DEMONSTRATIO MATHEMATICA Vol. XXXVI No 3 2003 N. Hussain, A. R. Khan COMMON FIXED POINTS AND BEST APPROXIMATION IN p-NORMED SPACES Abstract. A general common fixed point result is obtained for commuting maps on non-starshaped domain in a p-normed space. As applications, we obtain Brosowski- Meinardus type approximation theorems in p-normed spaces which are not necessarily lo- cally convex. Recent approximation results of a number of authors follow as a consequence of our results. 1. Introduction In 1963, Meinardus [13] combined the concepts of fixed point and best approximation in function spaces and observed that some of the useful properties of a function remain invariant under some suitable conditions. Afterwards in 1969, Brosowski [2] obtained the following generalization of Meinardus's result. THEOREM 1.1. Let X be a normed space and T : X —> X be a linear and nonexpansive operator. Let M be a T-invariant subset of X and u € F(T). If PM{U), the set of best approximations of u in M, is nonempty compact and convex, then there exists a y in PM{ U ) which is also a fixed point ofT. Using a fixed point theorem, Subrahmanyam [16] obtained the following generalization of the above mentioned theorem of Meinardus [13]. THEOREM 1.2. Let X be a normed space. IfT:X—*X is a nonexpansive operator with a fixed point u, leaving a finite dimensional subspace M of X invariant, then there exists a best approximation of u in M which is also a fixed point of T. In 1979, Singh [15] observed that the linearity of the operator T and the convexity of PM(U) in Theorem 1.1, can be relaxed and proved the following extension of it. THEOREM 1.3. Let X be a normed space, T : X —> X be a nonexpansive op- erator, M be a T-invariant subset of X andu € F(T). If PM(U) is nonempty Unauthenticated Download Date | 2/26/20 4:32 AM