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

Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte

Nonlinear qq-fractional differential equations with nonlocal and sub-strip type boundary conditions

This paper is concerned with new boundary value problems of nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions. Our results are new in the present setting and rely on the contraction mapping principle and a fixed point theorem due to O'Regan. Some illustrative examples are also presented

Fractional fundamental lemma and fractional integration by parts formula -- Applications to critical points of Bolza functionals and to linear boundary value problems

In the first part of the paper, we prove a fractional fundamental (du Bois-Reymond) lemma and a fractional variant of the integration by parts formula. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville fractional derivatives, derived in this paper too. In the second part of the paper, we use the previous results to give necessary optimality conditions of Euler-Lagrange type (with boundary conditions) for fractional Bolza functionals and to prove an existence result for solutions of linear fractional boundary value problems. In the last case we use a Hilbert structure and the Stampacchia theorem.Comment: This is a preprint of a paper whose final and definite form is published in Advances in Differential Equation

New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions

We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: x(0)=βx(θ), x(ξ)=α∫η1‍x(s)ds, and 0<θ<ξ<η<1. According to these conditions, the value of the unknown function at the left end point t=0 is proportional to its value at a nonlocal point θ while the value at an arbitrary (local) point ξ is proportional to the contribution due to a substrip of arbitrary length (1-η). These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented

Functional Calculus

The aim of this book is to present a broad overview of the theory and applications related to functional calculus. The book is based on two main subject areas: matrix calculus and applications of Hilbert spaces. Determinantal representations of the core inverse and its generalizations, new series formulas for matrix exponential series, results on fixed point theory, and chaotic graph operations and their fundamental group are contained under the umbrella of matrix calculus. In addition, numerical analysis of boundary value problems of fractional differential equations are also considered here. In addition, reproducing kernel Hilbert spaces, spectral theory as an application of Hilbert spaces, and an analysis of PM10 fluctuations and optimal control are all contained in the applications of Hilbert spaces. The concept of this book covers topics that will be of interest not only for students but also for researchers and professors in this field of mathematics. The authors of each chapter convey a strong emphasis on theoretical foundations in this book

Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations

Under certain conditions, solutions of the boundary value problem&nbsp;y(n)&nbsp;= f(x,y,y',...,y(n-1)), a &lt; x &lt; b, y(i-1)(x1) = yi, i=1,...,n-1, y(x2) ∑&nbsp;i=1mγi&nbsp;∫&nbsp;ξiηiy(x)dx=yn, a&lt;x1&lt;ξ1&lt;η1&lt;ξ2&lt;η2&lt;...&lt;ξm&lt;ηm&lt;x2&lt;b, are differentiated with respect to the boundary conditions