Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses

In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order $p$-Laplacian differential equations with instantaneous and non-instantaneous impulses: \begin{equation*} \left\{ {\begin{array}{l} -(\rho(t)\Phi_{p} (u'(t)))'+g(t)\Phi_{p}(u(t))=\lambda f_{j}(t,u(t)),\quad t\in(s_{j},t_{j+1}],\; j=0,1,...,m,\\ \Delta (\rho (t_{j})\Phi_{p}(u'(t_{j})))=\mu I_{j}(u(t_{j})), \quad j=1,2,...,m,\\ \rho (t)\Phi_{p} (u'(t))=\rho(t_{j}^{+}) \Phi_{p} (u'(t_{j}^{+})),\quad t\in(t_{j},s_{j}],\; j=1,2,...,m,\\ \rho(s_{j}^{+})\Phi_{p} (u'(s_{j}^{+}))=\rho(s_{j}^{-})\Phi_{p} (u'(s_{j}^{-})),\quad j=1,2,...,m,\\ u(0)=0, \quad u(1)=\zeta u(\eta), \end{array}} \right. \end{equation*} where \Phi_{p}(u): = |u|^{p-2}u, \; p > 1, \; 0 = s_{0} < t_{1} < s_{1} < t_{2} < ... < s_{m_{1}} < t_{m_{1}+1} = \eta < ... < s_{m} < t_{m+1} = 1, \; \zeta > 0, \; 0 < \eta < 1 , $\Delta (\rho (t_{j})\Phi_{p}(u'(t_{j}))) = \rho (t_{j}^{+})\Phi_{p}(u'(t_{j}^{+}))-\rho (t_{j}^{-})\Phi_{p}(u'(t_{j}^{-}))$ for $u'(t_{j}^{\pm}) = \lim\limits_{t\to t_{j}^{\pm}}u'(t)$, $j = 1, 2, ..., m$, and $f_{j}\in C((s_{j}, t_{j+1}]\times\mathbb{R}, \mathbb{R})$, $I_{j}\in C(\mathbb{R}, \mathbb{R})$. $\lambda\in (0, +\infty)$, $\mu\in\mathbb{R}$ are two parameters. $\rho(t)\geq 1$, $1\leq g(t)\leq c$ for $t\in (s_{j}, t_{j+1}]$, $\rho(t), \; g(t)\in L^{p}[0, 1]$, and $c$ is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results

A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.&nbsp

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)

Existence results for impulsive fractional differential equations with pp-Laplacian via variational methods

summary:This paper presents several sufficient conditions for the existence of at least one classical solution to impulsive fractional differential equations with a $p$-Laplacian and Dirichlet boundary conditions. Our technical approach is based on variational methods. Some recent results are extended and improved. Moreover, a concrete example of an application is presented

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

Symmetry and Complexity

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry

Problemas impulsivos de ordem superior não-lineares e funcionais

A abordagem teórica dos problemas do tipo impulsivo tem assumido importância crescente no mundo científico e industrial de hoje, pelas respostas que oferece aos problemas de pesquisa e produção industrial, nos quais as ocorrências de descontinuidades abruptas e saltos funcionais são decisivos nos respectivos contextos. Em suma, problemas impulsivos representam fenómenos em que ocorrem mudanças repentinas nas suas propriedades dinâmicas. Esses problemas são frequentes no estudo da dinâmica populacional, na otimização de processos de produção, na biologia, nos problemas de estudo do comportamento dos fatores ambientais, na medicina e na farmacologia, entre outros ramos da Ciência. Nesta teses são abordados problemas de ordem superior com valores na fronteira, com mudanças instantâneas na função incógnita e nas suas derivadas. Nos últimos anos, os operadores Laplaciano e as suas variantes como, por exemplo, p-Laplaciano e phi-Laplaciano, têm sido aplicados em várias situações mas poucas vezes no caso descontínuo. Os três primeiros capítulos da tese procuram contribuir para colmatar esta falha. O caso periódico e a situação em que o domínio de definição não é limitado, são mais delicados e exigem um tipo de abordagem diferente. Nos Capítulos 4, 5 e 6 apresentam-se técnicas e métodos topológicos que permitem abordar estes tipos de problemas; ABSTRACT: The theoretical approach of impulsive problems has assumed an increasing importance in the scientiÖc and industrial world today, due to its answers to the problems of research and industrial production, in which the occurrences of abrupt discontinuities and functional leaps are decisive in the respective contexts. In short, impulsive problems represent phenomena in which sudden changes in their dynamic properties occur. These problems are frequent in the study of population dynamics, in the optimization of production processes, in biology, in problems of studying the behavior of environmental factors, in medicine and pharmacology, among other branches of science. This thesis addresses problems of a higher order with boundary values problems with instantaneous changes in the unknown function and its derivatives. In recent years, Laplacian operators and their variants, such as p-Laplacian and Phi-Laplacian, have been applied in several situations, but rarely in the discontinuous case. The Örst three chapters of the thesis seek to contribute to Öll this gap. The periodic case and the situation in which the domain of deÖnition is not bounded, are more delicate and require a di§erent type of approach. In Chapters 4, 5 and 6, topological techniques and methods are presented that allow to approach these types of problems

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

A numerical method for fluid-structure interactions of slender rods in turbulent flow

This thesis presents a numerical method for the simulation of fluid-structure interaction (FSI) problems on high-performance computers. The proposed method is specifically tailored to interactions between Newtonian fluids and a large number of slender viscoelastic structures, the latter being modeled as Cosserat rods. From a numerical point of view, such kind of FSI requires special techniques to reach numerical stability. When using a partitioned fluid-structure coupling approach this is usually achieved by an iterative procedure, which drastically increases the computational effort. In the present work, an alternative coupling approach is developed based on an immersed boundary method (IBM). It is unconditionally stable and exempt from any global iteration between the fluid part and the structure part. The proposed FSI solver is employed to simulate the flow over a dense layer of vegetation elements, usually designated as canopy flow. The abstracted canopy model used in the simulation consists of 800 strip-shaped blades, which is the largest canopy-resolving simulation of this type done so far. To gain a deeper understanding of the physics of aquatic canopy flows the simulation data obtained are analyzed, e.g., concerning the existence and shape of coherent structures

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics