-integrability test for discrete equations via multiple scale expansions.
solutions of systems of finite difference equations.
Canards et râteaux
On étudie le phénomène de retard à la bifurcation dans des systèmes dynamiques discrets du plan. La distinction d’une courbe invariante par le système permet de ramener l’étude de ce phénomène à l’étude d’un objet. On démontre la présence du retard dans les systèmes analytiques oscillants. On fait état d’un nouveau phénomène découvert expérimentalement qui apparaît dans les systèmes non inversibles: la courbe invariante présente une succession de pôles exponentiellement étroits. On démontre la présence...
Cayley's problem
Newton's method for computation of a square root yields a difference equation which can be solved using the hyperbolic cotangent function. For the computation of the third root Newton's sequence presents a harder problem, which already Cayley was trying to solve. In the present paper two mutually inverse functions are defined in order to solve the difference equation, instead of the hyperbolic cotangent and its inverse. Several coefficients in the expansion around the fixed points are obtained,...
Čebyšev's inequality on time scales.
Characterizacion of the bivariate discrete distributions defined by a partial difference equations system.
Conditions under which the solutions of a partial difference equations system can be probability functions are examined.When the coefficients of the system are polynomials then the partial difference equations system satisfied by generating functions associated to these distributions are easily obtained; they give useful recurrence relations for the moments. Three examples are given as well.
Characterizations of Dominant and Dominated Solutions of Linear Recursions.
Classification and existence of positive solutions to nonlinear dynamic equations on time scales.
Classification of nonoscillatory solutions of higher order neutral type difference equations
The authors consider the difference equation where , , , , , and is a sequence of integers with and . They obtain results on the classification of the set of nonoscillatory solutions of () and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.
Classification of solutions of delay difference equations.
Classification rationnelle et confluence des systèmes aux différences singuliers réguliers
En choisissant des “caractères” et des “logarithmes”, méromorphes sur , construits à l’aide de la fonction Gamma d’Euler, et en utilisant des séries de factorielles convergentes, nous sommes en mesure, dans une première partie, de donner une “forme normale” pour les solutions d’un système aux différences singulier régulier. Nous pouvons alors définir une matrice de connexion d’un tel système. Nous étudions ensuite, suivant une idée de G.D. Birkhoff, le lien de celles-ci avec le problème de la classification...
Comparison and oscillation results for delay difference equations with oscillating coefficients.
Comparison results and linearized oscillations for higher-order difference equations.
Comparison theorems for half-linear second order difference equations
Conjugacy criteria for second order linear difference equations
We establish conditions which guarantee that the second order difference equation possesses a nontrivial solution with at least two generalized zero points in a given discrete interval
Conjugate sequences in a Fibonacci-Lucas sense and some identities for sums of powers of their elements.
Conjugation to a shift and the splitting of invariant manifolds
We give sufficient conditions for a diffeomorphism in the plane to be analytically conjugate to a shift in a complex neighborhood of a segment of an invariant curve. For a family of functions close to the identity uniform estimates are established. As a consequence an exponential upper estimate for splitting of separatrices is established for diffeomorphisms of the plane close to the identity. The constant in the exponent is related to the width of the analyticity domain of the limit flow separatrix....
Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs
This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical...
Construction of the general solution of planar linear discrete systems with constant coefficients and weak delay.
Construction of upper and lower solutions for singular discrete initial and boundary value problems via inequality theory.