158:
301:
Instead of considering collections of only points, set-valued analysis considers collections of sets. If a collection of sets is endowed with a topology, or inherits an appropriate topology from an underlying topological space, then the convergence of sets can be studied.
277:
443:. This among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.
455:
546:
197:
140:
111:
838:
513:
68:
Set-valued functions are also known as multivalued functions in some references, but herein and in many others references in
833:
353:
757:
436:
377:
580:
447:
381:
361:
54:
345:
to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point.
400:
349:
843:
743:
463:
446:
Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the
428:
393:
389:
322:
306:
162:
53:, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including
412:
291:
69:
318:
73:
50:
31:
728:
373:
365:
272:{\displaystyle \operatorname {argmax} _{x\in \mathbb {R} }\cos(x)=\{2\pi k\mid k\in \mathbb {Z} \}}
81:
690:
672:
642:
338:
753:
660:
458:, Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in
808:
519:
509:
788:
682:
634:
626:
589:
555:
440:
330:
295:
818:
813:
575:
459:
424:
342:
314:
310:
116:
87:
454:
spaces. Other selection theorems, like
Bressan-Colombo directional continuous selection,
348:
There exist set-valued extensions of the following concepts from point-valued analysis:
607:
408:
369:
326:
58:
827:
611:
560:
541:
537:
694:
157:
404:
334:
35:
17:
593:
766:
712:
686:
451:
432:
62:
638:
663:; P.V. Semenov (2008). "Ernest Michael and theory of continuous selections".
523:
793:
776:
503:
385:
357:
161:
This diagram represents a multi-valued, but not a proper (single-valued)
646:
482:
Some authors use the term ‘semicontinuous’ instead of ‘hemicontinuous’.
151:
191:
630:
677:
156:
49:) is a mathematical function that maps elements from one set, the
439:
for set-valued functions has been applied to prove existence of
777:"Hermite-Hadamard inequalities for convex set-valued functions"
739:, Grundl. der Math. Wiss. 264, Springer - Verlag, Berlin, 1984
729:
Topological Fixed Point
Principles for Boundary Value Problems
737:
Differential
Inclusions, Set-Valued Maps And Viability Theory
578:(1941). "A generalization of Brouwer's fixed point theorem".
508:. Pavel Vladimirovič. Semenov. Dordrecht: Kluwer Academic.
456:
Kuratowski and Ryll-Nardzewski measurable selection theorem
341:. In optimization theory, the convergence of approximating
84:
property, namely that the choice of an element in the set
30:
For multi-valued functions of mathematical analysis, see
305:
Much of set-valued analysis arose through the study of
194:
of a function is in general, multivalued. For example,
768:
Topological methods for set-valued nonlinear analysis
200:
119:
90:
399:One can distinguish multiple concepts generalizing
392:, while differential equations are generalized to
271:
134:
105:
775:Mitroi, F.-C.; Nikodem, K.; Wąsowicz, S. (2013).
721:Infinite dimensional analysis. Hitchhiker's guide
547:Journal of Mathematical Analysis and Applications
34:. For functions whose arguments are sets, see
759:Continuous Selections of Multivalued Mappings
505:Continuous selections of multivalued mappings
450:, which provides another characterisation of
8:
762:, Kluwer Academic Publishers, Dordrecht 1998
411:. There are also various generalizations of
266:
240:
113:defines a corresponding element in each set
27:Function whose values are sets (mathematics)
792:
723:, Springer-Verlag Berlin Heidelberg, 2006
676:
559:
262:
261:
213:
212:
205:
199:
118:
89:
765:E. U. Tarafdar and M. S. R. Chowdhury,
494:
475:
290:is the study of sets in the spirit of
7:
771:, World Scientific, Singapore, 2008
719:C. D. Aliprantis and K. C. Border,
542:"Integrals of Set-Valued Functions"
732:, Kluwer Academic Publishers, 2003
713:Multivalued Differential Equations
25:
169:is associated with two elements,
313:, partly as a generalization of
423:Set-valued functions arise in
409:upper and lower hemicontinuity
234:
228:
129:
123:
100:
94:
1:
726:J. Andres and L. Górniewicz,
594:10.1215/S0012-7094-41-00838-4
321:" is used by authors such as
735:J.-P. Aubin and A. Cellina,
561:10.1016/0022-247X(65)90049-1
437:Kakutani fixed-point theorem
687:10.1016/j.topol.2006.06.011
165:, because the element 3 in
860:
612:"Continuous Selections. I"
29:
839:Mathematical optimization
750:, Birkhäuser, Basel, 1990
716:, Walter de Gruyter, 1992
581:Duke Mathematical Journal
448:Michael selection theorem
382:topological degree theory
362:implicit function theorem
76:is a set-valued function
781:Demonstratio Mathematica
431:and related subjects as
464:differential inclusions
429:differential inclusions
394:differential inclusions
154:an ordinary function.
794:10.1515/dema-2013-0483
502:Repovš, Dušan (1998).
425:optimal control theory
323:R. Tyrrell Rockafellar
307:mathematical economics
273:
182:
136:
107:
51:domain of the function
619:Annals of Mathematics
292:mathematical analysis
274:
160:
137:
108:
70:mathematical analysis
834:Variational analysis
374:fixed-point theorems
366:contraction mappings
319:variational analysis
198:
135:{\displaystyle f(y)}
117:
106:{\displaystyle f(x)}
88:
74:multivalued function
32:Multivalued function
748:Set-Valued Analysis
415:to multifunctions.
388:are generalized to
288:Set-valued analysis
283:Set-valued analysis
150:, and thus defines
80:that has a further
43:set-valued function
18:Set-valued analysis
756:and P.V. Semenov,
639:10338.dmlcz/119700
462:and the theory of
339:Boris Mordukhovich
269:
183:
132:
103:
809:Selection theorem
621:. Second Series.
538:Aumann, Robert J.
384:. In particular,
16:(Redirected from
851:
798:
796:
742:J.-P. Aubin and
699:
698:
680:
657:
651:
650:
616:
604:
598:
597:
576:Kakutani, Shizuo
572:
566:
565:
563:
534:
528:
527:
499:
483:
480:
343:subdifferentials
331:Jonathan Borwein
296:general topology
278:
276:
275:
270:
265:
218:
217:
216:
149:
145:
141:
139:
138:
133:
112:
110:
109:
104:
79:
21:
859:
858:
854:
853:
852:
850:
849:
848:
824:
823:
819:Binary relation
814:Ursescu theorem
805:
774:
707:
705:Further reading
702:
659:
658:
654:
631:10.2307/1969615
614:
606:
605:
601:
574:
573:
569:
536:
535:
531:
516:
501:
500:
496:
492:
487:
486:
481:
477:
472:
460:optimal control
441:Nash equilibria
421:
354:differentiation
315:convex analysis
311:optimal control
285:
201:
196:
195:
188:
147:
143:
115:
114:
86:
85:
77:
39:
28:
23:
22:
15:
12:
11:
5:
857:
855:
847:
846:
844:Control theory
841:
836:
826:
825:
822:
821:
816:
811:
804:
801:
800:
799:
787:(4): 655–662.
772:
763:
751:
740:
733:
724:
717:
706:
703:
701:
700:
671:(8): 755–763.
652:
625:(2): 361–382.
608:Ernest Michael
599:
588:(3): 457–459.
567:
529:
514:
493:
491:
488:
485:
484:
474:
473:
471:
468:
420:
417:
403:, such as the
370:measure theory
327:Roger J-B Wets
284:
281:
268:
264:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
215:
211:
208:
204:
187:
184:
131:
128:
125:
122:
102:
99:
96:
93:
59:control theory
47:correspondence
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
856:
845:
842:
840:
837:
835:
832:
831:
829:
820:
817:
815:
812:
810:
807:
806:
802:
795:
790:
786:
782:
778:
773:
770:
769:
764:
761:
760:
755:
752:
749:
745:
744:H. Frankowska
741:
738:
734:
731:
730:
725:
722:
718:
715:
714:
710:K. Deimling,
709:
708:
704:
696:
692:
688:
684:
679:
674:
670:
666:
665:Topology Appl
662:
656:
653:
648:
644:
640:
636:
632:
628:
624:
620:
613:
609:
603:
600:
595:
591:
587:
583:
582:
577:
571:
568:
562:
557:
553:
549:
548:
543:
539:
533:
530:
525:
521:
517:
515:0-7923-5277-7
511:
507:
506:
498:
495:
489:
479:
476:
469:
467:
465:
461:
457:
453:
449:
444:
442:
438:
434:
430:
427:, especially
426:
418:
416:
414:
410:
407:property and
406:
402:
397:
395:
391:
387:
383:
379:
375:
371:
367:
363:
359:
355:
351:
346:
344:
340:
336:
332:
328:
324:
320:
316:
312:
308:
303:
299:
297:
293:
289:
282:
280:
258:
255:
252:
249:
246:
243:
237:
231:
225:
222:
219:
209:
206:
202:
193:
185:
180:
176:
172:
168:
164:
159:
155:
153:
126:
120:
97:
91:
83:
75:
71:
66:
64:
60:
56:
52:
48:
44:
37:
33:
19:
784:
780:
767:
758:
747:
736:
727:
720:
711:
668:
664:
661:Dušan Repovš
655:
622:
618:
610:(Mar 1956).
602:
585:
579:
570:
551:
545:
532:
504:
497:
478:
445:
435:, where the
422:
419:Applications
405:closed graph
398:
378:optimization
347:
335:Adrian Lewis
317:; the term "
304:
300:
287:
286:
189:
178:
174:
170:
166:
67:
55:optimization
46:
42:
40:
36:Set function
554:(1): 1–12.
452:paracompact
433:game theory
358:integration
63:game theory
828:Categories
490:References
401:continuity
390:inclusions
350:continuity
82:continuity
754:D. Repovš
678:0803.4473
386:equations
259:∈
253:∣
247:π
226:
220:
210:∈
146:close to
803:See also
695:14509315
540:(1965).
524:39739641
186:Examples
163:function
647:1969615
413:measure
337:, and
152:locally
693:
645:
522:
512:
380:, and
203:argmax
192:argmax
691:S2CID
673:arXiv
643:JSTOR
615:(PDF)
470:Notes
177:, in
520:OCLC
510:ISBN
333:and
325:and
309:and
294:and
190:The
173:and
142:for
72:, a
61:and
45:(or
789:doi
683:doi
669:155
635:hdl
627:doi
590:doi
556:doi
223:cos
65:.
830::
785:46
783:.
779:.
746:,
689:.
681:.
667:.
641:.
633:.
623:63
617:.
584:.
552:12
550:.
544:.
518:.
466:.
396:.
376:,
372:,
368:,
364:,
360:,
356:,
352:,
329:,
298:.
279:.
57:,
41:A
797:.
791::
697:.
685::
675::
649:.
637::
629::
596:.
592::
586:8
564:.
558::
526:.
267:}
263:Z
256:k
250:k
244:2
241:{
238:=
235:)
232:x
229:(
214:R
207:x
181:.
179:Y
175:c
171:b
167:X
148:x
144:y
130:)
127:y
124:(
121:f
101:)
98:x
95:(
92:f
78:f
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
