Appendix A Set-Valued Maps Let X and Y be two nonempty sets. A set-valued map or multivalued map or point- to-set map or multifunction T W X ! 2Y from X to Y is a map that associates with any x 2 X a subset T.x/ of Y;thesetT.x/ is called the image of x under T.Theset Dom.T/ Dfx 2 X W T.x/ ¤;gis called the domain of T. Actually, a set-valued map T is characterized by its graph, the subset of X Y defined by Graph.T/ Df.x; y/ W y 2 T.x/g: Indeed, if A is a nonempty subset of the product space X Y, then the graph of a set-valued map T is defined by y 2 T.x/ if and only if .x; y/ 2 A: The domain of T is the projection of Graph.F/ on X.Theimage of T is a subset of Y defined by [ [ Im.T/ D T.x/ D T.x/: x2X x2Dom.T/ It is the projection of Graph.T/ on Y. A set-valued map T from X to Y is called strict if Dom.T/ D X, that is, if the image T.x/ is nonempty for all x 2 X.LetK be a nonempty subset of X and T be a strict set-valued map from K to Y.Itmaybe useful to extend it to the set-valued map TK from X to Y defined by T.x/; when x 2 K; T .x/ D K ;; when x … K; whose domain Dom.TK/ is K. © Springer International Publishing AG 2018 487 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 488 A Set-Valued Maps Let T be a set-valued map from X to Y and K  X, then we denote by TjK the restriction of T to K. Definition A.1 Let T W X ! 2Y be a set-valued map. For a nonempty subset A of X, we write [ T.A/ D T.x/: x2A If A D;, we write T.;/ D;.ThesetT.A/ is called the image of A under the set-valued map T. Y Theorem A.1 Let fA˛g˛2 be a family of nonempty subsets of X and T W X ! 2 be a set-valued map. (a) If A1  A2!then T.A1/  T.A2/; [ [ (b) T A˛ D T .A˛/; ˛2 ! ˛2 \ \ (c) T A˛ T .A˛/; ˛2 ˛2 (d) T .X n A1/ à T.X/ n T .A1/. Definition A.2 Let T1 and T2 be two set-valued maps from X to Y. •Theunion of T1 and T2 is a set-valued map .T1 [ T2/ from X to Y defined by .T1 [ T2/.x/ D T1.x/ [ T2.x/; for all x 2 X: •Theintersection of T1 and T2 is a set-valued map .T1 \ T2/ from X to Y defined by .T1 \ T2/.x/ D T1.x/ \ T2.x/; for all x 2 X: •TheCartesian product of T1 and T2 is a set-valued map .T1 T2/ from X to Y Y defined by .T1 T2/.x/ D T1.x/ T2.x/; for all x 2 X: •IfT1 is a set-valued map from X to Y and T2 is another set-valued map from Y to Z, then the composition product of T2 by T1 is a set-valued map .T2 ı T1/ from X to Z defined by .T2 ı T1/.x/ D T2.T1.x//; for all x 2 X: A Set-Valued Maps 489 Theorem A.2 Let T1 and T2 be set-valued maps from X to Y and A be a nonempty subset of X. Then (a) .T1 [ T2/.A/ D T1.A/ [ T2.A/; (b) .T1 \ T2/.A/  T1.A/ \ T2.A/; (c) .T1 T2/.A/  T1.A/ T2.A/; (d) .T2 ı T1/.A/ D T2.T1.A//: Definition A.3 If T is a set-valued map from X to Y, then the inverse T1 of T is defined by T1. y/ Dfx 2 X W y 2 T.x/g; for all y 2 Y Further, let B be a subset of Y.Theupper inverse image T1.B/ and lower inverse 1 image TC .B/ of B under F are defined by T1.B/ Dfx 2 X W T.x/ \ B ¤;g and 1 TC .B/ Dfx 2 X W T.x/  Bg: 1 1 We also write T .;/ D;and TC .;/ D;. It is clear from the definition of inverse of T that .T1/1 D T and y 2 T.x/ if and only if x 2 T1. y/. We have the following relations between domain, graph and image of T and T1. Dom.T1/ D Im.T/; Im.T1/ D Dom.T/ and Graph.T1/ Df. y; x/ 2 Y X W .x; y/ 2 Graph.T/g: Theorem A.3 Let fB˛g˛2 be a family of nonempty subsets of Y, A  X and B  Y. Let T W X ! 2Y be a set-valued map. 1 1 (a) If B1  B2,thenT .B1/  T .B2/; 1 (b) A TC .T.A//; . 1. // (c) B T TC B! ; [ [ 1 1 (d) TC B˛ TC .B˛/; ˛2 ! ˛2 \ \ 1 1 (e) TC B˛ D TC .B˛/; ˛2 ˛2 (f) T1.T.A// A; 490 A Set-Valued Maps ! \ \ 1 1 (g) T B˛ T .B˛/; ˛2 ! ˛2 [ [ 1 1 (h) T B˛ D T .B˛/. ˛2 ˛2 Y Theorem A.4 Let T1; T2 W X ! 2 be set-valued maps such that .T1 \ T2/.x/ ¤; for all x 2 X and B  Y. Then 1 1 1 (a) .T1 [ T2/ .B/ D T1 .B/ [ T2 .B/; 1 1 1 (b) .T1 \ T2/ .B/ T1 .B/ \ T2 .B/; 1 1 1 (c) .T1 [ T2/C .B/ D T1C.B/ \ T2C.B/; 1 1 1 (d) .T1 \ T2/C .B/ T1C.B/ [ T2C.B/; Y Z Theorem A.5 Let T1 W X ! 2 and T2 W Y ! 2 be set-valued maps. Then for any set B  Z, we have . /1 . / 1 1. / (a) T2 ı T1 C B D T1C T2C B ; 1 1 1 (b) .T2 ı T1/ .B/ D T1 T2 .B/ . Y Z Theorem A.6 Let T1 W X ! 2 and T2 W X ! 2 be set-valued maps. Then for any sets B  Y and D  Z, we have 1 1 1 (a) .T1 T2/C .B D/ D T1C.B/ \ T2C.D/; 1 1 1 (b) .T1 T2/ .B D/ D T1 .B/ \ T2 .D/. For further details and applications of set-valued maps, we refer to [1–8]andthe references therein. Appendix B Some Algebraic Concepts Let K and D be nonempty subsets of a vector space X.Thealgebraic sum and algebraic difference of K and D are, respectively, defined as K C D Dfx C y W x 2 K and y 2 Dg and K D Dfx y W x 2 K and y 2 Dg: Let be any real number, then K is defined as K Dfx W x 2 Kg: We pointed out that K C K D 2K is not true in general for any nonempty subset K of a vector space X. Definition B.1 Let X be a vector space. The space of all linear mappings from X to R is called algebraic dual of X and it is denoted by X0. Definition B.2 A subset K of a vector space X is called • balanced if it is nonempty and ˛K K for all ˛ 2 Œ1; 1; • absolutely convex if it is convex and balanced; • absorbing or absorbent if for each x 2 X, there exists >0such that x 2 K for all jjÄ. Note that an absorbing set contains the zero of X. Theorem B.1 Let X and Y be vector spaces and f W X ! Y be a linear map. (a) If K is a balanced subset of X, then f .K/ is balanced. (b) If K is absorbing and f is onto, then f .K/ is absorbing. (c) Inverse image under f of absorbing or balanced subsets of Y are absorbing or balanced, respectively, subset of X. © Springer International Publishing AG 2018 491 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 492 B Some Algebraic Concepts Definition B.3 Let K be a nonempty subset of a vector space X. • The set cor.K/ Dfy 2 K W for every x 2 X there is a >0N with y C x 2 C for all 2 Œ0; N g is called the algebraic interior of K (or the core of K). •ThesetK is called algebraic open if K D cor.K/. • The set of all elements of X which neither belong to cor.K/ nor to cor.X n K/ is called the algebraic boundary of K. •Anelementx 2 X is called linearly accessible from K if there exists y 2 K, y ¤ x, such that y C .1 /x 2 K for all 2 .0; 1. The union of K and the set of all linearly accessible elements from K is called the algebraic closure of K and it is denoted by lin.K/ D K [fx 2 X W x is linearly accessible from Kg: The set K is called algebraic closed if K D lin.K/.