2011 Ð 5 ACTA MATHEMATICAE APPLICATAE SINICA May, ( MR(2000) ß Â 49J20; 47H10; 91A10

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1 À 34 À 3 Ù Ú ß Vol. 34 No Ð 5 ACTA MATHEMATICAE APPLICATAE SINICA May, 2011 Á É ÔÅ Ky Fan Ë ÍÒ ÇÙÚ ( ¾±» À ¾ ) (Ø À Ø ) ( dingtaopeng@126.com) Ü Ö Ë»«Æ Đ ĐÄ Ï Þ Å Ky Fan Â Ï Ò¹Ë Þ Å Ä Ï ¾Ïº»«Æ Ky Fan Ï Ò¹ Ç Ì ÙÞ (1) Ky Fan Ç Fan-Browder ÊÂÇ Ö (2) ÙÞ ÐÇ Nash Ö ÏÇ ÃÎ Ky Fan Ï Ç ÊÂÇ Nash Ö MR(2000) ß Â 49J20; 47H10; 91A10 Þ Â O177.9; O178; O225 1 ØÕ 1995 [1] ÆÕÊ Ky Fan Á Ò X Å φ : X X R y X, φ(y, y) 0, ß x X, y X, φ(x, y) 0. x µ Ý φ Ky Fan Á ²Òµ Ky Fan Á Ö Ky Fan Æ¹ Ñ Ê x Î Æ [2], Ú Ñµ Ky Fan ½ Ky Fan Ê Ì ß ÀÎ ½ Ò Ê µ ØÝ [3,4]. Ò Ky Fan [1 4] : 1.1 X Hausdorff ÀÎ Ø E ½ºÅ φ : X X R (1) Í y X, x φ(x, y) Ã «2010 Ñ Ñ 4 12 Ù 973 (2010CB732501) ÐØ ¹ ( ) ¼¹Ã

2 3 Ù ÓÇµ»«Æ Ä Þ Ky Fan ÆÛ Ì ÙÞ 527 (2) Í x X, y φ(x, y) Æ«(3) Í y X, φ(y, y) 0, x X, y X, φ(x, y) 0. ¼ ÕßØÝ µ Ú Ê ± Ú Á [5] ¼ÊÝ ½Î [6] Ý ÆÝ Î¹ ÐÄ ºÎ [5,7] Ý ÅÎ Ã Î [8] Ê ÄÈÊ Ky Fan [4,9] Í Ky Fan ÚØÝÅ ÐÄ ºÎ¼ Ý Ã ÎÒ ½Î ß Ý Ky Fan [10]. 1.2 (Ky Fan ) X Hausdorff ÀÎ Ø E ½Å Å Ü F : X 2 X x X, F(x) E Å Þ x 0 F(x 0 ) ºÅ ÚÍ X ½Å {x 1, x 2,, x n }, CO{x 1, x 2,, x n } n F(x i ), F(x) Ø. 2 Ky Fan ÛÓ x X Ý Ã Îß ºÎ¹ Ky Fan Á Î 2.1 X Hausdorff ÀÎ Ø E ½½Å Ý φ : X X R (1) Í y X, φ(y, y) 0; (2) Í x X, {y X : φ(x, y) > 0} ½Å«(3) y 0 X, cl X {x X : φ(x, y 0 ) 0} ºÅ Ú cl X B Å B X x X Î Í Ô y X, Å½ {x α } α I X, x α x, φ(x α, y) 0 Í² α I Â ( Í Ô y X, Ò x ÔÍ N(x ), x α N(x ), φ(x α, y) 0). ÆÕÅ Ü F : X 2 X F(y) = {x X : φ(x, y) 0}, y X. Ð¹ (1) Í y X, F(y) F KKM Ü Í ½Å {y 1, y 2,, y n } X, CO {y 1, y 2,, y n } Đ ½Å {y 1, y 2,,y n } X, α i 0, n F(y i ). (2.1) n α i = 1, x 0 = n α i y i n F(y i ). ÐÜ F ÆÕ φ(x 0, y i ) > 0 Í i = 1, 2,,n Â y i {y X : φ(x 0, y) > 0}. Ð¹ (2), Í x 0 X, Å {y X : φ(x 0, y) > 0} ½ ²Ò

3 528 Ù Þ 34 x 0 = n α i y i {y X : φ(x 0, y) > 0}, φ(x 0, x 0 ) > 0, ¹ (1) Î Ö¼ F KKM Ü «Å Ü F : X 2 X F(y) = cl X (F(y)), y X. Đ Í y X, F(y) Å ¹ (3) F(y0 ) º Ð F(y) F(y) Ò (2.1) Í ½Å {y 1, y 2,, y n } X, CO {y 1, y 2,,y n } n F(y i ). ½ F : X 2 X Ky Fan ( 1.2) ±¹ Ö¼ F(y) Ø. y X x F(y), x F(y) = cl X (F(y)) Í y X Â Í y X, y X Å½ {x α } α I F(y), x α x. Ð x α F(y), Â φ(x α, y) 0 Í² α I Â Ky Fan ÁÂ Æ 2.1 Á x µ Ky Fan Á ØÝÆ 2.1, Ky Fan Á Î 2.2 X Hausdorff ÀÎ Ø E ½½Å Æ Ý φ, ψ : X X R (1) Í y X, ψ(y, y) 0; (2) Í x X, {y X : ψ(x, y) > 0} ½Å«(3) y 0 X, cl X {x X : ψ(x, y 0 ) 0} º«(4) Í {y X : φ(x, y) > 0} Ø x X, y X, x int X {x X : ψ(x, y ) > 0}, Ú int X B Å B X ÁÅ x X, y X, φ(x, y) 0. Ð¹ (1) (3) Ý ψ : X X R Æ 2.1 ±¹ Ö¼Ý ψ Ky Fan Á x X. ¼ x Ý φ Ky Fan Á Đ y 0 X, φ(x, y 0 ) > 0, {y X : φ(x, y) > 0} Ø. Ð¹ (4), Í x X, y 0 X x int X {x X : ψ(x, y 0 ) > 0}, x Í N(x ), ψ(x, y 0 ) > 0 Í x N(x ) Â Î x Ý ψ Ky Fan Á ½ Í y 0 ß x Í N(x ), x α N(x ), ψ(x α, y 0 ) 0. ψ(x, y 0 ) > 0 Í x N(x ) Â Î 2.1 Æ 2.2 Ý φ ² ¹»Ð Ý ψ ½ ÊÍÝ φ ¾ ß 2.2 Ó Æ 2.2 Ò [4] Æ 3.3.2, Æ 3.3.6, Æ 3.3.7, [5] Æ 2, [6] Æ 3.2, [7] Æ 2.1 Ú Á 2.3 X Hausdorff ÀÎ Ø E ½½Å Ý φ : X X R (1) Í y X, φ(y, y) 0; (2) Í x X, {y X : φ(x, y) > 0} ½Å«

4 3 Ù ÓÇµ»«Æ Ä Þ Ky Fan ÆÛ Ì ÙÞ 529 (3) y 0 X cl X {x X : φ(x, y 0 ) 0} º«(4) Í {y X : φ(x, y) > 0} Ø x X, y X, x int X {x X : φ(x, y ) > 0}, x X, y X, φ(x, y) 0. Æ 2.2 ψ = φ 2.3 y φ(x, y) ½ Æ 2.3 ¹ (2) x φ(x, y) X Ã Æ 2.3 ¹ (4) ²ÒÆ 2.1, Æ 2.2, Æ 2.3 ½Ò Õ Ê Ky Fan ( 1.1): (1) ºÎ«(2) ÊÝ ½Î¹ «(3) ÊÝ Ã Î ß 2.4 X Hausdorff ÀÎ Ø E ½º½Å Ý φ : X X R (1) Í y X, φ(y, y) 0; (2) Í x X, {y X : φ(x, y) > 0} ½Å«(3) Í {y X : φ(x, y) > 0} Ø x X, y X, x int X {x X : φ(x, y ) > 0}, x X, y X, φ(x, y) 0. Ö X E º½Å Æ 2.3 ¹ (3) ¾Đ ²Ò Æ Â ÐÆ 2.2 [9] Æ [9] X Hausdorff ÀÎ Ø E ½½Å Æ Ý φ, ψ : X X R (1) φ(x, y) ψ(x, y), x, y X, Þ ψ(y, y) 0, y X; (2) Í y X, x φ(x, y) Ã «(3) Í x X, {y X : ψ(x, y) > 0} ½«(4) ºÅ K X ß y 0 K, x X\K, φ(x, y 0 ) > 0, x K, y X, φ(x, y) 0. Æ 2.2 ¹ (1), (2), (4) Đ ³¹ (3) Ð ¹ (1) ß (4), Å {x X : ψ(x, y 0 ) 0} {x X : φ(x, y 0 ) 0} K. K X ºÅ Ô cl X {x X : ψ(x, y 0 ) 0} K ºÅ ½ X ºÅ Ó Æ 2.2, x X, y X, φ(x, y) 0. x X\K φ(x, y 0 ) > 0, ²Ò x K. 2.5 X Hausdorff ÀÎ Ø E ½Å Ý φ : X X R (1) Í y X, φ(y, y) 0; (2) Í x X, {y X : φ(x, y) > 0} ½Å«(3) X Ì ½ºÅ {X n } n=1, X 1 X 2 X 3, X = n=1 X n, ÞÍ X {x n } n=1, x n X n, Þ n, x m X n, n 0 y n0 X n0, φ(x n0, y n0 ) > 0; (4) n = 1, 2,, Í {y X n : φ(x, y) > 0} Ø x X n, y n X n, x int Xn {x X n : φ(x, y ) > 0}.

5 530 Ù Þ 34 x X, y X, φ(x, y) 0. n = 1, 2,,X n ½ºÅ x X n, {y X n : φ(x, y) > 0} = X n {y X : φ(x, y) > 0} ½Å «Ð (1), (4), φ X n X n Æ 2.4 ±¹ Ö ¼ x n X n, y X n, φ(x n, y) 0. Ì N, {x n } n=1 X N, n, x m X n. Ó (3), n 0 y n0 X n0, φ(x n0, y n0 ) > 0, y X n, φ(x n, y) 0 Î Ö¼ N, {x n } n=1 X N. Ö X N ºÅ x n x X N X(n ). Ò x φ Ky Fan Á y 0 X, φ(x, y 0 ) > 0. Ö X 1 X 2 X 3, X = n=1 X n, Ô M N, y 0 X M, Þ n M, y 0 X n. Ö φ(x, y 0 ) > 0, Ô n M, {y X n : φ(x, y) > 0} Ø, Ð (4), y n X n x X n Í U(x ), x U(x ), φ(x, y n) > 0. Ö x n x (n ), Ô n ( n M) x n U(x ), ½ φ(x n, y n ) > 0, y X n, φ(x n, y) 0 Î Ô x φ Ky Fan Á 2.4 Æ 2.5 [11] Æ 2.1 Õ 3 Ì Ê Ky Fan Ï Æ Fan-Browder À Ä µ Ñ Æ 2.2 Æ³ Ë ØÝ Ñ Ky Fan ³ Æ ß Fan- Browder ÉÁÆ Õ 3.1 X Hausdorff ÀÎ Ø E ½½Å A, B X X (1) Í y X, (y, y) B; (2) Í x X, Å {y X : (x, y) B} ½«(3) y 0 X, cl X {x X : (x, y 0 ) B} º«(4) Í {y X : (x, y) A} Ø x X, y X x int X {x X : (x, y ) B}, x X, {x } X A. 3.1 Æ 2.2 Æ 3.1. ÆÕÆ Ý φ, ψ : X X R φ(x, y) = { 0, (x, y) A, 1, (x, y) A, ψ(x, y) = { 0, (x, y) B, 1, (x, y) B, x, y X. (a) Í y X, Ð¹ (1), ψ(y, y) = 0; (b) Í x X, Ð¹ (2), Å {y X : ψ(x, y) > 0} = {y X : ψ(x, y) = 1} = {y X : (x, y) B} ½«

6 3 Ù ÓÇµ»«Æ Ä Þ Ky Fan ÆÛ Ì ÙÞ 531 (c) Ó ¹ (3), Á y 0, Å cl X {x X : ψ(x, y 0 ) 0} = cl X {x X : ψ(x, y 0 ) = 0} = cl X {x X : (x, y 0 ) B} ºÅ«(d) x X {y X : φ(x, y) > 0} Ø, {y X : (x, y) A} Ø. Ð¹ (4), y X, x int X {x X : (x, y ) B}. {x X : (x, y ) B} = {x X : ψ(x, y ) = 1} = {x X : ψ(x, y ) > 0}, ²Ò x int X {x X : ψ(x, y ) > 0}. ¼Ý φ, ψ Æ 2.2 ±¹ ½ x X, y X, φ(x, y) 0, φ(x, y) = 0 Í² y X Â {x } X A. 3.2 X Hausdorff ÀÎ Ø E ½½Å Æ Å Ü M, N : X 2 X (1) Í y X, y N(y); (2) Í x X, N(x) ½«(3) y 0 X, cl X (X\N 1 (y 0 )) º«(4) Í M(x) Ø x X, y X x int X N 1 (y ), x X, M(x ) = Ø. 3.2 Æ 3.1 Æ 3.2. Å A = {(x, y) X X : y M(x)}, B = {(x, y) X X : y N(x)}. (a) ¹ (1) Í y X, (y, y) B; (b) Ð¹ (2), Í x X, Å {y X : (x, y) B} = {y X : y N(x)} = N(x) ½«(c) Ó ¹ (3), y 0» Å cl X {x X : (x, y 0 ) B} = cl X {x X : y 0 N(x)} = cl X (X\N 1 (y 0 )) º«(d) x X {y X : (x, y) A} Ø, A ÆÕ ¼ M(x) Ø. Ó ¹ (4), y X, x int X N 1 (y ). «B ÆÕ x int X {x X : (x, y ) B}. Ó Æ 3.1, x X, {x } X A. A ÆÕ ¼ y M(x ) Í² y X Â ²Ò M(x ) = Ø. 3.3 Æ 3.2 Æ 2.2. ÆÕÆ Å Ü M, N : X 2 X M(x) = {y X : φ(x, y) > 0}, N(x) = {y X : ψ(x, y) > 0}, x X, (a) Í y X, Ð¹ (1), ψ(y, y) 0, ÆÕ y N(y);

7 532 Ù Þ 34 º«(b) Í x X, Ð¹ (2), N(x) = {y X : ψ(x, y) > 0} ½«(c) Ó ¹ (3), y 0» Å cl X (X\N 1 (y 0 )) = cl X {x X : y 0 N(x)} = cl X {x X : ψ(x, y 0 ) 0} (d) x X M(x) Ø, {y X : φ(x, y) > 0} = M(x) Ø. Ó ¹ (4), y X, x int X {x X : ψ(x, y ) > 0}. ºÔ {x X : ψ(x, y ) > 0} = {x X : y N(x )} = N 1 (y ), ²Ò x int X N 1 (y ). ¼Æ 3.2 ¹ ± ²Ò x X, M(x ) = Ø, y M(x ) Í² y X Â ½ φ(x, y) 0 Í² y X Â ² Æ 2.2 Æ 3.1 Æ 3.2 Æ 2.2, ²ÒÆ 2.2, Æ 3.1, Æ 3.2 «3.3 X Hausdorff ÀÎ Ø E ½½Å A X X (1) Í y X, (y, y) A; (2) Í x X, Å {y X : (x, y) A} ½«(3) y 0 X, cl X {x X : (x, y 0 ) A} º«(4) Í {y X : (x, y) A} Ø x X, y X x int X {x X : (x, y ) A}, x X, {x } X A. Æ 3.1 A = B 3.4 X Hausdorff ÀÎ Ø E ½½Å A X X (1) Í y X, (y, y) A; (2) Í x X, Å {y X : (x, y) A} ½«(3) y 0 X, {x X : (x, y 0 ) A} º«(4) Í y X, Å {x X : (x, y) A} Å Å x X, {x } X A. Æ ¹ ÅÜÊÆ 3.3 ¹ ²Ò Â 3.1 Ky Fan ³ Æ [10] Â Æ 3.1, Æ 3.3 ßÆ 3.4 Ë ºÎ 3.5 X Hausdorff ÀÎ Ø E ½½Å Å Ü N : X 2 X (1) Í x X, N(x) ½«(2) y 0 X, cl X (X\N 1 (y 0 )) º«(3) Í x X, y X x int X N 1 (y ), x X, x N(x ). Æ Â Í y Y, y N(y). Å Ü M : X 2 X, Í x X, M(x) Ø (Á M(x) X, x X), Ð¹ (3), y X x int X N 1 (y ). «Ð¹ (1)(2) Æ 3.2 ¹ ± x X,

8 3 Ù ÓÇµ»«Æ Ä Þ Ky Fan ÆÛ Ì ÙÞ 533 M(x ) = Ø, M : X 2 X Å Ü Î ²Ò Â x X, x N(x ). 3.6 X Hausdorff ÀÎ Ø E ½½Å Å Ü N : X 2 X (1) Í x X, N(x) ½«(2) y 0 X, X\N 1 (y 0 ) º«(3) Í y X, N 1 (y) Å Å x X, x N(x ). Æ ¹ ÅÜÊÆ 3.5 ¹ ²Ò Â 3.7 X Hausdorff ÀÎ Ø E ½ºÅ Å Ü N : X 2 X (1) Í x X, N(x) ½Å«(2) Í x X, y X, x int X N 1 (y ), x X, x N(x ). Ö X º½Å Æ 3.5 ¹ (2) Đ «Ð¹ (1)(2) Æ 3.5 ¹ ± 3.2 Æ 3.5, Æ 3.6, Æ 3.7 Ò Á (1) ºÎ«(2) ßÅ Ü ( Ü ) ÁÁ ( ) «(3) ßÅ Ü ÁÁ «(4) Å Ü ( ) Ã Î ½ ÕÊ Fan-Browder ÉÁÆ [12]. 3.3 [8] ÛÆ 3.7, ØÝ³ Ñ Ê Tarafdar É ÁÆ [13] ß [9] Æ Nash È ÛÓ I = {1, 2,, n} Å i I, X i i ³ Å X = n X i. Í i I, Xî = X j, f i : X R i Ý Ú³ j I\i x = (x 1, x 2,,x n ) X i I, f(x i, x î ) = max u i X i f(u i, x î ), x µ ¼ n Nash Õ Á Ú x î = (x 1,, x i 1, x i+1,,x n ) X î, x = (x i, x î ). 4.1 i I, X i Hausdorff ÀÎ Ø E i ½½Å f i : X R (1) x X, { n y X : f i (y i, xî) > r(x) } ½Å Ú r(x) = n f i (x); { n (2) y 0 X, cl X x X : f i (yi 0, x ) n f î i (x i, xî) } º«(3) Í { y X : x int X { x X : n f i (y i, xî) > n n f i (y i, x ) > n f i (x î i, x )}. î ¼ Nash Õ Á f i (x i, xî) } Ø x X, y X

9 534 Ù Þ 34 ÆÕÝ φ : X X R φ(x, y) = n [f i (y i, xî) f i (x i, xî)], x = (x 1, x 2, x n ), y = (y 1, y 2,, y n ) X. Ó y X, φ(y, y) = 0; x X, {y X : φ(x, y) > 0} ½Å«y 0» Å cl X {x X : φ(x, y 0 ) 0} º«Í {y Y : φ(x, y) > 0} Ø x X, y Y, x int X {x X : φ(x, y ) > 0}. Ó Æ 2.3, x X, y X, φ(x, y) 0, n [f i (y i, x ) n f i (x î i, x )] 0 Í y = (y 1, y 2,, y n ) î Â i I, u i X i, Ï y = (u i, x ), î φ(x, y) = f i (u i, x ) f i(x î i, x ) 0, î f(x i, x ) = max f(u i, x ), i I. ²Ò î u i X i î x ¼ Nash Õ Á 4.2 i I, X i Hausdorff ÀÎ Ø E i ½½Å f i : X R (1) x X, y n f i (y i, xî) X Æ«{ n (2) y 0 X, cl X x X : f i (yi 0, x ) n f î i (x i, xî) } º«(3) n f i X Ã «(4) y X, x n f i (y i, xî) X Ã ¼ Nash Õ Á Ó Æ ¹ (1) ÅÜÆ 4.1 ¹ (1); Æ ¹ (3), (4) ÅÜÆ 4.1 ¹ (3), Ö¼ Æ Â ÐÆ 4.2 ² [4] Æ 4.2.4, ³ [14] Æ 14 ¼ ¼ 4.1 i I, X i Hausdorff ÀÎ Ø E i ½º½Å, f i : X R x X, y n f i (y i, xî) X Æ«n f i X Ã «y X, x n f i (y i, xî) X Ã ¼ Nash Õ Á 4.1 (² [4, 14]) X ß Y Hausdorff ÀÎ Ø Ý f : X Y R Ã Å Ü G : Y 2 X Ã Þ º Ý g(y) = max f(x, y) Y x G(y) Ã 4.3 i I, X i Hausdorff ÀÎ Ø E i ½ºÅ f i : X R (1) f i X Ã «(2) xî Xî, k = 1, 2,, {y i X i : f i (y i, xî) > M k (xî)} ½Å Ú M k (xî) = max u i X i f i (u i, xî) 1/k; (3) x X, y X, x int X {x X : f i (y i, x î ) > M k(x î )}. ¼ Nash Õ Á k = 1, 2,, ÆÕÅ Ü Ì N k : X 2 X

10 3 Ù ÓÇµ»«Æ Ä Þ Ky Fan ÆÛ Ì ÙÞ 535 N k (x) = n { } y i X i : f i (y i, xî) > max f i (u i, xî) 1/k, u i X i Ð (1), x X, M k (xî) ÔÕ«Ð (2), x X, N k (x) ½Å«ºÔ y X, Ð (3), x X, y x k N k (x k ), Ö X = n n N 1 k (y) = { } x X : f i (y i, xî) > max f i (u i, xî) 1/k, u i X i i I, X, x int X N 1 k (y ). Ó Æ 3.7, x k X, X i ºÅ x k x. f i (x k i, x k î ) > max u i X i f i (u i, x k î ) 1/k. Ö f i Ã Ð 4.1, xî max f i (u i, xî) Ã ²Ò f i (x i, u x ) max f i (u i, x ). i X i î u i X i î x i X i, ²Ò f i (x i, x ) = î max f i (x i, x ), x i X i î x ¼ Nash Õ Á ÐÆ 4.3 ² [4] Æ [14] Æ 16, ³ [15] Æ 2.1 ¼ 4.2 i I, Xi Hausdorff ÀÎ Ø E i ½ºÅ f i : X R f i X Ã «xî Xî, y i f i (y i, xî) X i Æ«x i X i, xî f i (y i, xî) Xî Ã ¼ Nash Õ Á ÝÑ ²É Æ É Ê Ñ Ê Ï Ô Ð [1] Tan K K, Yu J, Yuan X Z. The Stability of Ky Fan s Points. Proc. Amer. Math. Soc., 1995, 123: [2] Fan Ky. A Minmax Inequality and Its Applications. In: Inequalities III, edited by O. Shisha. New York: Academic Press, 1972 [3] Yuan X Z. Knaster-Kuratowski-Mazurkiewicz Theorem. Ky Fan Minimax Inequalities and Fixed Point Theorems. Nonlin. World, 1995, 2: [4] Á ¾Å º 2008 (Yu J. Game Theory and Nonlinear Analysis. Beijing: Science Press, 2008) [5] Bianchi M, Schaible S. Equilibrium Problems under Generalized Convexity and Generalized Monotoncity. J. Global Optim., 2004, 30: [6] Bianchi M, Pini R. Coercivity for Equilibrium Problems. J. Optim. Theory Appl., 2005, 124(1): [7] Fakhar M, Zafarani J. Equilibrium Problems in The Quasimonotone Case. J. Optim. Theory App1., 2005, 126(1):

11 536 Ù Þ 34 [8] ÔÈ ÅÉ È Æ ÈÀ Ü Í ÐÛ Đ Ð, 2009, 52(3): (Peng D T. New Existence Theorem for Vector Equilibrium Problem and Its Equivalent Version with Applications. Acta Math. Sinica, 2009, 52(3): [9] ¾ Û ¾Å º 2004 (Zhang C J. Set-valued Analysis and It s Applications in Economics. Beijing: Science Press, 2004) [10] Fan Ky. A Generalization of Tychonoff s Fixed-point Theorem. Math. Ann., 1961, 142: [11] Banach ± Ky Fan Ã Æ Đ Ð, 2008, 31(1): (Yu J. The Existence of Ky Fan s Points over Reflexive Banach Spaces. Acta Math. Appl. Sinica, 2008, 31(1): ) [12] Browder F E. The Fixed Point Theory of Multi-valued Mappings in Topological Vectoe Spaces. Math. Ann., 1968, 177: [13] Tarafdar E. Five Equivalent Theorems on a Convex Subset of a Topological Vector Space. Comment. Math. Univ. Carolinae, 1989, 30(2): [14] Nash Æ Ð È Đ, 2002, 22(3): (Yu J. The Existence and Stability of Nash Equilibrium. J. Sys. Sci. & Math. Scis., 2002, 22(3): ) [15] Tan K K, Yu J. Existence Theorem of Nash Equilibria for Non-cooperative N-person Games, Int. J. Game Theory, 1995, 24: Ky Fan s Inequalities for Discontinuous Functions on Non-compact Set and Its Equivalent Version with Their Applications PENG Dingtao (School of Science, Beijing Jiaotong University, Beijing ) (School of Science, Guizhou University, Guizhou, Guiyang ) ( dingtaopeng@126.com) Abstract The existence of weakly Ky Fan s point for the functions with no continuity on the non-compact set is proved. Based on this result, the Ky Fan s inequality is generalized to the functions with weak continuity, weak convexity and without compactness of the set. Author also give two equivalent versions for the result. As applications, (1) Ky Fan s section theorem and Fan-Browders s fixed point theorem are generalized; (2) some new existence theorems of Nash equilibria for n-person non-cooperative games are proved. Key words Ky Fan s inequality; existence; section theorem; fixed point theorem; Nash equilibrium MR(2000) Subject Classification 49J20; 47H10; 91A10 Chinese Library Classification O177.9; O178; O225