SOME FIXED POINT THEOREMS WITH APPLICATIONS TO BEST SIMULTANEOUS APPROXIMATIONS

For a subset K of a metric space (X,d) and x∈X, the set PK(x)={y∈K:d(x,y)=d(x;K)≡inf{d(x,k):k∈K}} is called the set of best K -approximant to x. An element g∘∈K is said to be a best simultaneous approximation of the pair y1,y2∈X ifmax{d(y1,g∘),d(y2,g∘)}=infg∈K max{d(y1, g),d(y2,g)}.Some results on T-invariant points for a set of best simultaneous approximation to a pair of points y1,y2 in a convex metric space (X,d) have been proved by imposing conditions on K and the self mapping Ton K . For self mappings T and S on K , results are also proved on both T- and S- invariant points for a set of best simultaneous approximation. The results proved in the paper generalize and extend some of the results of P. Vijayaraju [Indian J. Pure Appl. Math. 24(1993) 21-26]. Some results on best K -approximant are also deduced.