Nonlinear analysis, differential equations, and applications

Intro -- Preface -- Contents -- On Compound Superquadratic Functions -- 1 Introduction -- 2 Refinements of Jensen's Inequality via Compound Functions -- 3 The Quasi-Mean Ff( x,λ) -- References -- Best Hyers-Ulam Stability Constants on a Time Scale with Discrete Core and Continuous Periphery -- 1 Introduction -- 2 Time Scale with Discrete Core and Continuous Periphery -- 3 Best Constants for First-Order Equations with Constant Complex Coefficient -- 4 Connection with h-Difference Equations in the Case |1+hλ|> -- 1 -- 5 Conclusion and Future Directions -- References -- Invariance Solutions and Blow-Up Property for Edge Degenerate Pseudo-Hyperbolic Equations in Edge Sobolev Spaces -- 1 Introduction -- 2 Edge Sobolev Spaces -- 3 Some Auxiliary Results -- 4 Invariance of the Solutions -- 5 Global Existence and Finite-Time of the Solutions -- References -- ϕ4 Solitons in Kirchhoff Wave Equation -- 1 Introduction -- 2 SSB in ϕ4 Scalar Field Theory: The Kink Solitons -- 3 Kirchhoff Wave Function, Energy and Potential -- 4 The Kpd Produced from Classical Field Theory -- 5 SSB in the Kpd -- 6 Conclusions -- Appendix: Estimation of the Potential Density U(ϕ) -- References -- Estimates for Lipschitz and BMO Norms of Operators on Differential Forms -- 1 Introduction -- 2 Local Poincaré-Type Inequalities -- 3 Estimates for BMOs and locLipαs Norms -- 4 Applications -- References -- Application of Boundary Perturbations on Medical Monitoring and Imaging Techniques -- 1 Introduction -- 1.1 Boundary Perturbations. A Short Introduction -- 2 Mathematical Formulation of Medical Monitoring Techniques. Electroencephalography and Magnetoencephalography -- 2.1 The Influence of Geometric Variations on the Forward Problem -- 2.1.1 EEG -- 2.1.2 MEG -- 2.2 The Influence of Geometric Variations on the Inversion Algorithm -- 2.2.1 EEG -- 2.2.2 MEG -- 3 Example -- 3.1 EEG.

3.2 MEG -- 4 Conclusions and Discussion -- References -- Poynting-Robertson and Oblateness Effects on the Equilibrium Points of the Perturbed R3BP: Application on Cen X-4 Binary System -- 1 Introduction -- 2 Equations of Motion -- 3 Existence and Positions of Equilibrium Points -- 4 Stability of the Non-collinear Equilibrium Points -- 5 Numerical Application -- 6 Discussion and Conclusion -- References -- Localization and Perturbation of Complex Zeros of Solutions to Second Order Differential Equations with Polynomial Coefficients. A Survey -- 1 Introduction -- 2 Solution Estimates for ODEs -- 3 Singular Values of Compact Operators -- 4 Bounds for Zeros of Entire Functions via Taylor Coefficients -- 5 Bounds for Sums and Products of Zeros of an Entire Function via Its Order -- 6 Sums and Products of Zeros of Solutions to Homogeneous ODEs with Polynomial Coefficients -- 7 Applications of Theorem 3 -- 8 A Perturbation Bound for the Zeros of Entire Functions in Terms of Taylor Coefficients -- 9 Proof of Theorem 4 -- 10 A Perturbation Bound for the Zeros of an Entire Function via Its Order -- 11 An Estimate for the Difference of Two Solutions -- 12 Perturbations of the Zeros of Solutions to ODEs -- 13 Example to Theorem 6 -- References -- Dynamics of a Higher-Order Ginzburg-Landau-Type Equation -- 1 Introduction -- 2 Limit Set and Collapse -- 2.1 Existence of the Limit Set -- 2.2 Conditions for Collapse -- 3 Soliton Dynamics: Perturbative Approach -- 3.1 Perturbation Theory for Bright Solitons (s=-1) -- 3.2 Perturbation Theory for Dark Solitons (s=+1) -- 3.3 Solitons and Shock Waves in an Effective KdV-Burgers Picture -- References -- The Role of Differential Equations in Applied Statistics -- 1 Introduction -- 2 Theoretical Framework for the Nonlinear Design -- 3 Growth Curves -- 4 Differential Equations in Probability -- 4.1 Brownian Motion.

4.2 Pure Birth Process -- 4.3 The Birth-and-Death Process -- 5 Discussion -- Appendix 1 -- Appendix 2: Introduction to Heat Equation -- References -- Geometric Derivation and Analysis of Multi-Symplectic Numerical Schemes for Differential Equations -- 1 Introduction and Motivation -- 2 Review of Variational Integrators -- 3 Exponential Integrators -- 3.1 Exponential High Order Variational Integrators -- 3.2 Frequency Estimation for Mass Points Motion in Three Dimensions -- 4 Triangle and Square Discretization -- 4.1 Triangle Discretization -- 4.2 Square Discretization -- 5 Numerical Examples Using Triangle Discretization -- 5.1 Linear Wave Equation -- 5.2 Laplace Equation -- 5.3 Poisson Equation -- 6 Numerical Examples Using Square Discretization -- 6.1 Klein-Gordon -- 7 Analysis of the Proposed Schemes -- 7.1 Dispersion Analysis -- 7.2 Convergence Experiments -- 8 Summary and Conclusions -- Appendix -- References -- Non-radial Solutions of a Supercritical Equation in Expanding Domains: The Limit Case -- 1 Introduction -- 2 Notations and Some Background Material -- 3 Preliminary Results -- 4 Solution of the Problem (P) -- References -- Financial Contagion in Interbank Networks: The Case of Erdős -Rényi Network Model -- 1 Introduction -- 2 Related Literature -- 3 Erdős-Rényi Random Graph Model -- 3.1 The Mathematical Description of the Contagion Model -- 3.2 The Interbank Network -- 3.3 Shock Propagation and Contagion Dynamics -- 3.4 Monte Carlo simulations -- 4 Main Findings -- 4.1 Computer Experiments -- 4.2 Simulation Results -- 5 Conclusions -- References -- Higher Order Strongly m-convex Functions -- 1 Introduction -- 2 m-Convex Functions -- 3 Strongly m-Convex Functions -- 4 Properties of Strongly m-Convex Functions -- 5 Applications -- Conclusion -- References -- Characterizations of Higher Order Strongly Generalized Convex Functions.

1 Introduction -- 2 Formulations and Basic Facts -- 3 Generalized Convex Functions -- 4 Higher Order Strongly Generalized Convex Functions -- 5 Applications -- 5.1 Generalized Variational Inequalities -- Conclusion -- References -- A Note on Generalized Nash Games Played on Networks -- 1 Introduction -- 2 Network Games -- 2.1 Elements of Graph Theory and Game Classes -- 2.2 The Linear-Quadratic Model -- 3 Generalized Nash Equilibrium Problems on Networks -- 3.1 An Overview of GNEPs and the Variational Inequality Approach to their Solution -- 3.2 A Linear-Quadratic Network GNEP -- 4 Numerical Experiments -- 5 Conclusions and Further Research Perspectives -- References -- Piecewise Polynomial Inversion of the Radon Transform in Three Space Dimensions via Plane Integration and Applications in Positron Emission Tomography -- 1 Introduction -- 2 The Radon Transform in Two Space Dimensions -- 3 The Radon Transform in Three Space Dimensions -- 4 The Inversion of the Radon Transform in Three Space Dimensions via Plane Integration -- 5 Numerical Implementation of the Inversion of the Radon Transform in Three Space Dimensions via Piecewise Cubic Polynomials -- References -- Factorization and Solution of Linear and Nonlinear Second Order Differential Equations with Variable Coefficients and Mixed Conditions -- 1 Introduction -- 2 Preliminaries -- 3 Factorization Method for Linear Differential Equations -- 4 Factorization Method for Nonlinear Differential Equations -- 5 Conclusions -- References -- A General Framework for Studying Certain Generalized Topologically Open Sets in Relator Spaces -- 1 Motivations -- 2 Preliminaries -- 3 A Few Basic Facts on Relations -- 4 A Few Basic Facts on Relators -- 5 Structures Derived from Relators -- 6 Further Structures Derived from Relators -- 7 Closure Operations for Relators.

8 Some Further Theorems on the Operations and -- 9 Projection Operations for Relators -- 10 Reflexive, Non-partial and Non-degenerated Relators -- 11 Topological and Quasi-Topological Relators -- 12 Proximal and Quasi-Proximal Relators -- 13 A Few Basic Facts on Filtred Relators -- 14 A Few Basic Facts on Quasi-Filtered Relators -- 15 Some Further Theorems on Topologically Filtered Relators -- 16 Some More Particular Theorems on Topologically Filtered Relators -- 17 Some Generalized Topologically Open Sets -- 18 Some Further Theorems on Generalized Topologically Open Sets -- 19 A Further Family of Generalized Topologically Open Sets -- 20 Some Further Theorems on the Family AR -- 21 Characterizations of the Family AR -- 22 Some Further Characterizations of the Family AR -- 23 Lower and Upper Nearness Relations for Sets -- 24 Some Set-Theoretic Properties of the Relation `3́9`42`"̇613A``45`47`"603ALnR -- 25 Some Topological Properties of the Relation `3́9`42`"̇613A``45`47`"603ALnR -- 26 Some Further Topological Properties of the Relation `3́9`42`"̇613A``45`47`"603ALnR -- 27 Some Relation-Theoretic Properties of the Relation `3́9`42`"̇613A``45`47`"603ALnR -- 28 Lower and Upper Nearness Closures of Families of Sets -- 29 Some Set-Theoretic Properties of the Families A and Au -- 30 Intersection Properties of the Families A and Au -- 31 Some Algebraic and Topological Properties of the Operations and u -- 32 Nearness Closures of the Families TR, TRs and TRp -- 33 Some Further Theorems on the Families TRs and TRp -- 34 Some Basic Properties of the Families TRsu and TRp -- 35 Some Further Theorems on the Closure Operation -- 36 An Illustrating Diagram and Two Related Examples -- References -- Graphical Mean Curvature Flow -- 1 Introduction -- 2 Riemannian Submanifolds -- 2.1 Notation and Conventions -- 2.2 The Pull-back Bundle.