Special Issue "Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 30 November 2019.

Special Issue Editors

Prof. Constantin Buse
E-Mail Website
Guest Editor
Department of Mathematics, Polytechnic University of Timisoara, Romania
Interests: integral equations in Banach spaces; groups and semigroups of linear operators; qualitative theory of discrete and continuous evolution equations in Banach spaces; Hyers–Ulam stability and its connections with exponential dichotomy; long time behavior for solutions of abstract Cauchy problems in Banach spaces; fixed point theory and its application
Prof. Donal O'Regan
E-Mail Website
Guest Editor
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland
Interests: integral equations in Banach spaces; groups and semigroups of linear operators; qualitative theory of discrete and continuous evolution equations in Banach spaces; Hyers–Ulam stability and its connections with exponential dichotomy; long time behavior for solutions of abstract Cauchy problems in Banach spaces; fixed point theory and its application
Prof. Toka Diagana
E-Mail Website
Guest Editor
Department of Mathematical Sciences, University of Alabama in Huntsville, 301 Sparkman Drive, Huntsville, AL 35899, United States of America
Interests: integral equations in Banach spaces; groups and semigroups of linear operators; qualitative theory of discrete and continuous evolution equations in Banach spaces; Hyers–Ulam stability and its connections with exponential dichotomy; long time behavior for solutions of abstract Cauchy problems in Banach spaces; fixed point theory and its application

Special Issue Information

Dear Colleagues,

Nonlinear functional analysis is a branch of mathematical analysis that considers nonlinear mappings. This area is very popular mainly because many applications in functional analysis arise naturally in real-world problems. For example, operator theory arises in many applications in quantum mechanics, and new methods and results of functional analysis are now widely applied in mathematical physics, theoretical physics, and other areas of science. One of the main objectives of nonlinear analysis is to study differential and integral equations and nonlinear operators, and a popular area of focus is considering the local approximation of nonlinear operators by taking linear operators into account. As a result, the theory of approximation (in particular, fixed-point principles) and differential and integral calculus for functions that act between Banach space or more generally topological vector spaces are some of the basic tools of nonlinear functional analysis.

The present issue considers:

  1. The qualitative study of solutions for nonhomogeneous (difference and differential) equations having a forced term in some particular spaces of vector-valued functions (for example, Lebesgue–Bochner or Orlicz–Bochner spaces);
  2. Different integral conditions and their connections with stability of solutions for evolution equations;
  3. Connections between Hyers–Ulam stability (for reccurrences and differential equations and dynamical systems) and exponential dichotomy;
  4. The qualitative study of partial differential equations from the perspective of groups or semigroups of operators on Banach spaces.

Prof. Constantin Buse
Prof. Donal O'Regan
Prof. Toka Diagana
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • differential and difference equations
  • exponential dichotomy and Hyers–Ulam stability
  • functional calculus with matrices and operators
  • semigroups and groups of bounded linear operators in Banach spaces
  • qualitative theory for evolution equations
  • global problems concerning polynomials of matrices and operators

Published Papers (10 papers)

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Research

Open AccessArticle
A Modified Equation for Thickness of the Film Fabricated by Spin Coating
Symmetry 2019, 11(9), 1183; https://doi.org/10.3390/sym11091183 - 18 Sep 2019
Abstract
According to the equation for Newtonian fluids, the film thickness after spin coating is determined by five parameters: angular velocity, spin coating time, viscosity, density of the coating material, and initial thickness of the material before spin coating. The spin coating process is [...] Read more.
According to the equation for Newtonian fluids, the film thickness after spin coating is determined by five parameters: angular velocity, spin coating time, viscosity, density of the coating material, and initial thickness of the material before spin coating. The spin coating process is commonly controlled by adjusting only the angular velocity parameter and the coating time in the Newtonian expression. However, the measured coating thickness obtained is then compared to the theoretical thickness calculated from the Newtonian fluid equation. The measured coating thickness usually varies somewhat from the theoretical thickness; further details are described in Section 1. Thus, the Newtonian fluid equation must be modified to better represent the actual film thickness. In this paper, we derive a new formula for the spin coating film thickness, which is based on the equation for Newtonian fluids, but modified to better represent film thicknesses obtained experimentally. The statistical analysis is performed to verify our modifications. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
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Open AccessArticle
Pre-Schauder Bases in Topological Vector Spaces
Symmetry 2019, 11(8), 1026; https://doi.org/10.3390/sym11081026 - 09 Aug 2019
Abstract
A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n N in X such that for every x X there exists a unique sequence of scalars ( λ n ) n [...] Read more.
A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n N in X such that for every x X there exists a unique sequence of scalars ( λ n ) n N satisfying that x = n = 1 λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory
Symmetry 2019, 11(7), 928; https://doi.org/10.3390/sym11070928 - 16 Jul 2019
Abstract
We give a new and simplified definition of spectrumfor a nonlinear operator F acting in a real Banach space X, and study some of its features in terms of (qualitative and) quantitative properties of F such as the measure of noncompactness, [...] Read more.
We give a new and simplified definition of spectrumfor a nonlinear operator F acting in a real Banach space X, and study some of its features in terms of (qualitative and) quantitative properties of F such as the measure of noncompactness, α ( F ) , of F. Then, using as a main tool the Ekeland Variational Principle, we focus our attention on the spectral properties of F when F is a gradient operator in a real Hilbert space, and in particular on the role played by its Rayleigh quotient R ( F ) and by the best lower and upper bounds, m ( F ) and M ( F ) , of R ( F ) . Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients
Symmetry 2019, 11(4), 512; https://doi.org/10.3390/sym11040512 - 09 Apr 2019
Cited by 3
Abstract
Let q 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q [...] Read more.
Let q 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q 1 A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues
Symmetry 2019, 11(3), 339; https://doi.org/10.3390/sym11030339 - 07 Mar 2019
Cited by 1
Abstract
Let q 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : [...] Read more.
Let q 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q 1 A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x y ) ( y z ) ( z x ) 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n Z + is Hyers–Ulam stable if and only if ( | x | 1 ) ( | y | 1 ) ( | z | 1 ) 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w C : | w | = 1 } . Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
Nonoscillatory Solutions to Higher-Order Nonlinear Neutral Dynamic Equations
Symmetry 2019, 11(3), 302; https://doi.org/10.3390/sym11030302 - 28 Feb 2019
Abstract
For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of [...] Read more.
For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of the coefficients. Two examples are presented to demonstrate the efficiency of new results. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
β–Hyers–Ulam–Rassias Stability of Semilinear Nonautonomous Impulsive System
Symmetry 2019, 11(2), 231; https://doi.org/10.3390/sym11020231 - 15 Feb 2019
Cited by 5
Abstract
In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the β –Ulam stability, β –Hyers–Ulam stability and β –Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. [...] Read more.
In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the β –Ulam stability, β –Hyers–Ulam stability and β –Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. We use Grönwall type inequality and evolution family as a basic tool for our results. We present an example to demonstrate the application of the main result. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
A New Approach to the Solution of Non-Linear Integral Equations via Various FBe-Contractions
Symmetry 2019, 11(2), 206; https://doi.org/10.3390/sym11020206 - 12 Feb 2019
Cited by 2
Abstract
In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended F B e -contraction, the extended F B e -expanding contraction, and the extended generalized F B e -contraction. Thereafter, we [...] Read more.
In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended F B e -contraction, the extended F B e -expanding contraction, and the extended generalized F B e -contraction. Thereafter, we propose a simple and efficient solution for non-linear integral equations using the fixed point technique in the setting of a B e -metric space. Moreover, to address conceptual depth within this approach, we supply illustrative examples where necessary. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
Open AccessArticle
Optimal Control of Nonsmooth Production Systems with Deteriorating Items, Stock-Dependent Demand, with or without Backorders
Symmetry 2019, 11(2), 183; https://doi.org/10.3390/sym11020183 - 04 Feb 2019
Abstract
We propose a nonsmooth dynamic system integrating production and inventory where the items may deteriorate and the demand is stock-dependent. We aim to derive the optimal production rate. In our first model, backorders are not allowed, while in the second model they are. [...] Read more.
We propose a nonsmooth dynamic system integrating production and inventory where the items may deteriorate and the demand is stock-dependent. We aim to derive the optimal production rate. In our first model, backorders are not allowed, while in the second model they are. Using optimal control, necessary optimality conditions are obtained for general forms of the cost, demand, and deterioration rates and closed form solutions are derived for specific forms of these rates. Numerical simulations are presented and sensitivity of the solutions are examined. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
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Open AccessArticle
Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces
Symmetry 2019, 11(1), 28; https://doi.org/10.3390/sym11010028 - 30 Dec 2018
Abstract
We prove an existence result, in the separable Banach spaces setting, for second order differential inclusions of type sweeping process. This type of differential inclusion is defined in terms of normal cones and it covers many dynamic quasi-variational inequalities. In the present paper, [...] Read more.
We prove an existence result, in the separable Banach spaces setting, for second order differential inclusions of type sweeping process. This type of differential inclusion is defined in terms of normal cones and it covers many dynamic quasi-variational inequalities. In the present paper, we prove in the nonconvex case an existence result of this type of differential inclusions when the separable Banach space is assumed to be q-uniformly convex and 2-uniformly smooth. In our proofs we use recent results on uniformly generalized prox-regular sets. Part of the novelty of the paper is the use of the usual Lipschitz continuity of the set-valued mapping which is very easy to verify contrarily to the ones used in the previous works. An example is stated at the end of the paper, showing the application of our existence result. Full article
(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)
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