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

NONLINEAR SINGULAR STURM-LIOUVILLE PROBLEMS WITH IMPULSIVE CONDITIONS

In this paper, we consider a non-linear impulsive Sturm-Liouville problem on semiinfinite intervals in which the limit-circle case holds at infinity for THE Sturm-Liouville expression. We prove the existence and uniqueness theorems for this problem

Fractional Calculus - Theory and Applications

In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

Multiple Positive solutions of Sturm-Liouville problems for second order singular and impulsive differential equations

Abstract In this paper,we study the existence of multiple positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations.The proof is based on the theory of fixed point index in cones. Mathematics Subject Classification: 34B15 Keywords: Multiple positive solutions; Singular two-point boundary value problem; Second-order impulsive differential equations; Fixed point index in cones

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

Positive solutions of nonlinear fractional differential equations with integral boundary value conditions

AbstractIn this paper, we consider the existence of positive solutions for a class of nonlinear boundary-value problem of fractional differential equations with integral boundary conditions. Our analysis relies on known Guo–Krasnoselskii fixed point theorem

Existencia de tres soluciones para el sistema hamiltoniano fraccionario

In this paper we consider the fractional Hamiltonian system given by(0.1)                       −tDα T(0Dα t u(t)) = ∇F(t,u(t)), a.e t ∈ [0,T]                                 u(0) = u(T) = 0.where α ∈ (1/2,1), t ∈ [0,T], u ∈Rn, F : [0,T]×Rn →R is a given function and ∇F(t,u) is the gradient of F at u. The novelty of this paper is that, using a modified version of mountain pass theorem for functional bounded from below we prove the existence of at least three solutions for (0.2). En este artículo se considera un sistema Hamiltoniano dado por:(0.1)              −tDα T(0Dα t u(t)) = ∇F(t,u(t)), a.e t ∈ [0,T]                       u(0) = u(T) = 0.donde α ∈ (1/2,1), t ∈ [0,T], u ∈ Rn, F : [0,T]×Rn → R es una función dada y ∇F(t,u) es el gradiente de F en u. La novedad de este trabajo es que, usando una versión modificada del teorema del paso de montaña para funcional limitada desde abajo probamos la existencia de por lo menos tres soluciones para (0.1)

Mixed Boundary Value Problem for Nonlinear Fractional Volterra Integral Equation

In this paper we present the existence of solutions for a nonlinear fractional integral equation of Volterra type with mixed boundary conditions, some necessary hypotheses have been developed to prove the existence of solutions to the proposed equation. Krasnoselskii Theorem, Banach Contraction principle, and Leray-Schauder degree theory are the basic theorems used here to find the results. A simple example of the application of the main result is presented