Nonnegative solutions of nonlinear fractional Laplacian equations

The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows long-range interactions in space, and it is therefore important from the application point of view. The fractional Laplacian operator plays a similar role in the study of nonlocal diffusion operators as the Laplacian operator does in the local case. Therefore, the goal of this dissertation is a systematic treatment of steady state reaction-diffusion problems involving the fractional Laplacian as the diffusion operator on a bounded domain and to investigate existence (and nonexistence) results with respect to a bifurcation parameter. In particular, we establish existence results for positive solutions depending on the behavior of a nonlinear reaction term near the origin and at infinity. We use topological degree theory as well as the method of sub- and supersolutions to prove our existence results. In addition, using a moving plane argument, we establish that, for a class of steady state reaction-diffusion problems involving the fractional Laplacian, any nonnegative nontrivial solution in a ball must be positive, and hence radially symmetric and radially decreasing. Finally, we provide numerical bifurcation diagrams and the profiles of numerical positive solutions, corresponding to theoretical results, using finite element methods in one and two dimensions

Symmetry in Modeling and Analysis of Dynamic Systems

Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries

Fractional Differential Equations, Inclusions and Inequalities with Applications

During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering

Sistemas de equações diferenciais não lineares de ordem superior em domínios limitados ou não limitados

The Boundary value problems on bounded or unbounded intervals, involving two or more coupled systems of the nonlinear differen- tial equations with full nonlinearities are scarce and have gap in literature. The present work modestly try to fill this gap. The systems covered in the work are essentially of the second- order (except for the first chapter of the first part) with boundary constraints either in bounded or unbounded intervals presented in several forms and conditions (three points, mixed, with functional dependence, homoclinic and heteroclinic). The existence, and in some cases, the localization of the solu- tions is carried out in of Banach space and norms considered, fo- llowing arguments and approaches such as: Schauder’s fixed-point theorem or of Guo–Krasnosel’ski˘ı fixed-point theorem in cones, allied to Green’s function or its estimates, lower and upper solutions, convenient truncatures, the Nagumo condition presented in different forms, concept of equiconvergence, Carathéodory functions and sequences. On the other hand, parallel to the theoretical explanation of this work, there is a range of practical examples and applications involving real phenomena, focusing on the physics, mechanics, bio- logy, forestry, and dynamical systems; A falta ou a raridade de problemas de valor fronteira na literatura, quer em dom´ınios limitados ou ilimitados, envolvendo sistemas de duas ou mais equações n˜ao lineares acopladas com todas as n˜ao linearidades completas, levou à elaboração do presente trabalho. Os sistemas abordados no trabalho s˜ao essencialmente de segunda ordem (exceto o primeiro capítulo da primeira parte) com condições de fronteira em domínios limitados ou ilimitados, de diversos tipos (três pontos, mistas, com condições funcionais, homoclínicas e heteroclínicas). A existência e em alguns casos a localização das soluções dos sistemas è considerada em espaços de Banach, seguindo vários ar- gumentos e abordagens: o teorema de ponto fixo de Schauder ou de Guo–Krasnosel’ski˘ı em cones, aliados a funções de Green ou suas estimativas, sub e sobre-soluções, truncaturas convenientes, a condição de Nagumo apresentada sob várias formas, o conceito de equiconvergência e funções e sucess˜oes de Carath´eodory. Por outro lado, paralelamente àcomponente teórica do trabalho, encontra-se um leque de aplicações e exemplos práticos envolvendo fenómenos reais, com enfoque na física, mecânica, biologia, exploração florestal e sistemas dinâmico

Existence and Multiplicity of Solutions of Functional Differential Equations

The first part of the memory goes through those discoveries related to Green’s functions. In order to do that, first we recall some general results concerning involutions which will help us understand their remarkable analytic and algebraic properties. Chapter 1 will deal about this subject while Chapter 2 will give a brief overview on differential equations with involutions to set the reader in the appropriate research framework. In Chapter 3 we start working on the theory of Green’s functions for functional differential equations with involutions in the most simple cases: order one problems with constant coefficients and reflection. Here we solve the problem with different boundary conditions, studying the specific characteristics which appear when considering periodic, anti-periodic, initial or arbitrary linear boundary conditions. We also apply some very well known techniques (lower and upper solutions method or Krasnosel’skiĭ’s Fixed Point Theorem, for instance) in order to further derive results. Computing explicitly the Green’s function for a problem with nonconstant coefficients is not simple, not even in the case of ordinary differential equations. We face these obstacles in Chapter 4, where we reduce a new, more general problem containing nonconstant coefficients and arbitrary differentiable involutions, to the one studied in Chapter 3. To end this part of the work, we have Chapter 5, in which we deepen in the algebraic nature of reflections and extrapolate these properties to other algebras. In this way, we do not only generalize the results of Chapter 3 to the case of -th order problems and general twopoint boundary conditions, but also solve functional differential problems in which the Hilbert transform or other adequate operators are involved. The last chapters of this part are about applying the results we have proved so far to some related problems. First, in Chapter 6, setting again the spotlight on some interesting relation between an equation with reflection and an equation with a -Laplacian, we obtain some results concerning the periodicity of solutions of that first problem with reflection. Chapter 7 moves to a more practical setting. It is of the greatest interest to have adequate computer programs in order to derive the Green’s functions obtained in Chapter 5 for, in general, the computations involved are very convoluted. Being so, we present in this chapter such an algorithm, implemented in Mathematica. The reader can find in the appendix the exact code of the program. In the second part of the Thesis we use the fixed point index to solve four different kinds of problems increasing in complexity: a problem with reflection, a problem with deviated arguments (applied to a thermostat model), a problem with nonlinear Neumann boundary conditions and a problem with functional nonlinearities in both the equation and the boundary conditions. As we will see, the particularities of each problem make it impossible to take a common approach to all of the problems studied. Still, there will be important similarities in the different cases which will lead to comparable results