We derive conditions for the existence of fixed points of cone mappings without assuming scalabil... more We derive conditions for the existence of fixed points of cone mappings without assuming scalability of functions. Monotonicity and scalability are often inseparable in the literature in the context of searching for fixed points of interference mappings. In applications, such mappings are approximated by non-negative neural networks. It turns out, however, that the process of training non-negative networks requires imposing an artificial constraint on the weights of the model. However, in the case of specific non-negative data, it cannot be said that if the mapping is non-negative, it has only non-negative weights. Therefore, we considered the problem of the existence of fixed points for general neural networks, assuming the conditions of tangency conditions with respect to specific cones. This does not relax the physical assumptions, because even assuming that the input and output are to be non-negative, the weights can have (small, but) less than zero values. Such properties (often found in papers on the interpretability of weights of neural networks) lead to the weakening of the assumptions about the monotonicity or scalability of the mapping associated with the neural network. To the best of our knowledge, this paper is the first to study this phenomenon.
Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system... more Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$ with elastic impact on the boundary of $K$ is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulation it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.
The paper deals with semilinear differential inclusions with state-dependent impulses in Banach s... more The paper deals with semilinear differential inclusions with state-dependent impulses in Banach spaces. Defining a suitable Banach space in which all the solutions can be embedded we prove the first existence result for at least one global mild solution of the problem considered. Then we characterize this result by means of a new definition of Lyapunov pairs. To this aim we give sufficient conditions for the existence of Lyapunov pairs in terms of a new concept of contingent derivative.
In the paper we study the problem of existence of solutions to differential inclusions remaining ... more In the paper we study the problem of existence of solutions to differential inclusions remaining in a prescribed closed subset of a Euclidean space. We find some sufficient conditions for existence of such trajectories and their localization.
The soil biological conditions of two 5-year-old polyculture tree plantations in Amazonia were st... more The soil biological conditions of two 5-year-old polyculture tree plantations in Amazonia were studied comparatively to a 13-year-old secondary forest and a nearby undisturbed primary forest. The polycultures had been planted to regenerate the soil degraded by land preparation and a former rubber tree monoculture. Abundance and biomass of functional groups of soil meso-and macrofauna were measured at three-months-intervals over 2 years and litterbag experiments with fauna exclusion were carried out. This paper concentrates on the description of the structure of the soil fauna communities, forming the background for an evaluation of the decomposition processes in polyculture plantations. Decomposition rates were strongly determined by the macrofauna particularly in primary forest, where large earthworms, termites and ants dominated the soil fauna. In the plantations, where litter originated predominantly from the non-planted, adventitious vegetation, an abundant decomposer fauna was found, in which however other groups or species dominated. Although decomposition rates in the plantations were about 60 % lower and soil biological variables like organic matter-, nitrogen-content and water holding capacity were slightly lower than in the primary forest, conditions seem favourable for a manipulation of the soil fauna by management of secondary vegetation and litter quantities.
Sufficient conditions for the invariance of evolution problems governed by perturbations of (poss... more Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) m-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint functional are expressed in terms of the Dini derivative of that functional, outside the considered sublevel set in directions determined by the governing m-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper [7] is presented. Applications to nonlinear obstacle problems and agestructured population models are presented in spaces of continuous functions where advantages of that approach are taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct applications to problems with barriers are shown.
Journal of Mathematical Analysis and Applications, 2017
Abstract An R δ -structure of solution sets to some classes of impulsive differential inclusions ... more Abstract An R δ -structure of solution sets to some classes of impulsive differential inclusions with variable impulse times on the half-line is shown. Sufficient conditions on barriers are discussed in detail. They imply two different techniques: the inverse limit method and the one based on a definition of R δ -sets and used in a new suitable Frechet function space.
In the paper we construct a homotopy index on sleek sets for multivalued flows generated by diffe... more In the paper we construct a homotopy index on sleek sets for multivalued flows generated by differential inclusions using single-valued approximations. The index is described by behavior of a multivalued map (some tangency conditions) on a boundary of a given set. Several properties of the index are proved. Some results on existence of equilibria are also presented.
ABSTRACT In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems ... more ABSTRACT In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form
Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are exami... more Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are examined. The so-called impulsive index is introduced, as a topological tool, and its properties are studied. Some sufficient conditions for its homotopy property are discussed in detail. In a construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier where jumps occur. Several illustrative examples are added.
Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_2002_8_5B_2_431_0> L'utilizzo e la st... more Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_2002_8_5B_2_431_0> L'utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l'utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (
Zeitschrift für Analysis und ihre Anwendungen, 2000
Acydicity of solution sets-to asymptotic problems, when the value is prescribed either at the ori... more Acydicity 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.
A concept of generalized topological essentiality for a large class of multivalued maps in topolo... more 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 note is a study of fixed points of multivalued maps in Fréchet spaces. The approach presented in the paper allows to check not only the nonemptiness of the fixed point set but also its topological structure.
In many dynamical system problems, it is interesting not only to know that equilibria do exist bu... more In many dynamical system problems, it is interesting not only to know that equilibria do exist but also to know if the equilibria can be reached by at least one trajectory (possibly asymptotically). It is worth pointing out that this question is different from those of stability, or attractiveness of the equilibria. In the present article, our purpose is to
ABSTRACT In the paper the celebrated Ważewski retract method is developed for planar dynamic equa... more ABSTRACT In the paper the celebrated Ważewski retract method is developed for planar dynamic equations on an arbitrary time scale without a restrictive assumption that the whole boundary of a set of constraints, where we look for solutions, is a set of egress points. As a consequence, some recently published results are essentially generalized. One transparent example illustrating the main theorem is presented. The results are new and more general even for ordinary difference equations.
This paper is concerned with existence of equilibrium of a set-valued map in a given compact subs... more This paper is concerned with existence of equilibrium of a set-valued map in a given compact subset of a finite-dimensional space. Previously known conditions ensuring existence of equilibrium imply that the set is either invariant or viable for the differential inclusion generated by the set-valued map. We obtain some equilibrium existence results with conditions which imply neither invariance nor viability of the given set. The problem of existence of strict equilibria is also discussed.
... Downloads (6 Weeks), 0. Downloads (12 Months), 0. View colleagues of Lech Górniewicz. top of ... more ... Downloads (6 Weeks), 0. Downloads (12 Months), 0. View colleagues of Lech Górniewicz. top of page REFERENCES. ... Sb. 188 (12) (1997) 33-56 (in Russian). 15. {15} L. Górniewicz, Homological methods in fixed point theory of multivalued mappings, Dissertations Math. ...
We derive conditions for the existence of fixed points of cone mappings without assuming scalabil... more We derive conditions for the existence of fixed points of cone mappings without assuming scalability of functions. Monotonicity and scalability are often inseparable in the literature in the context of searching for fixed points of interference mappings. In applications, such mappings are approximated by non-negative neural networks. It turns out, however, that the process of training non-negative networks requires imposing an artificial constraint on the weights of the model. However, in the case of specific non-negative data, it cannot be said that if the mapping is non-negative, it has only non-negative weights. Therefore, we considered the problem of the existence of fixed points for general neural networks, assuming the conditions of tangency conditions with respect to specific cones. This does not relax the physical assumptions, because even assuming that the input and output are to be non-negative, the weights can have (small, but) less than zero values. Such properties (often found in papers on the interpretability of weights of neural networks) lead to the weakening of the assumptions about the monotonicity or scalability of the mapping associated with the neural network. To the best of our knowledge, this paper is the first to study this phenomenon.
Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system... more Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$ with elastic impact on the boundary of $K$ is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulation it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.
The paper deals with semilinear differential inclusions with state-dependent impulses in Banach s... more The paper deals with semilinear differential inclusions with state-dependent impulses in Banach spaces. Defining a suitable Banach space in which all the solutions can be embedded we prove the first existence result for at least one global mild solution of the problem considered. Then we characterize this result by means of a new definition of Lyapunov pairs. To this aim we give sufficient conditions for the existence of Lyapunov pairs in terms of a new concept of contingent derivative.
In the paper we study the problem of existence of solutions to differential inclusions remaining ... more In the paper we study the problem of existence of solutions to differential inclusions remaining in a prescribed closed subset of a Euclidean space. We find some sufficient conditions for existence of such trajectories and their localization.
The soil biological conditions of two 5-year-old polyculture tree plantations in Amazonia were st... more The soil biological conditions of two 5-year-old polyculture tree plantations in Amazonia were studied comparatively to a 13-year-old secondary forest and a nearby undisturbed primary forest. The polycultures had been planted to regenerate the soil degraded by land preparation and a former rubber tree monoculture. Abundance and biomass of functional groups of soil meso-and macrofauna were measured at three-months-intervals over 2 years and litterbag experiments with fauna exclusion were carried out. This paper concentrates on the description of the structure of the soil fauna communities, forming the background for an evaluation of the decomposition processes in polyculture plantations. Decomposition rates were strongly determined by the macrofauna particularly in primary forest, where large earthworms, termites and ants dominated the soil fauna. In the plantations, where litter originated predominantly from the non-planted, adventitious vegetation, an abundant decomposer fauna was found, in which however other groups or species dominated. Although decomposition rates in the plantations were about 60 % lower and soil biological variables like organic matter-, nitrogen-content and water holding capacity were slightly lower than in the primary forest, conditions seem favourable for a manipulation of the soil fauna by management of secondary vegetation and litter quantities.
Sufficient conditions for the invariance of evolution problems governed by perturbations of (poss... more Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) m-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint functional are expressed in terms of the Dini derivative of that functional, outside the considered sublevel set in directions determined by the governing m-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper [7] is presented. Applications to nonlinear obstacle problems and agestructured population models are presented in spaces of continuous functions where advantages of that approach are taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct applications to problems with barriers are shown.
Journal of Mathematical Analysis and Applications, 2017
Abstract An R δ -structure of solution sets to some classes of impulsive differential inclusions ... more Abstract An R δ -structure of solution sets to some classes of impulsive differential inclusions with variable impulse times on the half-line is shown. Sufficient conditions on barriers are discussed in detail. They imply two different techniques: the inverse limit method and the one based on a definition of R δ -sets and used in a new suitable Frechet function space.
In the paper we construct a homotopy index on sleek sets for multivalued flows generated by diffe... more In the paper we construct a homotopy index on sleek sets for multivalued flows generated by differential inclusions using single-valued approximations. The index is described by behavior of a multivalued map (some tangency conditions) on a boundary of a given set. Several properties of the index are proved. Some results on existence of equilibria are also presented.
ABSTRACT In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems ... more ABSTRACT In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form
Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are exami... more Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are examined. The so-called impulsive index is introduced, as a topological tool, and its properties are studied. Some sufficient conditions for its homotopy property are discussed in detail. In a construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier where jumps occur. Several illustrative examples are added.
Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_2002_8_5B_2_431_0> L'utilizzo e la st... more Unione Matematica Italiana <http://www.bdim.eu/item?id=BUMI_2002_8_5B_2_431_0> L'utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l'utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (
Zeitschrift für Analysis und ihre Anwendungen, 2000
Acydicity of solution sets-to asymptotic problems, when the value is prescribed either at the ori... more Acydicity 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.
A concept of generalized topological essentiality for a large class of multivalued maps in topolo... more 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 note is a study of fixed points of multivalued maps in Fréchet spaces. The approach presented in the paper allows to check not only the nonemptiness of the fixed point set but also its topological structure.
In many dynamical system problems, it is interesting not only to know that equilibria do exist bu... more In many dynamical system problems, it is interesting not only to know that equilibria do exist but also to know if the equilibria can be reached by at least one trajectory (possibly asymptotically). It is worth pointing out that this question is different from those of stability, or attractiveness of the equilibria. In the present article, our purpose is to
ABSTRACT In the paper the celebrated Ważewski retract method is developed for planar dynamic equa... more ABSTRACT In the paper the celebrated Ważewski retract method is developed for planar dynamic equations on an arbitrary time scale without a restrictive assumption that the whole boundary of a set of constraints, where we look for solutions, is a set of egress points. As a consequence, some recently published results are essentially generalized. One transparent example illustrating the main theorem is presented. The results are new and more general even for ordinary difference equations.
This paper is concerned with existence of equilibrium of a set-valued map in a given compact subs... more This paper is concerned with existence of equilibrium of a set-valued map in a given compact subset of a finite-dimensional space. Previously known conditions ensuring existence of equilibrium imply that the set is either invariant or viable for the differential inclusion generated by the set-valued map. We obtain some equilibrium existence results with conditions which imply neither invariance nor viability of the given set. The problem of existence of strict equilibria is also discussed.
... Downloads (6 Weeks), 0. Downloads (12 Months), 0. View colleagues of Lech Górniewicz. top of ... more ... Downloads (6 Weeks), 0. Downloads (12 Months), 0. View colleagues of Lech Górniewicz. top of page REFERENCES. ... Sb. 188 (12) (1997) 33-56 (in Russian). 15. {15} L. Górniewicz, Homological methods in fixed point theory of multivalued mappings, Dissertations Math. ...
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