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In this paper, we give combinatorial proofs of Baillon and Simons’ almost fixed point and fixed point theorems for discrete-valued mappings (J. Combin. Theory Ser. A 60 (1992) 147–154).

The main theme of this paper is that almost fixed point properties of discrete structures and fixed point properties of (topological) spaces are interdeducible via a suitable category which contains both graphs and spaces as objects. To carry out the program, we have to consider (almost) fixed points of multifunctions, and for this we need a preliminary discussion of power structures for graphs and simplicial complexes. Specific applications developed are: a “digital convexity” (discrete) version of Kakutani's fixed point theorem for convex-valued multifunctions; and fixed point properties of dendrites in terms of those of finite discrete trees.

In this paper, a sound notion of "continuous" multifunctions for discrete spaces is introduced. It turns out that a multifunction is continuous (= both upper and lower semi-continuous) if and only if it is "strong", or equivalently it takes (original) neighbours into (induced) neighbours w.r.t. Hausdorff metric. Generalizing various previous results, we show that any finite contractible graph has the almost fixed point property (afpp) for strong multifunctions. We then turn to the fixed clique property, or fcp (a desirable strengthening of afpp). We are able to establish this property only under much more explicitly geometric assumptions than in the case of afpp. Specifically, we have to restrict to convex-valued multifunctions, over spaces admitting a suitable notion of convexity; and, for the best results, to the standard n-dimensional digital space with (3n - 1)-adjacency. In this way we obtain what may be considered (with reservations, discussed in the Conclusion) as an analogue of the classical Kakutani fixed point theorem.

We extend some of our previous results on fixed points of graph multifunctions to posets. The posets of most interest here are the (finite) Khalimsky spaces, in their specialization order. Retracts of Khalimsky spaces coincide with Helly posets. Notions of convexity can be defined in these spaces, providing the basis for certain “geometric” fixed point theorems.

In this paper, the concepts of power graphs and power complexes are introduced. The multifunctions for graphs are denned and will be classified. The concept of simplicial mappings for complexes then is extended to multifunctions. A notion of weak convexity is defined in the intersection graphs of (3− 1)-adjacent n -dimensional real digital pictures based on the usual Euclidean convex closure operator. It is shown that any (3 − 1)-adjacent n -dimensional digital picture has the simplicial weak convex almost fixed point property, which may be considered as a digital version of the Kakutani fixed point theorem for convex-valued multifunctions.