Local and Global Lmscmzian Mappings on Ordered Metric Spaces
By M ~R A I TURINICI in Iagi (Romania) (Eingegangen am 19. 5. 1980) 0. Introduction An important problem concerning a wide class of mappings acting on a metric space is that of finding sufficient conditions in order that a "local" LIPSCHITZ property (in a sense precised by an appropriate context) should imply a "global" one on that subset. A first step in this direction was donein case of a normed space -by W. A. KIRK & W. 0. RAY [13], through a transfinite induction argument; later, in author's paper [15], the same problem was solved (in a similar framework) by a ZORN'S type procedure, due essentially to A. BR0NDsTED [4].
Under these lines, it is the main aim of this note to state and prove a lipschitzanness test for a class of closed mappings acting on a (complete) metric space, giving thus a direct extension of the above quoted KIRK-RAY'S, as well as author's paper and, in this context, it's not without importance to emphasize that the theoretical instrument in proving our main result is represented by a maximal element theorem extending the classical ones (A. BR0NDsTED [a], J. CARISTI [7]) as well as some recent generalizations (M. TURINICI [15], D. DOWNING & W. A. KIRK [9]
). It should also be noted that our main result has a number of important applications to nonlinear semigroups theory; these questions will be t,reated in a forthcoming paper under preparation.