The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. Let C be a nonempty convex subset of a normed linear space X , f : C → X a continuous function, g : C → C continuous, onto and almost quasi-convex. Assume that C has a nonempty compact convex subset D such that the set A = { y ∈ C : ‖ g ( x ) − f ( y ) ‖ ≥ ‖ g ( y ) − f ( y ) ‖       for       all       x ∈ D } is compact.

Then there is a point y 0 ∈ C such that ‖ g ( y 0 ) − f ( y 0 ) ‖ = d ( f ( y 0 ) , C ) .