A variant of a fixed point theorem of Browder‐Fan and Reich
V. M. Sehgal · International Journal of Mathematics and Mathematical Sciences · 1985
Let S be a convex, weakly compact subset of a locally convex Hausdorff space (E, τ) and f : S → E be a continuous multifunction from its weak topology ω to τ. let ρ be a continuous seminorm on (E, τ) and for subsets A, B of E let p(A, B) = inf{p(x − y) : x ϵ A, y ϵ B}. In this paper, sufficient conditions are developed for the existence of an x ϵ S satisfying p(x, fx) = p(fx, S). The result is then used to prove several fixed point theorems.