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2018 | Book

New Applications of γ-Quasiconvexity Read chapter Read first chapter

Applications of Nonlinear Analysis

Editor: Themistocles M. Rassias

Publisher: Springer International Publishing

Book Series : Springer Optimization and Its Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

New applications, research, and fundamental theories in nonlinear analysis are presented in this book. Each chapter provides a unique insight into a large domain of research focusing on functional equations, stability theory, approximation theory, inequalities, nonlinear functional analysis, and calculus of variations with applications to optimization theory.

Topics include:

Fixed point theory

Fixed-circle theory

Coupled fixed points

Nonlinear duality in Banach spaces

Jensen's integral inequality and applications

Nonlinear differential equations

Nonlinear integro-differential equations

Quasiconvexity, Stability of a Cauchy-Jensen additive mapping

Generalizations of metric spaces

Hilbert-type integral inequality, Solitons

Quadratic functional equations in fuzzy Banach spaces

Asymptotic orbits in Hill’sproblem

Time-domain electromagnetics

Inertial Mann algorithms

Mathematical modelling

RoboticsGraduate students and researchers will find this book helpful in comprehending current applications and developments in mathematical analysis. Research scientists and engineers studying essential modern methods and techniques to solve a variety of problems will find this book a valuable source filled with examples that illustrate concepts.

Table of Contents

Frontmatter
New Applications of γ-Quasiconvexity
Abstract
This survey deals with inequalities satisfied by γ-quasiconvex functions which are one of the many variants of convex functions. The γ-quasiconvex functions have already been dealt with by S. Abramovich, L.-E. Persson and N. Samko. Among the applications we demonstrate here are Jensen, Hardy, Hölder, Minkowski, Jensen-Steffensen and Slater-Pečarić inequalities. These inequalities can be seen as extensions and refinements of inequalities satisfied by convex functions.
Shoshana Abramovich
Criteria for Convergence of Iterates in a Compression-Expansion Fixed Point Theorem of Functional Type
Abstract
In this paper we show how one can use suitable k-contractive conditions to prove that iterates converge to a fixed point guaranteed by a compression-expansion fixed point theorem of functional type, even though the operator is not known to be invariant on the underlying set.
Richard I. Avery, Douglas R. Anderson, Johnny Henderson
On Lagrangian Duality in Infinite Dimension and Its Applications
Abstract
The aim of this contribution is to review some recent results on Lagrangian duality in infinite dimensional spaces which permit to deal with problems where the ordering cone describing the inequality constraints has empty topological interior. For instance, the topological interior of the cone of the nonnegative L p functions (p > 1) is empty, as it is the cone of nonnegative functions in many Sobolev spaces. To point out where the difficulty comes from, we first review the classical theory which requires the nonemptiness of the ordering cone and then describe the main results obtained by some authors in the last decade, based on what they called “Assumption S”. At last, we show how the new theory can be applied to extend a classical result by Rosen on Nash equilibria, from \(\mathbb {R}^n\) to infinite dimensional spaces.
Antonio Causa, Giandomenico Mastroeni, Fabio Raciti
Stability Analysis of the Inverse Problem of Parameter Identification in Mixed Variational Problems
Abstract
Numerous applications lead to inverse problems of parameter identification in mixed variational problems. These inverse problems are commonly studied as optimization problems, and there are a variety of optimization formulations. The known formulations include an output least-squares (OLS), an energy OLS (EOLS), and a modified OLS (MOLS). This work conducts a detailed study of various stability aspects of the inverse problem under data perturbation and gives new stability estimates for general inverse problems using the OLS, EOLS, and MOLS formulations. We present applications of our theoretical results.
M. Cho, A. A. Khan, T. Malysheva, M. Sama, L. White
Nonlinear Duality in Banach Spaces and Applications to Finance and Elasticity
Abstract
In this chapter we first present some theoretic concepts related to the strong duality in the infinite-dimensional setting. Then, we apply such results to the general financial equilibrium economy, studying also the dual formulation of the problem, analyzing both the sector’s and the system’s viewpoints and deriving the contagion phenomenon. Further, we provide an evolutionary Markowitz-type measure of the risk with a memory term. Finally, we apply Assumption S to the elastic-plastic torsion problem for linear operators and investigate the existence of Lagrange multipliers to the elastic-plastic torsion problem for nonlinear monotone operators, providing an example of the so-called Von Mises functions and searching for radial solutions.
G. Colajanni, Patrizia Daniele, Sofia Giuffrè, Antonino Maugeri
Selective Priorities in Processing of Big Data
Abstract
This paper investigates the method of selective priorities for the data amounts in the processing of big data. After defining the focal data sets, it is introduced the concept of program of data selection which specifies the data amount that a processor may take into account. Then, they are determined the relations of data selection preference and of rational choice for the data amounts. Subsequently, it is considered the case of several data processors and it is shown that there are cores and equilibriums of contrasts, the study of which may provide useful information.
Nicholas J. Daras
General Inertial Mann Algorithms and Their Convergence Analysis for Nonexpansive Mappings
Abstract
In this article, we introduce general inertial Mann algorithms for finding fixed points of nonexpansive mappings in Hilbert spaces, which includes some other algorithms as special cases. We reanalyze the accelerated Mann algorithm, which actually is an inertial type Mann algorithm. We investigate the convergence of the general inertial Mann algorithm, based on which, the strict convergence condition on the accelerated Mann algorithm is eliminated. Also, we apply the general inertial Mann algorithm to show the existence of solutions of the minimization problems by proposing a general inertial type gradient-projection algorithm. Finally, we give preliminary experiments to illustrate the advantage of the accelerated Mann algorithm.
Qiao-Li Dong, Yeol Je Cho, Themistocles M. Rassias
Reverses of Jensen’s Integral Inequality and Applications: A Survey of Recent Results
Abstract
Several new reverses of the celebrated Jensen’s inequality for convex functions and Lebesgue integral on measurable spaces are surveyed. Applications for weighted discrete means, to Hölder inequality, Cauchy-Bunyakovsky-Schwarz inequality and for f-divergence measures in information theory are also given. Finally, applications for functions of selfadjoint operators in Hilbert spaces with some examples of interest are also provided.
Silvestru Sever Dragomir
Ordering Structures and Their Applications
Abstract
Ordering structures play a fundamental role in many mathematical areas. These include important topics in optimization theory such as vector optimization and set optimization, but also other subjects as decision theory use ordering structures as well. Due to strong connections between ordering structures and cones in the considered space, order theory is also used every time two elements of a space, which is more general than the real line, are compared with each other. Therefore, also cone programming possessing restrictions defined using cones, and especially semidefinite optimization where the variables are symmetric matrices, make use of ordering structures. These structures may, on the one hand, be independent of the considered element of a given space or, on the other hand, vary for each element of this space. In the last case, we speak of variable ordering structures, which is one of the important topics in the newest research on vector optimization.
Gabriele Eichfelder, Maria Pilecka
An Overview on Singular Nonlinear Elliptic Boundary Value Problems
Abstract
We give a survey of old and recent results concerning existence and multiplicity of positive solutions (classical or weak) to nonlinear elliptic equations with singular nonlinear terms of the form
$$\displaystyle \left \{ \begin {array}{ll} -\varDelta _p u= f(x,u)+ u^{-\gamma }, & \mbox{ in }\ \varOmega \\ u>0, & \mbox{ in }\ \varOmega \\ u=0, & \mbox{ on }\ \partial \ \varOmega , \end {array} \right . $$
where Ω is a bounded domain in \(\mathbb {R}^N\) (N ≥ 2) with sufficiently smooth boundary ∂Ω, Δ p u = div(|∇u|p−2∇u) (1 < p < ∞), f : Ω × [0, +∞) → [0, +∞) is a Carathéodory function and γ > 0. In some cases and in order to control more carefully the nonlinearity, we need to multiply the singular term u −γ or f(⋅, u) by positive parameters. The main difficulty which arises in the study of such problems is the lack of differentiability of the corresponding energy functional which represents an obstacle to the application of classical critical point theory.
Francesca Faraci, George Smyrlis
The Pilgerschritt (Liedl) Transform on Manifolds
Abstract
Finding geodesic lines connecting two points on a manifold with linear connection is usually done by shooting methods. In the last seventieth R. Liedl hat the idea to choose an arbitrary path between these two points and transforming it in order to get the geodesic line. He worked out this method for Lie groups, and in this paper the generalization to manifolds is given together with a convergence theorem that locally this method can be achieved in order to approximate geodesic lines.
Wolfgang Förg-Rob
On Some Mathematical Models Arising in Lubrication Theory
Abstract
In this paper, we present a synthesis of some results related to the hydrodynamic lubrication problem from mechanical engineering. The hydrodynamic lubrication problem is related to the analysis of the physical phenomena arising when a viscous lubricant is forced to flow between two surfaces in relative motion. A question of great importance is to describe how cavitation can take place and influence the pressure distribution in the lubricant between the two moving surfaces.
D. Goeleven, R. Oujja
On the Spectrum of a Nonlinear Two Parameter Matrix Eigenvalue Problem
Abstract
We consider the nonlinear two parameter eigenvalue problem (T p − λ 1 A p1 − λ 2 A p2 − λ 1 λ 2 A p3)v p = 0, where λ 1, λ 2 ∈C; T p, A pk (p = 1, 2;k = 1, 2, 3) are matrices. Bounds for the spectral radius of that problem are suggested. Our main tool is the recent norm estimates for the resolvent of an operator on the tensor product of Euclidean spaces. In addition, we investigate perturbations of the considered problem and derive a Gershgorin type bounds for the spectrum. It is shown that the main result of the paper is sharp.
Michael Gil’
On the Properties of a Nonlocal Nonlinear Schrödinger Model and Its Soliton Solutions
Abstract
Nonlinear waves are normally described by means of certain nonlinear evolution equations. However, finding physically relevant exact solutions of these equations is, in general, particularly difficult. One of the most known nonlinear evolution equation is the nonlinear Schrödinger (NLS), a universal equation appearing in optics, Bose-Einstein condensates, water waves, plasmas, and many other disciplines. In optics, the NLS system is used to model a unique balance between the critical effects that govern propagation in dispersive nonlinear media, namely dispersion/diffraction and nonlinearity. This balance leads to the formation of solitons, namely robust localized waveforms that maintain their shape even when they interact. However, for several physically relevant contexts the standard NLS equation turns out to be an oversimplified description. This occurs in the case of nonlocal media, such as nematic liquid crystals, plasmas, and optical media exhibiting thermal nonlinearities. Here, we study the properties and soliton solutions of such a nonlocal NLS system, composed by a paraxial wave equation for the electric field envelope and a diffusion-type equation for the medium’s refractive index. The study of this problem is particularly interesting since remarkable properties of the traditional NLS—associated with complete integrability—are lost in the nonlocal case. Nevertheless, we identify cases where derivation of exact solutions is possible while, in other cases, we resort to multiscale expansions methods. The latter, allows us to reduce this systems to a known integrable equation with known solutions, which in turn, can be used to approximate the solutions of the initial system. By doing so, a plethora of solutions can be found; solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are predicted to occur.
Theodoros P. Horikis, Dimitrios J. Frantzeskakis
Stability of a Cauchy-Jensen Additive Mapping in Various Normed Spaces
Abstract
In this paper, using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation in various normed spaces.
Hassan Azadi Kenary, Choonkil Park, Themistocles M. Rassias, Jung Rye Lee
NAN-RN Approximately Generalized Additive Functional Equations
Abstract
The authors have studied the generalized Hyers-Ulam-Rassias stability of approximately generalized additive functional equations.
Hassan Azadi Kenary, Themistocles M. Rassias
On the HUR-Stability of Quadratic Functional Equations in Fuzzy Banach Spaces
Abstract
In this paper, we prove the Hyers-Ulam-Rassias stability of the following quadratic functional equations
$$\displaystyle f\left (\sum _{i=1}^n a_i x_i\right )+\sum _{i=1}^{n-1}\sum _{j=i+1}^nf(a_ix_i\pm a_jx_j)=(3n-2)\sum _{i=1}^n a_{i}^2 f(x_i), $$
where \(a_1,\cdots ,a_n \in \mathbb {Z}-\{0\}\) and l ∈{1, 2, ⋯ , n − 1}, a l ≠ 1 and a n = 1, where n is a positive integer greater or at least equal to two, in fuzzy Banach spaces.
The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300.
Hassan Azadi Kenary, Themistocles M. Rassias
Asymptotic Orbits in Hill’s Problem When the Larger Primary is a Source of Radiation
Abstract
A modification of the Hill problem when the larger primary is a source of radiation is considered and asymptotic motions around the collinear equilibrium points are studied. Our work focuses on the computation of homoclinic orbits to the collinear equilibrium points themselves or to the Lyapunov orbits emanating from each equilibrium point. These orbits depart asymptotically from an equilibrium point (or a Lyapunov orbit) and return to the same point (or orbit) asymptotically. In both cases, semi-analytical solutions have been obtained in order to determine appropriate initial conditions which have been used as suitable seed for the numerical computation of the asymptotic orbits with a predetermined accuracy. In addition, for homoclinic orbits to the Lyapunov periodic orbits, transversality is achieved by the construction of appropriate surface of section portraits of the unstable manifolds.
Vassilis S. Kalantonis, Angela E. Perdiou, Christos N. Douskos
Computations for Minors of Weighing Matrices with Application to the Growth Problem
Abstract
In this expository paper we survey some important results concerning the computations for minors of weighing matrices. The applicability to the growth problem for weighing matrices is highlighted. The history of the problem is presented, the importance of determinant calculations is stressed and the relevant open problems are discussed. Emphasis is laid on the contribution of determinant calculations to the study of the growth factor for weighing matrices after application of Gaussian Elimination with complete pivoting on them, which is an important scientific open problem in Numerical Analysis.
Christos D. Kravvaritis
Robots That Do Not Avoid Obstacles
Abstract
The motion planning problem is a fundamental problem in robotics, so that every autonomous robot should be able to deal with it. A number of solutions have been proposed and a probabilistic one seems to be quite reasonable. However, here we propose a more adoptive solution that uses fuzzy set theory and we expose this solution next to a sort survey on the recent theory of soft robots, for a future qualitative comparison between the two.
Kyriakos Papadopoulos, Apostolos Syropoulos
On the Exact Solution of Nonlinear Integro-Differential Equations
Abstract
A method for constructing exact explicit solutions to problems involving a nonlinear operator B defined as a perturbation of a linear correct operator \(\widehat {A}\) with linear bounded functionals and nonlinear continuous functionals is presented. The technique is general to deal with several kinds of nonlinear problems; it is easily programmable and suitable for large equations. The method is applied here to solve nonlinear integro-differential equations of Fredholm type.
I. N. Parasidis, E. Providas
Qualitative, Approximate and Numerical Approaches for the Solution of Nonlinear Differential Equations
Abstract
The differential equations that describe many realistic problems are nonlinear and most of these cannot be solved explicitly using standard analytic techniques. In such cases, qualitative, approximate or numerical techniques are employed, in order to obtain as much information as possible. The aim of the present chapter, is to give a description of the general ideas governing these techniques together with their advantages and limitations. This is achieved by implementing various methods to an initial value problem for a specific nonlinear ordinary differential equation, which combines both van der Pol and Duffing equations. This equation is solved using (a) the fourth order Runge-Kutta, the standard finite differences and the finite elements methods, (b) a nonstandard discretization technique based on functional analysis, (c) classical perturbation techniques and (d) the homotopy analysis method. Moreover, various results are given regarding the dynamic properties of its solution. Finally, this problem is connected with a Green function and this connection is again used for its numerical solution.
Eugenia N. Petropoulou, Michail A. Xenos
On a Hilbert-Type Integral Inequality in the Whole Plane
Abstract
By using methods of real analysis and weight functions, we prove a new Hilbert-type integral inequality in the whole plane with non-homogeneous kernel and a best possible constant factor. As applications, we also consider the equivalent forms, some particular cases and the operator expressions.
Michael Th. Rassias, Bicheng Yang
Four Conjectures in Nonlinear Analysis
Abstract
In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Ampère equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.
Biagio Ricceri
Corelations Are More Powerful Tools than Relations
Abstract
A subset R of a product set https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-89815-5_25/427052_1_En_25_IEq1_HTML.gif is called a relation on X to Y . While, a function U of one power set \(\mathcal {P}(X)\) to another \(\mathcal {P}(Y)\) is called a corelation on X to Y . Moreover, families \(\mathcal {R}\) and \(\mathcal {U}\) of relations and corelations on X to Y  are called relators and corelators on X to Y , respectively.
Relators on X have been proved to be more powerful tools than generalized proximities, closures, topologies, filters and convergences on X. Now, we shall show that corelators on X to Y  are more powerful tools than relators on X to Y . Therefore, corelators have to be studied before relators.
If \(\mathcal {U}\) is a corelation on X to Y , then instead of the notation \(\Subset _{\scriptscriptstyle \mathcal {U}}\) of Yu. M. Smirnov, for any A ⊆ X and B ⊆ Y , we shall write \(A\in \operatorname {\mathrm {Int}}_{\mathcal {U}}( B)\) if there exists \(U\in \mathcal {U}\) such that U(A) ⊆ B. Namely, thus we may also naturally write \( \operatorname {\mathrm {Cl}}_{\mathcal {U}}( B)= \mathcal {P}( X)\setminus \operatorname {\mathrm {Int}}_{\mathcal {U}}(Y\setminus B)\), and \(x\in \operatorname {\mathrm {int}}_{\mathcal {U}}( B)\) if \(\{ x\}\in \operatorname {\mathrm {Int}}_{\mathcal {U}}( B)\).
Moreover, we can also note that \( \operatorname {\mathrm {Int}}_{\mathcal {U}}\) is a relation on \(\mathcal {P}(Y)\) to \(\mathcal {P}( X)\) such that \( \operatorname {\mathrm {Int}}_{\mathcal {U}}=\bigcup _{ U\in \mathcal {U}} \, \operatorname {\mathrm {Int}}_{ U}\) with \( \operatorname {\mathrm {Int}}_{U}= \operatorname {\mathrm {Int}}_{\{U\}}\). Therefore, the properties of the relation \( \operatorname {\mathrm {Int}}_{\mathcal {U}}\) can be immediately derived from those of the relations \( \operatorname {\mathrm {Int}}_{U}\). This shows that corelations have to be studied before corelators.
For this, following the ideas of U. Höhle and T. Kubiak and the notations of B.A. Davey and H.A. Priestly, for any relation R and corelation U on X to Y , we define a corelation R ⊲ and a relation U ⊳ on X to Y  such that R ⊲(A) = R [ A ] and U ⊳(x) = U({x}) for all A ⊆ X and x ∈ X.
Here, for any two corelations U and V  on X to Y , we may naturally write U ≤ V  if U(A) ⊆ V (A) for all A ⊆ X. Thus, the maps ⊲ and ⊳ establish a Galois connection between relations and quasi-increasing corelations on X to Y  such that R ⊲⊳ = R, but U ⊳ ⊲ = U if and only if U is union-preserving.
Now, for any two corelations U on X to Y  and V  on Y  to Z, we may also naturally define U ∘ = U ⊳ ⊲, U −1 = U ⊲−1 ⊲ and V •U = (V ⊳ ∘ U ⊳)⊲. Moreover, for instance, for any relator \(\mathcal {R}\) on X to Y , we may also naturally define \( \operatorname {\mathrm {Int}}_{\mathcal {R}}= \operatorname {\mathrm {Int}}_{\mathcal {R}^{\triangleright }}\) and \( \operatorname {\mathrm {Int}}_{\mathcal {R}}= \operatorname {\mathrm {Int}}_{\mathcal {R}^{\triangleright }}\) with \(\mathcal {R}^{\triangleright }= \big \{R^{\triangleright }: \,\ R\in \mathcal {R}\,\big \}\).
Thus, in general \( \operatorname {\mathrm {Int}}_{\mathcal {U}}\) is a more general relation than \( \operatorname {\mathrm {Int}}_{\mathcal {R}}\). However, for instance, we already have \( \operatorname {\mathrm {int}}_{\mathcal {U}}= \operatorname {\mathrm {int}}_{\mathcal {U}^{\triangleleft }}\). Therefore, our former results on the relation \( \operatorname {\mathrm {int}}_{\mathcal {R}}\) and the families \(\mathcal {E}_{\mathcal {R}}=\big \{B\subseteq Y: \,\ \operatorname {\mathrm {int}}_{\mathcal {R}}( B) \ne \emptyset \,\big \}\) and \(\mathcal {T}_{\mathcal {R}}= \big \{A\subseteq X: \,\ A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}( A)\big \}\), whenever X = Y , will not be generalized.
Árpád Száz
Rational Contractions and Coupled Fixed Points
Abstract
The coupled fixed point statement in Nashine and Kadelburg (Nonlin Funct Anal Appl 17:471–489, 2012) is ultimately obtainable from fixed point principles involving rational contractions acting over appropriate metric spaces.
Mihai Turinici
A Multiple Hilbert-Type Integral Inequality in the Whole Space
Abstract
In this paper, by introducing some interval variables and using the weight functions and the way of real analysis, a multiple Hilbert-type integral inequality in the whole space with a best possible constant factor is given, which is an extension of some published results. The equivalent forms, the operator expressions with the norm, the equivalent reverses, a few particular cases and some examples with the particular kernels are also considered.
Bicheng Yang
Generalizations of Metric Spaces: From the Fixed-Point Theory to the Fixed-Circle Theory
Abstract
This paper is a research survey about the fixed-point (resp. fixed-circle) theory on metric and some generalized metric spaces. We obtain new generalizations of the well-known Rhoades’ contractive conditions, Ćiri ć’s fixed-point result and Nemytskii-Edelstein fixed-point theorem using the theory of an S b-metric space. We present some fixed-circle theorems on an S b -metric space as a generalization of the known fixed-circle (fixed-point) results on a metric and an S-metric space.
The content of this section is divided into the following:
1.
Introduction
 
2.
Some Generalized Metric Spaces
 
3.
New Generalizations of Rhoades’ Contractive Conditions
 
4.
Some Generalizations of Nemytskii-Edelstein and Ćirić’s Fixed-Point Theorems
 
5.
Some Fixed-Circle Theorems
 
Nihal Yılmaz Özgür, Nihal Taş
Finite-Difference Modeling of Nonlinear Phenomena in Time-Domain Electromagnetics: A Review
Abstract
Nonlinearities are likely to emerge in a wide range of electromagnetic (EM) problems, commonly described by Maxwell’s equations, which can be encountered in several real-world applications, in areas such as optical communications, etc. The necessity for efficiently analyzing this type of problems has led to the development of suitable computational approaches, among which schemes based on the finite-difference time-domain (FDTD) method play a prominent role. Unlike other numerical methods that perform reliably only if specific approximations are valid (e.g. wave propagation along a dominant direction), FDTD-based techniques can be applied in more generalized frameworks, and are capable of computing credible outcomes without requiring very complex algorithmic implementations. In the present chapter, we report various key contributions presented over the years regarding the nonlinear FDTD analysis of EM problems, as the original algorithm is suitable for linear cases only, and describe their basic formulation, features, and range of applicability. In essence, this work aspires to provide an updated look on existing finite-difference models for nonlinear problems, and offer to those not familiar with the subject a solid starting point for studying the corresponding electromagnetic phenomena.
Theodoros T. Zygiridis, Nikolaos V. Kantartzis
Metadata
Title
Applications of Nonlinear Analysis
Editor
Themistocles M. Rassias
Copyright Year
2018
Electronic ISBN
978-3-319-89815-5
Print ISBN
978-3-319-89814-8
DOI
https://doi.org/10.1007/978-3-319-89815-5

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