A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

Nabla fractional boundary value problem with a non-local boundary condition

In this work, we deal with the following two-point boundary value problem for a finite fractional nabla difference equation with non-local boundary condition: (−(∇ξρ(e) u(z) = p(z, u(z)), z ∈ Nfe+2, u(e) = g(u), u(f) = 0. Here e, f ∈ R, with f −e ∈ N3, 1 < ξ < 2, p : Nfe+2 ×R → R is a continuous function, the functional g ∈ C[Nfe → R] and ∇ξρ(e) denotes the ξth- order Riemann–Liouville backward (nabla) difference operator. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselskii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient conditions for the existence of at least one positive solution to the boundary value problem. Next, we discuss the uniqueness of the solution to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem respectively. Finally, we provide an example to illustrate the applicability of established results.Publisher's Versio

Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

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.Comment: 3 figures, 23 page

On the solvability of third-order three point systems of differential equations with dependence on the first derivative

This paper presents sufficient conditions for the solvability of the third order three point boundary value problem \begin{equation*} \left\{ \begin{array}{c} -u^{\prime \prime \prime }(t)=f(t,\,v(t),\,v^{\prime }(t)) \\ -v^{\prime \prime \prime }(t)=h(t,\,u(t),\,u^{\prime }(t)) \\ u(0)=u^{\prime }(0)=0,u^{\prime }(1)=\alpha u^{\prime }(\eta ) \\ v(0)=v^{\prime }(0)=0,v^{\prime }(1)=\alpha v^{\prime }(\eta ). \end{array} \right. \end{equation*} The arguments apply Green's function associated to the linear problem and the Guo--Krasnosel'ski\u{\i} theorem of compression-expansion cones. The dependence on the first derivatives is overcome by the construction of an adequate cone and suitable conditions of superlinearity/sublinearity near $0$ and $+\infty .$ Last section contains an example to illustrate the applicability of the theorem.Comment: 21 page

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

New Trends on Nonlocal and Functional Boundary Value Problems

In the last decades, boundary value problems with nonlocal and functional boundary conditions have become a rapidly growing area of research. The study of this type of problems not only has a theoretical interest that includes a huge variety of differential, integrodifferential, and abstract equations, but also is motivated by the fact that these problems can be used as a model for several phenomena in engineering, physics, and life sciences that standard boundary conditions cannot describe. In this framework, fall problems with feedback controls, such as the steady states of a thermostat, where a controller at one of its ends adds or removes heat depending upon the temperature registered in another point, or phenomena with functional dependence in the equation and/or in the boundary conditions, with delays or advances, maximum or minimum arguments, such as beams where the maximum (minimum) of the deflection is attained in some interior or endpoint of the beam. Topological and functional analysis tools, for example, degree theory, fixed point theorems, or variational principles, have played a key role in the developing of this subject. This volume contains a variety of contributions within this area of research. The articles deal with second and higher order boundary value problems with nonlocal and functional conditions for ordinary, impulsive, partial, and fractional differential equations on bounded and unbounded domains. In the contributions, existence, uniqueness, and asymptotic behaviour of solutions are considered by using several methods as fixed point theorems, spectral analysis, and oscillation theory

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)