1. PRELIMINARIES Let S be a non empty set and let a, b be elements of S. The functor a,b, ... yielding a function from N into S is defined by the condition (Def. 1). (Def. 1) Let i be a natural number. Then (i) if there exists a natural number k such that i = 2 · k, then (a,b, ...)(i) = a, and (ii) if it is not true that there exists a natural number k such that i = 2 · k, then (a,b, ...)(i) = ...

The classical degree function constructed earlier for pseudomonotone mappings has been used to develop a broader degree theory of classical type for the sum of a maximal monotone map from a reflexive Banach space to its dual together with a bounded pseudomonotone map. The proof uses the generalized Yosida approximation of the maximal monotone mapping.

In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

It is clear that a monotone map is pseudomonotone, while a pseudomonotone map is quasimonotone. The converse is not true. If T is pseudomonotone (quasimonotone) and w ∈ X*\{}, then T + w is not pseudomonotone (quasimonotone) in general. In the case of a single-valued linear map T defined on the whole space Rn, it is known that if T + w is quasimonotone, then T is monotone []. Many authors (se...

This paper concentrates on the existence of canonical cofinal maps of three types: continuous, generated by finitary monotone end-extension preserving maps, and generated by monotone finitary maps. The main theorems prove that every monotone cofinal map on an ultrafilter of a certain class of ultrafilters is actually canonical when restricted to some filter base. These theorems are then applied...

We shall consider slightly more general problems. Namely, we shall investigate the existence of continuous: piecewise monotone, piecewise strictly monotone, and piecewise linear n-th roots of interval maps which have a continuous n-th root. Here by an n-th root of f we mean a map g such that f = g (g is the n-th iterate of g). A continuous map f : I → J , where I, J are closed intervals, is pie...

We consider dissipative strongly competitive systems ẋi = xifi(x) of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface Σ, called the carrying simplex. In this note we give an amenable condition for Σ to be a C1 submanifold-with-corners. We also provide conditions, based on a recent work of M. Benäım (On invariant...

We study the set of periods of tree maps f : T −→ T which are monotone between any two consecutive points of a fixed periodic orbit P . This set is characterized in terms of some integers which depend only on the combinatorics of f |P and the topological structure of T . In particular, a type p ≥ 1 of P is defined as a generalization of the notion introduced by Baldwin in his characterization o...

The aim of this paper is to provide a brief review of the main results in the theory of discrete-time monotone dynamics.