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We present a version for the general readership of our article (Amster y Cid, 2022) in which we show how the complex square root allows us to prove in a very simple manner various topological results in the plane, as deep as the Brouwer Fixed Point Theorem and the Invariance of Domain Theorem, as well as the Fundamental Theorem of Algebra., Presentamos una versión divulgativa de nuestro artículo (Amster y Cid, 2022) en el que mostramos cómo la raíz cuadrada compleja nos permite demostrar de una forma muy sencilla diversos resultados topológicos en el plano, tan profundos como el Teorema de Brouwer y el Teorema de Invariancia del Dominio, así como del Teorema Fundamental del Álgebra.

We present some generalized Lipschitz conditions which imply uniqueness of solutions for scalar ODEs. We illustrate the applicability of our results with examples not covered by earlier Lipschitz–type uniqueness tests

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.

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.