Characterization of the Solvability of Generalized Constrained Variational Equations

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.This research is partially supported by the Junta de Andaluca Grant FQM359

Editorial: On the “Human” in Human-Artificial Intelligence Interaction

ST was supported by MIUR-Italian Ministry of University and Research (Department of Excellence Italian Law n.232, 11th December 2016) for University of Milan. ID was supported by Fondazione Umberto Veronesi.Artificial Intelligence or technologies able to perform tasks normally requiring human cognitive processes (e.g., reasoning, perception) are revolutionizing many fields such as healthcare and business. For example, medical doctors use artificial intelligence to analyze pathological data and patients’ genomic profiles to identify personalized treatment according to a precision medicine approach. In general, artificial intelligence represents an invaluable resource for any professional dealing with the need to understand data and make decisions.Ministry of Education, Universities and Research (MIUR)Fondazione Umberto Verones

An Iterative Algorithm for Approximating the Fixed Point of a Contractive Affine Operator

This research was partially supported by Junta de Andalucia, Project "Convex and numerical analysis", reference FQM359, and by the "Maria de Maeztu" Excellence Unit IMAG, reference CEX2020-001105-M, funded by MCIN/AEI/10.13039/501100011033/.First of all, in this paper we obtain a perturbed version of the geometric series theorem, which allows us to present an iterative numerical method to approximate the fixed point of a contractive affine operator. This result requires some approximations that we obtain using the projections associated with certain Schauder bases. Next, an algorithm is designed to approximate the solution of Fredholm’s linear integral equation, and we illustrate the behavior of the method with some numerical examples.Junta de Andalucia FQM359"Maria de Maeztu" Excellence Unit IMAG - MCIN/AEI CEX2020-001105-

Inverse Problems via the “Generalized Collage Theorem” for Vector-Valued Lax-Milgram-Based Variational Problems

We present an extended version of the Generalized Collage Theorem to deal with inverse problems for vector-valued Lax-Milgram systems. Numerical examples show how the method works in practical cases

Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces C[0,1]and C[0,1]2.This research is partially supported by M.E.C. (Spain) and FEDER, project MTM2006-12533, and by Junta de Andaluca Grant FQM359

Cyt c6-3: A new isoform of photosynthetic Cyt c6 exclusive to heterocyst-forming cyanobacteria

All known cyanobacteria contain Cyt c6, a small soluble electron carrier protein whose main function is to transfer electrons from the Cyt b6 f complex to PSI, although it is also involved in respiration. We have previously described a second isoform of this protein, the Cyt c6-like, whose function remains unknown. Here we describe a third isoform of Cyt c6 (here called Cytc6-3), which is only found in heterocyst- forming filamentous cyanobacteria. Cyt c6-3 is expressed in vegetative cells but is specifically repressed in heterocysts cells under diazotrophic growth conditions. Although there is a close structural similarity between Cyt c6-3 and Cyt c6 related to the general protein folding, Cyt c6-3 presents differential electrostatic surface features as compared with Cyt c6, its expression is not copper dependent and has a low reactivity towards PSI. According to the different expression pattern, functional reactivity and structural properties, Cyt c6-3 has to play an as yet to be defined regulatory role related to heterocyst differentiation.Fundación de Investigación de la Universidad de Sevilla FIUS05710000Ministerio de Economía y Competitividad BIO2012-35271, BIO2015-64169-PJunta de Andalucía PAIDI BIO-02

Using the generalized collage theorem for estimating unknown parameters in perturbed mixed variational equations

In this paper, we study a mixed variational problem subject to perturbations, where the noise term is modelled by means of a bilinear form that has to be understood to be “small” in some sense. Indeed, we consider a family of such problems and provide a result that guarantees existence and uniqueness of the solution. Moreover, a stability condition for the solutions yields a Generalized Collage Theorem, which extends previous results by the same authors. We introduce the corresponding Galerkin method and study its convergence. We also analyze the associated inverse problem and we show how to solve it by means of the mentioned Generalized Collage Theorem and the use of adequate Schauder bases. Numerical examples show how the method works in a practical context.Project MTM2016-80676-P (AEI/FEDER, UE)Junta de Andalucía Grant FQM35

Solving inverse problems for mixed-variational equations on perforated domains

The aim of this work is to analyze some conditions for the existence of solution of a perturbed mixed variational system and that of an associated inverse problem related to the collagebased approach, both on perforated domains or domains with holes. In addition, we study the influence of the size of the holes and state some convergence results. Finally, we conduct a computational study for solving some of those inverse problemsJunta de Andalucía, Project FQM359, and by the “María de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-MMCIN/AEI/10.13039/501100011033/.Universidad de Granada/CBU

A Discrete Characterization of the Solvability of Equilibrium Problems and Its Application to Game Theory

We state a characterization of the existence of equilibrium in terms of certain finite subsets under compactness and transfer upper semicontinuity conditions. In order to derive some consequences on game theory—Nash equilibrium and minimax inequalities—we introduce a weak convexity conceptJunta de Andalucia, Project FQM359Maria de Maeztu” Excellence Unit IMAGCEX2020-001105-M, funded by MCIN/AEI/10.13039/501100011033

Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

The authors present a method of numerical approximation of the fixed point of an operator, specifically the integral one associated with a nonlinear Fredholm integral equation, that uses strongly the properties of a classical Schauder basis in the Banach space .This research partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533 and by Junta de Andalucía Grant FQM359