643 research outputs found
A short course on positive solutions of systems of ODEs via fixed point index
We shall firstly study the existence of one positive solution of a model
problem for one equation via the classical Krasnosel'ski\u\i{} fixed-point
theorem. Secondly we investigate how to handle this problem via the fixed point
index theory for compact maps. Thirdly we illustrate how this approach can be
tailored in order to deal with non-trivial solutions for systems of ODEs
subject to local BCs. The case of nonlocal and nonlinear BCs will also be
investigated. Finally we present some applications to the existence of radial
solutions of some systems of elliptic PDEs.Comment: 52 pages 13 figure
Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications
We provide a theory to establish the existence of nonzero solutions of
perturbed Hammerstein integral equations with deviated arguments, being our
main ingredient the theory of fixed point index. Our approach is fairly general
and covers a variety of cases. We apply our results to a periodic boundary
value problem with reflections and to a thermostat problem. In the case of
reflections we also discuss the optimality of some constants that occur in our
theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page
First order systems of odes with nonlinear nonlocal boundary conditions
In this article, we prove an existence of solutions for a non-local boundary
value problem with nonlinearity in a nonlocal condition. Our method is based
upon the Mawhin's coincidence theory
Fifty Years of Stiffness
The notion of stiffness, which originated in several applications of a
different nature, has dominated the activities related to the numerical
treatment of differential problems for the last fifty years. Contrary to what
usually happens in Mathematics, its definition has been, for a long time, not
formally precise (actually, there are too many of them). Again, the needs of
applications, especially those arising in the construction of robust and
general purpose codes, require nowadays a formally precise definition. In this
paper, we review the evolution of such a notion and we also provide a precise
definition which encompasses all the previous ones.Comment: 24 pages, 11 figure
Positive operators and maximum principles for ordinary differential equations
We show an equivalence between a classical maximum principle in differential equations and positive operators on Banach Spaces. Then we shall exhibit many types of boundary value problems for which the maximum principle is valid. Finally, we shall present extended applications of the maximum principle that have arisen with the continued study of the qualitative properties of Green’s functions
Valuation of boundary-linked assets
This article studies the valuation of boundary-linked assets and their derivatives in continuous-time markets. Valuing boundary-linked assets requires the solution of a stochastic differential equation with boundary conditions, which, often, is not Markovian. We propose a wavelet-collocation algorithm for solving a Milstein approximation to the stochastic boundary problem. Its convergence properties are studied. Furthermore, we value boundary-linked derivatives using Malliavin calculus and Monte Carlo methods. We apply these ideas to value European call options of boundary-linked asset
ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO BOUNDARY VALUE PROBLEMS ON TIME SCALES
This work formulates existence, uniqueness, and uniqueness-implies-existence theorems for solutions to two-point vector boundary value problems on time scales. The methods used include maximum principles, a priori bounds on solutions, and the nonlinear alternative of Leray-Schauder
A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions
In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described.
*The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)
- …