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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 sets is considered in terms of absolute retracts. We also formulate a continuation principle for multivalued contractions as a nonlinear alternative based on the topological essentiality. Two illustrative examples about coupled multivalued fractals are supplied.

Fixed point theory is very useful in nonlinear analysis, diferential equations, differential and random differen- tial inclusions. It is well known that different types of fixed points implies the existence of specific solutions of the respective problem concerning differential equations or inclusions. There are several classifications of fixed points for single valued mappings. Recall that in 1949 M.K. Fort [19] introduced the notion of essential fixed points. In 1965 F.E. Browder [12], [13] introduced the notions of ejective and repulsive fixed points. In 1965 A.N. Sharkovsky [31] provided another classification of fixed points but only for continous mappings of subsets of the Euclidean space R n . For more information see also: [15], [18]-[22], [3], [25], [27], [31]. Note that for multivalued mappings these problems were considered only in a few papers (see: [2]-[8], [14], [23], [24], [32]) - always for admissible multivalued mappings of absolute neighbourhood retracts (ANR-s). In this paper ejective, repulsive and essential fixed points for admissible multivalued mappings of absolute neighbourhood multi retracts (ANMR-s) are studied. Let as remark that the class of MANR-s is much larger as the class of ANR-s (see: [32]). In order to study the above notions we generalize the fixed point index from the case of ANR-s onto the case of ANMR-s. Next using the above fixed point index we are able to prove several new results concerning repulsive ejective and essential fixed points of admissible multivalued mappings. Moreover, the random case is mentioned. For possible applications to differential and random di?erential inclusions see: [1], [2], [8]-[11], [16], [25], [26].

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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 systems have been investigated by numerous authors in Banach sequence spaces @?^~. But continuous model and corresponding infinite uncountable systems have been investigated by a few authors only. It is interesting that we can attack the above problems by using the same topological fixed-point theory tools as in the countable case. It is the study of infinite uncountable systems of parabolic reaction-diffusion-convection equations that is the subject matter of this paper. Motivation for considering such systems and examples thereof are presented. The paper is an introduction to the study of infinite uncountable systems of parabolic equations.

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 Frechet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, A R and R δ ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.

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. [email protected][email protected]?{t"1,...,t"m},y(t"k^+)-y(t"k^-)=I"k(y(t"k^-)),k=1,...,m,y(0)=y(b) where J=[0,b] and 0=t"0

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

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 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 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 Gorniewicz (1976); compare also Gorniewicz (1995)) on tori. Since the Poincare maps generated by the Marchaud vector fields are of this type (see (Andres, 1997)), we can obtain in such a way multiplicity results for differential inclusions. More precisely, the nontrivial Nielsen number gives a lower estimate of coincidence points (in particular, fixed points) corresponding to the desired solutions.

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 presented. Note that our applications generalize earlier existence results obtained both on the whole Euclidean space R n and on a compact proximate retract of R n .

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 Poincare translation operator. In a certain case, the well-known Wazewski principle can also be generalized in this way, because no transversality is required on the boundary.

This paper deals with a class of nonlinear control systems in R in presence of deterministic uncertainty. The uncertainty is modelled by a multivalued map F with nonempty, closed, convex values. Given a nonempty closed setK ⊂ R from a suitable class, which includes the convex sets, we solve the problem of finding a state feedback u(t, x) in such a way that K is invariant under any system dynamics f . As a system dynamics we consider any continuous selection of the uncertain controlled dynamics F . 0. Introduction. In this paper the invariance property of a given nonempty, closed set K ⊂ R under a nonlinear controlled dynamics is investigated. The dynamics of the control system is affected by a deterministic uncertainty which is modelled by a multivalued map F . Here we consider only the continuous selections f of F as possible system dynamics. Specifically, we consider a differential inclusion of the form (1) ẋ ∈ F (t, x, u), t ∈ [0, T ], u ∈ U ⊂ R, x ∈ R. Given a nonempty, closed set K ⊂ R, the problem that we want to study is that of determining a state feedback control u(t, x) such that K is invariant under any corresponding system dynamics f(t, x, u(t, x)) ∈ F (t, x, u(t, x)). That is, for any initial state x0 ∈ K, if x(t) is a solution of the Cauchy problem (2) { ẋ = f(t, x, u(t, x)) for a.a. t ∈ [0, T ], x(0) = x0, 1991 Mathematics Subject Classification: 93C10, 93C15, 93B52. The paper is in final form and no version of it will be published elsewhere.

The paper contains a survey of various results concerning the Schauder Fixed Point Theorem for metric spaces both in single-valued and multi-valued cases. A number of open problems is formulated.

Udostępnienie publikacji Wydawnictwa Uniwersytetu Łódzkiego finansowane w ramach projektu „Doskonałość naukowa kluczem do doskonałości kształcenia”. Projekt realizowany jest ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Wiedza Edukacja Rozwój; nr umowy: POWER.03.05.00-00-Z092/17-00.

The concept of bifurcation belongs to one of Poincare's pioneering studies. In 1885, Poincare introduced the notion of bifurcation, discussed the elements of the theory and described the phenomenon to an extremely detailed extent. The basic elements of the theory may be grouped into three areas: critical solutions, dynamic stability and structural stability. Our work concerns the area of structural stability, namely we use the topological degree theory methods to get local and global bifurcation results.

The paper introduces a definition of the generalized topological degree for the class of acyclic set-valued mappings acting between spheres of different dimensions. The presented theory is applied in order to obtain a bifurcation index which constitutes a homotopy invariant yielding global bifurcation of solutions to equations (inclusions) involving acyclic maps.

A hierarchical classification of different concepts of shape of compact connected sets in Rn (topological, Lipshitz, homotopic, Borsuk an homological shapes) is given. The most general among them is the homological shape. There is only a countable number of homological shapes for connected compact sets in Rn. In the case n = 2 even the number of different Borsuk shapes for connected compact sets is countable. Giving a probability distribution of shapes we can define a shape entropy, a mean shape and shape fluctuations. This enables a formulation of information thermodynamics of shape and its applications to different fields (physics – small systems, chemistry, biophysics, pattern recognition). The paper does not develop yet these applications, its aim is to clear the basic notions.