Variational approach to second-order impulsive dynamic equations on time scales

The aim of this paper is to employ variational techniques and critical point theory to prove some conditions for the existence of solutions to nonlinear impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also we will be interested in the solutions of the impulsive nonlinear problem with linear derivative dependence satisfying an impulsive condition.Comment: 17 page

Sucesións e series de funcións reais

Grao en MatemáticasEsta unidade didáctica forma parte do curso de Cálculo Diferencial e Integral,e está destinada a estudantes que inician o estudo da análise matemática. A súa finalidade é presentar de maneira gradual os conceptos fundamentais e as técnicas básicas das sucesións e series funcionais. As sucesións e series de funcións, que xa foron motivadas en temas anteriores do curso, posúen unha importancia e interese intrínsecos. Por un lado, con elas pódense definir novas funcións, polo outro, permiten aproximar funcións por outras máis sinxelas, de xeito que das propiedades destas se poidan inferir as da función orixinal, idea esta última que subxace en moitos dos métodos da análise. A representación de funcións mediante series de potencias xogou un importante papel no desenvolvemento da análise. Moito do traballo de Newton con derivación e integración foi realizado no contexto das series de potencias. Gregory (un dos primeiros matemáticos que traballaron con elas), Euler, Lagrange, Leibniz e Joham e Jacob Bernoulli usaron amplamente as series de potencias no cálculo. Os traballos de Fourier neste campo obrigaron a cuestionar o concepto de función existente na época, e motivaron a outros matemáticos, como a Cauchy e Dirichlet. Dende o punto de vista histórico o uso das series de potencias precede e motiva o estudo de sucesións e series funcionais. Disposición que podería ser mellor dende o punto de vista didáctico. Non obstante, para facilitar a presentación formal da unidade didáctica escollemos outra orde, que coincide coa formulación usual en manuais sobre o tema. Pretendemos desenvolver os temas coa motivación real que propiciou a súa orixe de modo que os estudantes sexan participantes activos da evolución das ideas e non queden como meros observadores pasivos dos resultados. Os conceptos e resultados da unidade con frecuencia irán precedidos dunha discusión xeométrica ou intuitiva para dar unha idea máis profunda dos resultados. Aínda que nos contidos da unidade non se inclúen as demostracións dos teoremas, estas serán feitas nas aulas; as demostracións dos resultados máis importantes considéranse parte esencial no desenvolvemento das ideas matemáticas.Universidade de Santiago de Compostela. Servizo de Normalización Lingüístic

Positive Solution of Singular BVPs for System of Dynamic Equations on Time Scales

This paper is devoted to derive some necessary and suficient conditions for the existence of positive solutions to a singular second order system of dynamic equations with Dirichlet boundary conditions. The results are obtained by employing the fixed-point theorems and the method of the lower and upper solutions.Comment: 19 page

Existence and Uniqueness of Positive Solution for Singular BVPs on Time Scales

This paper is devoted to derive some sufficient conditions for the existence and uniqueness of positive solutions to a singular second-order dynamic equation with Dirichlet boundary conditions.This research is partially supported by D.G.I. and F.E.D.E.R. project MTM2007-61724 and by the Xunta of Galicia and F.E.D.E.R. project PGIDIT06PXIB207023PR, SpainS

Phase portraits of a family of Kolmogorov systems depending on six parameters

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia), grant ED431C 2019/10We consider a general 3-dimensional Lotka-Volterra system with a rational first integral of degree two of the form H = xi yj zk. The restriction of this Lotka-Volterra system to each surface H(x, y, z) = h varying h ∈ R provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form xl ym est they reduce to the Kolmogorov systems x˙ = x (a0 − µ(c1x + c2z2 + c3z)), z˙ = z (c0 + c1x + c2z2 + c3z)). We classify the phase portraits in the Poincaré disc of all these Kolmogorov systems which depend on six parameters

Dynamics of a two prey and one predator system with indirect effect

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia), grant ED431C 2019/10We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper

Global phase portraits of a predator-prey system

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator–prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems

Basic properties of Sobolev's spaces on time scales

We study the theory of Sobolev's spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the Lebesgue Δ-measure; analogous properties to that valid for Sobolev's spaces of functions defined on an arbitrary open interval of the real numbers are derived.This research is partially supported by D.G.I. and F.E.D.E.R. project MTM 2004-06652- C03-01, and by Xunta of Galicia and F.E.D.E.R. project PGIDIT05PXIC20702PN, SpainS

Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations

This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equationThis research is partially supported by MEC and F.E.D.E.R. Project MTM2007-61724, and by Xunta of Galicia and F.E.D.E.R. Project PGIDIT05PXIC20702PN, SpainS

Stability of periodic solutions of first-order difference equations lying between lower and upper solutions

We prove that if there exists α≤β, a pair of lower and upper solutions of the first-order discrete periodic problem Δu(n)=f(n,u(n));n∈IN≡{0,…,N−1},u(0)=u(N), with f a continuous N-periodic function in its first variable and such that x+f(n,x) is strictly increasing in x, for every n∈IN, then, this problem has at least one solution such that its N-periodic extension to ℕ is stable. In several particular situations, we may claim that this solution is asymptotically stable