A Note on a Fixed Point Theorem in Hilbert Spaces

Thai Journal of Mathematics Volume 3 (2005) Number 2 : 219–221 A Note on a Fixed Point Theorem in Hilbert Spaces M. Imdad and Javid Ali Abstract : In this note we observe that a fixed point theorem (due to Nigam et al. [1]) proved in Hilbert spaces remains true in metric spaces. Keywords : Hilbert space, common fixed point and commuting mappings. 2000 Mathematics Subject Classification : 47H10, 54H25. 1 Introduction There exists an extensive literature on fixed point theorems for various contractive conditions whose comprehensive survey can be found in Rhoades [3, 4]. Recently, Nigam et al. [1] proved the following fixed point theorem in Hilbert spaces using the parallelogram identity. Theorem 1.1([1]) Let S and T be non-commuting self-mappings of a closed subset C of a Hilbert space H satisfying kST x−T Syk2 ≤ akx−yk2 +b[kx−ST xk2 +ky−T Syk2 ]+c[kx−T Syk2 +kST x−yk2 ] for all x, y ∈ C, and a, b, c being positive reals with 0 < a+2b+4c < 1, 0 < b+c < 1 and 0 < a + 2c < 1. Then ST and T S have a unique common fixed point. It is straightforward to note the following : (a) Theorem 1.1 is in fact a result for a pair of mappings T1 = ST and T2 = T S. (b) Theorem 1.1 reduces to a result for a single map in case S and T commute. (c) If the pair (S, T ) commutes at z and T z, then z remains the unique fixed point of S and T separately provided z is the unique common fixed point of ST and T S. To see this, consider ST (Sz) = S(T Sz) = Sz, T S(Sz) = ST (Sz) = S(T Sz) = Sz ST (T z) = T S(T z) = T (ST z) = T z, T S(T z) = T (ST z) = T z 220 Thai J. Math. 3(2005)/ M. Imdad and Javid Ali which show that Sz and T z are common fixed points of ST and T S. Now using the uniqueness of the common fixed point of ST and T S, one gets z = Sz = T z. The purpose of this note is to observe that even the following general form of Theorem 1.1 remains true in metric spaces. In doing this, we never need the specific properties of an inner product norm. In fact, a slightly improved form of Theorem 1.1 remains true in metric spaces which is as under : Theorem 1.2 Let T1 and T2 be self-mappings of a closed subset C of a complete metric space X satisfying kT1 x − T2 yk2 ≤ akx − yk2 + bkx − T1 xkky − T2 yk + ckx − T2 ykkT1 x − yk (1) for all x, y ∈ C and a, b, c non negative reals, with max{a + b, a + c} < 1. Then T1 and T2 have a unique common fixed point. Unfortunately, Theorem 1.2 is not a new result and is in fact a corollary to Theorem 3.1 of Park [2]. Here we state relevant portion of Theorem 3.1 of Park [2] which runs as follows : Theorem 1.3([2]) Let S and T be self-mappings of a metric space (X, d). If there exists a sequence {xi | i = 0, 1, 2, 3, . . .} ⊂ X, where x2n+1 = Sx2n , x2n+2 = T x2n+1 such that {xi } is complete, and if there exists a λ ∈ [0, 1) such that d(Sx, T y) ≤ λd(x, y) (2) for each distinct x, y ∈ {xi } satisfying either x = T y or y = Sx, then either: (i) S or T has a fixed point in {xi }, or (ii) {xi } converges to some z ∈ X and d(xi , z) ≤ λi d(x0 , x1 )/(1 − λ) for i > 0. To substantiate our claim, let us set y = T1 x in (1) so as to get kT1 x − T2 T1 xk2 ≤ akx − T1 xk2 + bkx − T1 xkkT1 x − T2 T1 xk + 0 ≤ (a + b) max{kx − T1 xk2 , kx − T1 xkkT1 x − T2 T1 xk} ≤ (a + b) max{kx − T1 xk2 , max{kx − T1 xk2 , kT1 x − T2 T1 xk2 }} = (a + b) max{kx − T1 xk2 , kT1 x − T2 T1 xk2 } = (a + b)kx − T1 xk2 , yielding thereby kT1 x − T2 T1 xk ≤ kkx − T1 xk, where k = (a + b)1/2 . (3) A Note on a Fixed Point Theorem in Hilbert Spaces 221 Thus inequality (3) implies (2). If Case (i) holds, then (1) implies that any fixed point of T1 is a fixed point of T2 and vice-versa. Otherwise, if (ii) is satisfied, then {xn } converges to u ∈ C (as C is closed). Using (1), kT1 u − x2n+2 k2 ≤ aku − x2n+1 k2 + bku − T1 ukkx2n+1 − x2n+2 k + cku − x2n+2 kkT1 u − x2n+1 k. Taking as n → ∞ yields kT1 u − uk2 ≤ 0, and u = T1 u. Now in view of (1) one gets u = T2 u and hence u is a common fixed point of T1 and T2 . The uniqueness of a common fixed point is a consequence of the condition (1). References [1] S. K. Nigam, N. P. S. Bawa and P. K. Shrivastava, Fixed point theorems in Hilbert space, Varāhmihir J. Math. Sci., 3(1)(2003), 113-118. [2] S. Park, Fixed points and periodic points of contractive pairs of maps, Proc. College Nat. Sci., Seoul Nat. Univ., 5(1)(1980), 9-22. [3] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226(1977), 257-290. [4] B. E. Rhoades, Contraction definitions revisited, Contemporary Math., 21 (1983), 189-205. (Received 2 July 2005) M. Imdad and Javid Ali Department of Mathematics Aligarh Muslim University Aligarh-202002, India. e-mail : [email protected], [email protected]