A positive fixed point theorem with applications to systems of Hammerstein integral equations

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.Comment: 11 page

Semilinear problems at resonance with nonlinearities depending only on the derivative

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGWe deal with second-order semilinear equations with a nonlinear term depending only on the derivative. Our main results extend or complement some previous work by J. MawhinAgencia Estatal de Investigación | Ref. PID2021-122442-10

The full power of the half-power

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGWe use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiledAgencia Estatal de Investigación | Ref. PID2021-122442NB-10

The Brouwer fixed point theorem and periodic solutions of differential equations

The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent.Universidade de Vigo/CISUGAgencia Estatal de Investigación | Ref. MTM2017-85054-C2-1-

Existence and multiplicity of solutions for a periodic hill’s equation with parametric dependence and singularities

We deal with the existence and multiplicity of solutions for the periodic boundary value problem positive parameter. The function f : is allowed to be singular, and the related Green’s function is nonnegative and can vanish at some points.This work was partially supported by FEDER and Ministerio de Educacion y Ciencia, Spain, project no. MTM2010-15314S

A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs

We prove the following result: if a continuous vector field F is Lipschitz when restricted to the hypersurfaces determined by a suitable foliation and a transversal condition is satisfied at the initial condition, then F determines a locally unique integral curve. We also present some illustrative examples and sufficient conditions in order to apply our main resultPartially supported by Xunta de Galicia (Spain), project EM2014/032 (both authors) and MINECO, project MTM2017-85054-C2-1-P (first author)S