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Jorge Rodríguez-López

Publications and source records attributed to Jorge Rodríguez-López.

11 recordsLinked to original sources
arXiv2026

Multiple non-collision periodic solutions for second order singular differential systems with parameters

In this paper, new existence and multiplicity results for positive periodic solutions of singular second-order problems under sublinear and superlinear growth assumptions at infinity are established. Two aspects of our results should be highlighted in comparison with the existing literature. On the one hand, they allow for the consideration of systems exhibiting mixed behavior: sublinear in one component and superlinear in another. On the other hand, in the superlinear case, the existence of up to four positive solutions is established, whereas only two were obtained in previous works. Our approach is based on a recently established vectorial version of the Krasnoselskii-Guo fixed point theorem in product spaces.

math.CA↗
arXiv2026

A unified approach to compression-expansion fixed point theorems for operators systems and applications

In this paper, we present some fixed point theorems for operator systems in the line of Krasnosel'skii's theorem in cones. The cone-compression and cone-expansion type conditions are imposed in a component-wise manner. Unlike related results in the literature, the operators are allowed to be defined in the Cartesian product of conical regions delimited by nonconvex sets. Our approach, based on the fixed point index, ensures the existence of a coexistence fixed point--that is, one with nontrivial components. As a first application, we establish several localization results for systems of integral equations between strictly star-shaped sets defined by functionals. These results cannot be derived solely from previous studies dealing with operators in annular regions. A second application concerns nonlinear systems involving the Φ-Laplacian.

math.FA↗
arXiv2026

Birkhoff-Kellogg type results in product spaces and their application to differential systems

We provide a new version of the well-known Birkhoff-Kellogg invariant-direction Theorem in product spaces. Our results concern operator systems and give the existence of component-wise eigenvalues, instead of scalar eigenvalues as in the classical case, that have corresponding eigenvectors with all components nontrivial and localized by their norm. We also show that, when applied to nonlinear eigenvalue problems for differential equations, this localization property of the eigenvectors provides, in turn, qualitative properties of the solutions. This is illustrated in two contexts of systems of PDEs and ODEs. We show the applicability of our theoretical results with two explicit examples.

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arXiv2026

An existence result in annular regions times conical shells and its application to nonlinear Poisson systems

We provide a new existence result for abstract nonlinear operator systems in normed spaces, by means of topological methods. The solution is located within the product of annular regions and conical shells. The theoretical result possesses a wide range of applicability, which, for concreteness, we illustrate in the context of systems of nonlinear Poisson equations subject to homogeneous Dirichlet boundary conditions. For the latter problem we obtain existence and localization of solutions having all components nontrivial. This is also illustrated with an explicit example in which we also furnish a numerically approximated solution, consistent with the theoretical results. We conclude with an application of our results to a reaction--diffusion Lotka--Volterra system with source terms for competing species.

math.AP↗
arXiv2025

On Krasnosel'skii-Precup fixed point theorem and eigenvalue criteria for existence of positive solutions

We provide an alternative approach to Krasnosel'skii-Precup fixed point theorem for operator systems based on the Leray-Schauder fixed point index in cones. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second--order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion.

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arXiv2025

Component-wise Krasnosel'skii type fixed point theorem in product spaces and applications

We present a version of Krasnosel'skii fixed point theorem for operators acting on Cartesian products of normed linear spaces, under cone-compression and cone-expansion conditions of norm type. Our approach, based on the fixed point index theory in cones, guarantees the existence of a coexistence fixed point - that is, one with nontrivial components. As an application, we prove the existence of periodic solutions with strictly positive components for a system of second-order differential equations. In particular, we address cases involving singular nonlinearities and hybrid terms, characterized by sublinear behavior in one component and superlinear behavior in the other.

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arXiv2024

A Birkhoff-Kellogg type theorem for discontinuous operators with applications

By means of fixed point index theory for multi-valued maps, we provide an analogue of the classical Birkhoff--Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in details this fact for a class of second order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory.

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arXiv2024

A hybrid Krasnosel'skiĭ-Schauder fixed point theorem for systems

We provide new results regarding the localization of the solutions of nonlinear operator systems. We make use of a combination of Krasnosel'ski\uı cone compression-expansion type methodologies and Schauder-type ones. In particular we establish a localization of the solution of the system within the product of a conical shell and of a closed convex set. By iterating this procedure we prove the existence of multiple solutions. We illustrate our theoretical results by applying them to the solvability of systems of Hammerstein integral equations. In the case of two specific boundary value problems and with given nonlinearities, we are also able to obtain a numerical solution, consistent with our theoretical results.

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arXiv2023

On discontinuous differential equations and the method of solution-regions

We adapt the method of solution regions to prove new existence and localization results for systems of discontinuous differential equations. Some assumptions concerning the definition of a solution region are relaxed and thus our results enlarge the applicability of this method even in the case of Carathéodory nonlinearities. Several examples are provided to illustrate the theory.

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arXiv2022

A fixed point index approach to Krasnosel'skii-Precup fixed point theorem in cones and applications

We present an alternative approach to the vector version of Krasnosel'skii 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 $(p_1,p_2)$-Laplacian systems.

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