De Blasi, Francesco Saverio

Pianigiani, Giulio

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeFrancesco Saverio De Blasi; Giulio PianigianiRemarks on Hausdorff continuous multifunction and selectionsCommentationes Mathematicae Universitatis Carolinae, Vol. 24 (1983), No. 3, 553--561Persistent URL: http://dml.cz/dmlcz/106254Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1983Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROUNAE 24,3 (1983) REMARKS ON HAUSDORFF CONTINUOUS MULTIFUNCTION AND SELECTIONS F. S. De BLASt, G. PIANIGIANI Abstract. Continuity properties of multifunctions and existence of continuous selections are investigated. Key words. Multifunctions, Hausdorff distance, selections. Classification: 54 C 60, 54 C 65. *• Introduction. Let X be a metric space and let Y be a real normed space. Denote by & the space of all closed convex bounded subsets of Y with nonempty interior endowed with Hausdorff distance. In this note we establish some proper-ties of multifunctions which are used in [1] in order to study the structure of the solution set of the Cauchy problem (*) x € 3F(t,x), x(0) «-• xQ. In [1] it is supposed that P: [0,1] x Y •• S is Hausdorff continuous and Y is a real refle-xive Banach space. The existence of solutions of (*) could be proved directly. However in CI3, we establish a more precise result stating that almost all (in the sense of the Baire category) solutions of x € F(t,x), x(0) * XQ are solu-tions of (*). in Section 2 we introduce the terminology and review some elementary properties of Hausdorff continuous multifunctions. In Section 3 we prove the exi-stence of (nontrivial) continuous multivalued selections for multifunctions F:X-»<8. Y 2. Notations and preliminaries. Let 2 be the family of nonempty subsets Y of the real normed space Y. We shall consider the following subfamilies of 2 : J = {A € 2 Y | A is bounded}, X « (A e 2 Y | A is closed bounded}, ^ « { A € 2 Y | A is closed convex bounded}, © « {A € 2 j A is closed convex bounded with nonempty interior], & = {A € 2 | A is open convex bounded}, XL « {A « 2 | A is convex with nonempty interior}. Let (X,e) be a metric space. For any set A c x we denote by int A, A, 3A respectively the interior, the closure, the boundary of A. If A c x is nonempty, diam A stands for the diameter of A. 553 For any u e X we put S(u,r) « {x e X | e(x,u) < r}, r > 0, S(u,r) « {x c x | e(x,u) < r), r i 0. For notational convenience the unit balls S(0,1), S(0,1) in Y are denoted by S, S . For any A, B e 3 define h(A,B) • inf {t > 0 | A c B + tS, B c A + ts). As is well known, h is a pseudometric in # , 2. while it is a metric (Hausdorff distance) in X • % i ̂ . For any u e X and A c x (A i- 0) , we set d(u,A)' - inf {e(u,a) | a e A ) . A multifunction F : X -*• 2 is said to be Hausdorff lower seaicontinuous "Hausdorff l.s.c." (resp. Hausdorff upper semicontinuous "Hausdorff u.s.c") at x0 e X if for every e > 0 there is a « > 0 such that F(x0) <= F(x) + es (resp. F(x) <- p(x0) + es) whenever x e S(x0,6). F is said to be Hausdorff continuous at x 0 if is Hausdorff l.s.c. and Hausdorff u.s.c. at x 0 . Proposition 2.1. Let F t X -+ tJ be Hausdorff continuous. Then so is the SMltltmvction F : X -• 2* (resp. 3F : X •+ X) given by F (x) » Y \ F ( x ) (resp. (dF)(x) - 3F(x)), x e X. Proof. It is routine to see that F is continuous. To prove that 3F is c continuous take x 0 e. X and let e > 0. There is a 6 > 0 such that for each x e S(x0,5) we have h(F(x), F(x0)) < E, h(F (x), F (x0)) < e. Since 3F(x) - F(x) n F (x) c (F(x0) + eS) n (F (x0) + es) - 3F(x0) + es , and 3F(x0) - F(x0) n F (x0) c (F(x) + eS) n (F (x) + eS) - 3F(x) + es it follows that 3F Is continuous. Lemma 2.2. Let A, B e (36 satisfy A n B 3 i(y0,r), r > 0. Let e > 0. Then A n (B + aS) c A n B + eS where o • er / diam A. Proof. Let y e A n (B + aS) and take y* e B such that |y-y| < 0. Sup-pose y j- y (the case y » y is trivial) and set u • y 0 + r(y - y ) / |y-y|. Clearly u c S(y0,r) c A. Since y and u lie in the convex set A, also v(t) • ty + (l-t)u (t e CO,IT) i* in A. Analogously v(t) - ty + (l-t)y0 (t c [0,1]) is in B. An easy computation shows that v(t*) •* v(t*) for t* • r/(r+|y-y|). Hence, denoting by y* the point v(t*) « v(t*), we have y* e A n B; furthermore |y-y*j « ( i . . . t * ) | u - y | - |u-y| |y-y|/( x+|y-y|) < (diam A)|y-y|/r < t . Thus y « y + ( y - y ) e y + eS c A n B + cS and the lemma i s proved. Proposition 2 . 3 . Let F : X + ft and G : X -> 6b be Hausdorff continuous 554 multifunctions such that F(x) n G(x) (x c x) has nonempty interior. Then the multifunction F n G : X -> Co given by (F n G) (X) » F(x) n G(x), x € X , is Haus-dorff continuous. Proof. Fix x 0 e X, 0 < e < 1, and take k » diam (F(x0) U G(x0)). From the hypotheses it follows that there is a 6 > 0 such that for each x € S(x0,6) we have: F(x) n G(x) => S(y0,r) (for some y 0 € Y and r > 0), and h(F(x), F(x0)) < a, h(G(x), G(x0)) < o, where a - er/(k+l). Hence, by virtue of Lempva 2.2, » we have F(x) n G(x) c (F(x0)+oS) n (G(xo)+0S) c (F(x0) +oS) n G(x0) + es c F( X 0 ) n G(x0) + 2eS , x *S(xfi ,«) . Analogously P(x0) t|G(x0) c F(X) n G(x) + 2eS . and the proof is complete. Proposition 2.4. Let F : X -> 6J> and G : X -> *€ be Hausdorff continuous and satisfy G(x) +rS c F(X) , x c X, for some r > 0. Then the multifunction F \ G : x **• ̂  given by (F\G) (x) « F(x)\G(x), x c X , is Hausdorff continuous. Proof. Let x0 e X and take 0 < e < r/2. Take 6 > 0 such that h(F(x), F(x0)) < e, h(G(x), G(x0)) < e for each x € S(x0,6). From this and the fact that G(x0) +rS c F(x0), G(x) + rS c F(x) it is not difficult to obtain h(F(x)\ G(x), F(x0)\G(x0)) < 2c. Remark 2.5. The statement of Proposition 2.1 fails if Vg is replaced by X . If in the^Pjroposition 2.3 the assumption that F(x) n G(x) have nonempty interior is replaced by F(x) n G(x) •»- 0 (x c X ) , the conclusion is no longer true. If in the Proposition 2.4 the hypothesis G(x) + rS c F(X) , x € X , is replaced by G (x) c F (x) , the conclusion' is not true in general. 3. Multivalued selections of multi functions. For each A € © let om -"' - ' * " " A sup {r>o|there is a c A such that S(a,r)cA} . Evidently, o > 0. Lemma 3.1. Let F:X->-(& be Hausdorff l.s.c. (resp. u.s.c). Then the fun-ction o : x -> » given by o (x) » a , x € X, is l.s.c. (resp. u.s.c). In particular 0_ is continuous whenever F is Hausdorff continuous. \ FProof. Let F be Hausdorff l.s.c. and, for a contradiction, suppose that o is not l.s.c Then there are x0 € X, € > 0, and a sequence {x } c x con-r w n verging to xQ such that o (x ) < o (x0) -e, nclN. Since F is Hausdorff J F n F u - 555 l . s . c , there i s n0 e IN such that F(x0) c F (X ) + (e /2 )S . We have o (x ) + v **o F n 0 e < o (x 0 ) , thus there are y e F(x0) and r e m , o (x ) + e < r * o ( x 0 ) , such t h a t S(y, r ) c F ( X 0 ) . Therefore S(y,o (x ) + e/2) + (e/2)S c s ( y , r ) c F (x 0 ) c F n0 F(x ) + (e/2)S and so S(y,o (x ) + e/2) c F ( X ). Hence o (x ) + e/2 <. n0 F n0 n0 F n0 o (x ), a contradiction, and o is l.s.c. If F is Hausdorff u.s.c. the F n0 F proof is similar. The last statement is obvious. Lemma 3.2. Let Aeft. For each 0<y<o put A ={aeA|s(a,y) c A} A y and let Aft «* A if y «- 0. Then A eft and, furthermore, we have u y (3.1) A * {a e A | d(a, 9A) 2. y} y (3.2) 9A « {a e A I d(a, 9A) - y} . y Proof. When y =- 0 we have A0 e & and (3.1), (3.2) are true. Suppose 0 < y < a.. From the definition of o there is a e A and y < r £ om such that A A A S(a,r) c A. Since S(a,r-y) + yS » S(a,r) c A it follows that S(a,r-y) c A V and so A has nonempty interior. Let us prove that A is convex, y y To this end let a1# a2 e A that is S(aj,y) c A, S(a2,y) c A. Since A is convex, for each t c [0,1] we have tS(alfy) + (l-t)S(a2,y) = S(taj + (l-t)a2,y) c A and hence ta«. + (l-t)a, e A . Clearly A is bounded and, as one can easily 1 z y y verify, also closed. Therefore A eft. Consider now (3.1). Let a e A . Then y V» S(a,y) c A and hence d(a, 9A) 2: y. Conversely, if a e A satisfies d(a, 9A) >. y, we have S(a,y) c A thus a e A . Therefore (3.1) is true. Let us prove (3.2). Denote by B the set on the right hand side of (3.2). Let a e 9A . Since a e A , from (3.1) we have d(a, 3A) 2: y. For a contradiction. y y suppose d(a, 9A) > r > y. Evidently S(a,r-y) + yS =- S(a,r) c A which implies that a e int A , a contradiction. Hence d(a, 9A) -= y and a e B . Conversely, y y let a e B . We have a e A for B c A . Suppose that a e int A that is y y y y y S(a,r) c A for some r > 0. Then S(a, y+r) *» S(a,r) + yS -A from which we y obtain d(a, 9A) 2: y + r, a contradiction. Therefore a e 9A and also (3,2) is true. Remark 3.3. Let A e 6S . For any 0 S y < o , put A0 • {ae A | d(a,3A)>y} . " A y Evidently A0 * int A thus A0 is nonempty open convex bounded, that is A° € SL. 556 -Remark 3.4. If A e (S*> and 0<y <a , we have A +VS c A. The inclusion A y can be strict. In fact simple examples show that A \ (A + ys) can have nonempty interior. Lemma 3.5. Let A c ® . Let 0 < y < a /2 and take 0 < e < dlam A. There is then 6 0>0, given by 60 « e(a /2 - y)/diam A (reap. 6Q « min {y,e(a / 2 - y ) / diam A>) such that, whenever 0-S6:S60, we have A c A .+ eS (resp. A * c u y y+6 * y-6 A + eS). Moreover, if 0 < y < a /4, we have h(A ,A) 5 (y diam A)/(a / 2 - y ) . y A y A Proof. Let A, y, e and 0 <. 6 <. <50 » c(a /2 - v)/diam A be as in the sta-* tement. From the definition of a,, there is a £ A such that S(a,o /2) c A. A A Since A and A „ are in & (in fact 0<y:Sy + 6<o /2) the inclusion A c y y+6 A y A . + eS ( 0 < ^ 6 n ) is true if we show that 3A c A . + eS. To this end, y+6 ° y y+6 let y € 3A and suppose that |y-a| -5 c. Since S(a,y+6) c S(a,y+(a /2 - y)) » • S(a,a /2) c A, we have a e A . and hence y « a + (y-a) e A . + eS. Now, A y+6 * M+6 suppose that y € 3A is such that |y-a| > e. Let y* «• (l-t*)y + t*a, where t* • e/ |y-a|, and observe that |y*-y| * e. Observe that S(a,a /2-y) +yS •» * S(a,a /2) c A whence S(a,a,/2-y) • a+ (a /2-y)S c A . Also y e A thus, A A A y y since A is convex, we have y (3.3) A D (l-t*)y + t*Ca+(aA/2-y)s3 - y* + t*(aA/2-y)S . This implies that dCjr-.MU * t . ( ^ - p ) - ^ ( ^ - p ) * E(aA/2-y) diamA Let v c 9A be arbitrary. From (3.3), y* c int A whence the segment [y*,v] meets 9A in a point u and we have |y*-v| » |y* - u| + |u - v|. Evidently, |y*-u| > d(y*,3A ) 2. 60 2. 6. On the other hand v € 9A and u € 3A thus |u-v| >- y. Hence |y*~v| 2; 6 + y and, since v £ 9A is arbitrary, we have d(y*,9A) *t y + 6, thus y* e A ,. Since y « y* + (y-y*) £ A , + eS, the proof y+o y+o of the inclusion A c A , + eS (0 < 6 £ 60) is complete. The argument to prove y y+6 that A C A + C S {0 & 6 & 6ft) is similar. It can be obtained (with few minor u-6 y u modifications) by replacing A and A „ in the above proof by A and A y pH r y-o y respectively. To prove the last statement of the proposition, suppose 0 < y < a /4. Let e «- (y diam A)/(a /2 - y) and observe that 0 < e < diam A. Moreover, 6Q • - 557 min {y,e(o /2-y)/diam A} «- y, thus we have A „ c A + eS, that is A c A y-60 y A + eS. Evidently A c A and so h(A ,A) £ e « (y diam A)/(a / 2 - y ) . This y y y A completes the proof. Lemma 3.6. [2, p. 1703. Let pj : X •+ IR and p 2 • X •*• 3R be an u.s.c. and a l.s.c. function such that pj (x) < P2 (x), x e X. Then there exists a con-tinuous function p : X -*- M such that pj (x) < p(x) < P2(x), x e X. Let F : X -+& be Hausdorff l.s.c. By Lemma 3.1, 0 is l.s.c. and posi-tive and by Lemma 3.6 there is a continuous function y : X •+ m satisfying 0 < y(x) < 0 (x)/2, x € X. For each x € X set F (x)={y€F(x) | d(y,3F(x)) >-y(x)}, x € X. Evidently, F , % (x) € ft thus the multifunction F : X •+ <8 y(x) y given by F (x) •» F , ,(x), x cX, is a multivalued selection of F. y y(x. Proposition 3.7. Let F : X •+ C5 be Hausdorff l.s.c. (resp. continuous) and let y : X •+ JR be continuous and satisfy 0 < y(x) < o (x)/2, x € X. Then F the multifunction F : X •*• % given by F (x) * F , „ (x), x € X, is also y y y(x) Hausdorff l.s.c. (resp. continuous). Moreover if 0 < y(x) < 0 (x)/4, x e X, we have h(F (x), F(x)) <- (y(x) diam F(x) )/(0p(x)/2 - y(x)). Proof. Let F be Hausdorff l.s.c. and suppose, for a contradiction, that F is not so. Then there are x0 € X, 0 < e < diam F(x0), and a sequence {x } c x converging to x0 such that F , . (x0) 4- F , (x ) + eS, n € W . n u y(x0) u yfx^) n Let {y } c y be such that (3.4) y € F ' x(x0) y i F (x ) + eS , n e M . n y(x0) uv n y(x ) n By Lemma 3.5 we have F . .(xn) c F , , , (x0) + eS where 60 = e(o (x0)/2 -y(x0) u y(x0J+60 u u F u y(x0))/diam F(x0). Hence, for each n c W , y c F (x0) + eS and so n y \X0)+OQ there is z e F , . „ (x0) satisfying |y -z | < e. Moreover, since y is n y(x0)+60 ° ' n n' continuous, there is k e H such that whenever n >. k we have y(x0) - <50/2 < y(x ) < yj(x0) + 60/2 and, in particular, y(x0) > y(x ) - <50/2. Consequently, V(H) * « , > U(xn) • V - «»>lch implUs tha t F ^ ^ (x„> <= ' ^ . e ^ W • n 2. k . Since F i s Hausdorff l . s . c . there i s ki >-k such t h a t F(x0) c F (X ) + (60 /2)S for a l l n >. k-_. Furthermore, for each n 2. kj we have z e p , % . ,-.<x0> which implies t ha t S(z , y(x ) + <W2) c F ( X 0 ) . Hence, z + y(xn)+60 /2 v n n v n + (y(x ) + 60 /2)S c F ( X 0 ) c F (X ) + (50 /2)S thus z + u ( x )S c F ( X ) , t ha t i s n w u n u n n n z € F (X ) , i f n >. k i . Then y * z + ( y - z ) € F , % (x ) + cs for n y t x ^ n Jn n ' n n u < x n ) n 558 -each n i kw in cont radic t ion t o ( 3 . 4 ) . Therefore F i s Bausdorff l . s . c . 1 y Now, suppose F Hausdorff cont inuous. To show t h a t so i s F i t i s s u f f i -c i e n t to prove t h a t F i s u . s . c . Arguing by cont radic t ion one finds x"0 € x , 0 < e < diam F ( x 0 ) , and a sequence {x } c X converging t o x0 such t h a t F , . (x ) i F , % (x0) + e s , n € IN . Let {y } c Y be such t h a t y(xn) n y(x0) u ' n (3.5) y € F / f x ) Y * F / * <*0> + eS , n c j * . Jn y(x ) n n y(x0) By Lemma 3.5 there is 60 > 0 given by 60 *- min {y(x0), e(a (x0)/2 - y(x0))/ diam F(x0)} such that F , . , (xg) c F . fx0) + es. By the continuity of u y(x0)-60 J y(x0) u y there is k e W such that y(x0) - 60/2 < y(x ) < y(x0) + 60/2 if n >- k. Thus, for each n >- k, y(x0) > y(x ) - 60/2 > y(x0) -6j »• 0 and hence F „ . . ,̂ (xn) c F , „ (xn). On the other hand by the Bausdorff continuity y(xn;-60/2 ° y(x0J-60 ° of F there is k j - k such that F(x ) c F(x0) + (60/2)S if n >- kx. Since y + y(x )S c F(x ) c F(X0) + (60/2)S, it follows that y + (y(x ) - 60/2)S cF(x0). n n n n n Hence for each n >. kt we have y € F ,.-.(x0) c F . . - (x0) c 1 Tn y(xn)-60/2 ° y(x0)-60 ° F , ,(x0) + eS, which contradicts (3.5). Therefore F is Hausdorff u.s.c. . y(x0) u y The last statement of the proposition follows from Lemma 3.5. This completes the proof. •Remark 3.8. Let F : X + ft be Hausdorff continuous. Let y : X -*-» be continuous and satisfy 0 < y(x) < o (x)/2, x c X. For each x € x, put F°, « {y e F(x) | d(y,3F(x)) > y(x)}. From Remark 3.3 it follows that y(x) F0, . (x) e X, thus the multifunction F° given by F°(x) » F°, ,(x) , xcX, maps y(x) y * J y y(x) X into 1. Since F°(x) • int F (x), by virtue of Propositions 3.7 and 2.15, it follows that F° is Hausdorff continuous. Observe that also the multifunction y 3F° : X + X given by OF0) (x) - 3F° (x), x e x, is Hausdorff continuous sin-y y y ce, by Proposition 2.5, x •+ 9F (x) is so and 3F (X) » 3F°(x), X e X. y y y ' Proposition 3.9. Let F : X -*• *U» be Hausdorff l.s.c. Then there exists a Hausdorff continuous multifunction G : X ->• % and a positive continuous function t : X +3R, satisfying G(x) + t(x)S c F(X), X € X. Proof. Let z e X. Since F(z) has nonempty interior there are G c §S and t > 0 such that G + 2t S c F(z). Furthermore, F is Hausdorff l.s.c. z z z thus there is 6 > 0 such that G + t S c F(x) for each x c s - (u £ X I z z z z ' e (u ,z) < 6 }. As {S } i s an open covering of the metr ic space X, t he re Z Z Z C X t r v xr 559 -is a partition of unity subordinated to ^S 2) 2 x • Hence there is a family *? of continuous functions p :X -*• [0,1], whose supports form a neighborhood fi-nite closed covering of X; furthermore the support of each p lies in S , and I p (x) * 1, x £ X. Set uzeX z t(x) - I p2(x) tz G0(x) « I pz(x) Gz , x £ X . Z£X Z£X Observe that t : X •+ » is continuous and positive while, as we shall see, G0 is Hausdorff continuous. To this end, fix x0 £ X and e > 0. For r0 > 0 small enough there is only a finite number of functions p £ *? (i-=l,2,...,k) zi whose supports meet S(x0,r0). By the continuity of P there is 0 < r < r0 zi such that k -1 IP„ (X> - K (x0>l K e C I h(G„ ' °>- ' x£S(x0,r) , Zi Zi 1-1 Zi where i - l,2,...,k. Then, for each x £ S(x0,r), we have k k h(G(x), G(x0)) - h ( I p (x)G , I p (x )G„ ) 1-1 Zi zi i-i zi Zi k k < I h(p (x)G , p (x0)G ) < I |p (x) -p (x0)| h(G, ,0) < e i-1 Zi Zi Zi Zi 1-1 Zi Zi 2i and G0 is Hausdorff continuous at x0. Moreover, we have G0(x) + t(x)S c p(x), x € X. In fact, take any x0 £ X and denote by p (i •* l,2,...,k) those zi functions in *? whose supports contain x0. Since x0 £ S we have G + t S c P(x0), i » l,2,...*,k , and thus zi k k G0(x0) +t(x0)S - I p (xQ)Gr + ( I P, (x0)t )S 1-1 zi zi i-i zi zi k k " I P, (x0>(G, +t S) c I p (x0)F(x0) - P(x0) . i-1 Zi Zi Zi i-1 Zi Then the multifunction G defined by G(x) - G0(x), x £ X, maps X into S , is Hausdorff continuous, and satisfies G(x) + t(x)S c F(X), x € X. This comple-tes the proof. Remark 3.10. The above argument shows that, if we retain the hypotheses - 560 (and notations) of Proposition 3.9, then F : X -> U admits a continuous single valued selection g : X ->- Y satisfying g(x) +t(x)S c F(x), x € X. For continuous F, from X to the nonempty open convex subsets of Y, the exi-stence of, continuous single valued selections follows from Michael C4, Theorem 8.5]. Observe that if in Proposition 3.9 F is supposed to be lower semiconti-nuous (that is, whenever V c y is open in Y then the set {x e X ( F(x) n V t 0) is open in X), the existence of continuous single valued selections may fail. In fact, as shown by Michael C3, Example 6.3], there exists a lower semiconti- » nuous multifunction, from CO,l] to the nonempty open convex subsets of a Banach space, which has no single valued continuous selections. This pathology is ruled out under the stronger hypothesis that F be Hausdorff l.s.c. References Cl] F.S. De Blasi, G. Pianigiani, A Baire category approach to the existence of solutions of multivalueđ differential equations in Banach spaces, Funkcial. Ekvac. (2) 25 (1982), 153-162. C2] J. Dugunđji, Topology, Allyn anđ Bacon, Boston 1966. CЗ] E. Michael, Continuous selections I, Ann. of Math. (2) 63 (1956), 361-382. C4] E. Michael, Continuous selections III, Ann.ofMath. (2) 65 (1957), 375-390. lstituto Matematico U. Dini lstituto Matematico Universita Viale Morgagni,67/A Via Mantica, 3 50134 Firenze, Italy 33100 Udine, Italy (Oblátům 27/11.1982) - 561 

Remarks on Hausdorff continuous multifunction and selections

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Remarks on Hausdorff continuous multifunction and selections

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