F. Adrián F. Tojo

Non-trivial solutions of local and non-local Neumann boundary-value problems

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

Differential equations with involutions

Involutions and differential equations.- General results for differential equations with involutions.- Order one problems with constant coefficients.- The non-constant case.- General linear equations.- A cone approximation to a problem with reflection.

On linear differential equations and systems with reflection

In this paper we develop a theory of linear differential systems analogous to the classical one for ODEs, including the obtaining of fundamental matrices, the development of a variation of parameters formula and the expression of the Greens functions. We also derive interesting results in the case of differential equations with reflection and generalize the Hyperbolic Phasor Addition Formula to the case of matrices.

General Results for Differential Equations with Involutions

This chapter is devoted to those results related to differential equations with reflection not directly associated with Green’s functions. The proofs of the results can be found in the bibliography cited for each case. We will not enter into detail with these results, but we summarize their nature for the convenience of the reader.

Differential systems with reflection and matrix invariants

In this work we derive important properties regarding matrix invariants which occur in the theory of differential equations with reflection.

Involutions and Differential Equations

Involutions, as we will see, have very special properties. This is due to their double nature, analytic and algebraic. This chapter is therefore divided in two sections that will explore the two kinds of properties, arriving at last to some parallelism between involutions and complex numbers for their capability to decompose certain polynomials (see Remark 1.3.6). In this chapter we recall results from several authors.

General Linear Equations

In this chapter we study differential problems in which the reflection operator and the Hilbert transform are involved. We reduce these problems to ODEs in order to solve them. Also, we describe a general method for obtaining the Green’s function of reducible functional differential equations and illustrate it with the case of homogeneous boundary value problems with reflection and several specific examples.

A Cone Approximation to a Problem with Reflection

In this chapter we continue this study and we prove new results regarding the existence of nontrivial solutions of Hammerstein integral equations with reflections of the form

Comparison results for first order linear operators with reflection and periodic boundary value conditions

\begin{aligned} u(t)=\int _{-T}^{T} k(t,s)g(s)f(s,u(s),u(-s))\mathrm {d}s,\quad t\in [-T,T], \end{aligned}