The theory of translation arcs, initiated by Brouwer, is a powerful tool to prove the existence of fixed and periodic points of planar maps. Most of the results obtained in this way are purely qualitative and do not provide any information on the location of the fixed points.
In the present paper, the authors consider homeomorphisms \(h\) of the plane satisfiying additional geometric conditions and they show that it is possible to obtain some bounds. First they assume that the map \(h\) satisfies the condition \[ \|h(x)-x\|\leq M \] for a fixed \(M>0\) and every point \(x\) in the plane. This condition leads to the estimate \[ \beta (n)\leq (n+1)M+\beta (1) \] where \(\beta (n)\) is a bound of the set of \(n\)-cycles. This inequality was already obtained by Campos and the reviewer, but only when \(n\) is a power of two. In a second result, the authors obtain a fixed point theorem for certain homeomorphisms of the type \(h(x)=x+K(x)\) where \(K\) is Lipschitz-continuous. Related results had been previously obtained by Aarao and Martelli and by Graff and Nowak-Przygodzki but in the previous papers \(K\) was a contraction. Some consequences in the theory of periodic differential equations are presented in the last section of the paper.
All the proofs are based on the use of a clever variant of the Arc Translation Lemma (Theorem 2.2).