Common fixed points and coincidences of mapping families on partially ordered sets

On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms

“…Observe that the recurrence equations of type (1) can be recovered from those of type (6). In fact, the former recurrence equations can be transformed into one of the following type…”

“…The asymptotic lower and upper bounds for a few celebrated algorithms, like Quicksort, Hanoi, Largetwo and Fibonnacci (see [5,2]), whose running time of computing holds the recurrence equation (6), were discussed by means of appropriate versions of the technique exposed in Theorem 3.1 and, thus, by means of the Banach fixed point theorem. Notice that in such versions the unique thing to be proved, additionally to original Schellekens' proof, was the contractive character of the mapping Proposition 3.10.…”

“…2. Suppose that there exist v, w ∈ X with v d f (v) and f (w (6) and to (8), are monotone and continuous, since they are contractive, i.e., they satisfy that…”

“…However, in many real applications one of two conditions can be unfulfilled. Motivated, in part, by this fact a few works have focused their efforts on generalized versions of Kleene's fixed point theorem recently (see, for instance, [4,6,7]). In the original version of the celebrated Kleene fixed point theorem, and also in the aforesaid references, the assumed conditions have a global character, i.e., each element of the partially ordered set (the mathematical model) must satisfy them.…”

A new approach to coincidence and common fixed points under a homotopy of families of mappings in $b$-metric spaces

“…Kamihigashi and Stachurski [8] and Batsari et al [3] established fixed point results of orderpreserving self mappings in spaces equipped with a transitive binary relation and some distance measures. Recently, Fomenko and Podoprikhin [6,10] generalized the results of Arutyunov [1,2] to the case of families of multivalued mappings in partially ordered sets. They [7,10] introduced the notion of order homotopy and proved that common fixed point and coincidence point properties are preserved under homotopies of families of mappings in ordered sets.…”