The paper deals with the general theme of what is known about the existence of fixed points and approximate fixed points for mappings which satisfy geometric conditions in product spaces. In particular it is shown that if X and Y are metric spaces each of which has the fixed point property for nonexpansive mappings, then the product space $(X \times Y )_\infty$ has the fixed point property for nonexpansive mappings satisfying various contractive conditions. It is also shown that the product space $H = (M \times K)_\infty$ has the approximate fixed point property for nonexpansive mappings wheneverM is a metric space which has the approximate fixed point property for such mappings and K is a bounded convex subset of a Banach space.
"FIXED POINTS AND APPROXIMATE FIXED POINTS IN PRODUCT SPACES." Taiwanese J. Math. 5 (2) 405 - 416, 2001. https://doi.org/10.11650/twjm/1500407346