A substantial number of problems in physics, chemical physics, and biology, are modeled through reaction-diffusion equations to describe temperature distribution or chemical substance concentration. For problems arising from ecology, sociology, or population dynamics, they describe the density of some populations or species. In this book the state variable is a concentration, or a density according to the cases. The reaction function may be complex and include time delays terms that model various situations involving maturation periods, resource regeneration times, or incubation periods. The dynamics may occur in heterogeneous media and may depend upon a small or large parameter, as well as the reaction term. From a purely formal perspective, these parameters are indexed by n. Therefore, reaction-diffusion equations give rise to sequences of Cauchy problems.The first part of the book is devoted to the convergence of these sequences in a sense made precise in the book. The second part is dedicated to the specific case when the reaction-diffusion problems depend on a small parameter ∊ₙ intended to tend towards 0. This parameter accounts for the size of small spatial and randomly distributed heterogeneities. The convergence results obtained in the first part, with additionally some probabilistic tools, are applied to this specific situation. The limit problems are illustrated through biological invasion, food-limited or prey-predator models where the interplay between environment heterogeneities in the individual evolution of propagation species plays an essential role. They provide a description in terms of deterministic and homogeneous reaction-diffusion equations, for which numerical schemes are possible.

From its origins in the minimization of integral functionals, the notion of variations has evolved greatly in connection with applications in optimization, equilibrium, and control. This book develops a unified framework and provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, and normal integrands.

From its origins in the minimization of integral functionals, the notion of variations has evolved greatly in connection with applications in optimization, equilibrium, and control. This book develops a unified framework and provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, and normal integrands.

This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory of nonexpansive operators. Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and interactions between the areas as the central focus, and it is illustrated by a large number of examples. The Hilbert space setting of the material offers a wide range of applications while avoiding the technical difficulties of general Banach spaces. The authors have also drawn upon recent advances and modern tools to simplify the proofs of key results making the book more accessible to a broader range of scholars and users. Combining a strong emphasis on applications with exceptionally lucid writing and an abundance of exercises, this text is of great value to a large audience including pure and applied mathematicians as well as researchers in engineering, data science, machine learning, physics, decision sciences, economics, and inverse problems. The second edition of Convex Analysis and Monotone Operator Theory in Hilbert Spaces greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated. Heinz H. Bauschke is a Full Professor of Mathematics at the Kelowna campus of the University of British Columbia, Canada. Patrick L. Combettes, IEEE Fellow, was on the faculty of the City University of New York and of Université Pierre et Marie Curie – Paris 6 before joining North Carolina State University as a Distinguished Professor of Mathematics in 2016.

This Volume contains the (refereed) papers presented at the 38th Conference of the School of Mathematics "G.Stampacchia" of the "E.Majorana" Centre for Scientific Culture of Erice (Sicily), held in Memory ofG. Stampacchia and J.-L. Lions in the period June 20 - July 2003. The presence of participants from Countries has greatly contributed to the success of the meeting. The School of Mathematics was dedicated to Stampacchia, not only for his great mathematical achievements, but also because He founded it. The core of the Conference has been the various features of the Variational Analysis and their motivations and applications to concrete problems. Variational Analysis encompasses a large area of modem Mathematics, such as the classical Calculus of Variations, the theories of perturbation, approximation, subgradient, subderivates, set convergence and Variational Inequalities, and all these topics have been deeply and intensely dealt during the Conference. In particular, Variational Inequalities, which have been initiated by Stampacchia, inspired by Signorini Problem and the related work of G. Fichera, have offered a very great possibility of applications to several fundamental problems of Mathematical Physics, Engineering, Statistics and Economics. The pioneer work of Stampacchia and Lions can be considered as the basic kernel around which Variational Analysis is going to be outlined and constructed. The Conference has dealt with both finite and infinite dimensional analysis, showing that to carry on these two aspects disjointly is unsuitable for both.

Since I started working in the area of nonlinear programming and, later on, variational inequality problems, I have frequently been surprised to find that many algorithms, however scattered in numerous journals, monographs and books, and described rather differently, are closely related to each other. This book is meant to help the reader understand and relate algorithms to each other in some intuitive fashion, and represents, in this respect, a consolidation of the field. The framework of algorithms presented in this book is called Cost Approxi mation. (The preface of the Ph.D. thesis [Pat93d] explains the background to the work that lead to the thesis, and ultimately to this book.) It describes, for a given formulation of a variational inequality or nonlinear programming problem, an algorithm by means of approximating mappings and problems, a principle for the update of the iteration points, and a merit function which guides and monitors the convergence of the algorithm. One purpose of this book is to offer this framework as an intuitively appeal ing tool for describing an algorithm. One of the advantages of the framework, or any reasonable framework for that matter, is that two algorithms may be easily related and compared through its use. This framework is particular in that it covers a vast number of methods, while still being fairly detailed; the level of abstraction is in fact the same as that of the original problem statement.