مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

Three results are given concerning relations of the form $f(x,y)=\\theta$ , in which $x$ and $y$ are variables in given spaces $U$ and $V$ , respectively, and $\\theta$ is the zero element of a third space $W$ . Such relations often arise in applications. Under reasonable hypotheses, and in a general normed linear space setting, one of the theorems provides necessary and sufficient conditions under which it is possible to globally and uniquely solve $f(x, y)= {\\theta}$ for $x$ in terms of $y$ , with the solution map continuous. Another theorem addresses the problem of determining conditions under which given any pair $(x_0, y_0)$ such that $f(x_0, y_0)=\\theta$ , there is a unique continuous map $g$ such that $x_0= g(y_0)$ and $f(g(y), y)=\\theta$ for all $y \\in V$ , with $g$ independent of sufficiently small changes in $(x_0, y_0)$ . The third result gives, under similar reasonable hypotheses, necessary and sufficient conditions under which a relation $f(x, y) =\\theta$ is equivalent to $x=g(y)$ for some homeomorphism $g$ of $V$ onto $U$ .