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We are concerned with existence, localization and multiplicity of positive radial solutions to Dirichlet problems with Ø-Laplacians in a ball, in both scalar and system cases. Our approach essentially relies on fixed point index computations and a main feature is that it avoids any Harnack type inequality. Applications to some problems involving operators with Uhlenbeck structure are discussed.

The main goal of this paper is to prove that for each n > 2, every sequence of integers satisfying Dold’s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving Rn-homeomorphism at an isolated stable fixed point. We use Conley index techniques even though stable fixed points are not isolated invariant sets., MICINN, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The author proves that if f is an orientation reversing homeomorphism of the plane and p is an isolated and stable fixed point of f then the fixed point index of f at p is equal to 1 . In the orientation preserving case this result was obtained by E. N. Dancer and R. Ortega [J. Dynam. Differential Equations 6 (1994), no. 4, 631–637. The proof is based on the prime ends compactifications method and a fixed point result by K. M. Kuperberg [Proc. Amer. Math. Soc. 112 (1991), no. 1, 223–229., MCyT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Caratheodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations., MEC, MEC, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Let U be an open subset of R2 and let f :U → R2 be a local homeomorphism. Let p € U be a non-repeller 4xed point of f suchth at {p} is an isolated invariant set. We introduce a particular class of index pairs for {p} that we call generalized 4ltration pairs. The computation of the 4xed point index of any iteration of f at p is quite easy once one knows a generalized 4ltration pair. The existence of generalized 4ltration pairs provides a short and elementary proof of a theorem of P. Le Calvez and J.C. Yoccoz (Ann. of Meth. 146 (1997) 241–293), and it also allows to compute the 4xed point index of any iteration of arbitrary local homeomorphisms., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Let f : U →R2 be a continuous map, where U is an open subset of R2. We consider a fixed point p of f which is neither a sink nor a source and such that p is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations ind(fn, p) n=1 is periodic,bounded by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem [Annals of Math., 146 (1997), 241-293] onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere., MSHE, MICINN, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Dedicated to Professor Jose M. Montesinos on the occasion of his 65th birthday, Let U subset of R-3 be an open set and f : U -> f(U) subset of R-3 be a homeomorphism. Let p is an element of U be a fixed point. It is known that if {p} is not an isolated invariant set, then the sequence of the fixed-point indices of the iterates of f at p, (i(f(n), p))(n >=) (1), is, in general, unbounded. The main goal of this paper is to show that when {p} is an isolated invariant set, the sequence (i(f(n), p))(n >= 1) is periodic. Conversely, we show that, for any periodic sequence of integers (I-n)(n >= 1) satisfying Dold's necessary congruences, there exists an orientation-preserving homeomorphism such that i(f(n), p) = I-n for every n >= 1. Finally we also present an application to the study of the local structure of the stable/unstable sets at p., MICINN, ANR, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Let U subset of R-2 be an open subset, f : U -> f (U) subset of R-2 be an orientation reversing homeomorphism and let 0 is an element of U be an isolated, as a periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices i(R2)(f, 0) = i(R2)(f(2), 0) = 1 then there exists an orientation preserving dissipative homeomorphism phi: R-2 -> R-2 such that f(2) = phi in a small neighbourhood of 0 and {0} is a global attractor for phi. As a corollary we have that for orientation reversing planar homeomorphisms a fixed point, which is an isolated fixed point for f(2), is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally, Ministerio de Ciencia e Innovación (MICINN), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U subset of X be an open subset and f:U -->X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K subset of U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors, DGES., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

In this paper we study a Mayer-Vietoris kind of formula for the fixed point index of maps of ENR triplets f : (X;X1,X2) → (X;X1,X2) having compact fixed point set. We prove it under some suitable conditions. For instance when (X;X1,X2) = (En;En+,En −). We use these results to generalize Poincar´e-Bendixson index formula for vector fields to continuos maps having a sectorial decomposition, to study the fixed point index i(f, 0) of orientation preserving homeomorphisms of E2 + and (E3;E3 +,E3 −) and the fixed point index in the invariant subspace., Ministerio de Economía y Competitividad (MINECO), Faculty of Mathematics and Computer Science of Jagiellonian University in Krakow, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let U c R2 be an open subset and let f :U → f (U) c R2 be a homeomorphism. Let M = M1 U· · ·U Mr C U be a disjoint union of discs that isolates the invariant compactum K. The aimof this paper is to study the dynamics of f in K and to use the fixed point index to detect, in a simple and geometric way, the existence of periodic orbits on which f follows a determined pattern. Our method allows us to compute the fixed point index of every iteration of f in a neighborhood of the periodic orbits following a given itinerary in classical and important semidynamical systems with chaotic dynamics., MCyT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

[EN] In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

[EN] In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

[EN] In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

[EN] In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

We present an alternative approach to the vector version of Krasnosel’skiĭ compression–expansion fixed point theorem due to Precup, which is based on the fixed point index. It allows us to obtain new general versions of this fixed point theorem and also multiplicity results. We emphasize that all of them are coexistence fixed point theorems for operator systems, that means that every component of the fixed points obtained is non-trivial. Finally, these coexistence fixed point theorems are applied to obtain results concerning the existence of positive solutions for systems of Hammerstein integral equations and radially symmetric solutions of (P1,P2) Laplacian systems

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving -Laplacian operators in a ball. In particular, -Laplacian and Minkowski-curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack-type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions

[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.

Let Y and X denote Ck vector fields on a possibly noncompact surface with empty boundary, 1 ≤ k< ∞. Say that YtracksX if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. Theorem Assume the Poincaré–Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of Ck vector fields that track X, then the elements of g have a common zero in K. Applications are made to the dynamics of attractors and transformation groups.

Let Y and X denote Ck vector fields on a possibly noncompact surface with empty boundary, 1 ≤ k< ∞. Say that YtracksX if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. Theorem Assume the Poincaré–Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of Ck vector fields that track X, then the elements of g have a common zero in K. Applications are made to the dynamics of attractors and transformation groups.

We characterize the sequences of fixed point indices {i(f(n) ,p)} n >= 1 of fixed points that are isolated as an invariant set for a continuous map f in the plane. In particular, we prove that the sequence is periodic and i(f(n) ,p) <= 1 for every n >= 0. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the 2-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree-d maps in the 2-sphere., MINECO, CNPq fellowship (Brazil), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, we obtain some properties of the first eigenvalue of a fractional differential equation. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence of positive solutions are established.

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results.

In this paper, we study the existence of positive solutions for the singular second order integral boundary value problem [unable to display problem], where c(t) is allowed to be singular at t=0,1 and f(u) may be singular at u=0. The existence of positive solutions for the above problem is established by applying the fixed point index theorems under some weaker conditions concerning the first eigenvalue corresponding to the relevant linear operator. The results obtained herein generalize and improve some known results including singular and non-singular cases.

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, UX be an open subset and f :U→X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the y perspace of X. For any compact isolated invariant set, KU, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors., DGES, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub