Ky Fan’s Best Approximation Theorem in Hilbert space

International Journal of Scientific and Research Publications, Volume 4, Issue 7, July 2014 ISSN 2250-3153 1 Ky Fan’s Best Approximation Theorem in Hilbert space Alok Asati *, A.D. Singh**, Madhuri Asati*** *&** Department of Mathematics, Govt. M. V. M., Bhopal (India) Department of Applied Science, SKSITS, Indore (India) *** Abstract- The aim of this paper is to prove a fixed point theorem using semicontractive mapping a well-known result of Ky Fan in Hilbert space. Keywords - Fixed point theorem, Ky Fan’s best approximation theorem, Semicontractive mapping. AMS (2010) Subject Classification - 47H10, 54H25. I. INTRODUCTION Fixed point theory has always been existing itself and its applications in new areas. The theory of approximation also played an important role. Ky Fan one of the great mathematician establish an existing theorem in 1969, which was known as Ky Fan’s best approximation theorem which has been of great importance in nonlinear analysis, minimax theory and approximation theory. Several interesting fixed point theorems have been proved by using Ky Fan’s best approximation theorem. This approach helps to find fixed point theorems under different boundary conditions. Most of the fixed point theorems are given for self maps that are for a function with domain and range are the same. In case a function does not have the same domain and range then we need a boundary condition to guarantee the existence of fixed point. Let X be a normed linear space and K be a nonempty subset of X. Let T : K  X be a function. We look for an x in K that satisfies the following equation x  T ( x)  d (T ( x), K)  inf{ y  T ( x) : y  K}        ( A) x in K exists, it is called a best approximation for T ( x) . We note that x K is a solution of (A) if and only if x is a fixed point of QK oT where QK is the metric projection on K. We refer to Carbone [2], Caristi [3], Cheney [4], Furi Et al. [6], If a solution Kuratowski [8], Nussbaum [10], Park [11], Schoneberg [14], Singh and Watson [15]. In this paper fixed point theorem has been established using the concept of semicontractive mapping which generalized the result of some standard result. II. PRELIMINARIES Lemma 2.1 [1] Suppose that H be a Hilbert space and K be a nonempty closed convex subset of X . A function called semicontractive if there exists a mapping D of H  H  K such that: (i) (ii) (iii) for each fixed x in K T ( x)  D( x, x), for each fixed x in K , D( x,) is compact, for each fixed x in K , D(, x) is nonexpansive. y  T ( y)  d (Ty, K) . Corollary 2.2[13] Suppose that K be a closed bounded and convex subset of H and suppose Then there exists a y  K such that T : K  H be a semicontractive. Definition 2.3[7] Suppose that K be a subset of a Hilbert space H for each x  K . Let the inward set of K at by I K ( x)  {x  r (t  x) : t  K, r  0}. A mapping T : K  H is x, I K ( x) be defined T : K  H is said to be inward if for each x  K, T (x) lies in I K (x) and it is weakly inward if T (x) lies in I K (x) . www.ijsrp.org International Journal of Scientific and Research Publications, Volume 4, Issue 7, July 2014 ISSN 2250-3153 2 Theorem 2.4 [9] Suppose that H be a Hilbert space and K be a nonempty closed convex subset of H , T a continuous semicontractive map of K into H . Let either (I QK oT)(K) is closed in H or (I QK oT)(clco(QK oT( K))), where QK , is the proximity map of H into K . If T ( K ) is bounded then there exists a point III. v in K such that v  T (v)  d (T (v), K ). MAIN RESULTS Theorem 3.1 Suppose that H be a Hilbert space and K be a nonempty closed convex subset of H and T be a continuous semicontractive map of K into H . Let either ( I  QK oT )(clco(QK oT ( K ))) or ( I  QK oT )( K ) is closed in H where QK is the proximity map of H into K . Suppose T ( K ) is bounded and T has a fixed point in K if and only if it satisfies one of the conditions below: [1]  y in I K ( x)  {x  t ( z  x) : for some z  K, some t  0} such that y  T ( x)  x  T ( x) , for x  K with x  T ( x). Proof: Consider that T satisfies condition. By using theorem 2.4  a point v in K such that v  T (v)  d (T (v), K ). If v  T (v), then  a y in I K (v) such that y  T (v)  v  T (v) . If y  K, which is a contradiction for supposition of v . Hence y  K, and  a z  K, such that y  v  t ( z  v) for some t  1 . 1 1 1 i.e. z  y  (1  )v  (1   ) y   v where   1  , 0    1 . t t t z  T (v)  (1   ) y  v  T (v)  (1   ) y  T ( y)   v  T (v) Hence  (1   ) v  T (v)   v  T (v)  v  T (v) Which contradicts the supposition of v . Hence   x  (1   )T ( x)  K. [2] There is a number v  T (v) real or complex depending on the vector space X respectively. For each x  K , such that Proof: Consider that T satisfies condition. Using theorem2.4  point fixed point in K , then 0  Therefore 0   1 and v in K such that v  T (v)  d (T (v), K ). Let T has no v  T (v) . For point v , there is a number  such that   1 and  v  (1   )T (v)  x  K . v  T (v)  d (T (v), K)  x  T (v)   v  T (v)  v  T (v) Which our supposition. Hence T has a fixed point in K . [3] If v  QK oT (v), where v be any point on the boundary of K , then v is a fixed point of T. Proof: Consider that T satisfies condition. Using theorem 2.4 then d (T (v), K )  0 and v is a fixed point of T. If T (v)  K then from T (v)  QK oT (v)  d (T (v), K )  T (v)  v , and the uniqueness of the nearest point, Hence  a point v in K such that v  T (v)  d (T (v), K ). If T (v)  K , QK oT (v)  v . Implies that v lies on the boundary of K , which contradicts our supposition. v is a fixed point of T. www.ijsrp.org International Journal of Scientific and Research Publications, Volume 4, Issue 7, July 2014 ISSN 2250-3153 [4]  3 x  K, T ( x)  clI K ( x), i.e. T is weakly inward. Proof: Consider that T satisfies condition. y  B(T ( x), x  T ( x) x  K, T ( x)  clI K ( x) . If x  T ( x) then there exists a point y in I K ( x) such that ) , where B{T ( x), 2 y  T ( x)  x  T ( x) . x  T ( x) 2 }, is an open ball with centre T ( x) and radius x  T ( x) 2 , Therefore Hence T has a fixed point in K . [5]  x On the boundary of K, T ( x)  y  x  y for some y in K . Proof: Proof of this condition is similar to the proof of condition (1). Corollary 3.2 Suppose that H be a Hilbert space and K be a nonempty closed convex subset of a Hilbert space H and T be a continuous 1-setcontraction map of K into H . If T ( K ) is bounded and T satisfies any one of the five conditions of Theorem 3.1.Then T has a fixed point in K. T : K  H be semicontractive mapping with bounded range such that for each x K , Tx  y  x  y , for some y  K. Then T has a fixed Corollary 3.3 Suppose that H be a Hilbert space and K be a closed convex subset of Hilbert space H . Suppose that point. REFERENCES 1. F.E Browder, “Semicontractive and semiaccretive nonlinear mappings in Banach spaces”, vol.74, Bull. Amer. Math. Soc., 1968, pp.660-665. 2. A. Carbone, “An extension of a best approximation theorem”, vol.19, Intern. J. Math and Math Sci., 1996, pp.711-716. 3. J. Caristi, “Fixed point theorems for mappings satisfying inward conditions”, vol.215, Trans. Amer. Math. Soc., 1976, pp.241-251. 4. E.W. Cheney, “Introduction to approximation theory”, Mc-Graw-Hill, New York, 1966. 5. Ky Fan, “Extensions of two fixed point theorems of F. E. Browder”, vol.112, Math. Z., 1969, pp. 234-240. 6. M. Furi, and A. Vignoli, “On a-nonexpansive mappings and fixed points”, Vol. 48(8), Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 1970, pp.195-198. 7. B. Halpern, and G. Bergman, “A Fixed point theorem for inward and outward maps”, vol.130, Trans. Amer. Math. Soc., 1968, pp. 353-358. 8. C. Kuratowski, “Sur les espaces complets”, vol.15, Fund. Math., 1930, pp.301-309. 9. T.C. Lin and C.L. Yen, “Applications of the proximity map to fixed point theorems in Hilbert space”, vol.52, J. Approx. Theory, 1988, pp.141-148. 10. R.D. Nussbaum, “The fixed point index and fixed point theorems for k-set-contractions”, vol. 9: (5&6), Doctoral dissertation, the univ. of Chicago, 1969. 11. S. Park, “On generalizations of Ky Fan’s theorem on best approximations”, Numer. Funct. Anal. Optim, 1987, pp.619-628. 12. S.Park, “Remarks on generalizations of best approximation theorems”,vol.16, Honam Math. J., 1994, pp. 27-39. 13. D.Roux, and S.P.Singh , “On a best approximation theorem”, vol. 19 , Jnanabha, 1989, pp. 1-19. 14. R. Schoneberg, “Some fixed point theorems for mappings of nonexpansive type”, vol. 17, Com. Math. Univ. Carolinae, 1976, pp.399-411. 15. S.P. Singh and B. Watson, “Proximity maps and Fixed Points”, vol.39, J. Approx, Theory, 1983, pp. 72-76. www.ijsrp.org