Recent Advances in Functional Analysis, Semigroup Theory and Difference-Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 August 2021) | Viewed by 11129

Special Issue Editors


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Departamento de Matemáticas, Universidad de Zaragoza, Spain
Interests: Functional analysis; semigroup theory; Banach algebra; number theory;

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Guest Editor
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: fractional calculus; dynamics on time scales; mathematical biology; calculus of variations; optimal control
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Special Issue Information

Functional analysis is a methodology that is used to explain the workings of a complex system, such as that of our physical world. There has been special interest in illustrating its connections with semigroup theory and differential–difference equations; both branches are powerful tools that can provide new and interesting results.

In this Special Issue, we will show the importance of these theories with numerous applications to theoretical and physical problems. We will consider fractional difference–differential equations (in a wide sense) rather than ordinary differential–difference derivatives, as they provide an excellent instrument to describe certain processes and systems with nonlocality and memory. Consequently, these equations are used in an ever-widening range of models of physical processes, and they have attracted much attention in recent years. This research will provide new challenges to the scientific community.

Prof. Dr. Pedro J. Miana
Prof. Dr. Delfim F. M. Torres
Guest Editors

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Keywords

  • semigroup theory
  • evolution equations
  • difference–differential equations
  • fractional powers
  • discrete and continuous Laplacians

Published Papers (6 papers)

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Research

14 pages, 524 KiB  
Article
On Hermite–Hadamard-Type Inequalities for Coordinated h-Convex Interval-Valued Functions
by Dafang Zhao, Guohui Zhao, Guoju Ye, Wei Liu and Silvestru Sever Dragomir
Mathematics 2021, 9(19), 2352; https://doi.org/10.3390/math9192352 - 22 Sep 2021
Cited by 14 | Viewed by 1725
Abstract
This paper is devoted to establishing some Hermite–Hadamard-type inequalities for interval-valued functions using the coordinated h-convexity, which is more general than classical convex functions. We also discuss the relationship between coordinated h-convexity and h-convexity. Furthermore, we introduce the concepts of [...] Read more.
This paper is devoted to establishing some Hermite–Hadamard-type inequalities for interval-valued functions using the coordinated h-convexity, which is more general than classical convex functions. We also discuss the relationship between coordinated h-convexity and h-convexity. Furthermore, we introduce the concepts of minimum expansion and maximum contraction of interval sequences. Based on these two new concepts, we establish some new Hermite–Hadamard-type inequalities, which generalize some known results in the literature. Additionally, some examples are given to illustrate our results. Full article
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12 pages, 277 KiB  
Article
Pontryagin Maximum Principle for Distributed-Order Fractional Systems
by Faïçal Ndaïrou and Delfim F. M. Torres
Mathematics 2021, 9(16), 1883; https://doi.org/10.3390/math9161883 - 8 Aug 2021
Cited by 4 | Viewed by 1899
Abstract
We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques [...] Read more.
We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example. Full article
15 pages, 311 KiB  
Article
Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces
by Pedro J. Miana and Natalia Romero
Mathematics 2021, 9(9), 984; https://doi.org/10.3390/math9090984 - 27 Apr 2021
Cited by 1 | Viewed by 1195
Abstract
Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula [...] Read more.
Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them. Full article
13 pages, 801 KiB  
Article
Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent
by Muhammad Zainul Abidin and Jiecheng Chen
Mathematics 2021, 9(5), 498; https://doi.org/10.3390/math9050498 - 28 Feb 2021
Cited by 6 | Viewed by 1779
Abstract
In this paper, we consider the generalized porous medium equation. For small initial data u0 belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the [...] Read more.
In this paper, we consider the generalized porous medium equation. For small initial data u0 belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution. Full article
15 pages, 320 KiB  
Article
(ω,c)-Periodic Mild Solutions to Non-Autonomous Abstract Differential Equations
by Luciano Abadias, Edgardo Alvarez and Rogelio Grau
Mathematics 2021, 9(5), 474; https://doi.org/10.3390/math9050474 - 25 Feb 2021
Cited by 6 | Viewed by 1533
Abstract
We investigate the semi-linear, non-autonomous, first-order abstract differential equation [...] Read more.
We investigate the semi-linear, non-autonomous, first-order abstract differential equation x(t)=A(t)x(t)+f(t,x(t),φ[α(t,x(t))]),tR. We obtain results on existence and uniqueness of (ω,c)-periodic (second-kind periodic) mild solutions, assuming that A(t) satisfies the so-called Acquistapace–Terreni conditions and the homogeneous associated problem has an integrable dichotomy. A new composition theorem and further regularity theorems are given. Full article
9 pages, 293 KiB  
Article
Lp-Lq-Well Posedness for the Moore–Gibson–Thompson Equation with Two Temperatures on Cylindrical Domains
by Carlos Lizama and Marina Murillo-Arcila
Mathematics 2020, 8(10), 1748; https://doi.org/10.3390/math8101748 - 12 Oct 2020
Cited by 1 | Viewed by 1288
Abstract
We examine the Cauchy problem for a model of linear acoustics, called the Moore–Gibson–Thompson equation, describing a sound propagation in thermo-viscous elastic media with two temperatures on cylindrical domains. For an adequate combination of the parameters of the model we prove Lp [...] Read more.
We examine the Cauchy problem for a model of linear acoustics, called the Moore–Gibson–Thompson equation, describing a sound propagation in thermo-viscous elastic media with two temperatures on cylindrical domains. For an adequate combination of the parameters of the model we prove Lp-Lq-well-posedness, and we provide maximal regularity estimates which are optimal thanks to the theory of operator-valued Fourier multipliers. Full article
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