a b s t r a c tThis work is devoted to the study of the first order operator x ′ (t) + m x(−t) coupled with periodic boundary value conditions. We describe the eigenvalues of the operator and obtain the expression of its related Green's function in the non resonant case. We also obtain the range of the values of the real parameter m for which the integral kernel, which provides the unique solution, has constant sign. In this way, we automatically establish maximum and anti-maximum principles for the equation. Some applications to the existence of nonlinear periodic boundary value problems are shown.

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Abstract. We prove new results on the existence, non-existence, localization and multiplicity of nontrivial 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 nonlocal boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of Hammerstein integral equations with reflections. We apply our results to a first-order periodic boundary value problem with reflections. MSC: Primary 34K10; secondary 34B15; 34K13

Abstract. We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.

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In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our knowledge, has never been used before. In order to obtain fixed points of the integral operator, we will consider the fixed point index theory and apply it to this new Banach space.Partially supported by Xunta de Galicia (Spain), project EM2014/032 and AIE Spain and FEDER, grants MTM2013-43014-P, MTM2016-75140-P.Supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain.1 consequence of the lack of compactness of the operator. In all of the cited references the authors solve this problem by means of the following relatively compactness criterion (see [1,13]) which involves some stability condition at ±∞: 1 ([13, Theorem 1]). Let E be a Banach space and ( , E) the space of all bounded continuous functions x : → E. For a set D ⊂ ( , E) to be relatively compact, it is necessary and sufficient that:for any t ∈ ; 2. for each a > 0, the family D a := {x| [−a,a] , x ∈ D} is equicontinuous; 3. D is stable at ±∞, that is, for any ǫ > 0, there exists T > 0 and δ > 0 such that if x(T ) − y(T ) ≤ δ, then x(t) − y(t) ≤ ǫ for t ≥ T and if x(−T ) − y(−T ) ≤ δ, then x(t) − y(t) ≤ ǫ for t ≤ −T , where x and y are arbitrary functions in D.

In this work 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.