# Lech GórniewiczInstitut de Mathématiques de Luminy | IML · University of Kqazimierz Wielki , Bydgoszcz, POLAND

Lech Górniewicz

Prof.dr hab. dr h.c.

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153

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Introduction

**Skills and Expertise**

## Publications

Publications (153)

In 1923 S. Lefschetz proved the famous fixed point theorem known as the Lefschetz fixed point theorem (comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery ([10]). They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces (absolute neighbourhood r...

The existence of fixed points and, in particular, coupled fixed points is investigated for multivalued contractions in complete metric spaces. Multivalued coupled fractals are furthermore explored as coupled fixed points of certain induced operators in hyperspaces, i.e. as coupled compact subsets of the original spaces. The structure of fixed point...

Deterministic as well as random Lefschetz-type fixed point theorems are formulated for multivalued spheric maps on various sorts of special retracts in a Euclidean space.

The fixed point index is newly generalized for a large class of spheric multivalued mappings on compact, connected Euclidean neighbourhood retracts. Sufficient conditions are then formulated for the existence of essential fixed points in terms of a nontrivial fixed point index and the Lefschetz number, provided the topological dimension of the set...

A new definition of essential fixed points is introduced for a large class of multivalued maps. Two abstract existence theorems are presented for approximable maps on compact ANR-spaces in terms of a nontrivial fixed point index, or a nontrivial Lefschetz number and a zero topological dimension of the fixed point set. The second one is applied to t...

The existence of essential fixed points is proved for compact self-maps of arbitrary absolute neighborhood retracts, provided the generalized Lefschetz number is nontrivial and the topological dimension of a fixed point set is equal to zero. Furthermore, continuous self-maps of some special compact absolute neighborhood retracts, whose Lefschetz nu...

In this paper, we establish some fixed point results of Krasnosel'skii type fixed point theorem for the sum of B + G, where B is expansive linear operator and G is a weakly continuous map or weakly-weakly upper semi-continuous multivalued operator. Finally, our results are used to prove the existence of solution for multivalued Dirihlet problem in...

In this survey, we present current results from the topological fixed point theory of multivalued mappings which were obtained by ourselves in the last five years (see Andres and Górniewicz in Fixed Point Theory 12(2):255-264, 2011; Topol. Methods Nonlinear Anal. 40:337-358, 2012; Libertas Mathematica 33(1):69-78, 2013; Int. J. Bifurc. Chaos 24(11)...

At first we shall generalize the fixed point index from the case of compact admissible mappings to the case of compact absorbing contraction mul-reversed not sign}tivalued mappings defined on arbitrary absolute neighbourhood retracts (ANR-spaces). Then we formulate, in terms of the Lefschetz number and the fixed point index, necessary conditions fo...

The Browder nonejective fixed point theorem [Browder, 1965] will be generalized in two directions. One generalization will be then applied to the existence of nonejective topological multivalued fractals on Peano's continua. An illustrative example will be supplied.

This monograph gives a systematic presentation of classical and recent
results obtained in the last couple of years. It comprehensively describes the
methods concerning the topological structure of fixed point sets and solution
sets for differential equations and inclusions. Many of the basic techniques
and results recently developed about this the...

We present a random topological degree effectively applicable mainly to periodic problems for random differential inclusions. These problems can be transformed to the existence problems of random fixed points or periodic orbits of the associated Poincaré translation operators. The solvability can be so guaranteed either directly by means of nontriv...

In this excellent book a comprehensive description of methods concerning
the topological structure of ﬁxed point sets and solution sets for diﬀerential equations and inclusions is presented. The book contains six chapters. Four main chapters and two supplementary chapters which containthe basic notions for a useful basis for the entirebook. Chapter...

In this survey paper, the authors present some existence results of mild
solutions and the topological structure of solutions sets for ﬁrst order
impulsive semilinear diﬀerential inclusions with initial and periodic
boundary conditions. This survey paper in organized in seven chapters.
Chapter one (Introduction) contains, as motivation, some models...

Infinite countable or uncountable systems of nonlinear ordinary and partial differential equations, which are discrete and continuous models, respectively, of real-world phenomena and processes were analysed in physics, biology, neuroscience, in studying of neural systems or in pure mathematics. Discrete models and corresponding infinite countable...

In this paper, we consider the existence of solutions as well as the topological
and geometric structure of solution sets for first-order impulsive differential
inclusions in some Fr´echet spaces. Both the initial and terminal problems
are considered. Using ingredients from topology and homology, the topological
structure of solution sets (closedne...

In this paper we prove existence results for first and second order impulsive functional and neutral functional differential inclusions in Banach spaces.

The aim of this paper is to present new fixed point theorems which can be used in the theory of dynamical systems. The main novelty consists especially in the fact that non-metric versions of these theorems are formulated. In the metric case, we give some simple applications to differential equations.

This paper deals with the existence of solutions to some classes of partial impulsive hyperbolic differential inclusions with variable times involving the Caputo fractional derivative. Our works will be considered by using the nonlinear alternative of Leray-Schauder type.

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions: {(y′−Ay)(t)∈F(t,y(t)), a.e. t∈J∖{t1,…,tm},y(tk+)−y(tk−)=Ik(y(tk−)),k=1,…,m,y(0)=y(b) where J=[0,b] and 0=t0t1⋯tmtm+1=b(m...

We investigate the existence and uniqueness of solutions of a class of partial impulsive hyperbolic differential equations with fixed time impulses involving the Caputo fractional derivative. Our main tool is a fixed point theorem.

A concept of generalized topological essentiality for a large class of multivalued maps in topological vector Klee admissible
spaces is presented. Some direct applications to differential equations are discussed. Using the inverse systems approach
the coincidence point sets of limit maps are examined. The main motivation as well as main aim of this...

In this paper, we first present an impulsive version of Filippov’s
Theorem for first-order semilinear functional differential inclusions with an infinitesimal generator of a C_0-semigroup on a separable Banach space E and a nonlinear set-valued map.
Then the convexified problem is considered and a Filippov–Wazewski result
is proved. Further to sev...

In this paper, we shall establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional and neutral functional differential equations with fractional order and infinite delay. Our approach is based on a nonlinear alternative of Leray-Schauder type.

In this paper we prove existence results for first and second order semilinear differential inclusions in Banach spaces with nonlocal conditions.

In this paper, we prove existence and controllability results for first- and second-order semilinear differential inclusions in Banach spaces with nonlocal conditions.

In this paper we generalize the Lefschetz fixed point theorem from the case of metric ANR-s to the case of acceptable subsets of Klee admissible spaces. The results presented in this paper were announced in an earlier publication of the authors.

We establish sufficient conditions for the existence of extremal mild solutions for perturbed neutral functional evolution inclusions in Banach spaces.

The Lefschetz number is constructed for two maps of noncompact nonorientable topological manifolds of the same dimension; it is proved that if it is nonzero, then there is a coincidence point of these maps. Bibliography: 15 titles.

In this chapter we would like to present a systematic study of the fixed point theory for multivalued maps by using homological
methods. Homological methods were initiated in 1946 by S. Eilenberg and D. Montgomery in their celebrated paper [EM]. Using
methods of homology we can obtain stronger results than those obtained by means of the approximati...

We reprove in an extremely simple way the classical theorem that time periodic dissipative systems imply the existence of harmonic periodic solutions, in the case of uniqueness. We will also show that, in the lack of uniqueness, the existence of harmonics is implied by uniform dissipativity. The localization of starting points and multiplicity of p...

We investigate the controllability of first-order semilinear functional and neutral functional differential equations in Banach spaces.

We prove controllability results for first and second order semilinear differential inclusions in Banach spaces with nonlocal conditions.

There are two significant sets of methods in the fixed point theory of multivalued mappings. The first are the so called homological methods, started in 1946 by S. Eilenberg and D. Montgomery ([EM]), and depend on using algebraic topology tools, e.g. homology theory, homotopy theory, etc. The second, started in 1935 by J. Von Neumann ([Neu]), are c...

In this chapter, we present a concise review of the requisite mathematical background.First we recall fundamental facts from geometric topology, later we discuss the part of homology theory related to the Vietoris mapping theorem and, finally, necessary information about the Lefschetz number.

The aim of this chapter is to give a systematic and unified account of topics in fixed point theory methods of differential inclusions which lie on the border line between topology and ordinary differential equations.

In this paper by using semigroup of evolution operators and fixed point argument we establish existence results for the controllability of semilinear functional and neutral functional differential inclusions in a Banach space with infinite delay when the right hand side has convex as well as nonconvex values.

Control problems appear in many branches of physics and technical science. In this paper we investigate the controllability of semilinear differential equations and inclusions via the semigroup theory in Banach spaces. All results are obtained by using fixed point theorems both for single and multivalued mappings.

In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.

Control problems appear in many branches of physics and technical sciences. In this paper weinvestigate the controllability of first order semilinear impulsive functional differential inclusions in the case where the right hand side is convex or nonconvex valued. All results are obtained by using fixed point results for multivalued mappings.

This paper discusses the topological structure of the set of solutions for a variety of Volterra equations and inclusions. Our results rely on the existence of a maximal solution for an appropriate ordinary differential equation. 1.

In this article we investigate the existence of solutions for second order impulsive hyperbolic differential inclusions in separable Banach spaces. By using suitable fixed point theorems, we study the case when the multi-valued map has convex and non-convex values.

The Lefschetz and the Nielsen periodic point theorems are developed for compact absorbing contractions on ANRs. These results are reformulated in terms of discrete multivalued semi-dynamical systems. They are also applied to Carathéodory differential inclusions on tori for obtaining the existence and multiplicity results for boundary value problems...

In this paper we prove controllability results for mild solutions defined on a compact real interval for first order dierential evolution inclusions in Banach spaces with non-local conditions. By using suitable fixed point theorems we study the case when the multi-valued map has convex as well as non-convex values.

In this section, we shall present current results concerning the Browder-Guptatype theorems for single-valued and multivalued mappings.

At first, we recall some basic properties of topological vector spaces and, in particular, locally convex spaces; for more details, see e.g. [Kö-M], [Lu-M], [RoRoM], [Ru-M], [Sch-M].

In this part, continuation principles in Chapter II.10 will be applied, jointly with some statements from the Nielsen Theory in Chapter I.11, to differential equations and inclusions. The results are based on our papers [AGG1], [AGJ2], [AndBa].

The Lefschetz Fixed Point Theorem for compact absorbing contraction morphisms (CAC-morphisms) of retracts of open subsets in admissible spaces in the sense of Klee is proved. Moreover, the relative version of the Lefschetz Fixed Point Theorem and the Lefschetz Periodic Theorem are considered. Additionally, a full classification of morphisms with co...

The topological structure of fixed point sets was investigated and technique for investigating the structure of fixed point sets of limit maps was developed. The possibility for obtaining Rδ-structure which is convenient for applications to differential inclusions was established. Several variants of multivalued generalizations of the Aronszajn the...

Given a nonempty, closed set K in a Hilbert space H, we consider the following two control problems involving the set K. 1. For a nonlinear control problem with state x epsilon H we study the existence of a state feedback control, taking values in a Hilbert space H-1, which "stabilizes" the set K in a precise sense, which is related to the viabilit...

We give a survey of recent results concerning the Banach contraction principle for multivalued mappings. Nevertheless, this survey contains also some new so far unpublished results. The following main problems are concerned:
(i)
existence of fixed points
(ii)
topological structure of the set of fixed points
(iii)
generalized essentiality.
Some appl...

The relative Lefschetz and Nielsen fixed-point theorems are generalized for compact absorbing contractions on ANR-spaces and nilman- ifolds. The nontrivial Lefschetz number implies the existence of a fixed- point in the closure of the complementary domain. The relative Nielsen numbers improve the lower estimate of the number of coincidences on the...

The Nielsen number is defined for a rather general class of multivalued maps on compact connected ANRs, including, e.g., admissible maps (in the sense of Górniewicz (1976); compare also Górniewicz (1995)) on tori. Since the Poincaré maps generated by the Marchaud vector fields are of this type (see (Andres, 1997)), we can obtain in such a way multi...

A class of multivalued mappings, called by us mixed semicontinuous mappings, which are upper semicontinuous in some points and lower semicontinuous in remaining points is considered. A general selection theorem for mixed semicontinuous maps is proved. Then applications to differential inclusions with mixed semicontinuous right-hand sides are presen...

Using methods from the multivalued analysis we show the existence of feedback controls which “stabilizes” a given closed set
\(
K \subseteq {\mathbb{R}^n}
\)
satisfying a suitable regularity property with respect to the dynamics of a nonlinear control system. For this, we study the structure and the properties of the external contingent Bouligand c...

Acyclicity of solution sets to asymptotic problems, when the value is prescribed either at the origin or at infinity, is proved for differential inclusions and discontinuous autonomous differential inclusions. Existence criteria showing that such sets are non-empty are obtained as well.

Starting from the famous Schauder fixed-point theorem, we present some Lefschetz-like and Nielsen-like generalizations for certain ad- missible (multivalued) self-maps on metric ANR-spaces. These fixed-point principles are applied for obtaining the existence and multiplicity results for boundary value problems.

A concept of topological essentiality is used to prove several existence results for nonlinear boundary value control problems. Some examples to illustrate the obtained results are presented.

We gather in this chapter the properties of multivalued maps (called also setvalued or multiple-valued maps) which are needed for the study of the fixed point theory and applications to nonlinear analysis. The first three sections deal with the concept of continuity. Then we consider the selection problem and the continuity of multivalued mappings...

BACKGROUND IN TOPOLOGY.- MULTIVALUED MAPPINGS.- APPROXIMATION METHODS IN FIXED POINT THEORY OF MULTIVALUED MAPPINGS.- HOMOLOGICAL METHODS IN FIXED POINT THEORY OF MULTIVALUED MAPPINGS.- CONSEQUENCES AND APPLICATIONS.- FIXED POINT THEORY APPROACH TO DIFFERENTIAL INCLUSIONS.- RECENT RESULTS.

. We shall consider periodic problems for ordinary di#erential equations of the form # x # (t) = f(t, x(t)), x(0) = x(a), (I) where f : [0, a] R n # R n satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of R n , the topological degree of,...

We give a survey of recent results concerning the Brouwer fixed point theorem for multivalued mappings. Some new results, open problems and concluding remarks are presented.

For a class of contractive multivalued maps defined on a complete absolute retract and with closed bounded values, the set of fixed points is proved to be an absolute retract. This result unifies and extends to arbitrary absolute retracts both Theorem 1 by B. Ricceri [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 81, No. 3, 28...

Further extension of the Levinson transformation theory is performed for partially dissipative periodic processes via the fixed point index. Thus, for example, the periodic problem for differential inclusions can be treated by means of the multivalued Poincaré translation operator. In a certain case, the well-known Ważewski principle can also be ge...

First, an analog of the deformation lemma is proved for functions f: M × P ® R, where M is an infinitedimensional Finsler manifold, P is a space of parameters, and f is continuous in both variables and locally Lipschitz in the first one. Taking P to be a compact manifold one derives a result generalizing both the Chang’s extension of the deformatio...

## Projects

Projects (3)

Dear Colleagues
I am pleased to inform you about our new journal Set-Valued Mathematics and Applications web page http://www.mukpublications.com/svm.php. Please provide information to interested specialists and also by preparing the materials in the journal. I request you to
kindly start inviting the articles for the coming issue of the journal.
Yours sincerely Yuri Zelinskyi