Drop theorems and lipschitzianness tests via maximality procedures

Abstract

An important geometrical method of studying operator equations on (finite or infinite dimensional) Banach spaces is that represented by the so-called drop theorems. As first results in this direction, we must quote the 1971 Browder's theorem [8] as well as the 1972 Dane]' result [12] proved by a Bishop--Phelps maximality argument [3] and, respectively, with the aid of the classical Cantor--Kuratowski intersection theorem [21, p. 318]. At the same time, a useful and effective metric instrument of studying the same class of problems is that represented by a local method known under the name "lipschitzianness test" and introduced by AL•MAN [1] as well as KIRK--RAY [20], with the aid of a transfinite induction principle. Under these lines, the main intention of the present note is to give a convex and a functional extension of the above quoted results. In this context it is not without importance to emphasize that the basic instrument in proving our generalized variants is represented by a maximality principle on (complete) metric spaces comparable with EK~LAND-BRONDSTED'S one [14], [6]. It should also be noted that the considered maximality principle (stated below) may be extended to a class of (complete) metrizable uniform spaces; this question will be treated elsewhere.