Maria do Rosário Grossinho

The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation

The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation M.R. Grossinho a;b;∗;1, St.A. Tersian c;2 a Departamento de Matem atica, ISEG, Universidade T ecnica de Lisboa, Rua do Quelhas, 6, 1200 Lisboa, Portugal b CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1699 Lisboa Codex, Portugal c Center of Applied Mathematics and Informatics, University of Rousse, 8 Studentska str., 7017 Rousse, Bulgaria

An introduction to minimax theorems and their applications to differential equations

Preface. 1. Minimization and Mountain-Pass Theorems. 2. Saddle-Point and Linking Theorems. 3. Applications to Elliptic Problems in Bounded Domains. 4. Periodic Solutions for Some Second-Order Differential Equations. 5. Dual Variational Method and Applications. 6. Minimax Theorems for Locally Lipschitz Functionals and Applications. 7. Homoclinic Solutions of Differential Equations. Notations. Index.

On the solvability of a boundary value problem for a fourth-order ordinary differential equation

We study the existence and multiplicity of nontrivial periodi cs olutions for a semilinear fourth-order ordinary differential equation arising in the study of spatial patterns for bistable systems. Variational tools such as the Brezis–Nirenberg theorem and Clark theorem are used in the proofs of the main results.

Nonlinear analysis and its applications to differential equations

An Overview of the Method of Lower and Upper Solutions for ODEs.- On the Long-time Behaviour of Solutions to the Navier-Stokes Equations of Compressible Flow.- Periodic Solutions of Systems with p-Laplacian-like Operators.- Mechanics on Riemannian Manifolds.- Twist Mappings, Invariant Curves and Periodic Differential Equations.- Variational Inequalities, Bifurcation and Applications.- Complex Dynamics in a Class of Reversible Equations.- Symmetry and Monotonicity Results for Solutions of Certain Elliptic PDEs on Manifolds.- Nielsen Number and Multiplicity Results for Multivalued Boundary Value Problems.- Bifurcation Theory and Application to Semilinear Problems near the Resonance Parameter.- Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces.- On the Method of Upper and Lower Solutions for First Order BVPs.- Nonlinear Optimal Control Problems for Diffusive Elliptic Equations of Logistic Type.- On The Use of Time-Maps in Nonlinear Boundary Value Problems.- Some Aspects of Nonlinear Spectral Theory.- Asymmetric Nonlinear Oscillators.- Hopf Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion.- Galerkin-Averaging Method in Infinite-Dimensional Spaces for Weakly Nonlinear Problems.- PBVPs for Ordinary Impulsive Differential Equations.- Homoclinic and Periodic Solutions for Some Classes of Second Order Differential Equations.- Global Bifurcation for Monge-Ampere Operators.- Remarks on Boundedness of Semilinear Oscillators.- The Dual Variational Method in Nonlocal Semilinear Tricomi Problems.- Symmetry Properties of Positive Solutions of Nonlinear Differential Equations Involving the p-Laplace Operator.- A Maximum Principle with Applications to the Forced Sine-Gordon Equation.- Lipschitzian Regularity Conditions for the Minimizing Trajectories of Optimal Control Problems.- Abstract Concentration Compactness and Elliptic Equations on Unbounded Domains.

On pairs of positive solutions for a singular boundary value

In this paper, we study the existence of multiple solutions of some singular boundary value problems where p can be equal to zero at t = 0 and t = T, the function g can be singular at t = 0, t = T and a h for u = 0. Such singularities generalize Emden-Fowler equations. We consider the case where the “slope” is larger than the first eigenvalue near 0 and near infinity. In between the slope is controlled from the existence of a strict upper solution. To prove our result we extend to the singular case the theory of lower and upper solutions a;nd its relation with the Leray-Schauder degree.

Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option pricing models including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary differential equation and implicit equation for the free boundary position. We finally present results of numerical approximation of the early exercise boundary, option price and their dependence on model parameters.

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.

Higher order nonlinear two‐point boundary value problems with sign‐type Nagumo condition

In this paper we present existence and location results for two‐point boundary value problems for third and fourth order fully nonlinear differential equations.Nonlinearities are assumed to satisfy a sign‐type Nagumo condition which allows an asymmetric unbounded behavior. The arguments make use of lower and upper solutions method and degree theory.

Minimization and Mountain-Pass Theorems

In this introductory chapter, we consider the concept on differentiability of mappings in Banach spaces, Frechet and Gâteaux derivatives, secondorder derivatives and general minimization theorems. Variational principles of Ekeland [Ek1] and Borwein & Preiss [BP] are proved and relations to the minimization problem are given. Deformation lemmata, Palais—Smale conditions and mountain-pass theorems are considered. The deformation approach and e—variational approach are applied to prove the mountainpass theorem and its various extensions.

Periodic Solutions for Some Second-Order Differential Equations

In this chapter, we apply variational methods to prove the existence of periodic solutions of some second-order non-linear differential equations, namely in resonance situations.