an application of fixed point theorem for s-convex function

D.B.Ojha et. al. / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3571-3575

AN APPLICATION OF FIXED POINT THEOREM FOR S-CONVEX FUNCTION D.B.OJHA Department of Mathematics, R.K.G.I.T.U.P.T.U., Delhi-Meerut Road, Ghaziabad, U.P./201003, INDIA

MANISH KUMAR MISHRA Research Scholar Singhania University)

Department of mathematics R.K.G.I.T. U.P.T.U Delhi-Meerut road Ghaziabad, U.P./201003, INDIA Abstract : In this paper, we showed an application of fixed point theorem in game theory with s-convex function. . Keywords: s-convex functions, fixed point. 1. Introduction Kakutani 1941[4] proved an extension of Brouwer’s fixed point theorem to upper semi-continuous set-valued mappings defined on compact convex subsets of  , which was extended to Banach spaces by H. F. Bohnenblust and S. Karlin 1950[1], and to locally convex spaces by Glicksberg 1952[3]. Nikaido 1968[6] gave a new proof of n

Kakutani’s theorem (in the case  ) based on the method of Schauder’s mappings. This proof is extended to Banach spaces in L. V. Kantorovich and G. P. Akilov 1964[5]. n

A locally convex space is a topological vector space

 X ,  admitting a neighbourhood

basis at 0 formed by

convex sets. It follows that every point in X admits a neighbourhood basis formed of convex sets and there is a neighbourhood basis at 0 formed by open convex symmetric sets. Let P be a family of semi norms on a vector space X and let F  P  :  f  P : f nonempty and finite . For f  f  P  and r  0, let

w'f  x, r    x '  X : p  f , p  x ' x   r

w f  x, r    x '  X : p  f , p  x ' x   r

1.1

If f   p , then we use the notation w p  x, r  and w p  x, r  to designate the open, respectively closed, p-ball. The family of sets

W '  x   w'f  x, r  : f  F  P  and r  0

forms a neighborhood basis of a locally convex topology  The family of sets

1.2

p on X.

W  x   w f  x, r  : f  F  P  and r  0 1.3 is also a neighbourhood basis at x for  p . If B is a convex symmetric absorbing subset of a vector space X, then the Minkowski functional pw : X   0,   defined by pw  x   inf   0 : x   w , x  X

1.4

is a semi norm on X and

 x  X : p  x   1   x  X : p  x   1. w

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w

1.5

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D.B.Ojha et. al. / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3571-3575 If X is a topological vector space and ‘w’ is an open convex symmetric neighbourhood of 0, then the semi norm pw is continuous,

w   x  X : pw  x   1 , cl w   x  X : pw  x   1.

If is a neighbourhood basis at 0 of a locally convex space

 X ,  ,

1.6 formed by open convex symmetric

neighbourhoods of 0, then P   pw : w  W  is a directed family of semi norms generating the topology



in

the way described above. Therefore, there are two equivalent ways of defining a locally convex space—as a topological vector space

 X ,  such that 0 admits a neighbourhood basis formed by convex sets, or as a pair

 X , P  where P is a family of semi norms

on X generating a locally convex topology on X. We consider only real

vector spaces. A directed set is a partially ordered set

 I , 

such that for every i1 , i2  I there exists i  I with i  i1 , and

i  i2 . A net in a set Z is a mapping  : I  Z . . If  J ,   is another directed set and there exists a non-decreasing mapping

 : I  Z . such that for every i  I there exists j  J with   j   i , then we say that   : J  Z

is a subnet of the net

 . One uses also the notation  zi : i  I  , where zi    i  ,

to designate the net



and

 z   : j  J  for a subnet. It is known that a subset K of a topological space T is compact if and only if every net  j

in K admits a subnet converging to an element of K (see R. Engelking 1989)[2]. If W  x  is a neighbourhood basis of a point x of a topological space

 X ,  then it becomes a directed set with respect to the order w1  w2  w2  w1. . If xB  X , B  W , then  xw : w  W  x   , is a net in X. We denote by V  x  the family of all neighbourhoods of a point x  X , and cl  Z  by the closure of a subset Z of X . It is a well known fact that convexity and its generalization plays important role in different part of mathematics, mainly in optimization theory. In our paper we deal with a common generalization of s-convexity, approximate convexity, and results of Bernstein and Doetsch 1915[7]. The concept of s-convexity and rational s-convexity was introduced by Breckner 1978[8]. In 1978 Breckner[8] and H. Hudzik and L. Maligranda 1994[9] it was proved that s-convex functions are nonnegative, when 0  s  1 , moreover the set of s-convex functions increases as s decreases. In the paper 1994 H. Hudzik and L. Maligranda[9] discussed a few results connecting with s-convex functions in second sense and some new results about Hadamard’s inequality for s-convex functions are discussed in (M. Alomari and M. Darus 1977-1989[10], 2008[11], U. S. Kirmaci 2007[13]), In 1999, S. S. Dragomir[12] et al. proved a variant of Hermite-Hadamard’s inequality for s-convex functions in second sense

2. Preliminaries:

A game is a triple U , V , W  where U,V are nonempty sets, whose elements are called strategies, and

W : U  V   is the gain function. There are two players, C and D , and W  x, y  represents the gain of

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D.B.Ojha et. al. / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3571-3575 the player  when he chooses the strategy x  U and the player D chooses the strategy y  V . The quantity

W  x, y  represents the gain of the player D in the same situation. The target of the player C is to maximize

D chooses a strategy that is the worst for α, that is, to choose x0  U such that

his gain when the player

inf W  x0 , y   max inf W  x, y  yV

xU

2.1

yV

Similarly, the player D chooses y0  V such that

sup W  x, y0   min sup W  x, y  yV

xU

2.2

xU

it follows

sup inf W  x, y   inf W  x0 , y   W  x0 , y0   sup W  x, y0   inf sup W  x, y  .

2.3

note that in general sup inf W  x, y   inf sup W  x, y 

2.4

xU yV

yV

xU

xU yV

yV xU

yV xU

If the equality holds in 2.4 , then, by 2.2

sup inf W  x, y   W  x0 , y0   inf sup W  x, y  . xU yV

2.5

yV xU

The common value in 2.5 is called the value of the game,  x0 , y0   U  V a solution of the game and x0 and y0 winning strategies. It follows that to prove the existence of a solution of a game we have to prove equality 2.4, that is, to prove a minimax theorem. 2.1 Lemma: If U, V are compact Hausdorff topological spaces and W : U  V   is continuous, then the functions

  x  : min W  x, y   min W  x,V  , x  U , yV

  y  : max W  x, y   max W U , y  , y  V , are continuous too. xU

2.6

The minimax result for s- convex function we will prove it the following. 2.2 Proposition: Let X, Y be topological spaces and F : X  Y a set-valued mapping. (a) If Y is regular, F is usc and for every x  X the set F(x) is nonempty and closed, then F has closed graph. (b) Conversely, if the space Y is compact Hausdorff and F is with closed graph, then F is usc. S.Cobzas 2006 [14] states and prove the Kakutani theorem in the locally convex case. An element of a set-valued mapping

F : X Y

if

x  F  x .

x X

is called a fixed point

If F is single valued then the usual notion of fixed point.

3.Main Result: 3.1 Theorem:

 X , P

let sets.

and Y , Q  be Housdoroff locally convex spaces and U  X , V  Y nonempty compact convex

Suppose that W : U  V   is continuous and

I. For every x  U the function W  x,. is s-convex and

II. For every y  V the function W ., y  is s-convex. Then

min max W  x, y   max min W  x, y  yV

xU

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xU

yV

3.1

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D.B.Ojha et. al. / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3571-3575 and the game U , V , W  has a solution. Proof : Let functions

  x   min W  x  V  and   y   min W U  y 

be as in lemma and let

M x   y  V : W  x, y     x  , N y   x  U : W  x, y     y  ,

3.2

for x  U and y  V . Since U,V are Housdorff compact spaces and the functions W ,  , are continuous, the sets M x and N y are nonempty and closed, for every  x, y   U  V . We will show that they are s-convex too. Let y1 , y2  M x , t   0,1 and y  1  t  y1  ty2 . Then, by (i)

  x   W  x, y   1  t  W  x, y1   t sW  x, y2   1  t    x   t s  x     x  s

s

3.3

showing that W  x, y     x  , that is, y  M x . Similarly, x1 , x2  N y , and t   0,1, we have by (ii),

  y   W  x, y   1  t  W  x1 , y   t sW  x2 , y   1  t    y   t s  y     y  s

s

3.4

showing that W  x, y     x  , that is, x  N y Let C  U  V and define F : C  C by F  x, y   N y  M x ,  x, y   C. it follows that F  x, y  is a nonempty closed convex subset of C for every  x, y   C . If we show that F has closed graph, then by Section 2.2 it is usc, so that, by Section 2.3 (S. COBZAS 2006) F has a fixed point  x0 , y0  . We have

 x0 , y0   F  x0 , y0   x0  N y , y0  M x 0

3.5

0

But

x0  N y0  W  x0 , y0   max W  x, y0   inf max W  x, y  . yV

xU

xU

y0  M x0  W  x0 , y0   max W  x0 , y   sup min W  x, y  . yU

xU

3.6

yV

Taking into account these last two inequalities and 11,we get

W  x0 , y0   sup min W  x, y   inf max W  x, y   W  x0 , y0  yV

xU

xU

3.7

yV

implying

max min W  x, y   W  x0 , y0   min max W  x, y  xU

yV

yV

3.8

xU

It remained to show that the graph GF of F, given by

GF 

 x, y, u, v   C :  u, v   F  x, y  , 2

3.9

  x , y  : i  I  is a net in C converging to  x, y   C , and  u , v   F  x , y  : i  I , are such that the net   u , v  : i  I  converges to  u, v   C . We have to show C 2 . Suppose that

is closed in i

i

i

i

i

i

i

i

that  u , v   F  x, y  .We have

 ui , vi   F  xi , yi   K  ui , yi     yi  , K  xi , vi     xi  Passing to limits for i  I , and taking into account the continuity of the functions

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W ,

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D.B.Ojha et. al. / International Journal of Engineering Science and Technology Vol. 2(8), 2010, 3571-3575 and , we get W  u , y     y  , and W  x, v     x  , that is, F  u , v   N y  M x  F  x, y  . The proof is complete. 4.Conclusion: If the gain function with two strategies for a game is non empty compact s-convex set on Hausdorff locally convex spaces with minmax equal to maxmin condition , then game has a solution. Which showed as an application of the Kakutani fixed point theorem is to game theory for s-convex function.

5. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

H. F. Bohnenblust and S. Karlin 1950, On a theorem of Ville, Contributions to the Theory of Games,Annals of Mathematics Studies, no. 24, Princeton University Press, New Jersey, 1950, pp. 155–160 R. Engelking 1989, General Topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann, Berlin, 1989. L. Glicksberg 1952, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proceedings of the American Mathematical Society 3 (1952), no. 1,170–174. S. Kakutani 1941, A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8 (1941), 457–459. L. V. Kantorovich and G. P. Akilov 1964, Functional Analysis, 3rd ed., Nauka, Moscow, 1984, Englishtranslation of the 1959 edition: Macmillan, New York 1964. H. Nikaid ˆo 1968, Convex Structures and Economic Theory, Mathematics in Science and Engineering,vol. 51, Academic Press, New York, 1968 Bernstein F. and G. Doetsch 1915, Zur Theorie der konvexen Funktionen, Math. Annalen 76 , 514–526. Breckner W. W. and G. Orb´an 1978, Continuity properties of rationally s-convex mappings with values in ordered topological liner space, ”Babes-Bolyai” University, Kolozsv´ar, Hudzik H. and L. Maligranda 1994, Some remarks on si-convex functions, Aequationes Math. 48 ,100–111 Alomari M. and M. Darus 2008, On Co-ordinated s¡convex functions, Inter. Math. Forum, 3(40) (2008) 1977-1989. Alomari M. and M. Darus 2008, Hadamard-type inequalities for s¡convex functions, Inter. Math. Forum, 3(40) 1965-1970. Dragomir S. S. 1999, S. Fitzpatrick, The Hadamard’s inequality for s¡convex functions in the second sense,Demonstratio Math. 32 (4) (1999) 687-696. Kirmaci U. S. 2007 et al., Hadamard-type inequalities for s-convex functions, Appl. Math. Comp., 193,26-35. S. Cobzas 2006, Fixed Point Theorems In Locally Convex Spaces —The Schauder Mapping Method, Fixed Point Theory and Applications, Volume 2006, Article ID 57950, Pages 1–13

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