Fixed Point Theorems for Set-Valued Maps with Contractive Orbits

This paper studies set-valued maps that are lower semicontinuous when restricted to orbits in Hausdorff spaces. We introduce two notions of contractive orbits for such maps. The first defines contraction in terms of the topology of the underlying space, while the second is based on a generalized distance function. Fixed point theorems are established for both classes of mappings. We also show that the Hausdorff assumption is essential, as the results generally fail without it. As an application, we generalize Cantor's intersection theorem for sequences of closed nested sets with diameters converging to zero. We derive fixed point theorems for set-valued maps in Hausdorff locally convex vector spaces. The results in premetric spaces we apply to establish fixed point theorems for set-valued maps regular with respect to a generalized distance function.