In the present paper we established a fixed point theorem in compact metric space and another result is proved for pseudo compact tichnov space. Our results are generalization form of many known results. Keywords: Compact metric spaces, Pseudo compact tichnov spaces, fixed point AMS subject classification: 47H10, 54H2

Chourey, Bharati

Sharma, R.K.

English

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Journal of Information Engineering and Applications                                                                                                                       www.iiste.org ISSN 2224-5782 (print) ISSN 2225-0506 (online) Vol.4, No.11, 2014  12 Fixed Point Theoremsrelated To Compact Metric Spaces R.K. Sharma & Bharati Chourey* Department of Mathematics, BUIT, Bhopal *Research Scholar, Department of Mathematics, BUIT, Bhopal Email: bharatichourey@gmail.com Abstract In the present paper we established a fixed point theorem in compact metric space and another result is proved for pseudo compact tichnov space. Our results are generalization form of many known results. Keywords: Compact metric spaces, Pseudo compact tichnov spaces, fixed point  AMS subject classification: 47H10, 54H25  2. Introduction & Preliminaries: There are several genralizations of classical contraction mapping theroem of Banach [1].In 1961 Edelstein [4] established the existence of a unique fixed point of a self map T of a compact metric space satisfying the inequality d(T(x), T(y))  <d (x, y) which is genralization of Banach. In the past  few years a number of  authers such as  Fisher [5], Soni [11,12]  have stablished a number of intresting results on compact metric spaces. More recently Fisher and Namdeo [6], Popa and Telci [10], Sahu [13] described some valuble results in compact metric spaces. Jain and Dixit [7], Pathak [9], Khan, S. and Sharma [8] worked on pseudo-compact Tichonov spaces. Recently Bhardwaj etl.[2,3] also worked for thease spaces. 3. Main Results Theorem 3.1: Let F be a continuous mappings of a compact metric space X into itself satisfying the condition; (3.1)  <    (3.1)      For all x, y ϵ X, x≠y and ≤2, where  are non negative real numbers, then F has a unique fixed point.  PROOF: First we define a function T as follows: T(x) =d(x,y(x)), for all x ϵ X. Since d and F are continuous on X, T is also continuous X. From compactness of X, there exists a point P∈X, such that (3.1.1)      T(P) =inf{T(x): x ∈X}             If T(P) ≠ 0, it follows that P≠F(p)  And so T(F(P))=d(F(P), (p))                d(F(P)),F(F(P))< d(P, )+ +  therefore,                d(F(P)), (P)) < d(F(P), ) + d(F(P), (P)) + (  + ) d(P,F(P)) That is, d(F(P), ) [1- ] < ( )d(P,F(P))              d(F(P), (P)) < ( )/ [1- ] d(P,F(P)) That is, T(F(P)) < ST(P)                         Where S= (  + )/ [1- ] ≤ 1                                     ≤2    Which is a contradiction to the condition (1.2) and hence P=F(P),consequently P is a fixed point of F.  Uniqueness: Now we shall prove the uniqueness of P. Let if possible Q≠P be another fixed point of F. Now d(P,Q)= d(F(P)),F(Q)) d(F(P),F(Q)) < d(P, )+ + + +d(P,Q) That is d(P,Q)< (  + )d(P,Q) Which is a contradiction because ≤2 Hence P is a unique point of F. Theorem 3.2: Let P be a pseudo compact Tichonov space and μ be a non-negative real number valued continuous function over (P×P) satisfying. [3.2.1] μ(x, x) = 0, for all x ∈ P and          μ(x, y) = μ(x, z)+ μ(z, y) for all x,y and z ∈ P           Let  T:P → P is a continuous map satisfying; [3.2.2] Journal of Information Engineering and Applications                                                                                                                       www.iiste.org ISSN 2224-5782 (print) ISSN 2225-0506 (online) Vol.4, No.11, 2014  13         μ( , )  = μ(x,y)[1+ μ(x, )+ μ(y, )+ μ(x,y)+min{ μ(x, ), μ(y, ), μ(x,y)}] for all distinct x,y ∈ P, then T has a unique fixed point in P. Proof: We define d: P→ R by ф(P) = μ (Tp, p) for all p ∈ P, when R is a set of real numbers clearly ф is continuous, being the composite  of two functions T and μ, since P is pseudo compact Tichonov space; every real valued  continuous function over P is bounded and attains its bounds. Thus there exists a point say  V∈ P, such that ф(V) = inf{ ф (P) : p ∈ P} It is clear that ф(P) ∁ R. We now affirm that v is a fixed point for T. If not, let us  that ≠ v. so by (2.2) ф( )= μ( v, )            = μ(T (TV), TV) ф(T(v))< ( , v)[1+ μ( ,v) ]             = μ( ,v) This implies ф(T(v))= μ( v, ) <  μ( , v) A contradiction, so T(v) = v i.e. v  P is a fixed point for T. To prove the uniqueness of  v, if possible, let w  P be another fixed point for T,i.e = w and w ≠ v So by, [3.2.2] μ(v, w) = μ( , ) < μ(v, w)[1+ μ(v, )+ μ(w, )+ μ(v, w) + min{ μ(v, ), μ(w, ), μ(v, w)}] μ(v, w) < μ(v, w) Again it is a contradiction. Hence v ∈ P is a unique fixed point for T in P. This completes the proof. References  1. Banach, S. “Surles operation dans les ensembles abstraits et leur application aux equations integrals”   Fund. Math. 3(1922) 133-181. 2. Bhardwaj, R.K., Sharma, A.K., Rajput, S.S., Yadava, R.N. and Dhagat, V.B. “Some fixed point theorems in compact spaces and pseudo compact Tichonov spaces” International Journal of Mathematical Analysis 2 No 11(2008) 543-550. 3. Bhardwaj, R.K., Choudhary, S., Rajput, S.S. and Yadava, R.N.  “Some fixed point theorems in compact spaces” International Journal of Mathematical Analysis 2 No 11(2008) 551-555. 4. Edelstein, M. “An extension of Banach’s contraction principale” Pro. Amer, Math. Soc.12 (1961)7-10. 5. Fisher, B. “On three fixed point mappings for compact metric spaces” Indian J. Pure and Appl. Math. 8(1977) 479-481. 6. Fisher B. and Namdeo, R.K. “Releted fixed point theorem for two pairs of set valued mappings   on compact metric space”   Indian J. of Mathematics 46 Nos.2, 3(2004) 161-171. 7. Jain, R.K. and Dixit, S.P. “Some results on fixed points in Pseudo compact Tichonov spaces” Indian J. pure and Appl. Math15 (1986) 455-458. 8. Khan, S. and Sharma, P.L. “Some results on fixed points in Pseudo compact Tichonov spaces” Acta Ciencia Indica 17 (1991) 483-488. 9. Pathak, H.K. “Some theorems on fixed points in Pseudo compact Tichonov spaces” Indian J. Pure and Appl. Math. 15(1986) 180-186. 10. Popa, V. and Telci, M. “Fixed point theorems for mappings satisfying implicit relations on Two Metric spaces” Mathemattica Balkanica, New series 20(2006) 143-151. 11. Soni, G.K. “Fixed point theorems for mappings in Pseudo compact Tichonov spaces” Acta Ciencia Indica 17 (1991) 479-482. 12. Soni, G.K. “Fixed point theorems in compact metric spaces” Acta Ciencia Indica 26 (2000) 295- 298. 13. Sahu, M.K. “Some fixed point theorems on compact metric spaces” International Review of Pure and Applied Mathematics 2 (2006) 151-154.      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Fixed Point Theoremsrelated To Compact Metric Spaces

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Fixed Point Theoremsrelated To Compact Metric Spaces

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