Annual lectures dedicated to the memory of Professor Andrzej Lasota.

A short biography of Professor Andrzej Lasota.

This is an account of my scientific and personal friendship with Prof. Andrzej (Andy) Aleksander Lasota from 1977 until his death 28 December, 2006. It is a tale that fascinates me because of the intertwined links between many people both East and West of several generations, and it illustrates what I feel is the strength and beauty of the personal side of the scientific endeavor.
This contribution is almost identical to the paper “Adventures in Poland: Having fun and doing research with Andrzej Lasota”, Matematyka Stosowana 8 (2007), 5–32. It is in no way to be considered a new contribution, but is rather a record of the second Annual Lecture Commemorating Professor Andrzej Lasota given in Katowice at Uniwersytet Slaski on 16 January, 2009.

We study the well-posedness of the fixed point problem for asymptotically regular self-mappings of a complete metric space (X,d) which satisfy the contractive condition (2.1) described below. This contractive condition is a variant of the contractive condition considered in [6]. The results of this paper provide some improvements and extensions to the results of Ciric [6], Sharma and Yuel [19], and Guay and Singh [7]. This work is inspired and motivated by the paper [6].

We prove that every Jensen convex function mapping a real linear Polish space into ℝ bounded above on a nonzero Christensen measurable set is convex.

We solve functional equation of the form
f(x+y−xy) + f(xy) = 2f((x+y)/2)
in the class of functions transforming the space of all reals into itself. We also prove that this equation is stable in the Hyers-Ulam’s sense.

We study the existence of positive periodic solutions of the equations
x⁽ⁿ⁾(t) − p(t)x(t) + μf(t, x(t), x'(t), . . . , x⁽ⁿ⁻¹⁾(t)) = 0,
x⁽ⁿ⁾(t) + p(t)x(t) = μf(t, x(t), x'(t), . . . , x⁽ⁿ⁻¹⁾(t)),
where n≥2, μ>0, p: (-∞,∞)→(0,∞) is continuous and 1–periodic, f is a continuous function and 1–periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.

Let X^⚹ be a tree of words over the changing alphabet (X₀,X₁, . . .) with X_i = {0, 1, . . . ,m_i − 1}, m_i > 1. We consider the group Aut(X^⚹) of automorphisms of a tree X^⚹. A cyclic automorphism of X^⚹ is called constant if its root permutations at any two words from the same level of X^⚹ coincide. In this paper we introduce the notion of a balanced automorphism which is obtained from a constant automorphism by changing root permutations at all words ending with an odd letter for their inverses. We show that the set of all balanced automorphisms forms a subgroup of Aut(X^⚹) if and only if 2∤m_i implies m_i+1 = 2 for i = 0, 1, . . . . We study, depending on a branch index of a tree, the algebraic properties of this subgroup.

Report of Meeting: The Ninth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities, Będlewo (Poland), February 4–7, 2009.