A fixed point index approach to Krasnosel’skiĭ-Precup fixed point theorem in cones and applications

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 systemsJorge Rodríguez–López was partially supported by Xunta de Galicia (Spain), project ED431C 2019/02 and AEI, Spain and FEDER , grant PID2020-113275GB-I00. The author thanks the referee for useful comments which led to the improvement of his paper and for the suggested additional referencesS

Compression–Expansion Fixed Point Theorems for Decomposable Maps and Applications to Discontinuous ϕ-Laplacian problems

In this paper, we prove new compression–expansion type fixed point theorems in cones for the so-called decomposable maps, that is, compositions of two upper semicontinuous multivalued maps. As an application, we obtain existence and localization of positive solutions for a differential equation with ϕ-Laplacian and discontinuous nonlinearity subject to multi-point boundary conditions. As far as we are aware, the existence results are new even in the classical case of continuous nonlinearitiesJorge Rodríguez-López was partially supported by Xunta de Galicia ED431C 2019/02S

Topological methods for discontinuous operators and applications

Topological methods are crucial in nonlinear analysis, especially in the study of existence of solutions to diverse boundary value problems. As a well–known fact, continuity is a basic assumption in the classical theory and the clearest limitation of its applicability. That is why most discontinuous differential equations fall outside its scope because the corresponding fixed point operators are not continuous. The main goal of this thesis is to introduce a new definition of topological degree which applies for a wide class of non necessarily continuous operators. This generalization is based on the degree theory for upper semicontinuous multivalued mappings. As a consequence, new fixed point theorems for this class of discontinuous operators are derived. This new theory for discontinuous operators is combined with classical techniques in nonlinear analysis in order to obtain existence, localization and multiplicity results for discontinuous differential equations

Positive solutions for φ-Laplacian equations with discontinuous state-dependent forcing terms

This paper concerns the existence, localization and multiplicity of positive solutions for a -Laplacian problem with a perturbed term that may have discontinuities in the state variable. First, the initial discontinuous differential equation is replaced by a differential inclusion with an upper semicontinuous term. Next, the existence and localization of a positive solution of the inclusion is obtained via a compression-expansion fixed point theorem for a composition of two multivalued maps, and finally, a suitable control of discontinuities allows to prove that any solution of the inclusion is a solution in the sense of Carathéodory of the initial discontinuous equation. No monotonicity assumptions on the nonlinearity are requiredS

Positive solutions for phi-Laplace equations with discontinuous state-dependent forcing terms

This paper concerns the existence, localization and multiplicity of positive solutions for a φ-Laplacian problem with a perturbed term that may have discontinuities in the state variable. First, the initial discontinuous differential equation is replaced by a differential inclusion with an upper semicontinuous term. Next, the existence and localization of a positive solution of the inclusion is obtained via a compression-expansion fixed point theorem for a composition of two multivalued maps, and finally, a suitable control of discontinuities allows to prove that any solution of the inclusion is a solution in the sense of Carathéodory of the initial discontinuous equation. No monotonicity assumptions on the nonlinearity are required

Positive radial solutions for Dirichlet problems via a Harnack-type inequality

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 solutionsJorge Rodríguez-López was partially supported by the Institute of Advanced Studies in Science and Technology of Babeş-Bolyai University of Cluj-Napoca (Romania) under a Postdoctoral Advanced Fellowship, project CNFIS-FDI-2021-0061 and by Xunta de Galicia (Spain), project ED431C 2019/02 and AIE, Spain, and FEDER, grant PID2020-113275GB-I00S

Uniqueness criteria for ordinary differential equations with a generalized transversality condition at the initial condition

In this paper, we present some uniqueness results for systems of ordinary differential equations. All of them are linked by a weak transversality condition at the initial condition, which generalizes those in the previous literature. Several examples are also provided to illustrate our results

New Lipschitz–type conditions for uniqueness of solutions of ordinary differential equations

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 testsRodrigo López Pouso and Jorge Rodríguez López were partially supported by grant ED431C 2019/02, Xunta de Galicia (Spain), and by Ministerio de Ciencia y Tecnología (Spain), grant PID2020-113275GB-I00S