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2021 | Buch

# Nonlinear Analysis and Global Optimization

herausgegeben von: Themistocles M. Rassias, Dr. Panos M. Pardalos

Buchreihe : Springer Optimization and Its Applications

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### Über dieses Buch

This contributed volume discusses aspects of nonlinear analysis in which optimization plays an important role, as well as topics which are applied to the study of optimization problems. Topics include set-valued analysis, mixed concave-convex sub-superlinear Schroedinger equation, Schroedinger equations in nonlinear optics, exponentially convex functions, optimal lot size under the occurrence of imperfect quality items, generalized equilibrium problems, artificial topologies on a relativistic spacetime, equilibrium points in the restricted three-body problem, optimization models for networks of organ transplants, network curvature measures, error analysis through energy minimization and stability problems, Ekeland variational principles in 2-local Branciari metric spaces, frictional dynamic problems, norm estimates for composite operators, operator factorization and solution of second-order nonlinear difference equations, degenerate Kirchhoff-type inclusion problems, and more.

### Inhaltsverzeichnis

##### Friction Models in the Framework of Set-Valued and Convex Analysis
Abstract
It is well known that modeling friction forces is a complex problem and constitutes an important topic in both mechanical engineering and applied mathematics. In this paper, we show how the approach of Moreau and Panagiotopoulos can be used to develop a suitable methodology for the formulation and the mathematical analysis of various friction models in nonsmooth mechanics. We study 11 widespread engineering friction models in the context of modern set-valued and convex analysis. The stability analysis (in the sense of Lyapunov) of a two-degree-of-freedom mechanical system with dry friction is also discussed.
Samir Adly, Daniel Goeleven, Rachid Oujja
##### A Survey on Markov’s Theorem on Zeros of Orthogonal Polynomials
Abstract
This manuscript is an extended version of the paper by the same authors who appeared in Castillo et al. (Appl Math Comput 339:390–397, 2018). It briefly surveys a Markov’s result dating back to the end of the nineteenth century, which is related to zeros of orthogonal polynomials.
Kenier Castillo, Marisa de Souza Costa, Fernando Rodrigo Rafaeli
##### A Review of Two Network Curvature Measures
Abstract
The curvature of higher-dimensional geometric shapes and topological spaces is a natural and powerful generalization of its simpler counterpart in planes and other two-dimensional spaces. Curvature plays a fundamental role in physics, mathematics, and many other areas. However, graphs are discrete objects that do not necessarily have an associated natural geometric embedding. There are many ways in which curvature definitions of a continuous surface or other similar space can be adapted to graphs depending on what kind of local or global properties the measure is desired to reflect. In this chapter, we review two such measures, namely the Gromov-hyperbolic curvature measure and a geometric measure based on topological associations to higher-dimensional complexes.
Tanima Chatterjee, Bhaskar DasGupta, Réka Albert
##### A Frictional Dynamic Thermal Contact Problem with Normal Compliance and Damage
Abstract
We study a class of non-clamped dynamical problems for visco-elastic materials, the contact condition is modeled by a normal compliance, with friction, damage and heat exchange. The weak formulation leads to a general system defined by a second-order quasi-variational evolution inequality on the displacement field coupled with a nonlinear evolutional inequality on temperature field and a parabolic variational inequality on the damage field. We present and establish an existence and uniqueness result of different fields, by using general results on evolution variational inequalities, with monotone operators and fixed point methods. Then, we present a fully discrete numerical scheme of approximation and derive an error estimate. Finally, various numerical computations are developed.
Oanh Chau, Adrien Petrov, Arnaud Heibig, Manuel Monteiro Marques
##### Mixed Concave–Convex Sub-Superlinear Schrödinger Equation: Survey and Development of Some New Cases
Abstract
The present chapter is concerned with a whole review of the well known Schrödinger equation in a mixed case of nonlinearities. We precisely consider a general nonlinear model characterized by a superposition of linear, sub-linear, super-linear sometimes concave–convex power laws on the form f(u) = |u|p−1 u ±|u|p−1 u. In a first part, we develop theoretical results on existence, uniqueness, classification as well as the behavior of the solutions of the ground state radial problem according to the power laws and the initial value. Next, in a second part, some examples are developed with graphical illustrations to confirm the theoretical results exposed previously. The graphs show coherent states between the theoretical findings and the numerical illustrations.The chapter in its whole aim is a review of existing results about the studied problems reminiscent of some few cases that are not previously developed. We aim thus it will constitute a good reference especially for beginners in the field of nonlinear analysis of PDEs.
Riadh Chteoui, Anouar Ben Mabrouk, Carlo Cattani
##### An Optimization Model for a Network of Organ Transplants with Uncertain Availability
Abstract
Thanks to advances in modern medicine and the presence of an increasingly efficient organizational network, nowadays transplantation can save thousands of lives every year. In our paper we present a supply chain model with transplant centers and donor hospitals, where we assume that the medical teams move to the hospitals, take the organs, and go back to the transplant centers, using the most suitable transport mode. Since the availability of organs in each donor hospital is unknown a priori, we introduce a random variable which gives us an expected value of such an availability. The aim of the model is to obtain a social optimum in which we intend to minimize the total costs, given by transport costs of both teams and organs, as well as those of transplant patients, the costs of removal, of transplantation and of post-transplantation, the costs of disposal of diseased or non-functioning organs and of the damaged ones, and the penalties. We deduce the associated variational inequality formulation and an existing result for the solution. Finally, we present some numerical examples.
Gabriella Colajanni, Patrizia Daniele
##### Algebraic Based Techniques as Decision Making Tools
Abstract
This study explores the use of some well-established algebraic structures as tools in multicriteria decision making. Under a rigorous axiomatic foundation, a complex decision problem is decomposed into a multilevel hierarchic structure of objective, criteria, and alternatives. A priority is derived for each element of the hierarchy, allowing comparisons based on linear rankings, weak orders, and other order structures. Consensus rules are provided for the final ranking of the alternatives.
M. Couceiro, G. Meletiou, K. Skouri
##### Norm Estimates for the Composite Operators
Abstract
In this paper, we obtain both local and global L p norm inequalities and imbedding inequalities for the composition of the homotopy operator, differential operator, and Green’s operator applied to differential forms. These inequalities can be used to study the integrability of the composition of the operators.
Shusen Ding, Guannan Shi, Yuming Xing
##### A Variational Inequality Based Stochastic Approximation for Inverse Problems in Stochastic Partial Differential Equations
Abstract
The primary objective of this work is to study the inverse problem of identifying a parameter in partial differential equations with random data. We explore the nonlinear inverse problem in a variational inequality framework. We propose a projected-gradient-type stochastic approximation scheme for general variational inequalities and give a complete convergence analysis under weaker conditions on the random noise than those commonly imposed in the available literature. The proposed iterative scheme is tested on the inverse problem of parameter identification. We provide a derivative characterization of the solution map, which is used in computing the derivative of the objective map. By employing a finite element based discretization scheme, we derive the discrete formulas necessary to test the developed stochastic approximation scheme. Preliminary numerical results show the efficacy of the developed framework.
Rachel Hawks, Baasansuren Jadamba, Akhtar A. Khan, Miguel Sama, Yidan Yang
##### An Iterative Method for a Common Solution of Split Generalized Equilibrium Problems and Fixed Points of a Finite Family of Nonexpansive Mapping
Abstract
In this paper, we introduce and analyze a general iterative algorithm for finding an approximate element of the common set of solutions of the split generalized equilibrium problem and the set of common fixed points of a finite family of nonexpansive mapping in the setting of real Hilbert space. Under appropriate conditions, we derive the strong convergence results for this method. Preliminary numerical experiments are included to verify the theoretical assertions of the proposed method. The results presented in this paper extend and improve some well-known results in the literature.
Ihssane Hay, Abdellah Bnouhachem, Themistocles M. Rassias
##### Periodic Solutions Around the Out-of-Plane Equilibrium Points in the Restricted Three-Body Problem with Radiation and Angular Velocity Variation
Abstract
In the present work, we study the motion of an infinitesimal body near the out-of-plane equilibrium points of the restricted three-body problem in which the angular velocity of the two primary bodies is considered in the case where both of them are sources of radiation. Firstly, these equilibria are determined numerically, and then the influence of the system parameters on their positions is examined. Due to the symmetry of the problem, these points appear in pairs and, depending on the parameter values, their number may be zero, two, or four. The linear stability of the out-of-plane equilibrium points is also studied, and it is found that there are cases where they can be stable. In addition, periodic motion around them is investigated both analytically and numerically. Specifically, the Lindstedt–Poincaré method is used in order to obtain a second order analytical solution, while the families emanating from the out-of-plane equilibrium points are finally computed numerically either in case where the corresponding equilibrium points are stable or unstable. For the numerical computation of a three-dimensional periodic orbit, we apply known unconstrained optimization methods to an objective function that is formed by the respective periodicity conditions that have to be fulfilled.
Vassilis S. Kalantonis, Aguda Ekele Vincent, Jessica Mrumun Gyegwe, Efstathios A. Perdios
##### Optimal Lot Size with Partial Backlogging Under the Occurrence of Imperfect Quality Items
Abstract
In this study, a continuous review inventory system with deterministic demand, partial backlogging, and imperfect quality items is considered. More precisely, the fraction of imperfect quality items is assumed as a random variable with a known distribution function. The order quantity is subjected to a 100%, error-free, screening process, with finite screening rate. After inspection, the imperfect quality items can be classified into two categories: low quality items and defective items. The demand rate is constant and manifests even during screening period. The demand during the stockout period is satisfied partially as soon as stock is available and before the new demand is met. Perfect and imperfect quality items are charged with different holding cost, giving the chance of different treatment for the two categories of products. The objective is to find the order quantity that maximizes the total profit of the system per unit time. Beyond, the analytical properties are established, the impact of imperfect quality and holding costs differentiation are examined and the behavior of the relative error using the EOQ with partial backlogging solution is displayed graphically. Finally, it is shown that this model can be reduced to other models existing in the literature.
G. Karakatsoulis, K. Skouri
##### Error Analysis Through Energy Minimization and Stability Properties of Exponential Integrators
Abstract
In this article, the stability property and the error analysis of higher-order exponential variational integration are examined and discussed. Toward this purpose, at first we recall the derivation of these integrators and then address the eigenvalue problem of the amplification matrix for advantageous choices of the number of intermediate points employed. Obviously, the latter determines the order of the numerical accuracy of the method. Following a linear stability analysis process we show that the methods with at least one intermediate point are unconditionally stable. Finally, we explore the behavior of the energy errors of the presented schemes in prominent numerical examples and point out their excellent efficiency in long term integration.
Odysseas Kosmas, Dimitrios Vlachos
##### A Degenerate Kirchhoff-Type Inclusion Problem with Nonlocal Operator
Abstract
The chapter focuses on a Kirchhoff-type elliptic inclusion problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points and for which the multivalued term is in the form of the generalized gradient of a locally Lipschitz function. The corresponding elliptic equation has been treated in (Liu et al., Existence of solutions to Kirchhoff-type problem with vanishing nonlocal term and fractional p-Laplacian). Multiple nontrivial solutions are obtained by applying the nonsmooth critical point theory combined with truncation techniques.
Dumitru Motreanu
##### Competition for Medical Supplies Under Stochastic Demand in the Covid-19 Pandemic: A Generalized Nash Equilibrium Framework
Abstract
The Covid-19 pandemic has negatively impacted virtually all economic and social activities across the globe. Presently, since there is still no vaccine and no curative treatments for this disease, medical supplies in the form of Personal Protective Equipment and ventilators are sorely needed for healthcare workers and certain patients, respectively. The fact that this healthcare disaster is not limited in time and space has resulted in intense global competition for medical supplies. In this paper, we construct the first Generalized Nash Equilibrium model with stochastic demands to model competition among organizations at demand points for medical supplies. The model includes multiple supply points and multiple demand points, along with prices of the medical items and generalized costs associated with transportation. The theoretical constructs are provided and a Variational Equilibrium utilized to enable alternative variational inequality formulations. A qualitative analysis is presented and an algorithm proposed, along with convergence results. Illustrative examples are detailed as well as numerical examples that are solved with the implemented algorithm. The results reveal the impacts of the addition of supply points as well as of demand points on the medical item product flows. The formalism may be adapted to multiple medical items both in the near term and in the longer term (such as for vaccines).
Anna Nagurney, Mojtaba Salarpour, June Dong, Pritha Dutta
##### Relative Strongly Exponentially Convex Functions

In this paper, we define and consider some new concepts of the strongly exponentially convex functions involving an arbitrary negative bifunction. Some properties of these strongly exponentially convex functions are investigated under suitable conditions. It is shown that the difference of strongly exponentially convex functions and strongly exponentially affine functions is again an exponentially convex function. Results obtained in this paper can be viewed as refinement and improvement of previously known results

Muhammad Aslam Noor, Khalida Inayat Noor, Themistocles M. Rassias
##### Properties of Exponentially m-Convex Functions
Abstract
In this paper, we define and introduce some new concepts of the exponentially m-convex functions involving a fixed constant m ∈ (0, 1]. We investigate several properties of the exponentially m-convex functions and discuss their relations with convex functions. Optimality conditions are characterized by a class of variational inequalities. Several interesting results characterizing the exponentially m-convex functions are obtained. Results obtained in this paper can be viewed as significant improvement of previously known results.
Muhammad Aslam Noor, Khalida Inayat Noor
##### Natural vs. Artificial Topologies on a Relativistic Spacetime
Abstract
Consider a set M equipped with a structure ∗. We call a natural topology T , on (M, ∗), the topology induced by ∗. For example, a natural topology for a metric space (X, d) is a topology T d induced by the metric d, and for a linearly ordered set (X, <), a natural topology should be the topology T < that is induced by the order < . This fundamental property, for a topology to be called “natural,” has been largely ignored while studying topological properties of spacetime manifolds (M, g), where g is the Lorentz “metric,” and the manifold topology T M has been used as a natural topology, ignoring the spacetime “metric” g. In this survey, we review critically candidate topologies for a relativistic spacetime manifold, and we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Göbel, Hawking-King-McCarthy and others, and we examine what should be meant by the term “natural topology” for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Gao–Wald type of “time dilation” are such examples.
##### On the Approximation of Monotone Variational Inequalities in L p Spaces with Probability Measure

In this paper we propose an approximation procedure for a class of monotone variational inequalities in probabilistic Lebesgue spaces. The implementation of the functional approximation in L p, with p > 2, leads to a finite dimensional variational inequality whose structure is different from the one obtained in the case p = 2, already treated in the literature. The proposed computational scheme is applied to the random traffic equilibrium problem with polynomial cost functions.

Mauro Passacantando, Fabio Raciti
##### Operator Factorization and Solution of Second-Order Nonlinear Difference Equations with Variable Coefficients and Multipoint Constraints
Abstract
A method for constructing solutions to boundary value problems for a class of second-order nonlinear difference equations with variable coefficients together with multipoint conditions is presented. The technique is based on the decomposition of the nonlinear difference equation into linear components of the same or lower order and the factorization of the associated second-order linear difference operators. The efficiency of the procedure is demonstrated by considering several examples.
E. Providas
##### An Invitation to the Study of a Uniqueness Problem
Abstract
In this very short chapter, we provide a strong motivation for the study of the following problem: given a real normed space E, a closed, convex, unbounded set X ⊆ E, and a function f : X → X, find suitable conditions under which, for each y ∈ X, the function
$$\displaystyle x\to \|x-f(x)\|-\|y-f(x)\|$$
has at most one global minimum in X.
Biagio Ricceri
##### Schrödinger Equations in Nonlinear Optics
Abstract
Using global optimization, we are able to find nontrivial solutions of the nonlinear steady-state Schrödinger equation arising in optics for wide ranges of the parameters. Our results hold in arbitrary dimensions.
Martin Schechter
##### Ekeland Variational Principles in 2-Local Branciari Metric Spaces
Abstract
An Ekeland Variational Principle is stated over a class of local and 2-local Branciari metric spaces, and its relationships with the Dependent Choice Principle are discussed. Applications to Caristi–Kirk fixed point theorems over such a setting are also being considered.
Mihai Turinici
Titel
Nonlinear Analysis and Global Optimization
herausgegeben von
Themistocles M. Rassias
Dr. Panos M. Pardalos