Description:
<jats:title>Abstract</jats:title><jats:p>Kakutani's Theorem states that every point convex and use multifunction <jats:italic>ϕ</jats:italic> defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which <jats:italic>ϕ</jats:italic> must satisfy if <jats:italic>c</jats:italic> is the unique fixed point of <jats:italic>ϕ</jats:italic>. It is e.g. shown that if the width of <jats:italic>ϕ</jats:italic>(<jats:italic>c</jats:italic>) is greater than zero, then <jats:italic>ϕ</jats:italic> cannot be lsc at <jats:italic>c</jats:italic>, and if in addition <jats:italic>c</jats:italic> lies on the boundary of <jats:italic>ϕ</jats:italic>(<jats:italic>c</jats:italic>), then there exists a sequence {x<jats:sub>k</jats:sub>} which converges to <jats:italic>c</jats:italic> and for which the width of the sets <jats:italic>ϕ</jats:italic>(x<jats:sub>k</jats:sub>) converges to zero. If the width of <jats:italic>ϕ</jats:italic>(<jats:italic>c</jats:italic>) is zero, then the width of <jats:italic>ϕ</jats:italic>(x<jats:sub>k</jats:sub>) converges to zero whenever the sequence {x<jats:sub>k</jats:sub>} converges to <jats:italic>c</jats:italic>, but in this case <jats:italic>ϕ</jats:italic> can be lsc at <jats:italic>c</jats:italic>.</jats:p>