The **Nonlinear seminar** of the OCTD is held on **Tuesdays** at **10:00 am** in room **B3-49**, in the Faculty building in **Umultowska street 87** in **Poznań**.

The seminar is conducted by prof. UAM dr hab. Dariusz Bugajewski.

The seminar is devoted to discussing problems in nonlinear analysis. Instead of presenting known results, we prefer to discuss open problems connected with the current research of the seminar members.

We pay particular attention to the following topics.

- Fixed point theorems for different classes of mappings acting in linear vector spaces, hyperconvex metric spaces or absolute retracts.
- Almost periodic and almost automorphic functions and their generalizations and applications to differential and integral equations.
- Functions of bounded variation with applications to the theory of nonlinear integral and superposition operators.
- Existence and uniqueness of solutions to nonlinear integral equations in spaces of functions of bounded variation (in various senses).
- Existence and topological structure of solutions sets of nonlinear differential and integral equations in finite-dimensional and abstract spaces.
- Selected topics from general topology: measures of noncompactness, hyperconvex spaces, R-trees and others.
- Selected topics in the theory of nonabsolute convergent integrals.
- Selected problems of formal analysis.

Here are the titles and abstracts of the seminars:

Jin Haixia**The Aleksandrov problem in non-Archimedean 2-fuzzy 2-normed space**

We will introduce the concept of isometry which is suitable to represent the notion of area preserving mapping in non-Archimedean 2-fuzzy 2-normed space. Then we obtain some results by applying the Ben's theorem for the Aleksandrow problem in non-Archimedean 2-fuzzy 2-normed space.

Shunxin Zhao**Wigner’s theorem in s space**

Wigner theorem is the cornerstone of the mathematical formula of quantum mechanics, it has promoted the research of basic theory of quantum mechanics. In this talk, we give a certain pair of functional equations between two real spaces s, that we called “phase isometry”. It is obtained that all such solutions are phase equivalent to real linear isometries in the space s.

Zhongqin Gao**Omega model for a jump-diffusion process with a two-step premium rate**

In this talk, a jump-diffusion Omega model with a two-step premium rate is studied. In this model, the surplus process is a perturbation of a compound Poisson process by a Brown motion. Firstly, using the strong Markov property and Taylor formula, the integro-differential equations for the Gerber-Shiu expected discounted penalty function and the bankruptcy probability are derived. Secondly, for a constant bankruptcy rate function, the renewal equations satisfied by the Gerber-Shiu expected discounted penalty function are obtained, and by iteration, the closed-form solutions of the function are also given. Further, the explicit solutions of the Gerber-Shiu expected discounted penalty function are obtained when the individual claim size is subject to exponential distribution. Finally, a numerical example is presented to illustrate some properties of the model.

Xihong Jin

(Tianjin University of Technology, Tianjin, China)**A variety of Tingley problem in -spaces**

During the seminar we are going to prove that for every surjective phaseisometry between the unit spheres of real -type spaces, its positive homogeneous extension is a phase-isometry which is phase-equivalent to a linear isometry.

**The seminar starts at 12:45.**

Michał Wojtal**Existence of square-mean almost periodic solutions to some stochastic differential equations**

We will present certain class of stochastic hyperbolic differential equations with infinite delay. Next, we show existence of square-mean almost periodic solutions to equations of this type using Krasnoselskii's Fixed Point Theorem. Presentation is based on paper "Existence of square-mean almost periodic solutions to some stochastic hyperbolic differential equations with infinite delay" by P. Bezandry and T. Diagana.

Adam Nawrocki**Autonomous superposition operator defined on the space of generalized almost periodic functions**

(Continuation of the 2018-11-13 Nonlinear Seminar)

Adam Nawrocki**Autonomous superposition operator defined on the space of generalized almost periodic functions**

During the seminar we consider the autonomous superposition operator defined on the space of almost periodic functions in view of the Lebesgue measure. In particular, we give connection of this class with the class of the Stepanov almost periodic functions.

Adam Nawrocki**Some remarks concerning the convolution of a certain almost periodic function**

(Continuation of the 2018-10-23 Nonlinear Seminar)

Adam Nawrocki**Some remarks concerning the convolution of a certain almost periodic function**

(Continuation of the 2018-10-23 Nonlinear Seminar)

**The seminar has been canceled and is rescheduled for Tuesday, November 6, 2018.**

Adam Nawrocki**Some remarks concerning the convolution of a certain almost periodic function**

For we define the family of the functions

dla .

Moreover for let if and if . Then the sets and are dense subsets of . During the seminar we give properties of these sets from the point of view of set theory, measure theory and topology.

**The seminar starts at 10:45.**

Kosma Kasprzak**About calculating certain limits**

(Continuation of the 2018-10-09 Nonlinear Seminar)

Kosma Kasprzak**About calculating certain limits**

In the first part of the talk we will show two ways of calculating the following limit: . In the second part we are going to present some extensions of that result as well as other results connected with this limit.

Simon Reinwand (Uniwersitet w Würzburgu)**On some Disparities of Multiplication and Composition Operators in BV Spaces**

Although multiplication and composition of two functions are first grade operations, the corresponding multiplication and (autonomous) composition operator exhibit many weird and difficult to handle properties in BV spaces. We give criteria for acting conditions, as well as for injectivity, surjectivity, bijectivity and compactness in BV spaces. This leads naturally to a comprehensive study of multiplier sets, of which we will give a short overview. While most of them are easy to determine, some of them - especially those related to classical functions like Darboux and continuous functions - are unknown to our knowledge.

Moreover, we introduce a new type of convergence for composition operators in BV which leads to new a new proof for old continuity criteria. We hope that those ideas can be generalized to other BV-type spaces and give the relevant ideas and conjectures.

Apart from recalling known and discussing new results we put a particular emphasis on examples and counter examples.

Ruidong Wang**The Tingley's Problem on two dimensional normed spaces**

(Continuation of the 2018-06-12 Nonlinear Seminar)

Ruidong Wang**The Tingley's Problem on two dimensional normed spaces**

In this seminar I will talk about the metric properties of the unit sphere of a two dimensional normed space and the Tingley's problem on two dimensional normed spaces.

dr Jacek Gulgowski (Gdańsk University)**Integral variation and piecewise polynomial approximation**

The concept of integral variation appears to be related to some problems considered in the approximation theory. The concept of polynomials of best approximation leads to certain norm that is equivalent to the q-integral p-variation norm.

Xiao-Xiong Gan (Morgan State University, Baltimore MD, USA)**From Formal Power Series to Formal Analysis**

For any , a formal power series on a ring is defined to be a mapping from to . A formal power series in from to is usually denoted as a sequence or as a power series , where for every . The set of all formal power series on is denoted by .

If considering a formal power series as a sequence, what is the difference between and ?

If considering a formal power series as a power series, what is the difference between formal power series and the traditional power series? What is the relationship between formal power series and traditional power series?

What is *formal analysis*?

This talk tries to answer those questions and brings discussion of all kinds of questions about formal analysis, a relatively new mathematical subject.

Michał Wojtal**Introduction to stochastic integral**

(Continuation of the seminars from 2018-01-16, 2018-01-30, 2018-02-27 and 2018-03-27)

Ruidong Wang**On isometric extension problem on the unit sphere of normed spaces**

(Continuation of the 2018-04-10 and 2018-04-17 seminars)

Ruidong Wang**On isometric extension problem on the unit sphere of normed spaces**

(Continuation of the 2018-04-10 Nonlinear Seminar)

Ruidong Wang**On isometric extension problem on the unit sphere of normed spaces**

The isometric extension problem has been studied by a number of mathematicians, and there are many deep and interesting partial results, even this problem has not been solved till now.

Michał Wojtal**Introduction to stochastic integral**

(Continuation of the seminars from 2018-01-16, 2018-01-30 and 2018-02-27)

Marcin Borkowski**On partial metric spaces**

(Continuation of the 2017-11-28 and 2018-03-06 seminars)

Marcin Borkowski**On partial metric spaces**

(Continuation of the 2017-11-28 Nonlinear Seminar)

Michał Wojtal**Introduction to stochastic integral**

(Continuation of the seminars from 2018-01-16 and 2018-01-30)

Michał Wojtal**Introduction to stochastic integral**

(Continuation of the 2018-01-16)

Michał Wojtal**Introduction to stochastic integral**

During the talk we will discuss the construction of stochastic integral and it's basic properties.

Piotr Kasprzak**On a certain fixed point theorem in Banach algebras**

During the seminar we are going to discuss a certain version of the Krasnoselskii fixed point theorem for the sum of two operators in Banach algebras.

Jian Meng (Tianjin University of Technology, China)**On some binomial difference sequence spaces**

The matrix domains of Cesàro mean, Riesz mean , Nörlund mean and Euler mean of order *r* in the sequence spaces , , *c* and have been studied by some authors. The Binomial mean is one of the most important methods in the summability theory, but the matrix domain of binomial mean has not been studied, until now. The aim of this talk is to introduce the binomial difference sequence spaces. We prove some basic properties of these spaces and some inclusion relations. Moreover, we obtain the Schauder bases of some of these spaces and compute the α-, β- and γ-duals of these sequence spaces.

dr Jacek Gulgowski**Continuity of nonautonomous superposition operator in Lambda BV spaces**

We are going to present sufficient conditions for the continuity of the nonautonomous superposition operator acting in Lambda BV spaces. The conditions cover generators of class in two variables, but allow also for much broader class of functions.

Marcin Borkowski**On partial metric spaces**

The talk will be devoted to basic properties of so-called partial metric spaces and to proving a variant of the Banach contraction principle for such spaces. The talk is based on a paper /Parrtial metric topology/ by S. G. Matthews.

Piotr Kasprzak**On weak topology in the space **

The aim of the seminar is to present a characterization of the weak convergence in the space of functions of bounded Jordan variation. We are also going to discuss the weak compactness in such spaces.

(Continuation of the 2017-06-06 Nonlinear Seminar).

Piotr Kasprzak**On weak topology in the space **

The aim of the seminar is to present a characterization of the weak convergence in the space of functions of bounded Jordan variation. We are also going to discuss the weak compactness in such spaces.

(Continuation of the 2017-06-06 Nonlinear Seminar).

Simon Reinwand (University of Würzburg)**On Functions with Primitives**

The problem of characterizing those functions, which possess a classical primitive has been investigated for decades. As far as we know, it is still an open problem if there is a “natural” characterization without involving integrals.

In this talk we give a quick overview about different characterizations while keeping our focus on probably the most natural one involving the Henstock-Kurzweil Integration Theory. While being a slight generalization of the Riemann Integral, it provides not only the most comprehensive version of the classical Fundamental Theorem of Calculus, but also yields a full characterization of the class of functions, which have a primitive.

We will discuss this class in more detail with respect to its size, its relation to other function classes, and some of its algebraic properties. Apart from recalling known and discussing new results we put a particular emphasis on examples and counter examples.

While answering questions regarding multiplication and change of variables we will point out connections between functions with primitives, Darboux-functions, HK-integrable functions and functions of bounded variation.

Piotr Kasprzak**On weak topology in the space **

The aim of the seminar is to present a characterization of the weak convergence in the space of functions of bounded Jordan variation. We are also going to discuss the weak compactness in such spaces.

Marcin Borkowski**On -hyperconvex quasi-metric spaces**

(Continuation of the seminars from 2017-04-04, 2017-04-11 and 2017-05-09)

Marcin Borkowski**On -hyperconvex quasi-metric spaces**

(Continuation of the seminars from 2017-04-04 and 2017-04-11)

Marcin Borkowski**On -hyperconvex quasi-metric spaces**

(Continuation of the seminar from 2017-04-04)

Marcin Borkowski**On -hyperconvex quasi-metric spaces**

The aim of the talk is to introduce the definition and basic properties

of quasi-metric spaces and the notion of -hyperconvexity in

these spaces.

Marcin Borkowski**On some metric spaces**

(Continuation of the seminars from 2016-10-04, 2016-10-11 and 2017-02-28)

Michał Goliński**Formal power series (Part 1)**

(Continuation of the 2017-03-07)

Michał Goliński**Formal power series (Part 1)**

At the seminar we will discuss the following statement of M. Artin:

Let be analytic equation ( is analytic). We will show that if the equation has a solution in the space of formal series than it has also analytic solution which differs from coefficients with large powers.

Marcin Borkowski**On some metric spaces**

(Continuation of the seminars from 2016-10-04 and 2016-10-11)

Piotr Maćkowiak**Action and continuity of superposition operators**

(Continuation of the 2017-01-24)

Piotr Maćkowiak**Action and continuity of superposition operators**

(Continuation of the 2016-11-15)

Marcin Borkowski**On some measures of efficiency of financial projects**

(Continuation of the 2017-01-03)

Meng Liu**On the Generalized Hyers-Ulam-Rassias Stability of an k-Dimensional Quadratic Functional Equation**

In this lecture, we investigate the generalized Hyers-Ulam-Rassias stability of an k- dimensional quadratic functional equation. At the same time, we will generalize the theorem of Jae-Hyeong Bae and Kil-Woung Jun. We also get Hyers-Ulam-Rassias stability of the k-dimensional quadratic functional equation in restricted domains.

Marcin Borkowski**On some measures of efficiency of financial projects**

(Continuation of the 2016-12-13)

Meng Liu**Generalized stability of an AQ-functional equation inquasi-(2;p)-Banach spaces**

The stability of functional equation is a vital subject of functional

analysis. We introduce and investigate the general solution of a new

functional equation -AQ functional equation and discuss its Generalized

Hyers-Ulam-Rassias stability under the conditions such as even, odd,

approximately even and approximately odd in quasi-(2;p)-Banach spaces.

Marcin Borkowski**On some measures of efficiency of financial projects**

A classic – and the most well-known – measure of efficiency of financial

projects is the net present value (NPV). However, it has certain issues

connected with choosing the “right” discount rate. In the paper by

Teichroew et al. (1965) it was shown how to compute net future value

using two rates: one for financing and one for reinvesting. The authors

mention how to use their method to compute an analogue of the NPV.

Unfortunately, their method is wrong. The first part of the talk will

be devoted to showing their error and comparing their method to

a correct one due to A. Kulakova (2010). Then, we will prove a number

of properties of the generalized net present value defined by the latter

author.

dr Giselle Monteiro

(Slovak Academy of Sciences)**Extremal solutions for measure differential equations**

We investigate the existence of greatest and least solutions for the so-called measure differential equations - namely, integral equations in which the Stieltjes type integral is in the sense of Kurzweil. Based on these results, we derive new theorems about extremal solutions for impulsive systems.

**Acknowledgement**: Research financed by the SASPRO Programme, co-financed by the European Union and the Slovak Academy of Sciences.

dr Jacek Gulgowski**On functions of bounded -integral -variation**

During the seminar we are going to discuss functions of bounded -integral -variation. In particular, we are going to present some of their basic properties. We are also going to discuss the relationship of the space of such functions with Sobolev spaces. Finally, we are going to show some results on superposition operator and nonlinear integral operator acting in the space of functions of bounded -integral -variation.

Piotr Maćkowiak**Action and continuity of superposition operators**

During the seminar we will present results from the book R.M. Dudley, R. Norvaisa, Concrete functional calculus (Springer 2011) concerning conditions for the action and continuity of superposition operators acting in the space of functions of bounded variation in the Jordan sense.

Monika Naskręcka

**Necessary and sufficient conditions of stability some particular differential equations system**

(Continuation of the 2016-10-25)

Monika Naskręcka

**Necessary and sufficient conditions of stability some particular differential equations system**

(Continuation of the 2016-10-18)

Monika Naskręcka**Necessary and sufficient conditions of stability some particular differential equations system**

During this seminar we will present lemmas and theorems which are necessary and sufficient conditions of stability some particular differential equations system with discontinuous right-hand side, which describe the dynamics of prices and stocks.

Marcin Borkowski**On some metric spaces**

During the talk we will present certain construction of metric spaces,

described by Száz and generalizing the *linking construction* known

from Aksoy and Maurizi's paper. We will then proceed to examine some of

its properties, in particular connected with measures of noncompactness.

Marcin Borkowski

**On hyperconvex hull of diversities**

(Continuation of the 2016-06-07)

Marcin Borkowski**On hyperconvex hull of diversities**

The talk will be devoted to problems with introducing the analogue of

hyperconvex hull (equivalently, tight span) of a diversity.

Lech Pasicki**Classical fixed point theorems**

Author’s extensions of the well-known fixed point theorems will be presented. The theorem of Banach, the theorem of Caristi (also with the variational results), and the theorem of Browder-Göhde-Kirk for nonexpansive mappings are involved. Our versions extend these theorems, as their specific assumptions are weaker, and the spaces under consideration are more general.

Adam Nawrocki**Levitan almost periodic solutions to some linear differential equation**

(Continuation of the 2016-04-19)

Adam Nawrocki**Levitan almost periodic solutions to some linear differential equation**

During this lecture we will show that a Levitan almost periodic (briefly: LAP) solution to the equation ' usually can be expressed by means of convolution. We will show that in this way we can get an unbounded LAP solution. We will give an example showing that the convolution in not always a good tool to look for LAP solutions.

Marcin Borkowski**Another look at hyperconvex hulls**

The talk will be devoted to a certain generalization of the notion of

a metric space, inspired by applications to taxonomy, in which one can

introsuce an analogue of the notion of hyperconvexity. We will start

with an a bit atypical angle on Isbell's hyperconvex hull, and then we

will proceed to describe some problems with transferring his ideas to

the generalization in question.

Albert Kubzdela**On some properties of some measure of non-compactness in non-Archimedean analysis**

Let be a locally compact non-Archimedean field, and a non-Archimedean Banach space over . We define non-Archimedean equivalents of several well-known measures of non-compactness defined on , we describe their properties and we present quantitative versions of several classic statements on weak compactness (theorems of Krein, Gantmacher and Grothendieck) for non-Archimedean Banach spaces.

Adam Nawrocki**The autonomous superposition operator defined on the space of almost periodic functions in view of the Lebesgue measure**

(Continuation of the 2016-03-15)

Adam Nawrocki**The autonomous superposition operator defined on the space of almost periodic functions in view of the Lebesgue measure**

During this lecture we shall discuss autonomuous superposition operators generated by mappings and acting in the space of -almost periodic functions.

Piotr Maćkowiak**Continuity of the non-autonomous superposition operator and around**

(Continuation of the 2016-01-26 and 2016-03-01 Nonlinear Seminar)

Piotr Maćkowiak**Continuity of the non-autonomous superposition operator and around**

(Continuation of the 2016-01-26)

Piotr Maćkowiak**Continuity of the non-autonomous superposition operator and around**

We shall present an example of a function which is lipschitzian and such that the (non-autonomous) superposition operator it generates is not continuous (as a mapping from to ). We shall show necessary and sufficient conditions for a non-autonomous superposition operator to be continuous (in ). We shall also prove some results for -class generators.

Marcin Borkowski**On some fixed point theorem in ultrametric spaces**

(Continuation of the 2015-12-15 and 2016-01-12 Nonlinear Seminar)

Marcin Borkowski**On some fixed point theorem in ultrametric spaces**

(Continuation of the 2015-12-15 Nonlinear Seminar)

Marcin Borkowski**On some fixed point theorem in ultrametric spaces**

We will discuss a theorem establishing the existence of a fixed point and a (generalized) sequence approximating this fixed point. The talk is based in a paper by S. Priess-Crampe and P. Ribenboim from 2013.

Mateusz Krukowski**Darbo type theorem for quasi-measures of non-compactness.**

This presentation introduces the notion of a quasi-measure of non-compactness. Such functions need not to be invariant in a convex hull in contrast to classical measures of non-compactness. Using the compactness criterion in the space ( are Banach spaces with ), we construct a quasi-measure of non-compactness and study its properties. Furthermore we prove the analog of the Darbo theorem using the measure of non-convexity. We also show the application of the shown tools to finding a fixed point of the Hammerstein operator.

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-11-17 Nonlinear Seminar)

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-10-20, 2015-11-03 and 2015-11-10 Nonlinear Seminar.)

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-10-20 and 2015-10-27 Nonlinear Seminar)

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-10-20 Nonlinear Seminar)

Elżbieta Gajecka**Resampling methods in the time domain for time series with periodic and almost periodic structure.**

(The seminar will be held in room B3-38)

Statistical reasoning, in the case of time series, based on asymptotic distributions cannot always be a basis for effective statistical procedures. However unknown distributions of estimators or statistics can be approximated by the so-called resampling procedures. The idea of resampling methods is obtaining replications of the estimator and calculating the empirical distribution from those replications. The main question, to be answered, is whether this empirical distribution, called the resampling distribution, is close to the real distribution.

Intense research is being conducted towards resampling methods in nonstationary time series, in particular series with periodic and almost periodic structure. During this seminar various methods of resampling, especially subsampling, will be shown. The advantage of subsampling is its insensitivity to the shape of the asymptotic distribution. We will also show conditions of compatibility of resampling methods in the domain of time for -mixing or weakly dependent time series with periodic or almost periodic structure. Especially weak dependency gives new tools to analyse statistical procedures for very general data generating time series, together with periodic series with long memory and heavy tails. An example of a periodic model with heavy tails and long memory will be shown.

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

This presentation concerns an article in which the problem of choosing the consumption path for a consumer using hyperbolic discounting (he treats the present and the whole future differently) in a model with random rate of return is discussed. The model, along with assumptions and the most important properties of the solution are presented.

Piotr Kasprzak**Abel's integral equation**

(Continuation of the 2015-10-06 Nonlinear Seminar.)

Piotr Kasprzak**Abel's integral equation**

The aim of the talk is to discuss mathematical background for the theory of Abel's integral equation and to present some of its real world applications.

Adam Nawrocki**Doubly periodic functions**

(Continuation of the 2015-05-05 and 2015-04-28 Nonlinear Seminar)

Prof. Xiao-Xiong Gan

(Mathematics Department, Morgan State University, Baltimore, USA)**Some Remarks on Minkowski's Inequality**

The classical Minkowski's inequality has two different forms based on the values of the positive number :

- If , then ;
- If , then .

We provide a one-form Minkowski's inequality for all . A non-conjugate Hölder's inequality is also introduced. Some applications of the one-form Minkowski's inequality and non-conjugate Hölder's inequality are introduced too.

**The lecture will be delivered in English**

Patryk Kołacki**Sliding mode control in automatic control systems**

(Continuation of the 2015-05-12 Nonlinear Seminar)

Patryk Kołacki**Sliding mode control in automatic control systems**

In this seminar, I'm going to present you on the most popular control laws, i.e. sliding mode control. There will be discussed it's basic, canonical form and it's most popular applications. I also want to show a few examples which let you understand the idea of this control law.

Adam Nawrocki**Doubly periodic functions**

(Continuation of the 2015-04-28 Nonlinear Seminar)

Adam Nawrocki**Doubly periodic functions**

During this seminar we shall discuss the notion of doubly periodic functions and almost periodic functions in the context of elliptic functions.

dr Przemysław Chełminiak**What connects a biological cell with a partial diffusion equation.**

Tracking individual molecules in living biological cells has become possible today by a new imaging technique called single-molecule spectroscopy. Interestingly, a diffusion of such tiny nano-molecules as proteins, nucleic acids or semiflexible polymers within a dense cellular environment is slower than ordinary Brownian diffusion of the micro-particles suspended in water. Their spatial development inside the cell, characterized by the time or ensemble averaged mean squared displacements, does not grow linearly in time. Moreover, these two averaging procedures are not equivalent when the measurement time is long compared to the characteristic time scale of the diffusion process. Therefore, the erratic motion of the subcellular molecules can not be perceived as a typical Brownian motion, but it must be considered in terms of the anomalous diffusion (subdiffusion). From a formal point of view, the anomalous diffusion can be described by a widely applicable stochastic process known as the continuous-time random walk model. We apply this method to construct the so called fractional Fokker-Planck equation which is an extended version of the subdiffusion equation in an external force field and close to thermal equilibrium. Using the methods of fractional calculus we find a few special solutions of this fractional partial differential equation and show that the mean squared displacement of a subdiffusing particle scales with time according to the power-law.

dr Giselle Monteiro**New convergence theorem for the abstract Kurzweil-Stieltjes integral**

In theory of Riemann integral, the impact of the Bounded Convergence Theorem, also called Arzela or Arzela-Osgood or Osgood Theorem, is comparable to the importance of the Lebesgue Dominated convergence Theorem in the theory of the Lebesgue integration. In this talk we are concerned with the abstract Kurzweil-Stieltjes integral, that is, the Stieltjes type integral for functions with values in a Banach space introduced by S. Schwabik. Our aim is to present the Bounded Convergence Theorem in this general setting.

Piotr Maćkowiak**A combinatorial lemma ant it's consequences**

(Continuation of the 2015-03-17 Nonlinear Seminar)

Marcin Borkowski**On a generalisation of a metric space**

(Continuation of the 2015-03-10 Nonlinear Seminar)

Piotr Maćkowiak**A combinatorial lemma ant it's consequences**

During this seminar we shall introduce a combinatorial lemma stating the existence for particular triangulations of the -dimensional simplex with a particular labeling of vertices of such triangulations with numbers of a chain of simplices connecting a "vartex" with a "base" such that for any simplex from that chain each element of the set is a label of some vertex of that simplex. We shall show, that (simple) consequences of this lemma are:

- the existence of zeros of the multidimensional function of excess demand
- the existence of continuum zeros of the parametrised multidimensional function of excess demand
- the well-known Sperner lemma

Marcin Borkowski**On a generalisation of a metric space**

During this seminar we shall give the axioms of a , which generalises the notion of metric spaces. We shall give examples of and prove some of their properties.

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

(Continuation of the 2015-01-13, 2015-01-20 and 2015-01-27 Nonlinear Seminar)

dr Jacek Gulgowski**Compactness in spaces of functions of bounded variation.**

The studies on the existence of fixed points of mappings in Banach spaces often involve methods connected with the Schauder fixed point theorem or the Leray-Schauder degree. Both approaches require the considered mappings to be completely continuous, and therefore, if one wants to work in spaces of functions of bounded variation, it is crucial to understand the mutual relations between and spaces. During the seminar we are going to discuss certain results on compact embeddings of such spaces and characterise the complete continuity of some nonlinear integral operators acting in spaces.

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

(Continuation of the 2015-01-13 and 2015-01-20 Nonlinear Seminar.)

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

(Continuation of the 2015-01-13 Nonlinear Seminar.)

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

During this seminar we shall introduce the definition of almost periodic patterns, their basic properties and connection to quasicrystals.

Monika Naskręcka**The existence of solutions of a certain system of differential equations with discontinuous right-hand side.**

During this seminar we will discuss the existence of solutions of a certain system of differential equations with discontinuous right-hand side. This system depicts the dynamics of supplies in a model of competitive economy.

Marcin Borkowski**On a certain fixed point theorem for ultrametric spaces**

We will discuss some variant of the Banach Contraction Principle, announced by N. Ackerman, for ultrametric spaces with metrics with values in a complete lattice.

(Continuation of the 2014-11-18 and 2014-11-04 Nonlinear Seminar.)

dr Przemysław Chełminiak

**Stochastic dynamics of self-organizing critical nets**

Self-organizing critical nets have been an object of intense research for over 15 years. They possess many peculiar properties, namely: fractality, self-similarity, topological criticality, little world effect and they are scale-free, which follow from their stochastic evolution. It occurs that many of such nets exist in an unstable state on the boundary of two phases, small-fractal world. Methods of construction and analysing critical nest will be shown, along with basic properties of probability distributions of stationary streams following stochastic dynamics of such nets.

Marcin Borkowski**On a certain fixed point theorem for ultrametric spaces**

We will discuss some variant of the Banach Contraction Principle, announced by N. Ackerman, for ultrametric spaces with metrics with values in a complete lattice.

(Continuation of the 2014-11-04 Nonlinear Seminar.)

**On a certain fixed point theorem for ultrametric spaces**

We will discuss some variant of the Banach Contraction Principle, announced by N. Ackerman, for ultrametric spaces with metrics with values in a complete lattice.

Dr Mateusz Maciejewski**Existence of positive solutions of systems of parabolic equations with non-local initial conditions.**

The topic of this speech is the existence of positive solutions of systems of parabolic equations with the boundary Dirichlet conditions and non-local initial conditions. For this purpose I shall show a fixed point theorem, which proof is based on Granas' fixed-point index. Thanks to the possibility of localisation of solutions I shall receive a theorem about existence of more than one solution.

Marcin Borkowski**Fixed point theorems with a boundary condition**

Piotr Kasprzak**The game of Hex and the Brouwer fixed-point theorem.**

During the seminar we are going to porve show that the celebrated Boruwer fixed-point theorem is a consequence of the fat that Hex cannot end in a draw.

Marcin Borkowski**Fixed point theorems with a boundary condition**

In 1998, Espínola and López proved a certain fixed point theorem for a mapping transforming an admissible subset of a hyperconvex metric space into the whole space, assuming that this mapping sends the boundary of into . We will show how to simplify their proof considerably, at the same time relaxing its assumptions. In particular, this will enable proving a new fixed point theorem, generalizing a classical result of Baillon from 1988.

prof. Gennaro Infante**Nontrivial solutions of perturbed Hammerstein integral equations**

We discuss the existence of positive solutions of some perturbed Hammerstein integral equations by means of classical fixed point index theory.We apply our results to some nonlocal BVPs that arise in some heat flow problems to prove the existence of multiple positive solutions under suitable conditions. We also illustrate how this approach might be applied to different problems.

Dariusz Bugajewski**BV-solutions of nonlinear equations**

During the seminar we are going to prove the existence of BV-solutions to several nonlinear equations using the fixed point approach.

Adam Nawrocki**The convolution operator defined on the space of Levitan almost periodic functions.**

(Continuation of the 2014-05-27 Nonlinear Seminar)

Adam Nawrocki**The convolution operator defined on the space of Levitan almost periodic functions.**

It is known, that the convolution of a bounded, Levitan almost periodic function with a function is a bounded, Levitan almost periodic function. During this lecture we shall discuss the result we will receive when unbounded Levitan almost periodic functions are used.

Adam Nawrocki**Asymptotic behaviour of a certain almost periodic function with regard to the Lebesgue measure**

The function

is a classical example of an unbounded and continuous -almost periodic function. For this function we have

During this lecture we will discuss the idea of the proof of the equality above, which uses diophantic approximations. We shall furthermore show, that the limit

does not exist.

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2014-04-29, 2014-03-25, 2014-01-28, 2014-01-21, 2013-11-05 and 2013-10-29 Nonlinear Seminar)

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2014-03-25, 2014-01-28, 2014-01-21, 2013-11-05

and 2013-10-29 Nonlinear Seminar)

Adam Nawrocki**Asymptotic behaviour of some -almost periodic functions**

The function

is a classical example of an unbounded and continuous -almost periodic function. For this function we have

During this lecture we will consider if changing the number to another irrational number may have relevant impact on the behaviour of this function. We shall construct such a number that the limit

will not exist.

The lecture will be preceded by a short speech by Marcin Borkowski.

dr Michał Goliński**About some properties of the space of almost periodic functions **

We will be talking about some properties of the space of the almost periodic functions, in particular the properties of the weak topology on this space (Cauchy sequences, completeness, reflexivity).

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2013-10-29 and 2013-11-05 Nonlinear Seminar.)

Marcin Borkowski**On some application of Mönch theorem**

(Continuation of the 2014-02-25 and 2014-03-04 Nonlinear Seminar.)

Dr Jacek Gulgowski

(Institute of Mathematics, Gdańsk University)**On the continuity of superposition operator in the spaces of functions of bounded variation**

It is a well-know fact that the superposition of a function satisfying a local Lipschitz condition with a function of bounded variation leads to the function which is also of bounded variation. During the seminar we are going to present a few results concerning the continuity of such a superposition operator . Furthermore, we are going to discuss several generalizations of the concept of bounded variation in the sense of Jordan and investigate the issue of continuity of the superposition operator , where denotes the space of functions of bounded variation in the generalized sense.

Marcin Borkowski**On some application of Mönch theorem.**

(Continuation of the 2014-02-25 Nonlinear Seminar.)

Marcin Borkowski**On some application of Mönch theorem.**

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2014-01-21 Nonlinear Seminar)

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2013-10-29 and 2013-11-05 Nonlinear Seminar)

prof. UW dr hab. Urszula Foryś (Warsaw University)**Modelling of biochemical reactions with delays**

Nowadays, in the description of biochemical reactions not only ordinary differential equations but also equations with delays are applied. Time delays can be used to reflect different time scales at which modelled reactions occur. During our (together with M. Bodnarem, J. Miękisz and J. Poleszczuk) research we have found that simple, intuitive methods of introducing delay can lead to improper conclusions, such as oscillatory dynamics of the simple protein delayed degradation process. I shall show that the simplest model proposed by Bartsun et al. (2005) should be modified. It occurs that to obtain oscillatory dynamics one needs to consider much more complex reactions channel, with negative feedback and dimerization.

The seminar is to be held in room **B1-37**.

dr Aneta Sikorska-Nowak**Nonlinear Sturm – Liouville dynamic equation**

Will be presented the necessary concepts of dynamic equations on time scales, and nonlinear dynamic equation of Sturm-Liouville.

Marcin Karczewski**Boltzmann kinetic equation**

(Continuation of the 2013-12-10 Nonlinear Seminar)

Marcin Karczewski**Boltzmann kinetic equation**

The aim of the seminar is to discuss the Boltzmann kinetic equation and its physical consequences. After gaining intuition we shall return to the earlier discussed VHP gas model.

Piotr Kasprzak**Application of formal power series to differential equations**

During the seminar we are going to discuss the Frobenius Method, that is, we are going to seek solutions to certain differential equations (not necessarily of integer-order) in the form of formal power (or Laurent) series. Several illustrative examples will be provided.

dr Aneta Sikorska-Nowak**Asymptotic stability of the integro-differential equation with delay**

(Continuation of the 2013-11-19 Nonlinear Seminar)

dr Aneta Sikorska-Nowak**Asymptotic stability of the integro-differential equation with delay**

Certainly, the Lyapunov direct method has been, for more than 100 years, the main tool for the study of stability properties of ordinary, functional, integro-differential and partial differential equations. Nevertheless, the applications of this method to problem of stability in differential equations with delay has encountered serious difficulties if the delay is unbounded or if the equation has unbounded terms. The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov’s direct method.

I will present the results on the asymptotic stability of solutions of nonlinear integro-differential equations with delay using Sadovskii Fixed Point Theorem and properties of the measure of noncompactness.

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2013-10-29 Nonlinear Seminar)

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

Equilibrium and stability are the most frequently investigated properties in the market equilibrium models. On global and local markets we usually observe the lack of equilibrium. Nevertheless, the original model does not consider the possibility of stocks. However, we observe this situation, when there exists an excess supply. We should consider then how the accumulated stocks affect the equilibrium (or disequilibrium) and the stability of the model. During the seminar I will present the competitive economy model and the model with stocks. I will explore how these stocks influence the path of the price trajectory, equilibrium conditions and price stability.

Piotr Kasprzak**Application of formal power series to differential equations**

(Continuation of the 2013-05-07 Nonlinear Seminar)

Piotr Kasprzak**Application of formal power series to differential equations**

During the seminar we are going to discuss the Frobenius Method, that is, we are going to seek solutions to certain differential equations (not necessarily of integer-order) in the form of formal power (or Laurent) series. Several illustrative examples will be provided.

Dr. hab. Aleksander Ćwiszewski**TITLE soon**

Prof. dr hab. Bronisław Jakubczyk (Institute of Mathematics, Polish Academy of Sciences)**Series of formally noncommutative variables versus functions and vector fields on manifolds**

Taylor coefficients of a function can be calculated as iterated derivatives along constant vector fields (versors), computed at a fixed point. If a constant vector fields is replaced by any fixed set of vector fields on a manifold, a similar operation leads to the definition of a formal series with noncommutative variables.

We will show some interesting facts connected with such series, in particular the so-called *realization theorem* and its consequences. In particular, we will prove that a controlled and observed dynamical system on a manifold can be represented by such a series. Another consequence is a new test of convergence of a series of noncommutative variables.

Jędrzej Sadowski**About integrate-and-fire models**

We will talk about two integrate-and-fire models.

The first one will be the neon bulb circuit, the second a japanese water toy.

Piotr Zdanowicz**Basic properties of formal series**

During the seminar we will discuss and prove some properties of formal series.

(Continuation of the 2012-11-20 and 2012-11-27 Nonlinear Seminar)

Marcin Borkowski**Some properties of the Henstock-Kurzweil Integral**

(Continuation of the 2013-01-22 and 2013-01-15 Nonlinear Seminar)

Marcin Borkowski**Some properties of the Henstock-Kurzweil Integral**

(Continuation of the 2013-01-15 Nonlinear Seminar)

Marcin Borkowski**Some properties of the Henstock-Kurzweil Integral**

During this seminar we will discuss some properties of the Henstock-Kurzweil Integral for an unbounded domain. In particular we shall prove that the functions of significantly bounded variation are multipliers of HK-integrable functions.

Monika Naskręcka**Homeomorphism of the space of polynomials and the space of their roots**

(Continuation of the 2012-12-11 and 2012-12-18 Nonlinear Seminar)

Monika Naskręcka**Homeomorphism of the space of polynomials and the space of their roots**

(Continuation of the 2012-12-11 Nonlinear Seminar)

Monika Naskręcka**Homeomorphism of the space of polynomials and the space of their roots**

During this seminar an elementary, topological proof of continuity of the roots of complex polynomials with respect to their roots will be shown. For normed polynomials with complex coefficients and properly defined metric spaces this theorem can be expressed as a homeomorphism of corresponding metric spaces of coeffecients and roots.

Piotr Maćkowiak**The existence of surplus demand function zeros**

(Continuation of the 2012-10-09 and 2012-10-16 Nonlinear Seminars.)

Piotr Zdanowicz**Basic properties of formal series**

(Continuation of the 2012-11-20 Nonlinear Seminar)

Piotr Zdanowicz**Basic properties of formal series**

Piotr Kasprzak**On a certain class of functions of -bounded variation**

(Continuation of the 2012-10-30 Nonlinear Seminar)

The seminar will take place at 8:30 in the Faculty Council meeting room (A1-33/34)

Justyna Signerska**Firing map for integrate-and-fire models with an almost periodic drive**

Consider the *integrate-and-fire* system , , in which the continuous dynamics is interrupted by the threshold-and-reset behaviour: if , meaning that once the value of a dynamical variable reaches a certain threshold it is immediately reset to a resting value and the system evolves again according to the differential equation. The question is to describe the time series of consecutive resets (spikes) as iterations of some map , called the *firing map*, and the sequence of interspike-intervals as a sequence of displacements along a trajectory of this map. The problem appears in various applications, e.g. in modelling of an action potential (spiking) by a neuron.

However, so far properties of the firing map were analytically investigated only for the function , which is smooth enough and often also periodic in . We present a complete description of the properties of the firing map arising from the most popular models: (Leaky Integrate-and-Fire) and (Perfect Integrator), when the function is only locally integrable and/or almost periodic. In particular, we prove that a Stepanov almost periodic function induces the firing map with the uniformly (Bohr) almost periodic displacement . In this way we provide a formal framework for next studying of the interspike-intervals in almost-periodically driven integrate-and-fire models.

Piotr Kasprzak**On a certain class of functions of -bounded variation**

During the seminar we are going to define a certain class of functions of -bounded variation and prove some properties of such functions. Furthermore, we are going to show that endowed with a certain functional the class in questions becomes a strictly convex Banach space. We are also going to provide several non-trivial examples illustrating our considerations.

Piotr Kasprzak**On a certain class of functions of -bounded variation**

During the seminar we are going to define a certain class of functions of -bounded variation and prove some properties of such functions. Furthermore, we are going to show that endowed with a certain functional the class in questions becomes a strictly convex Banach space. We are also going to provide several non-trivial examples illustrating our considerations.

Piotr Maćkowiak**The existence of surplus demand function zeros**

(Continuation of the 2012-10-09 Nonlinear Seminar)

Piotr Maćkowiak**The existence of surplus demand function zeros**

Piotr Kasprzak**On a certain class of functions of bounded variation**

(Continuation of the 2012-04-24 Nonlinear Seminar)

Michał Burzyński**Alternative proofs of Arrow's impossibility theorem**

(Continuation of the 2012-05-08 Nonlinear Seminar)

Michał Burzyński**Alternative proofs of Arrow's impossibility theorem**

(Continuation of the 2012-05-08 Nonlinear Seminar)

Michał Burzyński**Alternative proofs of Arrow's impossibility theorem**

Arrow's impossibility theorem states, that if there are at least three potential social states, there is no such social welfare function (that describes the society as a whole with given individual preferences), which satisfies the following four conditions: has an unbounded domain; the strong relation of social preferences satisfies the weak Pareto principle; satisfies the property of independence of irrelevant alternatives and in a society with preferences described with the function there is no dictatorship. The original proof, given by K. Arrow, consists of two steps: to show the existence of a deciding unit and to prove, it must be a dictator. The first of the alternative proofs is based on using the Condorcet preference. The second proof applies the first step of Arrow's proof and points, that all social decisions are made in the same way. The last proof is of graphic character.

Piotr Kasprzak**On a certain class of functions of bounded variation**

We are going to define a relatively large class of functions of bounded variation defined on an open subset of , which in particular contains the Sobolev space . Furthermore, we are going to discuss the connection between the classical variation in the sense of Jordan and the generalized one.

Piotr Kasprzak**On the compactness criterion in the space of bounded real-valued continuous functions defined on a non-compact domain**

During the seminar we are going to discuss a certain compactness criterion in the space of bounded real-valued continuous functions defined on a non-compact domain related to the well-known Arzela-Ascoli theorem.

Piotr Maćkowiak**Exploding points**

We will present and prove theorem of the exploding point. This theorem states that for any function defined on a compact set X, containing the cube K centered at zero in its interior, with values in X\K, which is the identity on the boundary of the set X, there exists a point c in X, that in any surroundings of c, there is a point x such that the value of f(x) and f(c) are located on opposite walls of the cube K.

Adam Nawrocki**An Application of Continued Fractions**

During this seminar we will discuss basic properties of Continued Fractions. Next we shall use Continued Fractions to calculate the limit of:

(Continuation of the 2012-03-13 Nonlinear Seminar)

Adam Nawrocki**An Application of Continued Fractions**

During this seminar we will discuss basic properties of Continued Fractions. Next we shall use Continued Fractions to calculate the limit of:

Marcin Wachowiak**Applications of Theorem Hopf-Lefschetz fixed point**

(Continuation of the 2011-11-22 Nonlinear Seminar and 2011-11-29 Nonlinear Seminar and 2011-12-06 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar, 2012-01-03 Nonlinear Seminar, 2012-02-14 Nonlinear Seminar and 2012-02-21 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar and 2012-01-03 Nonlinear Seminar and 2012-02-14 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar and 2012-01-03 Nonlinear Seminar).

Daria Bugajewska**On differential and inetgral equations in the spaces of functions of Lambda-bounded variation.**

(Continuation of the 2012-01-10 Nonlinear Seminar and 2012-01-17 Nonlinear Seminar).

Daria Bugajewska**On differential and inetgral equations in the spaces of functions of Lambda-bounded variation.**

(Continuation of the 2012-01-10 Nonlinear Seminar).

Daria Bugajewska**On differential and inetgral equations in the spaces of functions of Lambda-bounded variation.**

We are going to consider linear differential and nonlinear integral equations in the

spaces of functions of Lambda-bounded variation. We will state the existence and the existence and uniqueness results in this space as well as in its subspace consisting of contninuous functions of Lambda-bounded variation.

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

We will show differend ways to define N-almost periodic functions and prove their equivalence.

Marcin Wachowiak**Applications of Theorem Hopf-Lefschetz fixed point**

(Continuation of the 2011-11-22 Nonlinear Seminar and 2011-11-29 Nonlinear Seminar and 2011-12-06 Nonlinear Seminar).

Marcin Wachowiak**Applications of Theorem Hopf-Lefschetz fixed point**

(Continuation of the 2011-11-22 Nonlinear Seminar and 2011-11-29 Nonlinear Seminar).

Marcin Wachowiak**Applications of Theorem Hopf-Lefschetz fixed point**

(Continuation of the 2011-11-22 Nonlinear Seminar).

Marcin Wachowiak**Applications of Theorem Hopf-Lefschetz fixed point**

In the first part of the discussion we introduce the necessary concepts and facts about the homology and cohomology, then we will discuss the Hopf-Lefschetz theorem and its applications

Marcin Borkowski, Piotr Maćkowiak (Poznań University of Economics)**On superposition of formal power series**

(Continuation of the 2011-11-08 Nonlinear Seminar)

Marcin Borkowski, Piotr Maćkowiak (Poznań University of Economics)**On superposition of formal power series**

We will present a new proof of a criterion for existence of superposition of formal power series given by Gan and Knox. We will also give some properties concerning the behavior of such series on the boundary of their convergence circle.

Adam Burchardt**Schauder fixed point, revisited**

(Continuation of the 2011-10-18, 2011-10-11 and 2011-10-04 nonlinear seminars).

Adam Burchardt**Schauder fixed point, revisited**

(Continuation of the 2011-10-11 and 2011-10-04 nonlinear seminars).

Adam Burchardt**Schauder fixed point, revisited**

(Continuation of the 2011-10-04 Nonlinear Seminar).

Adam Burchardt**Schauder fixed point, revisited**

During the seminar we are going to discuss the following generalization of the classical Schauder fixed point theorem due to R. Cauty:

*Let be a convex subset of a topological vector space . If is a continuous function such that is contained in a compact subset of , then has a fixed point.*

The talk is based on the following paper: R. Cauty, *Solution du problème de point fixe de Schauder*, Fund. Math. **170** (2001), 231-246.

Marcin Borkowski**Some remarks on convergence of power series at the boundary of the convergence circle**

We will present a theorem concerning convergence of power series—together with all its derivatives—at the boundary of the convergence circle.

Joanna Wypych**On superposition of formal power series**

(Continuation of the 2011-05-10 and 2011-05-17 seminars)

Joanna Wypych**On superposition of formal power series**

(Continuation of the 2011-05-10 Nonlinear Seminar)

Joanna Wypych**On superposition of formal power series**

In the beginning we will present basic definitions concerning formal power series and their properties. Then we will discuss two theorems on superposition of such series. The talk is based on a paper *On composition of formal power series* by Xiao-Xiong Gan and Nathaniel Knox.

Piotr Kasprzak**On the equivalent characterizations of Levitan almost periodic functions**

(Continuation of the 2011-02-22 Nonlinear Seminar)

Marcin Borkowski**On formal power series**

(Continuation of the 2010-12-21 and 2011-03-15 Nonlinear Seminar seminars)

Adam Nawrocki**Examples of Henstock-Kurzweil integrable functions**

We will present methods of proving Henstock-Kurzweil integrability of real functions.

Piotr Kasprzak**On a new class of almost periodic functions**

During the talk not only will we define the so-called quasi almost automorphic functions but also we will prove some basic properties of these functions.

Joanna Wypych**Formal power series**

(Continuation of the 2011-01-18 and 2011-03-08 seminars)

Marcin Borkowski**On formal power series**

(Continuation of the 2010-12-21 Nonlinear seminar).

Joanna Wypych**Formal power series**

(Continuation of the 2011-01-18 Nonlinear Seminar)

**On the dual space to the space of Henstock-Kurzweil integrable functions**

During the talk we will present the result of Alexiewicz which gives a characterization of the dual space to the space of Henstock-Kurzweil integrable functions defined on a bounded interval.

Piotr Kasprzak**On the equivalent characterizations of Levitan almost periodic functions**

Joanna Wypych**Formal power series**

We will discuss basic properties of formal power series and the Krull topology. Among others, we will present theorems on the distributive laws and invertibility of a series. The talk will be based on the book by S. Łojasiewicz and J. Stasica *Formal analysis and analytic functions* (in Polish).

Marcin Wachowiak**A fixed point theorem for mappings satisfying the interior condition**

(Continuation of the 2010-11-23, 2010-11-30 and 2010-12-07 Nonlinear seminars).

Wojciech Jankowski**On a certain fixed point theorem for discontinuous mappings**

Marcin Borkowski**On formal power series**

The aim of the talk is to give an introduction into the theory of formal power series. After the basic definitions and properties we will show some application to combinatorics. Finally, we will show an example of a formal differential equation.

**On the compactness criterion in the space of bounded real-valued continuous functions defined on a non-compact domain**