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Arcwise Connectedness of Closed Efficient Point Sets
Journal of Mathematical Analysis and Applications 247, 377᎐383 Ž2000. doi:10.1006rjmaa.2000.6814, available online at http:rrwww.idealibrary.com on
Arcwise Connectedness of Closed Efficient Point Sets E. K. Makarov 1 Institute of Mathematics, National Academy of Sciences of Belarus, Surgano¨ a 11, Minsk 220072, Belarus
N. N. Rachkovski Belorussian State Pedagogical Uni¨ ersity, So¨ yetskaya, 18, Minsk 220809, Belarus
and Wen Song Harbin Normal Uni¨ ersity, Harbin 150080, People’s Republic of China, and Systems Research Institute, Newelska 6, 01-447 Warsaw, Poland Submitted by George Leitmann Received February 16, 1999
1. INTRODUCTION One of the most important problems of vector optimization is to investigate the structure of efficient point sets. For various applications, the possibility of continuous moving from one optimal solution to any other along optimal alternatives only is of special interest. This possibility is guaranteed if the efficient set is arcwise connected or at least connected. It is well known that in a finite dimensional space the efficient point set of a closed convex cone-compact set with respect to an order defined by a closed convex cone is connected, as well as the efficient solution set for concave objective functions Žsee w1, 12, 13x.. Several authors proved analogous theorems for some subclasses of quasiconcave objective functions Žsee w3, 7, 9, 14, 15, 19, 20x.. Many of the above results were also 1
E-mail: [email protected]. 377 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
378
MAKAROV, RACHKOVSKI, AND SONG
generalized to the infinite dimensional space settings by Luc w10, 11x, Fu w4x, Gong w5, 6x, and Song w16᎐18x. However, the results concerning strong topological properties, e.g., contractibility or arcwise connectivity, are rather seldom. Contractibility of the efficient point set and arcwise connectivity of the weak efficient point set are studied in w11x under some stronger assumptions, and contractibility of the efficient solution set for a strongly quasiconcave function Žcalled strictly quasiconcave function in w9, 10x. is studied in w9, 10x. In this paper we prove that the efficient point set MaxŽ Q < K . of a compact convex set Q ; X in a Hausdorff topological vector space X ordered by a closed convex pointed cone K ; X is arcwise connected if the set MaxŽ Q < K . is closed.
2. PRELIMINARIES Let X be a Hausdorff topological vector space, Q ; X, and K ; X a closed convex pointed cone. We denote the interior, the closure, and the convex hull of an arbitrary set A ; X by int A, cl A, and conv A, respectively. A point x g Q is said to be an efficient point of Q with respect to K Ž x g MaxŽ Q < K .. if Ž Q y x . l K s 04 , and it is said to be the least element of Q with respect to K if Q ; x q K. Let F be a set-valued mapping from a topological space W to X. By dom F and graph F we denote the domain and graph of F, i.e., dom F s w g W < F Ž w . / ⭋ 4
and
graph F s Ž w, x . g W = X < x g F Ž w . 4 . Let P be a nonempty subset of X, x 1 , x 2 g Q, and 0 F 1 - 2 F 1. Consider the set-valued map F: w 1 , 2 x § Q given by F Ž t . s ŽŽŽ 2 y t .rŽ 2 y 1 .. x 1 q ŽŽ t y 1 .rŽ 2 y 1 .. x 2 q P . l Q. LEMMA 2.1. Let P be a con¨ ex cone and t 1 g Ž 1 , 2 ., z1 g F Ž t 1 .. Suppose that the set-¨ alued map G: w 1 , 2 x § Q is gi¨ en by
¡
GŽ t . s
~ž ¢ž
t1 y t t1 y 1
2 y t 2 y t1
x1 q z1 q
t y 1 t1 y 1 t y t1
2 y t1
Then GŽ t . ; F Ž t . for all t g w 1 , 2 x.
z1 q P l Q,
if t g w 1 , t 1 x ;
x 2 q P l Q,
if t g w t 1 , 2 x .
/ /
379
CONNECTEDNESS OF EFFICIENT POINT SETS
Proof. Since t 1 g w 1 , 2 x, there exists ␣ g w0, 1x such that t 1 s ␣ 1 q Ž1 y ␣ . 2 . Since z1 g F Ž t 1 ., we have z1 g Ž ␣ x 1 q Ž1 y ␣ . x 2 q P . l Q. Hence, for every t g w 1 , t 1 x, we have t1 y t
GŽ t . ; ; s
t1 y 1
ž ž
x1 q
t y 1 t1 y 1
t1 y t q ␣ Ž t y 1 . t1 y 1
2 y t 2 y 1
x1 q
Ž ␣ x1 q Ž 1 y ␣ . x 2 q P . q P
x1 q Ž 1 y ␣ .
t y 1
2 y 1
t y 1 t1 y 1
lQ
x2 q P l Q
/
x2 q P l Q s F Ž t . .
/
For t g w t 1 , 2 x, the inclusion GŽ t . ; F Ž t . can be proved similarly. LEMMA 2.2. closed set.
If P and Q are closed sets, then F is closed, i.e., graph F is a
Proof. Define a continuous mapping p: w 1 , 2 x = X ª ⺢ = X by pŽ t, y . s Ž t, y y ŽŽŽ 2 y t .rŽ 2 y 1 .. x 1 q ŽŽ t y 1 .rŽ 2 y 1 .. x 2 ... Then graph F s py1 Žw 1 , 2 x = P . l Žw 1 , 2 x = Q .. Since the sets w 1 , 2 x = P and w 1 , 2 x = Q are closed, graph F is closed. LEMMA 2.3. Let Q1 > Q2 > . . . be a sequence of compact subsets of X, and x k g Q k be the least element of Q k with respect to K, for each k g ⺞. Then the sequence x k 4k g ⺞ con¨ erges to the unique least element of Q0 [ l Q k : k g ⺞4 / ⭋ with respect to K. Proof. Since X is Hausdorff and each Q k is compact, Q k 4k g ⺞ is a decreasing sequence of nonempty closed sets in compact space Q1. By w2, Proposition I-11.4x, we have Q0 / ⭋ and by w2, Proposition I-11.5x, x k 4k g ⺞ has an adherent point x 0 g Q0 . Suppose that there exists z g Q0 such that z f x 0 q K. Since x 0 q K is closed, there exists a neighbourhood U of zero in X such that Ž z y U . l Ž x 0 q K . s ⭋. Hence, z f x 0 q U q K. Take k g ⺞ such that x k g x 0 q U. Then x k q K ; x 0 q U q K and, therefore, z f x k q K. On the other hand, we have z g Q0 ; Q k ; x k q K. By this contradiction, we assert that Q0 ; x 0 q K. Hence, x 0 is the least element of Q0 . Since K is pointed, the set Q0 has the unique least element x 0 . Therefore, x 0 is the unique adherent point of x k 4k g ⺞ . By w2, Proposition I-11.5x, we have x k ª x 0 . 3. MAIN RESULT Recall that a set A of a topological space is said to be arcwise connected if for every two points x, y g A there exists a continuous function : w0, 1x ª A such that Ž0. s x, Ž1. s y.
380
MAKAROV, RACHKOVSKI, AND SONG
THEOREM 3.1. Let K ; X be a closed con¨ ex pointed cone and Q ; X be compact and con¨ ex. If the set MaxŽ Q < K . is closed, then it is arcwise connected. Proof. Take any x 1 , x 2 g MaxŽ Q < K .. The set-valued map F0 : w0, 1x § Q given by F0 Ž t . s ŽŽ1 y t . x 1 q tx 2 q K . l Q has compact nonempty values for each t g w0, 1x. By w8, Theorem 6.3 and Lemma 6.2x, we have ⭋ / 1 MaxŽ F0 Ž t .< K . ; MaxŽ Q < K . for all t g w0, 1x. For each t m s mr2, m s 1 .< 1 Ž Ž . 0, 1, 2, we choose an arbitrary z m g Max F0 t m K . Since F0 Ž t 01 . s x 1 4 and F0 Ž t 21 . s x 2 4 , we have z 01 s x 1 and z 21 s x 2 . Now we can define the set-valued map F1 : w0, 1x § Q by setting F1Ž t . s ŽŽ1 y 2 t . z 01 q 2 tz11 q K . l Q for t g w0, 1r2x and F1Ž t . s ŽŽ2 y 2 t . z11 q Ž2 t y 1. z 21 q K . l Q for t g w1r2, 1x. By Lemma 2.1, we have F0 Ž t . > F1Ž t . for all t g w0, 1x. It can be easily seen that F1Ž t . is compact and nonempty for each t g w0, 1x, and ⭋ / MaxŽ F1Ž t .< K . ; MaxŽ Q < K .. Let us now iterate this process. Suppose that for some k g ⺞ we have already defined the set-valued mappings Fi : w0, 1x § Q, i s 0, 1, . . . , k by putting i i Fi Ž t . s 2 i Ž t mq1 y t . z mi q 2 i Ž t y t mi . z mq1 qK lQ i for t g t mi , t mq1 ,
where t mi s 2yi m, m s 0, 1, . . . , 2 i. We also suppose that F0 Ž t . > F1Ž t . > ⭈⭈⭈ > Fk Ž t . for all t g w0, 1x and Fi Ž t mi . s z mi 4 ; MaxŽ Q < K . for m s 0, 1, . . . , 2 i, i s 0, 1, . . . , k. It can be easily seen that Fi Ž t . is compact and nonempty for each t g w0, 1x, i s 0, . . . , k, and ⭋ / MaxŽ Fi Ž t .< K . ; MaxŽ Q < K .. Thus, for each t mkq 1 s 2yky1 m, m g 0, 1, . . . , 2 kq 1 4 , we can choose an arbitrary z mkq 1 g MaxŽ Fk Ž t mkq 1 .< K . ; MaxŽ Q < K .. Note that 1 k k z 2kq n s z n , n s 0, 1, . . . , 2 , by assumption. Now we can define the set-valued mapping Fkq 1 : w0, 1x § Q by putting kq1 kq1 Fkq 1 Ž t . s 2 kq 1 Ž t mq1 y t . z mkq1 q 2 kq1 Ž t y t mkq1 . z mq1 qK lQ kq1 for t g t mkq 1 , t mq1 , kq 1 . Ž kq 1 . Ž kq 1 where m s 0, 1, . . . , 2 kq 1 y 1. Since 2 kq1 Ž t mq 1 y t s t mq1 y t r t mq1 kq 1 . y t m , by Lemma 2.1, we obtain Fk Ž t . > Fkq1Ž t . for all t g w0, 1x. It can be easily seen that Fkq 1Ž t . is compact and nonempty for each t g w0, 1x and ⭋ / MaxŽ Fkq 1Ž t .< K . ; MaxŽ Q < K .. Obviously, Fkq1Ž t mkq 1 . s z mkq 1 4 ; MaxŽ Q < K . for m s 0, 1, . . . , 2 kq 1. Define the set-valued mapping F : w0, 1x § Q by
F Ž t . s l Fk Ž t . : k g ⺞ 4 .
CONNECTEDNESS OF EFFICIENT POINT SETS
381
Note that for every t g w0, 1x and k g ⺞, Fk Ž t . is compact and nonempty. Moreover, we have Fk Ž t . > Fkq1Ž t .. Thus, we can assert w2, Proposition I-11.4x that F Ž t . s l Fk Ž t . : k g ⺞4 / ⭋ for all t g w0, 1x and, therefore, dom F s w0, 1x. Let T s j 2yk m : m s 0, 1, . . . , 2 k , k g ⺞4 ; w0, 1x. Obviously, cl T s w0, 1x. If t g T, then t s t mk for some m s 0, 1, . . . , 2 k , k g ⺞ and, hence, F0 Ž t . > F1Ž t . > ⭈⭈⭈ > Fk Ž t . s Fkq1Ž t . s ⭈⭈⭈ s F Ž t . s z mk 4 ; MaxŽ Q < K ., i.e., Ž t mk , z mk . g graph F. Now suppose that t g w0, 1x _ T. Then for each k g ⺞ we can find k k k x mŽ k . g 0, 1, . . . , 2 k y 14 such that t g w t mŽ k . , t mŽ k .q1 . Clearly, t mŽ k . ª t k and t mŽ k .q1 ª t as k ª q⬁. By the definition of Fk , we have Fk Ž t . s k kŽ k Ž2 k Ž t mŽ . k . k . t y t mŽ k s k .q 1 y t z mŽ k . q 2 k . z mŽ k .q 1 q K l Q. Let kŽ k k k 2 t mŽ k .q1 y t . g w0, 1x and x k s  k z mŽ k . q Ž1 y  k . z mŽ . Then Fk Ž t . k .q1 s Ž x k q K . l Q. Since x k is the least element of Fk Ž t . with respect to K for each k g ⺞, by Lemma 2.3, we obtain x k ª x 0 g l Fk Ž t . : k g ⺞4 s F Ž t . / ⭋ and F Ž t . ; x 0 q K. By compactness, there exist some subnets  kŽ .4 g M of  k 4k g ⺞ conkŽ . 4 k 4 verging to  0 g w0, 1x, z mŽ kŽ .. g M of z mŽ k . k g ⺞ converging to ¨ g Q, kŽ . k and z mŽ kŽ ..q 14 g M of z mŽ k .q14k g ⺞ converging to w g Q. Let us denote kŽ . kŽ . ␣ [  kŽ . , ¨ [ z mŽ kŽ .. , and w [ z mŽ kŽ ..q 1 , g M. Hence, x kŽ . s ␣¨ q Ž1 y ␣ . w converges to x 0 s  0¨ q Ž1 y  0 . w g Q. By Lemma 2.2, for every k g ⺞, graph Fk is closed. Hence, graph F s kŽ . kŽ . . Ž t mŽ . l graph Fk : k g ⺞4 is closed. Since Ž t mŽ kŽ .. , ¨ , kŽ ..q 1 , w g graph F for any g M, we have Ž t, ¨ . g graph F and Ž t, w . g graph F. Hence, ¨ , w g x 0 q K. Since x 0 s  0¨ q Ž1 y  0 . w and K is pointed, we assert that x 0 g ¨ , w4 . Note that ¨ , w4 g cl MaxŽ Q < K . s MaxŽ Q < K .. Therefore, x 0 4 s Q l Ž x 0 q K . > Q l F Ž t . s F Ž t .. Thus, F is a single-valued Žpoint-valued. map. By Lemma 2.2, graph F is closed. Since graph F ; w0, 1x = Q and Q is compact, we assert that graph F is compact as a closed subset of a compact set. Thus, F is continuous. The proof is completed. The assumption of convexity on Q in Theorem 3.1 is essential. EXAMPLE 3.1. Suppose that X s ⺢ 2 , Q s Ž x, y . g ⺢ 2 < x s 0, y g 2 w0, 1x4 j Ž x, y . g ⺢ 2 < x g w0, 1x, y s 04 and K s ⺢q [ Ž x, y . g ⺢ 2 < x G 4 0, y G 0 . Clearly, Q is compact and not convex and MaxŽ Q < K . s Ž1, 0., Ž0, 1.4 is closed, but not arcwise connected. Let now X be a locally convex space. Then we can use the following sufficient conditions for the set MaxŽ Q < K . to be closed given by Fu w4x. Let A be a subset of X. A point x g A is said to be an F-point if for every open set U with x g ŽU l A. y K, there exists a neighborhood W of
382
MAKAROV, RACHKOVSKI, AND SONG
x such that W l A ; Ž U l A. y K . The set A is said to be an F-set if every point of A is an F-point. THEOREM 3.2 w4x. Let Q be a subset of a locally con¨ ex space X and K be a closed con¨ ex cone such that there exists a closed con¨ ex cone S with K _ 04 ; int S and 0 f int S. If Q is a compact F-set, then MaxŽ Q < K . is closed. EXAMPLE 3.2. Let X s ⺢ 3 , Q s convŽŽ x, y, z . g ⺢ 3 < x 2 q y 2 F 1, z s 04 j Ž1, 0, 1.4., K s Ž x, y, z . g ⺢ 3 < x s y s 0, z G 04 . Clearly, MaxŽ Q < K . is arcwise connected, but not closed. From Example 3.2, we can see that the assumption that MaxŽ Q < K . is closed is not essential for MaxŽ Q < K . to be arcwise connected. So it is natural to ask if the conclusion of Theorem 3.1 is true without the assumption that MaxŽ Q < K . is closed.
REFERENCES 1. G. R. Bitran and T. L. Magnanti, The structure of admissible points with respect to cone dominance, J. Optim. Theory Appl. 29 Ž1979., 573᎐614. 2. G. Choquet, ‘‘Topology,’’ Academic Press, New YorkrLondon, 1966. 3. A. Daniilidis, N. Hadjisavvas, and S. Schaible, Connectedness of the efficient set for three-objective quasiconcave maximization problems, J. Optim. Theory Appl. 93 Ž1997., 517᎐524. 4. W. T. Fu, On the closedness and connectedness of the set of efficient points, J. Math. Res. Exposition 14 Ž1994., 89᎐96. wIn Chinesex 5. X. H. Gong, Connectedness of efficient solution sets for set-valued maps in normed spaces, J. Optim. Theory Appl. 83 Ž1994., 83᎐96. 6. X. H. Gong, Connectedness of the efficient solution sets of a convex vector optimization in normed spaces, Nonlinear Anal. 23 Ž1994., 1105᎐1114. 7. Y. D. Hu and E. J. Sun, Connectedness of the efficient set in strictly quasiconcave vector maximization, J. Optim. Theory Appl. 78 Ž1993., 613᎐622. 8. J. Jahn, ‘‘Mathematical Vector Optimization in Partially Ordered Linear Spaces,’’ Verlag Peter Lang, 1986. 9. D. T. Luc, Connectedness of the efficient point sets in quasiconcave vector maximization, J. Math. Anal. Appl. 122 Ž1987., 346᎐354. 10. D. T. Luc, ‘‘Theory of Vector Optimization,’’ Springer-Verlag, New YorkrBerlin, 1989. 11. D. T. Luc, Contractibility of efficient point sets in normed spaces, Nonlinear Anal. 15 Ž1990., 527᎐535. 12. P. H. Naccache, Connectedness of the set of nondominated outcomes in multicriteria optimization, J. Optim. Theory Appl. 25 Ž1978., 459᎐467. 13. J. W. Nieuwenhuis, Some results about nondominated solutions, J. Optim. Theory Appl. 36 Ž1982., 289᎐301. ´ 14. S. Schaible, Bicriteria quasiconcave programs, Cahiers Centre Etudes Rech. Oper. ´ 25 Ž1983., 93᎐101.
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15. E. U. Choo, S. Schaible, and K. P. Chew, Connectedness of the efficient set in ´ three-criteria quasiconcave programming, Cahiers Centre Etudes Rech. Oper. ´ 27 Ž1985., 213᎐220. 16. W. Song, A note on connectivity of efficient solution sets, Arch. Math. 65 Ž1995., 540᎐545. 17. W. Song, Connectivity of efficient solution sets in vector optimization of set-valued mappings, Optimization 39 Ž1997., 1᎐11. 18. W. Song, On the connectivity of efficient point sets, Appl. Math. ŽWarsaw. 25 Ž1998., 121᎐127. 19. E. J. Sun, On the connectedness of the efficient set for strictly quasiconvex vector minimization problems, J. Optim. Theory Appl. 89 Ž1996., 475᎐481. 20. A. R. Warburton, Quasiconcave vector maximization: Connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives, J. Optim. Theory Appl. 40 Ž1983., 537᎐557.
Arcwise Connectedness of Closed Efficient Point Sets E. K. Makarov 1 Institute of Mathematics, National Academy of Sciences of Belarus, Surgano¨ a 11, Minsk 220072, Belarus
N. N. Rachkovski Belorussian State Pedagogical Uni¨ ersity, So¨ yetskaya, 18, Minsk 220809, Belarus
and Wen Song Harbin Normal Uni¨ ersity, Harbin 150080, People’s Republic of China, and Systems Research Institute, Newelska 6, 01-447 Warsaw, Poland Submitted by George Leitmann Received February 16, 1999
1. INTRODUCTION One of the most important problems of vector optimization is to investigate the structure of efficient point sets. For various applications, the possibility of continuous moving from one optimal solution to any other along optimal alternatives only is of special interest. This possibility is guaranteed if the efficient set is arcwise connected or at least connected. It is well known that in a finite dimensional space the efficient point set of a closed convex cone-compact set with respect to an order defined by a closed convex cone is connected, as well as the efficient solution set for concave objective functions Žsee w1, 12, 13x.. Several authors proved analogous theorems for some subclasses of quasiconcave objective functions Žsee w3, 7, 9, 14, 15, 19, 20x.. Many of the above results were also 1
E-mail: [email protected]. 377 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
378
MAKAROV, RACHKOVSKI, AND SONG
generalized to the infinite dimensional space settings by Luc w10, 11x, Fu w4x, Gong w5, 6x, and Song w16᎐18x. However, the results concerning strong topological properties, e.g., contractibility or arcwise connectivity, are rather seldom. Contractibility of the efficient point set and arcwise connectivity of the weak efficient point set are studied in w11x under some stronger assumptions, and contractibility of the efficient solution set for a strongly quasiconcave function Žcalled strictly quasiconcave function in w9, 10x. is studied in w9, 10x. In this paper we prove that the efficient point set MaxŽ Q < K . of a compact convex set Q ; X in a Hausdorff topological vector space X ordered by a closed convex pointed cone K ; X is arcwise connected if the set MaxŽ Q < K . is closed.
2. PRELIMINARIES Let X be a Hausdorff topological vector space, Q ; X, and K ; X a closed convex pointed cone. We denote the interior, the closure, and the convex hull of an arbitrary set A ; X by int A, cl A, and conv A, respectively. A point x g Q is said to be an efficient point of Q with respect to K Ž x g MaxŽ Q < K .. if Ž Q y x . l K s 04 , and it is said to be the least element of Q with respect to K if Q ; x q K. Let F be a set-valued mapping from a topological space W to X. By dom F and graph F we denote the domain and graph of F, i.e., dom F s w g W < F Ž w . / ⭋ 4
and
graph F s Ž w, x . g W = X < x g F Ž w . 4 . Let P be a nonempty subset of X, x 1 , x 2 g Q, and 0 F 1 - 2 F 1. Consider the set-valued map F: w 1 , 2 x § Q given by F Ž t . s ŽŽŽ 2 y t .rŽ 2 y 1 .. x 1 q ŽŽ t y 1 .rŽ 2 y 1 .. x 2 q P . l Q. LEMMA 2.1. Let P be a con¨ ex cone and t 1 g Ž 1 , 2 ., z1 g F Ž t 1 .. Suppose that the set-¨ alued map G: w 1 , 2 x § Q is gi¨ en by
¡
GŽ t . s
~ž ¢ž
t1 y t t1 y 1
2 y t 2 y t1
x1 q z1 q
t y 1 t1 y 1 t y t1
2 y t1
Then GŽ t . ; F Ž t . for all t g w 1 , 2 x.
z1 q P l Q,
if t g w 1 , t 1 x ;
x 2 q P l Q,
if t g w t 1 , 2 x .
/ /
379
CONNECTEDNESS OF EFFICIENT POINT SETS
Proof. Since t 1 g w 1 , 2 x, there exists ␣ g w0, 1x such that t 1 s ␣ 1 q Ž1 y ␣ . 2 . Since z1 g F Ž t 1 ., we have z1 g Ž ␣ x 1 q Ž1 y ␣ . x 2 q P . l Q. Hence, for every t g w 1 , t 1 x, we have t1 y t
GŽ t . ; ; s
t1 y 1
ž ž
x1 q
t y 1 t1 y 1
t1 y t q ␣ Ž t y 1 . t1 y 1
2 y t 2 y 1
x1 q
Ž ␣ x1 q Ž 1 y ␣ . x 2 q P . q P
x1 q Ž 1 y ␣ .
t y 1
2 y 1
t y 1 t1 y 1
lQ
x2 q P l Q
/
x2 q P l Q s F Ž t . .
/
For t g w t 1 , 2 x, the inclusion GŽ t . ; F Ž t . can be proved similarly. LEMMA 2.2. closed set.
If P and Q are closed sets, then F is closed, i.e., graph F is a
Proof. Define a continuous mapping p: w 1 , 2 x = X ª ⺢ = X by pŽ t, y . s Ž t, y y ŽŽŽ 2 y t .rŽ 2 y 1 .. x 1 q ŽŽ t y 1 .rŽ 2 y 1 .. x 2 ... Then graph F s py1 Žw 1 , 2 x = P . l Žw 1 , 2 x = Q .. Since the sets w 1 , 2 x = P and w 1 , 2 x = Q are closed, graph F is closed. LEMMA 2.3. Let Q1 > Q2 > . . . be a sequence of compact subsets of X, and x k g Q k be the least element of Q k with respect to K, for each k g ⺞. Then the sequence x k 4k g ⺞ con¨ erges to the unique least element of Q0 [ l Q k : k g ⺞4 / ⭋ with respect to K. Proof. Since X is Hausdorff and each Q k is compact, Q k 4k g ⺞ is a decreasing sequence of nonempty closed sets in compact space Q1. By w2, Proposition I-11.4x, we have Q0 / ⭋ and by w2, Proposition I-11.5x, x k 4k g ⺞ has an adherent point x 0 g Q0 . Suppose that there exists z g Q0 such that z f x 0 q K. Since x 0 q K is closed, there exists a neighbourhood U of zero in X such that Ž z y U . l Ž x 0 q K . s ⭋. Hence, z f x 0 q U q K. Take k g ⺞ such that x k g x 0 q U. Then x k q K ; x 0 q U q K and, therefore, z f x k q K. On the other hand, we have z g Q0 ; Q k ; x k q K. By this contradiction, we assert that Q0 ; x 0 q K. Hence, x 0 is the least element of Q0 . Since K is pointed, the set Q0 has the unique least element x 0 . Therefore, x 0 is the unique adherent point of x k 4k g ⺞ . By w2, Proposition I-11.5x, we have x k ª x 0 . 3. MAIN RESULT Recall that a set A of a topological space is said to be arcwise connected if for every two points x, y g A there exists a continuous function : w0, 1x ª A such that Ž0. s x, Ž1. s y.
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THEOREM 3.1. Let K ; X be a closed con¨ ex pointed cone and Q ; X be compact and con¨ ex. If the set MaxŽ Q < K . is closed, then it is arcwise connected. Proof. Take any x 1 , x 2 g MaxŽ Q < K .. The set-valued map F0 : w0, 1x § Q given by F0 Ž t . s ŽŽ1 y t . x 1 q tx 2 q K . l Q has compact nonempty values for each t g w0, 1x. By w8, Theorem 6.3 and Lemma 6.2x, we have ⭋ / 1 MaxŽ F0 Ž t .< K . ; MaxŽ Q < K . for all t g w0, 1x. For each t m s mr2, m s 1 .< 1 Ž Ž . 0, 1, 2, we choose an arbitrary z m g Max F0 t m K . Since F0 Ž t 01 . s x 1 4 and F0 Ž t 21 . s x 2 4 , we have z 01 s x 1 and z 21 s x 2 . Now we can define the set-valued map F1 : w0, 1x § Q by setting F1Ž t . s ŽŽ1 y 2 t . z 01 q 2 tz11 q K . l Q for t g w0, 1r2x and F1Ž t . s ŽŽ2 y 2 t . z11 q Ž2 t y 1. z 21 q K . l Q for t g w1r2, 1x. By Lemma 2.1, we have F0 Ž t . > F1Ž t . for all t g w0, 1x. It can be easily seen that F1Ž t . is compact and nonempty for each t g w0, 1x, and ⭋ / MaxŽ F1Ž t .< K . ; MaxŽ Q < K .. Let us now iterate this process. Suppose that for some k g ⺞ we have already defined the set-valued mappings Fi : w0, 1x § Q, i s 0, 1, . . . , k by putting i i Fi Ž t . s 2 i Ž t mq1 y t . z mi q 2 i Ž t y t mi . z mq1 qK lQ i for t g t mi , t mq1 ,
where t mi s 2yi m, m s 0, 1, . . . , 2 i. We also suppose that F0 Ž t . > F1Ž t . > ⭈⭈⭈ > Fk Ž t . for all t g w0, 1x and Fi Ž t mi . s z mi 4 ; MaxŽ Q < K . for m s 0, 1, . . . , 2 i, i s 0, 1, . . . , k. It can be easily seen that Fi Ž t . is compact and nonempty for each t g w0, 1x, i s 0, . . . , k, and ⭋ / MaxŽ Fi Ž t .< K . ; MaxŽ Q < K .. Thus, for each t mkq 1 s 2yky1 m, m g 0, 1, . . . , 2 kq 1 4 , we can choose an arbitrary z mkq 1 g MaxŽ Fk Ž t mkq 1 .< K . ; MaxŽ Q < K .. Note that 1 k k z 2kq n s z n , n s 0, 1, . . . , 2 , by assumption. Now we can define the set-valued mapping Fkq 1 : w0, 1x § Q by putting kq1 kq1 Fkq 1 Ž t . s 2 kq 1 Ž t mq1 y t . z mkq1 q 2 kq1 Ž t y t mkq1 . z mq1 qK lQ kq1 for t g t mkq 1 , t mq1 , kq 1 . Ž kq 1 . Ž kq 1 where m s 0, 1, . . . , 2 kq 1 y 1. Since 2 kq1 Ž t mq 1 y t s t mq1 y t r t mq1 kq 1 . y t m , by Lemma 2.1, we obtain Fk Ž t . > Fkq1Ž t . for all t g w0, 1x. It can be easily seen that Fkq 1Ž t . is compact and nonempty for each t g w0, 1x and ⭋ / MaxŽ Fkq 1Ž t .< K . ; MaxŽ Q < K .. Obviously, Fkq1Ž t mkq 1 . s z mkq 1 4 ; MaxŽ Q < K . for m s 0, 1, . . . , 2 kq 1. Define the set-valued mapping F : w0, 1x § Q by
F Ž t . s l Fk Ž t . : k g ⺞ 4 .
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Note that for every t g w0, 1x and k g ⺞, Fk Ž t . is compact and nonempty. Moreover, we have Fk Ž t . > Fkq1Ž t .. Thus, we can assert w2, Proposition I-11.4x that F Ž t . s l Fk Ž t . : k g ⺞4 / ⭋ for all t g w0, 1x and, therefore, dom F s w0, 1x. Let T s j 2yk m : m s 0, 1, . . . , 2 k , k g ⺞4 ; w0, 1x. Obviously, cl T s w0, 1x. If t g T, then t s t mk for some m s 0, 1, . . . , 2 k , k g ⺞ and, hence, F0 Ž t . > F1Ž t . > ⭈⭈⭈ > Fk Ž t . s Fkq1Ž t . s ⭈⭈⭈ s F Ž t . s z mk 4 ; MaxŽ Q < K ., i.e., Ž t mk , z mk . g graph F. Now suppose that t g w0, 1x _ T. Then for each k g ⺞ we can find k k k x mŽ k . g 0, 1, . . . , 2 k y 14 such that t g w t mŽ k . , t mŽ k .q1 . Clearly, t mŽ k . ª t k and t mŽ k .q1 ª t as k ª q⬁. By the definition of Fk , we have Fk Ž t . s k kŽ k Ž2 k Ž t mŽ . k . k . t y t mŽ k s k .q 1 y t z mŽ k . q 2 k . z mŽ k .q 1 q K l Q. Let kŽ k k k 2 t mŽ k .q1 y t . g w0, 1x and x k s  k z mŽ k . q Ž1 y  k . z mŽ . Then Fk Ž t . k .q1 s Ž x k q K . l Q. Since x k is the least element of Fk Ž t . with respect to K for each k g ⺞, by Lemma 2.3, we obtain x k ª x 0 g l Fk Ž t . : k g ⺞4 s F Ž t . / ⭋ and F Ž t . ; x 0 q K. By compactness, there exist some subnets  kŽ .4 g M of  k 4k g ⺞ conkŽ . 4 k 4 verging to  0 g w0, 1x, z mŽ kŽ .. g M of z mŽ k . k g ⺞ converging to ¨ g Q, kŽ . k and z mŽ kŽ ..q 14 g M of z mŽ k .q14k g ⺞ converging to w g Q. Let us denote kŽ . kŽ . ␣ [  kŽ . , ¨ [ z mŽ kŽ .. , and w [ z mŽ kŽ ..q 1 , g M. Hence, x kŽ . s ␣¨ q Ž1 y ␣ . w converges to x 0 s  0¨ q Ž1 y  0 . w g Q. By Lemma 2.2, for every k g ⺞, graph Fk is closed. Hence, graph F s kŽ . kŽ . . Ž t mŽ . l graph Fk : k g ⺞4 is closed. Since Ž t mŽ kŽ .. , ¨ , kŽ ..q 1 , w g graph F for any g M, we have Ž t, ¨ . g graph F and Ž t, w . g graph F. Hence, ¨ , w g x 0 q K. Since x 0 s  0¨ q Ž1 y  0 . w and K is pointed, we assert that x 0 g ¨ , w4 . Note that ¨ , w4 g cl MaxŽ Q < K . s MaxŽ Q < K .. Therefore, x 0 4 s Q l Ž x 0 q K . > Q l F Ž t . s F Ž t .. Thus, F is a single-valued Žpoint-valued. map. By Lemma 2.2, graph F is closed. Since graph F ; w0, 1x = Q and Q is compact, we assert that graph F is compact as a closed subset of a compact set. Thus, F is continuous. The proof is completed. The assumption of convexity on Q in Theorem 3.1 is essential. EXAMPLE 3.1. Suppose that X s ⺢ 2 , Q s Ž x, y . g ⺢ 2 < x s 0, y g 2 w0, 1x4 j Ž x, y . g ⺢ 2 < x g w0, 1x, y s 04 and K s ⺢q [ Ž x, y . g ⺢ 2 < x G 4 0, y G 0 . Clearly, Q is compact and not convex and MaxŽ Q < K . s Ž1, 0., Ž0, 1.4 is closed, but not arcwise connected. Let now X be a locally convex space. Then we can use the following sufficient conditions for the set MaxŽ Q < K . to be closed given by Fu w4x. Let A be a subset of X. A point x g A is said to be an F-point if for every open set U with x g ŽU l A. y K, there exists a neighborhood W of
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x such that W l A ; Ž U l A. y K . The set A is said to be an F-set if every point of A is an F-point. THEOREM 3.2 w4x. Let Q be a subset of a locally con¨ ex space X and K be a closed con¨ ex cone such that there exists a closed con¨ ex cone S with K _ 04 ; int S and 0 f int S. If Q is a compact F-set, then MaxŽ Q < K . is closed. EXAMPLE 3.2. Let X s ⺢ 3 , Q s convŽŽ x, y, z . g ⺢ 3 < x 2 q y 2 F 1, z s 04 j Ž1, 0, 1.4., K s Ž x, y, z . g ⺢ 3 < x s y s 0, z G 04 . Clearly, MaxŽ Q < K . is arcwise connected, but not closed. From Example 3.2, we can see that the assumption that MaxŽ Q < K . is closed is not essential for MaxŽ Q < K . to be arcwise connected. So it is natural to ask if the conclusion of Theorem 3.1 is true without the assumption that MaxŽ Q < K . is closed.
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