, Extreme Points 287

but one, say Am = 1, and then x = xm E A c co A; or at least two of the As are positive, say A .. - 1 and Am' and then IX = Al + ... + Am-I> 0, 1 - IX = Am > 0, and x = IX(L'i- 1(As/lX)xs) + (1 - IX)Xm. Here, Xm E A, and the expression in parenthesis is a point of Am' and hence of co A by the induction hypothesis; and therefore x E co A. We have proved that Am C co A for every m, and hence A c co A.

(b) Let us prove that A is a convex set, and that A = co A. Indeed, if x, YEA and z = IXX + (1 - IX)Y, 0::; IX::; 1, then x = P1VI + ... + PmVm, Y = qlwi + ... + qMWM with Vs E A, Ps ~ 0, s = 1, ... , m, Ws E A, qs ~ 0, s = 1, ... , M, and PI + ... + Pm = 1, ql + ... + qM = 1. As a consequence, we have also

z = IXX + (1 - IX)Y = IXPIV I + ... + IXPmVm + (1 - lX)qlWI + ... + (1 - lX)qMWM'

Thus, z is a point of ii, and A is convex, A c ii c co A. Since co A is the smallest convex set containing A, we conclude that A = co A.

8.4.iii (THEOREM: CARArnEoOORY [I]). For any subset A of R", every point of the convex hull of A is the convex combination of at most n + 1 suitable points of A. In symbols,

co A = A = A"+I'

Proof. It is enough we prove that any convex combination of m ~ n + 2 points of R" is also a convex combination of at most n + 1 of the same points. For this it is enough to prove that any convex combination of m ~ n + 2 points of R" is also the convex combination of at most m - 1 of the same points. Let VI"", Vm be the m points of R", and x any convex combination of them. It is not restrictive to assume x = 0; hence

As ~ 0, s = 1, ... , m, Al + ... + Am = 1.

If some of the numbers As are zero, then x = 0 is the convex combination of fewer than m points Vs' Thus, we assume As > 0, s = 1, ... , m. Analogously, we can assume all Vs i= 0 and distinct. For m ~ n + 2, we have m - 1 ~ n + 1, and there must be a linear combination

(8.4.2)

of the m - 1 nonzero vectors Vb"', vm with coefficients c l , ... , Cm-I real and not all zero. Indeed, if Vi = (v~, ... , v~), then the system of n linear homogeneous algebraic equations in the m - 1 ~ n + 1 unknowns CI , ... , Cm - I ,

s = 1, ... , n,

must have a solution C1, ••• , Cm-I with numbers Ci not all zero. Thus, for every real IX we also have from (8.4.1) and (8.4.2)

0= (AI + IXcdvl + ... + (Am-I + IXCm- 1)Vm- 1 + Amvm,

and for all IX of a maximal interval ri < IX < JJ, ri < 0, JJ > 0, we still have

Thus, for IX = ri and IX = lJ, at least one of the numbers Al + IXC 1, ... , Am-l + IXCm-1 is zero, say for IX = ri, and Al + ric i = 0, As + rics ~ 0, s = 2, ... , m - 1. Then,

0= (A z + riCZ)vl + ... + (Am-l + riCm- 1)Vm- 1 + Amvm,

C = (Az + riCz) + ... + (Am-I + ricm-d + Am > O.


For qs = (As + rics)/C, S = 2, ... , m - 1, qm = Am/C, all numbers qs are nonnegative with q2 + ... + qm = 1, and 0 = q2V2 + ... + qmvm, that is, x = 0 is a convex combination ofm - 1 ofthe original m vectors VI' ... , Vm' Theorem (S.4.iii) is thereby proved. D

B. The Closed Convex Hull of a Set

I[ we denote by cl A and co A, respectively, the closure and the convex hull of a set A in X, then we may well consider also the sets cl co A, co cl A.

8.4.iv. For any set A in R", the sets A, cl A, co A, co cl A are all contained in cl co A.

Proof. It is enough to prove that co cl A c cl co A. Indeed, by (S.4.iii), any point x E co cl A can be written in the form x = L~ AiXi where 0 :::;; Ai :::;; 1, i = 1, ... , v = n + 1, L~ Ai = 1, and Xi E cl A, i = 1, ... , v. Hence, there are v sequences [Xik, k = 1, 2, ... ] of points X ik E A, k = 1, 2, ... , with X ik -> Xi as k -+ 00, i = 1, ... ,v. I[ we take X k = L; AiX ik , then

Hence, IXk - xl-+ 0 as k -+ 00, or X k -+ x, where now X k E co A. Thus, x E cl co A. We have proved that every point x E co cl A is also a point of cl co A, or co cl A c cl co A.

D

The set cl co A is often called the closed convex hull of A. Note that co cl A may well be actually smaller than cl co A. For instance, if A =

[(x, y)iO < x < + 00, y = ± 1/x], then co cI A = [(x, y)iO < x < + 00, - 00 < y < + 00],

while cl co A = [(x, y)IO:::;; x < + 00, - 00 < y < + 00]. However, if A is a bounded subset of R", then co cl A = cl co A. The proof of this last statement is left as an exercise for the reader. Also note that for any compact subset A of R" we have co A = co(aA), where aA denotes the boundary of A.

Note that if f(t) = (ft. ... J"), t E G, is any L-integrable function on a measurable set G c R, 0 < meas G < 00 (that is, each component}; is L-integrable), and if the values f(t) of f belong to some subset A of R", then the mean value of f on G belongs to cl co A. In other words, f E (L( G) )", f(t) E A c R", implies

(meas G)-I fG f(t) dt E cl co A.

Concerning the last statement, let us consider first any step function on G (cf. McShane [I, p. 54]), that is, a function f: G -> A c R" with values in A such that f is constant on each set Gi , i = 1, ... , N, of a finite decomposition of G into disjoint measurable subsets G;, each of positive measure. Then, if}; denotes the constant value of f on Gi, then by the definition of L-integral of a step function we have

N

fG f(t) dt = i~1 fi(meas G;).

I[ Yi = meas Gdmeas G, then 0 < Yi :::;; 1, i = 1, ... , N, Lf= I Yi = 1, and the mean value m(f) of f, or m(f) = (meas G) - I J G f( t) dt = Lf= I Y i}; E co A, is a convex combination of the values}; E A, and m(f) E co A. For any measurable L-integrable function

8.4 Convexity, Caratheodory's Theorem, Extreme Points 289

f: G -> A c: Rn, the L-integral of f is the limit of the integral defined on suitable step functions J;., and then m(f) = limk~oo m(fk) E cl co A. Thus, m(f) E A if A is already a closed convex set.

C. Supporting Hyperspaces

If I(x) denotes any real valued nonzero linear function on a real linear space X, then the set S = [x I/(x) = c] where I has a constant value c is said to be a hyperspace of X. Then a hyperspace S divides X into two half spaces, say S+ = [x I/(x) :?: c] and S- = [x l1(x) :s; c]. A hyperspace S is said to be a supporting hyperspace for a convex set K if K is contained in one of the two half spaces S+ or S-. For the sake of simplicity we shall assume from now on that X is a linear topological space.

8.4.v (EXISTENCE OF SUPPORTING HYPERSPACES) (cf. Dunford and Schwartz [I, pp. 412, 418]). For any convex subset K of a linear topological space X, possessing interior points, and any point Xo E X - K, there is some supporting hyperspace for K through xo, or S: I(x) = I(xo), where I is a continuous linear real valued nonzero functional on X.

A supporting hyperspace through a point need not be unique, as we can see by considering a convex polygon K in R2 and taking for Xo one of its vertices or any exterior point. Note that if a convex set K has interior points, then the set Ko = int K of all its interior points and the closure cl K of K are also convex.

If K c: X = Rn and K has dimension :S;n - 1, then K is contained in a hyperspace S of X = Rn, which also is a supporting hyperspace for K through each of its points. If K c: X = Rn has interior points, then Ko is convex, and each supporting space for Ko through any point Xo E bd K is also a supporting space for K through Xo.

S.4.vi. If K is any convex subset of X, then cl K = nS+, where the intersection is taken over all half spaces S+ containing K.

Proof. Since cl K c: S+ for every S+ as above, we certainly have cl K c: nS+. Let us prove the opposite relation, cl K => nS+. Let Xo denote any point Xo E nS+. Let x be any fixed point of K, and let s be the closed segment s = [xo, x]. Then s n cl K is a closed subset of s, and we denote by Xl the point of s n cl K closest to Xo. If Xl = Xo

then Xo = Xl E cl X. Now suppose, if possible, that Xo -# X l' Then there must be a circular neighborhood

U of Xo not containing points of K, since otherwise Xo E cl K, and Xl = Xo, which is not the case. Then the entire half-closed, half-open segment [XOx l ) does not contain points of K; in particular x = r I(XO + Xl) ¢ K. Then there is a hyperplane I through x with I(x) = c, I(x) > c, and I(xo) < c. In other words, Xo ¢ S+; hence Xo ¢ nS+, a contradiction. Thus, Xo = Xl> and Xo E cl K. We have proved that nS+ c: cl K, and (8.4.vi) is proved. D

Remark. Statement (8.4.vi) has also a more general version: For any two disjoint convex subsets K and L of the linear topological space X of which one at least has interior points, there is some separating hyperspace S: I(x) = c, where I is a continuous linear real valued nonzero functional on X, and I(x) :?: c for X E K, I(x) :s; c for X E L [Dunford and Schwa:tz, loco cit.]


D. Extreme Points

If K is a convex compact subset of any linear space X, a point x E K is said to be an extreme point of K provided: if x = C!(XI + (1 - OC)X2 for some two points Xl' x2 of K and ° < oc < 1, then Xl = X 2 , and hence x = Xl = X2. The set of all extreme points of K is denoted extr K.

For instance, if K is a convex polygon in R2, then the extreme points are the vertices of K. If K is a ball in R n, then all its boundary points are extreme points.

The existence and main properties of extreme points are specified in a statement of Krein-Milman which holds in any locally convex linear topological Hausdorff space X:

S.4.vii (KREIN-MILMAN) (cf. Dunford and Schwartz I, vol. 1. p. 440). Any convex compact nonempty subset K of a locally convex linear topological Hausdorff space X possesses extreme points, and K = co extr K.

Another interesting property of the extreme points is as follows:

S.4.viii (cf. Dunford and Schwartz, loco cit.). If K is any compact subset of X, then extr co K c K.

Note that the set extr K of a convex compact set need not be a closed set, as the following example in R3 shows. Let us consider the disk S = [(x, y)l(x - 1)2 + y2 :$; 1, Z = 0] and let K be the solid body we obtain by projecting S from the points (0,0,1) and (0,0, - 1). Then extr K is made up of the points (0,0, 1) and (0,0, -1) and all (x, y, z) with (x - 1)2 + y2 = 1, Z = 0, x> 0. This set is not closed. (It can be proved that extr K of a compact set K is a Gd-set.)

8.5 Upper Semicontinuity Properties of Set Valued Functions

A. Upper Semicontinuity by Set Inclusion, and Properties (K) and (Q)

We have already encountered sets U(t, x), Q(t, x) depending on the pair (t, x), that is, set valued functions, or multifunctions, (t, x) -+ U(t, x), (t, x) -+ Q(t, x), where the independent variable (t, x) ranges over a given set A of the tx-space Rn + 1. To simplify notation, we should now denote the independent variable simply by x, and consider set valued functions, or multifunctions, x -+ Q(x), x E A c W, Q(x) eRn. This is even more to the point in that we shall have occasion later to consider set valued maps t -+ Z(t), or x -+ Z(x). On the other hand, our considerations in the present section hold for any set valued map x -+ Q(x), where x ranges in any metric space X, and the sets Q(x) are subsets of any topological space, or linear topological space, Y. We shall use therefore this terminology in the present section. In Section 8.6 we shall

8.5 Upper Semicontinuity Properties of Set Valued Functions 291

need Y to be a finite dimensional space, and we shall return there to the notation x --+ Q(x), x E A c RV , Q(x) eRn.

First, let us assume that both X and Yare given metric spaces. Given a metric space X, a point Xo E X, and a number 6 > 0, we denote

by N o(xo), called the 6-neighborhood of Xo in X, the set of all x E X at a distance ::;6 from xo; thus No(xo) c X. Let x --+ Q(x) denote a set valued map, or multifunction, from a metric space X to a metric space Y, that is, Q(x) c Y for every x E X. Given Xo E X and 6 > 0, we denote by Q(xo; 6) the union of all sets Q(x) with x E No(xo), or Q(xo;6) = U[Q(x), x E No(xo)]. Also, given e > 0, we denote by [Q(XO)]e the e-neighborhood of Q(xo), i.e. the set of all points of Y at a distance ::; e from Q(xo), that is, at a distance ::; e from points of Q(xo).

We say that a set valued map x --+ Q(x), x E X, Q(x) c Y, is upper semicontinuous by set inclusion at Xo provided, given e > ° there is some 6 > 0, 6 = 6(xo,e), such that Q(x) c [Q(XO)]e for all x E No(x o), that is, Q(xo;6) c [Q(XO)]e' We say that x --+ Q(x) is upper semicontinuous by set inclusion in X if it has this property at every point Xo E X. For the sake of brevity, we may simply say that the sets Q(x) have such a property. In simple words, we could say that, for upper semicontinuity at Xo, the nearby sets Q(x) cannot be much "larger" than Q(xo), though some or even all of them could be much "smaller".

8.S.i. If X and Yare metric spaces, X is compact, and f: X --+ Y is a single valued continuous map, then the set valued function f- 1(y), y E Yo = f(X), is upper semicontinuous by set inclusion.

Proof· If this were not the case, then there would be Xo E X, Yo = f(x o) E Yo, e > 0, and sequences [xk ] in X and [Yk] in Yo with Yk = f(x k ), Yk --+ Yo as k --+ 00, and dist{xk,j-1(yo)} ~ e. By the compactness of X there is some x E X and a subsequence, say still [xk], with X k --+ X as k --+ 00. By the continuity of f, f(x) = Yo, or X E f- 1(yo) with Xk --+ x, dist{x,j-1(yo)} ~ e, a contradiction. 0

Note that (S.S.i) does not necessarily hold if X is not compact, as the following example shows. Take X = Y = R, f(x) = xe-X, so that f(O) = 0, f- 1(0) = {O}, and for Xk --+ + 00, Yk = f(xk) --+ 0. The points Xk escape any given neighborhood of ° in X, and the sets f-1(y) are not upper semi continuous by set inclusion at y = 0.

The property of upper semicontinuity by set inclusion is well suited for compact sets and spaces. For the "unbounded" case, other properties, essentially more general, have been proposed.

Given any set Z in a linear topological space Y, we shall denote by cl Z, bd Z, co Z the closure of Z, the boundary of Z, and the convex hull of Z respectively. Thus, cl co Z denotes the closure of the convex hull of Z, or briefly, the closed convex hull of Z (cf. Section 8.4B).


Let x --+ Q(x), x E X, Q(X) c: Y, be a set valued map from a metric space X to a linear topological space Y. Let Xo be a point of X. Kuratowski's concept of upper semicontinuity, or property (K), is relevant. We say that the map x --+ Q(x) has property (K) at Xo provided Q(xo) = no cl Q(xo; c5), that is,

(8.5.1 ) Q(xo) = n cl U Q(x). 0> ° XEN,,(xo)

Here Q(xo), as the intersection of closed sets, is certainly a closed set. We shall need also the following variant. We say that the map x --+ Q(x)

has property (Q) at Xo provided Q(xo) = nil cl co Q(xo; c5), that is,

(8.5.2) Q(xo) = n cl co U Q(x). 0> 0 XEN,,(xo)

Here Q(xo), as the intersection of closed convex sets, is certainly closed and convex.

Again, we shall say that the map x --+ Q(x) has property (K) [or property (Q)] in X if it has property (K) [or (Q)] at every point of X. For brevity, we may also say that the sets Q(x) have property (K) [or (Q)] at x o, or in X. The indication "with respect to x" may be needed if the sets depend also on other parameters.

Remark 1. Note that both in (8.5.1) and (8.5.2) the inclusion c: is trivial, since the second member always contains the entire set Q(xo). Thus, what is actually required in (8.5.1) and (8.5.2) is that the inclusion => hold.

Note that both in (8.5.1) and (8.5.2) we do not exclude the case of sets Q empty. Then (8.5.1) becomes Q(xo) = 0 = no cl Q(xo;c5), and (8.5.2) becomes Q(xo) = 0 = no cl co Q(xo; c5).

Also, note that properties (K) and (Q) are often expressed in terms of "a given sequence Xk, Zb k = 1, 2, ... ," with Xk E X, Xk --+ Xo in X, Zk E Q(Xk) c: Y, and in that case all is required is that, say, for property (Q),

Q(xo) => hOI cl co {.9h Zs}.

For instance, suppose that G is a given measurable subset of points t E W, and for every t E G that x --+ Q(t, x) is a given set valued function from Rn to R r and that xo(t), Xk(t), Zk(t), t E G, k = 1, 2, ... , are given measurable functions with values in Rn and Rr respectively, with Xk(t) --+ xo(t) pointwise a.e. in G, and Zk(t) E Q(t, Xk(t» for all k and t E G. Then all that need be required for a global property (Q) in G with respect to the sequence [xk, Zk] is that for almost all t E G we have

(8.5.3) Q(t,xo(t» => hOI clCO{.Vh Zs(t)}.

If xk(t) --+ xo(t) in measure, we need only require that (8.5.3) hold for a suitable subsequence [ZkJ. Below we shall refer to the definitions (8.5.1-2).


8.5.ii. Property (Q) implies property (K).

Indeed,

Q(xo) c: n cl Q(xo; b) c: n cl co Q(xo; b) = Q(xo), 0>0 0>0

and thus equality holds throughout in this relation.

8.5.iii. Let x --+ Q(x), X E X, Q(x) c: Y, be a set valued map, and let M = [(x, y) 1 x E X, Y E Q(x)] c: X x Y, that is, M is the graph of the set valued map. Then x --+ Q(x) has property (K) in X if and only if M is closed in the product space X x Y.

Proof. Suppose that the sets Q(x) have property (K) in X, and let us prove that M is closed. Let (x, y) be a point of accumulation of M. Then there is [(Xk, Yk)] with (Xb Yk) E M, (xk, Yk) --+ (x, y), Yk E Q(xd. Thus, Xk --+ x in X, and Yk --+ Y in Y with Yk E Q(Xk)' Thus, y E flo> 0 cl Q(x; b) = Q(x), or (x, y) E

M, and M is closed. Conversely, assume that M is closed in the product space, and let us prove that the sets Q(x) have property (K) at every point x E X. Let x be any point of X, and take any y E no> 0 cl Q(x; b). Then there is [(Xk,Yk)] with Xk --+ x in X, Yk --+ yin Y, with Yk E Q(Xk; bk), bk --+ 0, and the distance of Xk from x is ::;; bk. Then (Xk, Yk) E M, (Xk, Yk) --+ (x, y) in X x Y, where M is closed. We conclude that (x, y) E M, or y E Q(x), and property (K) holds at X. Here we have assumed that flo> 0 cl Q(x; b) is not empty. If this set is empty, then certainly Q(x) is empty, and property (K) holds at X.

Remark 2. We have noticed in Remark 1 that (8.5.iii) (direct and inverse) holds even if some of the sets Q(x) are empty (but property (K) is verified at every x E X). If A denotes the set of all x E X where Q(x) is not empty, then (8.S.iii) can be reworded by saying that M is closed in X x Y if and only the map x --+ Q(x) has property (K) in cl A (in A if A is closed). The requirement involving the closure of A is needed, as the following example shows: x E A = [0 < x < 1] c: R, Q(x) = {O} c: R. The sets Q have property (K) in A, but the graph M = [(x, 0)10 < x < 1] is not closed in R2. Statement (8.5.iii) with this remark and the parallel statement (8.5.v) below will be summarized in Remark 3. As a parenthetical remark, note that if we take Q(x) = [YIY ~ x- 1(1 - X)-l], 0 < X < 1, Q(x) the empty set otherwise, then A is open in R but M is closed in R2, and the map x --+ Q(x) has property (K) at all x E cl A(and at all x E R).

Concerning the relations between upper semi continuity by set inclusion and properties (K) and (Q), we shall limit ourselves to set valued maps x--+ Q(x), X E A c: W, Q(x) c: W.

8.5.iv. Let x --+ Q(x), X E A c: W, Q(x) c: Rn be a set valued map which is upper semicontinuous by set inclusion at Xo E A. If the set Q(xo) is closed, then the


sets Q(x) have property (K) at xo; if the set Q(xo) is closed and convex, the sets Q(x) have property (Q) at Xo.

Indeed, given e > 0, there is 1J > 0 such that

Q(xo) c Q(xo; 1J) c [Q(xo)] •. Hence

Q(xo) c n cl Q(xo; 1J) c cl[Q(xo)]., 0>0

and since Q(xo) is closed and e arbitrary, we also have

Q(xo) c n cl Q(xo; 1J) c Q(xo), 0>0

and thus equality holds throughout in this relation. Analogous proof holds for property (Q) if the set Q(xo) is closed and convex.

B. The Function F and Related Sets Q(x) and M

First let U be a given subset of Rn, and F(u), U E U, be a real valued function, finite everywhere in U. The epigraph of F, or epi F, is defined as the set

Q = epi F = [(ZO, u)izO ~ F(u), U E U] C R"+1.

We shall think of F as defined everywhere in Rn by taking F = + 00 for U E Rn - V and then we say that F is an extended function. This does not change the set Q = epi F.

8.5.v. If F is an extended function, then epi F is closed in Rn + 1 if and only if F is lower semicontinuous in Rn. This is certainly the case if V is closed in Rn and F is lower semicontinuous on U.

Proof. Let us assume that F is lower semicontinuous in R" and let us prove that epi F is closed. Let (z, u) E cl(epi F). Then there is a sequence (Zk' Uk), k = 1, 2, ... , of points (Zk' Uk) E epi F with Zk --+ Z, Uk --+ U as k --+ 00, with U E Rn and Zk Z F(Uk). By lower semi continuity F(u):::; lim inf F(Uk) :::; lim Zk = z, and (z, u) E epi F, that is, epi F is closed. Assume that epi F is closed, and let us prove that F is lower semicontinuous. Negate. Then there are points u, Ub U2' ... , such that Uk --+ U, F(Uk) --+ F(u) - e for some e > O. Then (F(Uk), Uk) E epi F, (F(Uk), Uk) --+ (F(u) - e, u), and thus (FU - e, u) ¢ epi F, that is, epi F is not closed, a contradiction. Here we have assumed F(u) finite. Analogous argument holds if F(u) = + 00 or F(u) = - 00. 0

Remark 3. Note that if U is not closed, the mere lower semicontinuity of F in U does not imply the lower semicontinuity of F in R". For instance, for U = [ui-oo < U < 0] c R, F(u) = 0 ifu < 0, F(u) = +00 ifu ~ 0, then F is lower semicontinuous in U but not in Rn, and epi F is not closed. Note the following slight extension of (8.S.v): If F1(u), ... , F~(u) are IX extended


functions on Rn, and Q denotes the set Q = [(z, u) = (zt, ... , t', u l , ... , un)1 Zi ~ Fi{u), i = 1, ... , Q(, U E W], then Q is closed in w+rz if and only if the Q( functions F 10 ••• , Fa are lower semicontinuous in Rn. We leave the proof of this statement as an exercise for the reader.

Remark 4. Let A be any subset of RV , let x -+ U{x), x E A c RV, U{x) c W,

and M = [(X,U)IXEA, UE U{x)] C Rv + n• Let F{x,u) be a real valued function defined on M, finite everywhere on M, and let

it = [(x, zo, u)lx E A, u E U{x), Zo ~ F{x, u)].

For any x E A, let Q{x) denote the set Q{x) = [(ZO, u) I ZO ~ F{x, u), u E U{x)] c W + l. Again, let us think of F as defined everywhere in W + n by taking F = + 00 in Rv + n - M so that Q{x) is the empty set for x E W - A. Combining (8.5.iii) and (8.5.v), we can now state that it is closed if and only if the extended function F{x, u) is lower semi-continuous in W+ n, and if and only if the sets Q{x), x E W, have property (K) in W (that is, the set valued map x -+ Q{x) has property (K) in R', or equivalently in cl A).

C. The Functions 10' 1 and the Related Function T and Sets Q(x) and M.

Let A be a closed subset of W, let x -+ U{x), X E A c RV, U{x) c Rm, and M = [(x, u)lx E A, u E U(x)] c Rv+m. Letfo(x, u) andf(x, u) = (fl, ... ,In) be functions defined on M, and for every x E A, let Q(x) = f(x, U(x» = [zlz = f(x, u), u E U(x)] eRn, and, as before, let M 0= [(x, z)lx E A, z E Q(x)] = [(x, z)lx E A, z = f(x, u), u E U(x)] C Rv+n. Again, for every x E A, let Q(x) denote the set Q(x) = [(zo, z) I ZO ~ fo(x, u), z = f(x, u), U E U(x)] C W + l, and let ito = [(x,zO,z)lx E A, (ZO,z) E Q(x)]. In other words, M is the graph of the sets U(x), M ° the graph of the sets Q(x), and ito the graph of the sets Q(x). Finally, for every x E A and z E Q(x), let T(x, z) be the extended real function

T(x, z) = inf[ ZO I (ZO, z) E Q(x)] = inf[ ZO I (x, zO, z) E ito]

= inf[ ZO I ZO ~ fo(x, u), z = f(x, u), U E U(x)].

Whenever inf above can be replaced by min, then (T(x, z), z) E Q(x). This is certainly the case if the sets Q(x) are closed (in particular if they have property (K) in the closed set A), and then ito = epi T(x, z) and Q(x) = epiz T(x, z) for every x E A. For ito = epi T(x, z), and by combining (8.5.iii) and (8.5.v), we can say that ito is closed in Rv+ n+ 1 if and only if the extended function T(x, z) is lower semicontinuous in Rv+n, and if and only if the sets Q(x) have property (K) in A (that is, the set valued map x -+ Q(x) has property (K) in A).

Remark 5. The following example shows that M closed (but not compact) and fo, f continuous on M does not imply that ito is closed, that T is lower


semicontinuous, or that the sets Q(x) have property (K). Take v = 1, m = 2, f(x, u, v) = lui, fo(x, u, v) = -Ixllvl + 4 -lX2V2, X E A = R, (u, v) E U = R2, so that M=R3 is certamly closed. Note that I/J(C)=-aC+4- la2C2 has minimum -1 at C = 2/a if a > 0, and I/J == 0 if a = O. Thus, Q(x) = [(ZO, z)lzO ~ -1, z ~ 0] if x =f. 0, Q(O) = [(ZO, z)lzO ~ 0, z ~ 0] if x = O. The set Nt ° is not closed, the function T is not lower semicontinuous, and the sets Q(x) do not have property (K) at x = O.

For M compact, the following simple statements hold, which we will have occasion to use.

8.S.vi. (a) Let x -+ U(x), X E A c W, U(x) c Rm, and M = [(x, u)lx E A, u E

U(x)] C W+ m• Let f(x, u) = (fl' ... ,fn) be a function defined in M, and for every x E A let Q(x) = f(x,U(x)) = [zlz = f(x, u), u E U(x)] c W. Also, let M ° = [(x, z)lx E A, z E Q(x)] = [(x, z)lx E A, z = f(x, u), u E U(x)] C w+ n. If M is compact and f continuous on M, then the sets Q(x) are all compact and contained in a fixed ball in Rn, are upper semicontinuous by set inclusion, and hence have property (K) in A, and if convex, also property (Q) in A. Moreover, M ° is compact.

(b) Let A, U(x), M, f, Q(x), and M ° be as above. Let fo(x, u) be a real valued function on M, and for every x E A let Q(x) = [(ZO, z) I ZO ~ fo(x, u), z = f(x, u), u E U(x)], and let Nt ° = [(x, zO, z) I x E A, (ZO, z) E Q(x)] = [(x, zO, z) I x E A, ZO ~ fo(x, u), z = f(x, u), u E U(x)] c RV + 1 +n. If M is compact and /0, / are continuous on M, then all sets Q(x) are closed with compact projection Q(x) on the z-space Rn; they are upper semicontinuous by set inclusion, and hence have property (K) in A, and if convex, also property (Q) in A. Moreover, Nt ° is closed with compact projection Moon the xz-space RV + n.

(c) For (x,z) E M o take

T(x, z) = inf[ ZO I (ZO, z) E Q(x)] = inf[ ZO I (x, zO, z) E Nt 0].

If M is compact and fo, f continuous on M, then inf can be replaced by min, and Nt ° = [(x, zO, z) I ZO ~ T(x, z), (x, z) E M oJ. Moreover, T is lower semicontinuous on Mo. If T is defined in all of w+ n by taking T = + 00 in W+ n -

M 0, then T is certainly lower semicontinuous in RV + n.

Proof. The compactness of M implies the compactness of A and of every set U(x). Indeed, U(x) is the intersection of M with the hyperspace x = x in W+ m and A is the projection of M on the hyperspace u = 0 of Rv+m. For any Xo E A let us consider the set Q(xo) = f(x o, U(xo)). Then Q(xo), as the continuous image of a compact set U(xo), is closed and compact.

Let us prove that the sets Q(x) are upper semicontinuous by set inclusion at Xo. Suppose this is not true. Then there are an 8 > 0, points Xk E A with Xk -+ Xo, and points Zk E Q(Xk), all Zk at a distance ~ 8 from Q(xo). But Zk = f(Xb Uk) for some Uk E U(Xk), so (Xb Uk) E M, a compact set. Thus there is


some Uo E Rm, and a subsequence, say still [k] for brevity, with (Xb Uk)--+ (XO, uo) in W+ n• But (xo, uo) EM since M is closed, so Uo E U(xo), and Zo = f(xo, uo) E Q(xo), and by the continuity off, also Zk = f(Xb Uk) --+ f(xo, UO) = Zo, with Zo E Q(xo), a contradiction. Thus, the sets Q(x) are upper semicontinuous by set inclusion, and hence have properties (K) and (Q) by (8.5.iv). This proves part (a) of (8.5.vi). We leave the proof of parts (b) and (c) as an exercise for the reader. 0

Remark 6. Under the assumption that f is continuous on M, then M compact as in (8.5.vi) implies that all sets Q(x) are compact and all contained in some ball in Rn, but the converse of course is not true. It is left as an exercise for the reader to see which parts of(8.5.vi) are still valid under the sole hypothesis that the sets Q(x) are all compact and contained in a fixed ball in W.

Remark 7. By using here the same notation as in (8.5.vi), the first conclusion in (c) can be summed up by saying that Mo is the epigraph of T, or Mo = epi T. The second conclusion in (c) cannot be improved, that is, T may not be continuous on A, but only lower semi continuous as stated. This is shown by the following example. Let v = n = m = 1, and take A = [xiO ~ x ~ 1], U(x) = {O} if 0 < x ~ 1, U(O) = {O} u {1}, so that M = [(x, u) 1 U = 0 if 0 < x ~ 1, U = 0 and U = 1 if x = 0]. Let f(x, u) = x, fo(x, u) = - U, so that Q(x) = [z = x, 0 ~ x ~ 1], Mo = [(x,z)IO ~ x ~ 1, z = x]. Finally, Q(x) = [ZO ~ 0, z = x] if 0 < x ~ 1, Q(O) = [ZO ~ -1, z = 0], Mo = [(x,zO,z)1 z = x, ZO ~ 0 if 0 < x ~ 1, ZO ~ -1 if x = 0], and T(x, z) = T(x, x) = 0 if o < x ~ 1 and T(O,O) = -1. Here T is lower semicontinuous but not continuous, M is compact, and fo, f are continuous on M.

8.5.vii. If A is closed, M is closed, fo(x, u), f(x, u) are continuous on M, and either fo --+ + 00, or Ifl--+ + 00 uniformly on A as lul--+ + 00, then the set M 0

is closed, that is, the set valued function x --+ Q(x), x E A c RV , Q(x) c Rn+ 1,

has property (K) in A.

The proof is left as an exercise for the reader. Here is an example of unbounded closed and convex sets possessing

properties (K) and (Q) but not upper semi continuous by set inclusion:

Q(t) = [(x,.y) Ix ~ 0, 0 ~ y ~ tx], 0 ~ t ~ 1.

Here t --+ Q(t), t E [0,1] c R, Q(t) c R2, each Q(t) is a cone in R2 (an angle), and obviously, for t > to ~ 0, Q(t) is not contained in any [Q(to)]" no matter how close t is to to.

Remark 8. Here is a situation under which property (Q) holds in the weak form as in Remark 1 of this section. Let A be a given subset of the tx-space W+', let Ao be the projection of A on the t-space RV, and for t E Ao let A(t) = [x I(t, x) E A]. Let G c Ao c W be a given measurable subset of finite measure of Ao, and let x(t), ¢k(t), 'k(t), t E G, k = 1, 2, ... , be given measurable functions.


8.S.viii. If (a) for almost all t E G, we have x(t) E A(t), Q(t, x(t)) is closed and convex, and ~k(t) E Q(t, x(t)), k = 1, 2, . ~ . ; and (b) the differences t5k(t) = ~k(t) - ~k(t), t E G, k= 1, 2, ... , approach zero pointwise a.e. in G as k -+ 00, then

Q(t,x(t))::J hOI cl co {Vh ~s(tl}. t E G (a.e.).

For instance, we may assume that ~k(t) E Q(t, xk(t)), that xk(t) -+ x(t) pointwise a.e. in G as k -+ 00, and we shall see in Section 13.4 specific hypotheses under which 15k -+ 0 as needed.

Proof· Given 1] > 0, there are a compact subset K of G and an integer ko such that meas( G - K) ~ 1] and I~k(t) - ~k(t)1 ~ 1] for all t E K, k ~ ko. Since ~k(t) E Q(t, x(t)) and Q(t,x(t)) is closed and convex, we also have

clCO{Vh ~s(t)} c Q(t,x(t)),

and for t E K and h ~ ko also

Hence,

cl co {Vh ~s(t)} c [Q(t, x(t) )]w

hOI cl co {Vh ~s(t)} c [Q(t,x(t))]~, and this is true for all t E K with meas(G - K) ~ 1]. Since 1] is arbitrary, we easily derive from this that for almost all t E G we have

o

8.6 The Elementary Closure Theorem

We consider here an orient or field equation as defined in Section 1.12

(8.6.1) x'(t) E Q(t, x(t)), x(t) = (Xl, ... ,xn ), (t,x(t)) E A,

where A is a given subset of the tx-space Rn + I, and we assume that to every (t, x) E A a subset Q(t, x) of Rn is assigned. Then a solution of (8.6.1) is an AC n-vector function x(t) = (Xl, ... ,xn ), tl S t S t2, satisfying (8.6.1) a.e. in [tl' t2J.

We are here interested in the following question: Given a sequence Xk(t), tl s t S t2, k = 1,2, ... ,of AC solutions of(8.6.1) convergent in some mode of convergence toward an AC function x(t), tl S t S t2, can we conclude that x is a solution of (8.6.1)? In this section we give sufficient conditions for the question to have an affirmative answer in connection with uniform convergence, namely the mode (a) of convergence in Section 2.14. In Chapter 10 we will see that the question has an affirmative answer also in connection

8.6 The Elementary Closure Theorem 299

with the mode (b) of convergence in Section 2.14 under weaker assumptions. An example at the end of this section will show that the question may not always have a positive answer.

8.6.i (A CLOSURE THEOREM). Let A be a closed subset of the tx-space R" + 1,

for every (t,x) E A let Q(t, x) be a given subset of points z = (ZI, ... ,z") E R", and let Xk(t) = (xt ... ,xi:), tlk ~ t ~ t2k, k = 1,2, ... , be a sequence of AC solutions of the orientor field (8.6.1) convergent in the p-metric to an AC function x(t) = (xl, ... ,x"), tl ~ t ~ t2. Let us assume that for almost all IE [t 1,t2] the sets Q(t, x) have property (Q) with respect to (t,x) at (I,x(I». Then x(t), tl ~ t ~ t2, is also a solution of the orientor field (8.6.1).

In other words, we know that each Xk(t), tlk ~ t ~ t2b k = 1,2, ... , is AC, that (t, Xk(t» E A for every t E [tlk, t2k], and that dXk/dt E Q(t, Xk(t» a.e. in [tlk' t2k]; we know that P(Xb x) ..... 0, hence t lk ..... t 1, t2k ..... t2, as k ..... 00, and that x(t) is AC in [t b t2], and we want to prove that (t,x(t» E A for all t E [t1,t2], and that dx/dt E Q(t,x(t)) a.e. in [t 1,t2].

Proof of (8.6.i). The vector functions x'(t), tl ~ t ~ t2, x~(t), tlk ~ t ~ t2k , are defined a.e. in [tl' t2] and [tlk' t2k] respectively, k = 1,2, ... , and are L-integrable in the respective intervals (that is, each component is Lintegrable).

Now P(Xk'X) ..... 0, hence t lk ..... t1, t2k ..... t2 and

max[ixk(t) - x(t)l, - 00 < t < + 00] ..... 0

as k ..... 00 (after extension of X k and x to (- 00, + 00) by continuity and constancy of these functions outside their intervals of definition). Thus if t E (t1, t 2), or t1 < t < t 2, then tlk < t < t2k for all k sufficiently large, (t, xk(t» E A for the same k, and Xk(t) ..... x(t) as k ..... 00. Therefore, we have (t, x(t)) E A for all tl < t < t2 since A is closed. Because x(t) is continuous in [tb t2] and hence continuous at tl and t2, and again A is closed, we conclude that (t,x(t» E A for every tl ~ t ~ t2.

For almost all t E [tl' t2] the derivative x'(t) exists and is finite. Let to be such a point with tl < to < t2. Then, there is a G > 0 with tl < to - G < to + G < t2, and for some ko and all k ~ ko, also tlk < to - G < to + G < t2k. Let Xo = x(to). We have Xk(t) ..... x(t) uniformly in [to - G, to + G], and all functions x(t), Xk(t) are continuous in the same interval. Thus, they are equicontinuous in [to - G, to + G]. Given Il > 0, there is ~ > 0 such that t, t' E [to - G, to + G], It - t'l ~~, k ~ ko implies

(8.6.2) Ix(t) - x(t')1 ~ 1l/2, IXk(t) - xk(t')1 ~ 1l/2.

We can assume 0 < ~ < G, ~ ~ Il. For any h, 0 < h <~, let us consider the averages

(8.6.3) mh = h- 1 f: x'(to + s)ds = h- 1 [x(to + h) - x(to)],


(8.6.4) mhk = h- 1 f: x~(to + s)ds = h-1[Xk(to + h) - Xk(tO)]·

Given r > 0, we can take h so small that

(8.6.5)

Having so fixed h, let us take kl ;::: ko so large that

(8.6.6) Imhk - mhl ~ r, IXk(tO) - x(to)1 ~ 8/2

for all k ;::: k1. This is possible because xk(t) -+ x(t) as k -+ 00 both at t = to and t = to + h. Finally, for ° ~ s ~ h,

(8.6.7) IXk(to + s) - x(to)1 ~ Ixk(tO + s) - xk(to)1 + Ixk(tO) - x(to)1

(8.6.8)

~ 8/2 + 8/2 = 8,

I(to + s) - tol = s ~ h ~ b ~ 8,

x~(to + s) E Q(to + s, xk(tO + s)) a.e.

Hence, for almost all s, ° ~ s ~ h, x~(to + s) E Q(to,xo;28) and consequently

x~(to + s) E cl co Q(to, xo; 28) a.e. in [0, h J. The average mhk as defined by (8.6.4) is then also a point of the same closed and convex set, or

mhk E cleo Q(to,Xo;28)

for the chosen h and every k;::: k1. By the relations (8.6.5) and (8.6.6) we derive

and hence

X'(to) E [cleo Q(to,Xo;28)J2r.

Here r is an arbitrary number, and the set in brackets is closed; hence

x'(to) En [cleo Q(to,Xo;28)J2r = cleo Q(to,Xo;28), r

for every 8 > 0. Thus, by property (Q),

X'(to) En cleo Q(to,Xo;28) = Q(to,Xo)· e

We have proved that for almost all t E [t1> t2J, we have dx/dt E Q(t, x(t)). The closure theorem (8.6.i) is thereby proved. D

The following example illustrates the closure theorem (8.6.i). Let n = 1, A=R2, Q=Q(t,x)=[zl-l~z~lJ, and Xk(t), O~t~l, k=1,2, ... , be defined by Xk(t) = t- ik- 1 ifik- 1 ~ t~ ik- 1 + (2k)-t, xk(t)=(i+ 1)k- 1-t ifik- 1 + (2k)-1 ~ t ~ (i + 1)k- 1, i = 1,2, ... , k - 1. Then Xk(t) -+ xo(t) = ° uniformly in [0,1 J. On the other hand, x~(t) = ± 1 according as t is an interior point of one or the other of the two sets of intervals above, x~(t) = 0,

8.7 Some Fatou-Like Lemmas 301

and x~(t), x~(t) E Q for almost all t. Here Q is a closed convex set. If we had taken Q = Q(t,x) = [zlz = -1 and z = + 1], then obviously x~(t) E Q while x~(t) ¢ Q. Here Q is closed but not convex.

8.7 Some Fatou-Like Lemmas

8.7.i (FAlOU'S LEMMA). Iff/k(t) 2 0, a ~ t ~ b, k = 1,2, ... , is a sequence of nonnegative L-integrable functions, and

liminf Sob f/k(t)dt = i < + 00, k~ro

then f/(t) = liminfk~ro f/k(t), a ~ t ~ b, is L-integrable, and S: f/(t)dt ~ i. Under the same hypotheses, if the functions f/k(t) converge in measure toward a function f/o(t) [or they converge pointwise a.e. to f/o(t)], then f/o(t) 2 0 is L-integrable, and again S: f/o(t) dt ~ i.

We refer for this lemma to McShane [I, p. 167]' The same statement holds if f/k(t) 2 -t/l(t) for all t and k, where t/I 2 0 is a fixed L-integrable function. It is enough to apply the statement above to the functions f/k(t) + t/I(t) 2 O. The same statement (S.7.i) holds even if t/I(t), t/lk(t), k = 1,2, ... , are L-integrable functions with f/k(t) 2 -t/lk(t), t/lk(t)-+ t/I(t) as k -+ 00 a.e. in [a, b], and S: t/lk(t) dt -+ S: t/I(t) dt as k -+ 00. Again, it is enough to apply (S.7.i) to the functions f/k(t) + t/lk(t) 2 O.

Under the conditions of Fatou's lemma, let us consider for each h > 0 the same function f/ above and the following functions iik and ii:

(8.7.1) f/(t) = lim inf f/k(t),

k~ro

iih(t) = lim inf h -1 ~r +h f/k(S) ds, k~ro

ii(t) = lim inf iih(t), a ~ t ~ b, h-O+

where in the second relation we understand that f/k(S) = 0 for s outside [a, b].

8.7.ii (A VARIANT OF FAlOU'S LEMMA). Under the conditions of (S.7.i), for almost all t E [a, b] we have 0 ~ f/(t) ~ ii(t), all functions iih and ii are L-integrable, and S: iih(t) dt ~ i, S: ii(t) dt ~ i.

Proof·

(a) Since f/k(t) 2 0, we certainly have f/(t) 2 O. Let us define f/k and iih to be equal to zero for t 2 b. Then by Fatou's lemma in the interval [t, t + h] we have

jr+h ()d I' . f jr+h Jr f/ s s ~ Imm Jr f/k(s)ds, k~ro

and by multiplication by h- I

h- I jr+h Jr f/(s) ds ~ iih(t)

for all t E [a, b] and any h > O. Since f/ is L-integrable, for almost all t, f/(t) is the derivative of its indefinite integral. In other words, for almost all t, there is ho = ho(t,8) > 0 such


that for 0 < h ::;; ho(t, e) we have

Ih- 1 f+h '1(s)ds - '1(t) I ::;; e,

and hence '1(t) - e ::;; iih(t), 0< h::;; ho(t, e).

As h -+ 0, by keeping t fixed, we have '1(t) - e ::;; i'f(t), and this holds for every e > O. Thus, '1(t) ::;; i'f(t) for almost all t.

(b) For every k and for every h, 0 < h < b - a, we have now

S: dt(h- 1 f+h '1k(S)dS) = h- 1 [La+h(S - a)'1k(s)ds + S:+h h'1k(S)dS]

::;; S: '1k(s)ds;

thus, each function h -1 S:+h '1k(S) ds is L-integrable in [a, b], and by Fatou's lemma we have

and

f i'fh(t)dt ::;; liminf f '1k(t)dt::;; i. a k-oo a

Finally, again by Fatou's lemma,

f.b if(t) dt ::;; lim inf f.b rih(t) dt ::;; i, a h-O+ a

and (8.7.ii) is thereby proved.

8.8 Lower Closure Theorems with Respect to Uniform Convergence

Problems of control of the Lagrange and Bolza types are usually reduced to orientor fields of the form

(8.8.1) ('1 (t), x'(t» E Q(t, x(t», x(t) = (Xl, ... , xn),

where S:~ '1(t) dt is the value of the functional, and the subsets Q(t, x) of Rn+ 1 have the property (a) in (8.8.i) below. The problem of closure in Section 8.6 is replaced here by the following question which is a combination of closure and lower semicontinuity: Given a sequence '1k(t), Xk(t), tl ::;; t ::;; t2 , k = 1, 2, ... , of functions, '1k(t) L-integrable, Xk(t) AC, satisfying (8.8.1) (a.e.), with Xk(t) converging in some mode of convergence toward an AC functionx(t), tl ::;; t::;; t2 , is there an L-integrable function '1 (t), tl ::;; t::;; t2 ,

such that the pair 11, x satisfies (8.8.1) a.e. in [t1o t2] and such that S:~ '1(t) dt ::;; lim infk_ oo

J:~ '1k(t)dt? This problem is often called a problem of "lower closure". In this section we discuss it in relation to uniform convergence of the trajectories.

8.8 Lower Closure Theorems with Respect to Uniform Convergence 303

8.8.i (A LoWER CLOSURE THEoREM). Let A be a closed subset of the tx-space R" + 1 and for every (t, x) E A let Q(t, x) be a given subset of points z = (Zo, z) = (ZO, z1, ... , z") E

Rn+ \ with the following properties: (a) if (ZO, z) E Q(t, x) and ZO ~ zO', then (ZO', z) E Q(t, x); (b) there is a real valued function "'(t) ~ 0, t E R, locally integrable, such that if (ZO, z) E Q(t, x) then ZO ~ -"'(t). Let f/k(t), Xk(t), tlk ~ t ~ t2b k = 1, 2, ... , be a sequence of junctions, f/k(t) scalar, L-integrable, Xk(t) = (xL ... ,xi:) AC in [t lk,t2k], such that

tlk ~ t ~ t2k (a.e.), k = 1,2, ... ,

- 00 < lim inff.'2k f/k(t)dt = i < + 00, k-+oo tlk

and such that the junctions Xk converge in the p-metric to an AC function x(t) = (Xl, ... , xn), tl ~ t ~ t2 • Let us assume that for almost aliI E [tlot2] the sets Q(t,x) have property (Q) with respect to (t, x) at (I, x(I)). Then there is a real valued L-integrable function f/(t), tl ~ t ~ t2, such that (t,x(t)) E A, (f/(t),x'(t)) E Q(t,x(t)), tl ~ t ~ t2 (a.e.), and

- 00 < f.'2 f/(t)dt ~ i < + 00. 11

Proof. First, we extend the functions f/k(t) by taking them equal to zero for t ~ t2k and t ~ tlk . Then, these functions are all defined in [t1> t2], and we construct the functions iMt) and if(t), tl ~ t ~ t2, as in Lemma (8.7.ii). Here f/k(t) ~ -"'(t) for all t and k; hence, if f/o(t) = lim inf f/k(t), we have if(t) ~ f/o(t) ~ -"'(t) for all t, where if is the function defined in (8.7.1). For almost all t E (t1> t2 ) the derivative x'(t) exists and is finite, and f/o(t) and if(t) are finite. Let to be such a point, tl < to < t2. Then there is a (J > 0 with tl < to - (J < to + (J < t2 , and for some ko and for all k > ko, also tlk < to - (J < to + (J < t2k•

Let Xo = x(to), Xo = x'(to). We have xit) -> x(t) uniformly in [to - (J, to + (J]. Given e > 0, there is (j > 0 such that t, t' E [to - (J, to + (J], It - t'l ~ (j, k ~ ko implies

(8.8.2) Ix(t) - x(t')1 ~ e/2, IXk(t) - Xk(t') I ~ e/2.

We can assume 0 < (j < (J, (j ~ e. For any h, 0 < h ~ (j, we consider the averages

(8.8.3) mh = h- 1 f: x'(to + s)ds = h- 1 [x(to + h) - x(to)],

(8.8.4) mhk = h- 1 s: x;'(to + s)ds = h- 1[Xk(tO + h) - xk(tO)].

Given T > 0, we know that for h > 0 sufficiently small we have Imh - xol < T. On the other hand, if(to) is finite, and if(to) = lim inf ifh(tO) as h -> O. Thus, we can choose h in such a way that

(8.8.5)

Having so fixed h, let us take kl ~ ko so large that

(8.8.6) Imhk - mhl ~ T, IXk(tO) - x(to)1 ~ e/2

for all k ~ k1• This is possible because xk(t) -> x(t) as k -> 00 both at t = to and t = to + h. Finally, for 0 ~ s ~ h,

Ixk(tO + s) - x(to)1 ~ IXk(tO + s) - xk(to)1 + IXk(tO) - xol ~ e/2 + e/2 = e,

I(to + s) - tol = s ~ h ~ (j ~ e,

(f/k(tO + s), x;'(to + s)) E Q(to + s, Xk(tO + s») a.e.,

(f/k(tO + s), x;'(to + s)) E Q(to,xo;2B),


where Q(to, Xo; 2e) is the union of all sets Q(t, x) with (t, x) E A, It - tol ::;; e, Ix - xol ::;; e. Finally, by the remark at the end of Section 8.4B concerning the mean value of vector valued functions, we have

(8.8.7) (h- 1 foh rJk(to + s)ds, h-1 S: x;'(to + S)dS) E clco Q(to,xo;2e).

Concerning the first term in the parentheses in this relation, we know that lin} inf h -1 J~ rJk(tO + s) ds = ;ih(to) as k -+ 00. Thus, there are infinitely many k such that

Ih- 1 S: rJk(tO + s)ds - ;ih(to)1 ::;; •

and by comparison with (8.8.5) also

Ih- 1 S: rJk(tO + s)ds - iWo)1 < 2. for infinitely many k.

The second term in parenthesis in (8.8.7) is the average mhk' and by (8.8.5) and (8.8.6) we derive that

Imkh - xol ::;; 2. for all k sufficiently large.

Thus, (8.8.7) yields (i7(to), xo) E [cl co Q(to, Xo; 2e)]4T

Here. > 0 is arbitrary, and thus

(i7(to),xo) E clco Q(to,xo;2e).

Here e > 0 is also arbitrary, and by property (Q) we derive that

(i7(to, x'(to)) E Q(to, x(to))·

Here to is any point of(tl> t2) not in a set of measure zero. We have proved that

(i7(t), x'(t)) E Q(t, x(t)),

and from Lemma (8.7.ii) we know that

f l2 i7(t) dt ::;; i.

JI' The lower closure theorem (8.8.i) is thereby proved. o

We may remark here that the various scalar functions we have been dealing with are in the relation

- I/!(t) ::;; T(t, x(t), x'(t)) ::;; rJ(t) ::;; i7(t),

where T(t, x, z) = inf[ ZO I (ZO, z) E Q(t, x)] is the scalar function defined in Section 8.5C.

Remark 1. Note that in the closure theorem (8.6.i) and in the lower closure theorem (8.8.i) we assume that Xk -+ x in the p-metric, that is, uniformly (mode (a) of Section 2.14), and no requirement is made concerning the derivatives x;'. In this situation, the requirement in (8.6.i) that the sets Q(t, x) have property (Q) with respect to (t, x) (and the analogous requirement on the sets Q(t, x) in statement (8.8.i)) cannot be reduced. This is shown by the following example. In Sections 10.6 and 10.8 (statements (1O.6.i) and

8.8 Lower Closure Theorems with Respect to Uniform Convergence 305

(10.8.i» we shall assume weak convergence of the derivatives and we will be able to dispense with explicitly requiring property (Q). However, in the proofs we shall still make use of a "reduced property (Q) with respect to x only" of certain auxiliary sets, and such property (Q) will be a consequence of the other assumptions.

Let n = I and A = [0, I] x R, let C be a closed Cantor subset of [0, I] whose measure, meas C, is positive, and let C = [0, 1] - C. Then C is the countable union of disjoint subintervals of [0,1]' Let a(t) be a continuous function on C which is positive and integrable on C and which tends to + 00 whenever t tends to an end of any interval component of C. Let m = 1, and define

Q(t, x) = U(t, x) = U(t) = { - I} if t E C,

= {u E Rlu ~ a(t)} if t E C,

and take Q(t, x) = [(ZO, z) I ZO ~ 0, Z E Q(t, x)], (t, x) E A. Let us extend the function a by taking a(t) = 0 for t E C, and consider the decomposition of [0,1] into k intervals of equal length: Jk = [tk,s-l,tks], s = 1, ... ,k, tks = slk. Define ~k(t) by taking ~it) =

a(t) + vk(t), where Vk(t) = -1 if t E C, and vk(t) = meas(C n Jd/meas(C n Jk) if t E

C n Jks ' Then ~k(t) is integrable in [0,1], and ~k(t) E U(t) for every t E [0,1] and k. Let Xk(t) = S~ ~k(T)dT, 0::::; t::::; 1, or

xk(t) = x(t) + Yk(t) = f~ a(T)dT + f~ vk(T)dT.

Here Yk(tks ) = 0 for all sand k, and IYk(t) I ::::; 11k. Hence, Xk -+ x uniformly on [0,1] as k -+ 00, and Xk and x are AC, with x'(t) = a(t), t E [0,1] (a.e.). We also take "k(t) = 0, ,,(t) = 0, t E [0,1]' Now x'(t) = 0 a.e. in C, while U(t) = { -I} for t E C. Thus x'(t) rt Q(t), (,,(t), x'(t» rt Q(t) on a subset C of positive measure in [0,1]'

In this example Q(t), Q(t) have property (Q) (with respect to t) on the set C as well as on the set C, but not in [0,1]'

8.8.ii (A LoWER CWSURE THEOREM). Let A be a closed subset of the tx-space R" + 1, and for every (t, x) E A let Q(t, x) be a given nonempty subset of points z = (zo, z) = (zO, zl, ... , z") E R"+ 1, with the following property: (a) if (ZO, z) E Q(t, x) and ZO ::::; zO', then (ZO', z) E

Q(t,x). Let "k(t), xk(t), t lk ::::; t::::; t2k, k = 1,2, ... , be a sequence of functions, "k(t) real valued and L-integrable, xk(t) = (xl, ... , x~) AC in [tlk' t2k]' such that

(t,xk(t» E A, ("k(t),X~(t» E Q(t,Xk(t», tlk ::::; t::::; t2k>

- 00 < lim inf i t2k "k(t) dt ::::; i < + 00, k~oo Jt1k

k = 1,2, ... ,

and the functions Xk converge in the p-metric to an AC junction x(t) = (Xl, ... , x"), tl ::::; t ::::; t2. Let us assume that (b) for every T E [t1> t2]' the sets Q(t, x) have property (Q) with respect to (t, x) at (I, x(T». Then there is a real valued L-integrable function ,,(t), tl ::::; t::::; t2, such that (t,x(t» E A, (,,(t),x'(t» E Q(t,x(t», t1 ::::; t::::; t2 (a.e.), - 00 < s:~ ,,(t) dt ::::; i.

Proof. For (t, x) E A let Q(t, x) denote the projection of Q(t, x) on the z-space R", and note that, for

T(t, x, z) = inf[" I (", z) E Q(t, x)]

we have - 00 ::::; T(t, x, z) < + 00 for z E Q(t, x), and T(t, x, z) = + 00 for z E R" - Q(t, x). ForT E [t 1,t2], x = x(T), we have (T,x) E A, and the sets Q(f,x), Q(Y,x) are not empty.


If x'(I) exists and is finite, then we take z = x'(Y); if x'(Y) does not exist, or is infinite, then we take for z any point z E Q(I, x). By (17.5.i), T(t, x, z) is seminorrnal at (I, x), and thus certainly property (X) of Section 17.3 must hold at (y, x, z). For e = 1, then there are numbers i) > 0, f real, D = (D1> ... ,Dn) ERn, such that, for h(z) = f + D . z we have

T(t, x, z) ~ h(z) for all (t, x) E A, It - II ~ i), Ix - xl ~ i), z ERn.

Let p, 0 < p ~ i)/2, be a number such that It - II ~ p, t E [tl' tz), implies Ix(t) - x(I)1 ~ i)/2. Now we consider the open intervals (Y - p, I + p) as an open cover of [tl' t2]. By the Borel covering theorem there is a finite system Ii' Pi, bi, ri' bi, hi(z) = ri + bi . Z,

i = 1, ... , N, such that the intervals (Ii- Pi' Ii + Pi), i = 1, ... , N, cover [t1,t2]. Let P = min Pi' 15 = min bi. Let r denote the graph of x, or r = [(t,x) I tl ~ t ~ t2, X = x(t)], and let rp be the p-neighborhood of r in Rn+l. Let a denote the maximum of tl - P and of those Ii - Pi which are < t1; let b denote the minimum of t2 + P and of those Ii + Pi which are> t2. Then a < tl < t2 < b. Now we use the end points of the intervals (Ii - Pi,ti + p;), i = 1, ... ,N, to define a finite partition, say a = '0 < '1 < ... < 'M + 1 = b, of [a, b]. Let us prove that if (t, x) E r p n A, a ~ t ~ b, then t E ['S, 'S+ 1] for some s, ['" 'S+ 1] is contained in some [Ii - P;,Ti + p;], and It - Id ~ bi, Ix - X(Ii) I ~ b. Indeed, either tl ~ t ~ t2, and then It - Iii ~ Pi ~ 15;/2 < bi, Ix - x(t)1 ~ Pi ~ 15;/2, Ix(t) - x (1;) I ~ 15;/2, and Ix - x (I;) I ~ 15;/2 + 15;/2 = bi; or a ~ t ~ t1> and then again It - Iii < 15;, Ix - x(tdl ~ P ~ 15/2, Ix(tl) - x(Yi)1 ~ 15;/2, and Ix - x(l;) I ~ b/2 + 15;/2 ~ bi; or finally t2 < t ~ b, and the analogous argument holds. If we denote ri and bi by rs and b" we conclude that

(8.8.8) (t,x) E rp n A, a ~ t ~ b, implies 's ~ t ~ 's+1 for some s,

and T(t, x,z) ;::: rs + bs . z for all z ERn. For any s, and 's ~ t ~ 's+ 1, (t,X) E rp n A, let Q(S)(t, x) denote the set

Then (ZO, z) E Q(S) implies ZO ~ 0. Moreover, for 's ~ t ~ 's+ 1> (t, x) E rp n A, the sets Q(S)(t,x) satisfy property (Q) with respect to (t,x) at every (y,x(Y».

Let xlS), x(s) denote the restrictions of Xb x on the interval ['" 's+ 1], and let Iks> Is be the respective intervals of definition. Since xk -> x in the p-metric, then also xlS) -> x(s) in the p-metric as k -> 00, S = 0, 1, ... ,M, and for k sufficiently large we have (t, xk(t» E r P' [t lk, t2k] c [a, b]. Finally, for 11~)(t) = I1k(t) - hs[ x;,(t)], we have

If is = lim infk Shs 11~)(t) dt, s = 0, 1, ... , M, we have l1~s)(t) ~ 0, is ~ 0, and by (8.8.i) there is an L-integrable function I1(S)(t), t E Is> with

s = 0,1, ... , M.

Let l1(t), tl ~ t ~ t2, be the function and Ll ks the numbers defined by

l1(t) = l1(s)(t) + hs[ x'(t)], t E Is> Ll ks = r hs[ x~(t)] dt - r hs[ x'(t)] dt. Jlks J1s

Bibliographical Notes

Then (1](t),x'(t)) E Q(t~x(t)), t E [t l ,t2] (a. e.), and we also have

(S.S.9) f,'2 1](t)dt = L f (1](S)(t) + hs[x'(t)])dt t1 J1s

s

~ L is + L t. hs[ x'(t)] dt s s

~ L lim inf [f 1]k(t) dt - f hs[ x~(t)] dt] + L f hs[ x'(t)] dt k Jlks J1ks JIs

s s

~ liminf[L f 1]k(t)dt - L Ll kS] ~ i + liminfjL Llksj. k s J1ks S k s

307

Now, hs(z) = rs + bs . z, and we take R = maxlrsl, B = maxslbsl. Then, for s = 1,2, ... , M - 1, we have I ks = Is = [r., 's+ I], and

~ Ibsll(Xk('S+I) - X('s+I)) - (Xk('s) - x('s))1

~ 2Bp(Xk, x).

For s = 0 we have Ikl = [t\k,'I], II = [tl"I], and

ILlkl1 ~ Iblll(xk('I) - X('I)) - (Xk(t\k) - x(td)1 ~ 2Bp(Xb X),

and analogously ILlkMI ~ 2Bp(Xb x). Thus ILs Llksl ~ 2(M + I)Bp(xk' x) approaches zero as k -+ 00, and from (S.S.9) we derive S:; 1](t) dt ~ i. This proves Theorem (S.S.ii). 0

Remark 2. Theorems (S.S.i) and (S.S.ii) are independent. In (S.S.i) we assume ZO ~ - .p(t), .p E L, for every (zo,z) E Q(t,x), and property (Q) is required for all (t,x(t)) but a set of points t of measure zero. In (S.S.ii) no lower bound .p is known, but property (Q) is required at all t E [t l , tzJ. Note that in (8.8.i) we could have specified that property (Q) is required for almost all I, but at any other point 7 we require that either (a) there are a number b > 0 and a linear function h(z) = r + b . z, r E R, b = (b l, ... , bn) ERn, such that (t, x) E A, It - 71 ~ b, Ix - X (7) I ~ b imply T(t, x, z) ~ h(z) for all z E Rn or equivalently (t, x) E A, It - 71 ~ b, Ix - X (7) I ~ b, (ZO, z) E Q(t, x) implies ZO ~ h(z); or (b) there are a number b > 0 and an L-integrable function .p(t) ~ 0,7- b ~ t ~ 7 + b, such that (t,x) E A, It - 71 ~ b, Ix - x(7)1 ~ b implies T(t, x, z) ~ -.p(t).

Bibliographical Notes

As stated in Section S.1, for the concepts of a-space and a-convergence we refer to V. Volterra and J. Peres [I]. The very general theorems (S.l.i) and (S.l.ii) are based on mere a-convergence, and therefore certainly apply to the two modes of convergence of interest here: uniform convergence, and weak convergence in HI,I. As soon as we deal with a Banach space, namely a reflexive Banach space X, and a-convergence is the weak convergence in X, then the much stronger theorems (S.l.iii-vi) hold, which we also have easily proved in Section S.1. (For further developments along this line we refer to J. L. Lions [I]).


The implicit function theorem, in the forms (8.2.ii) and (8.2.iii), is due to E. J. McShane and R. B. Warfield [1]. The selection theorems as presented in Section 8.3 are due to K. Kuratowski and C. Ryll-Nardzewski [1]. On implicit function theorems and measurable selection theorems we mention here the work of M. Q. Jacobs [1,2], A. P. Robertson [1], N. U. Ahmed and K. L. Teo [1], J. K. Cole [1], C.1. Himmelberg, M. Q. Jacobs, and F. S. Van Vleck [1], F. V. Chong [1], A. Plis [1-3], T. Wazewski [1-4], A. F. Filippov [2], S. K. Zaremba [1], and the recent monographs ofe. Castaing and M. Valadier [I] and of C. Berge [I]. A bibliography on this subject has been collected by D. H. Wagner [1], and a supplement by A. D. loffe [2].

In Section 8.3 we have presented forms of the theorem ofG. Scorza-Dragoni [1] in connection with the implicit function theorem in optimal control theory. For further extensions of this theorem we refer to G. S. Goodman [2], and on the same general topic we mention here the work of E. Baiada [1].

The concept of upper semicontinuity by set inclusion can be traced in F. Hausdorff [I]. K. Kuratowski introduced his concept of upper semicontinuity for closed set valued functions, property (K), in [1] in 1932. The variant called property (Q) for closed convex set valued functions was proposed by Cesari [6] in 1966. The preliminary properties in Section 8.5, and the more specific properties which will be proved in Section 10.5, were proved by Cesari in [6, 8, 13]'

Property (Q) was used by 1. D. Schuur and S. N. Chow [1] to prove the existence and main properties of solutions of the Cauchy problem x'(t) E Q(t, x(t)), x(to) = xo, in Banach spaces.

Property (Q) was used by Cesari [12-23], L. Cesari and D. E. Cowles [1], M. B. Suryanarayana [5-7], and L. Cesari and M. B. Suryanarayana [1-8] to prove theorems of lower semicontinuity and existence in problems of optimization with ordinary and partial differential equations, with single valued as well as multivalued functionals (Pareto problems). Some of the results will be presented in Chapter 10 of this book, where also more bibliographic references will be given.

Property (Q) was used by R. F. Baum [1-4] in problems of optimization with ordinary differential equations in infinite intervals (infinite horizon in economics), and in problems of optimization with partial differential equations where the controls are functions in Rk and the state variables are functions in R" with k < n.

Property (Q) has been used by T. S. Angell [1-4] in problems of optimization with functional differential equations and in problems with lags.

Property (Q) can be thought of as a generalization of Minty's and Brezis's maximal monotonicity property as proved by M. B. Suryanarayana [8, 10], and we shall present a proof of this result in Section 17.8. As such, property (Q) has been recently used by S. H. Hou [3] for the proof of existence theorems for boundary value problems for ordinary and partial differential equations (controllability), and by T. S. Angell [8,9] for nonlinear Volterra equations and hereditary systems.

The elementary closure theorem (8.6.i) was proved by L. Cesari [6]. The Fatou-like lemma (8.7.ii) and subsequent elementary proof of the lower closure theorem (8.8.i) appear here for the first time.

optimization—theory and applications || the implicit function theorem and the elementary closure...

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