WE INTRODUCE A GENERALIZED NOTION OF BEST APPROXIMATION AND ALSO INVESTIGATE SOME BASIC PROPERTIES OF THIS NOTION. SOME ILLUSTRATIVE EXAMPLES ARE PRESENTED.

As a counterpart to BEST APPROXIMATION in normed linear spaces, BEST simultaneous APPROXIMATION was introduced. In this paper, we shall consider the relation between simultaneous proximinality M in X and Lp(S,M) in Lp(S,X). Finally we give some results in simultaneous Chebyshev subspaces.

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IN THIS PAPER WE INTRODUCE THE NOTATION OF T-BEST APPROXIMATIVELY COMPACT SETS, T-BEST APPROXIMATION POINTS, T-PROXIMINAL SETS, T-BOUNDEDLY COMPACT SETS AND T-BEST PROXIMITY PAIR IN FUZZY METRIC SPACES. THE RESULTS DERIVED IN THIS PAPER ARE MORE GENERAL THAN THE CORRESPONDING RESULTS OF METRIC SPACES, FUZZY METRIC SPACES, FUZZY NORMED SPACES AND PROBABILISTIC METRIC SPACES.

IN THIS PAPER WE INTRODUCE A NOTION OF T-BEST APPROX-IMATIVELY COMPACT SETS, T-BEST APPROXIMATION POINTS, T-PROXIMINAL SETS AND T-BOUNDEDLY COMPACT SETS IN FUZZY METRIC SPACES. THE RESULTS OBTAINED IN THIS PAPER ARE RELATED TO THE CORRESPONDING RESULTS IN METRIC SPACES, FUZZY METRIC SPACES, FUZZY NORMED SPACES AND PROBABILISTIC METRIC SPACES.

IN THIS PAPER WE INTRODUCE THE CONCEPT OF ASYMPTOTIC BEST APPROXIMATION OF MAPS AND CONSIDER RELATION BETWEEN IT AND FIXED POINT THEORY IN THE BANACH SPACE.