A note on continuous dependence of solutions of dynamic equations on time scales

We give a precise formulation and a proof as constructive as possible of the widely accepted claim that solutions of a dynamic equation depend continuously on the base time scale. Our approach to this problem is via Euler polygons which opens possibilities for development of numerical analysis of dynamic equations on time scales.     

A continuous dependence result for ultraparabolic equations in option pricing

We prove continuous dependence results for solution to the Cauchy problem related to degenerate parabolic equations arising in the valuation of financial derivatives. These results are crucial in some standard calibration procedure for recent stochastic volatility and interest rates models.     

Continuous dependence on the geometry and on the initial time for a class of parabolic problems I

In this paper, we investigate the continuous dependence on the geometry and the initial time for solutions u( x , t) of a class of nonlinear parabolic initial‐boundary value problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Continuous dependence on the geometry and on the initial time for a class of parabolic problems II

Extending the investigations initiated in an earlier paper, the authors deal in this paper with the solutions of another class of initial‐boundary value problems for which continuous dependence inequalities on the geometry and the initial time are established. Copyright © 2007 John Wiley & Sons, Ltd.     

Stationary solutions to phase field crystal equations

This paper is devoted to the analytical and numerical study of stationary solutions of the one‐dimensional Phase Field Crystal Equation. This new model recently introduced by K. Elder and M. Grant describes phase transformations at the atomic level on large time scales. By using bifurcation methods, we investigate the quantitative and qualitative properties of these solutions: multiplicity, stability, and periodicity. Quite unusual bifurcation diagrams are obtained by numerical simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

A global existence and uniqueness result for a stochastic Allen–Cahn equation with constraint

This paper addresses the analysis of a time noise‐driven Allen–Cahn equation modelling the evolution of damage in continuum media in the presence of stochastic dynamics. The nonlinear character of the equation is mainly due to a multivoque maximal monotone operator representing a constraint on the damage variable, which is forced to take physically admissible values. By a Yosida approximation and a time‐discretization procedure, we prove a result of global‐in‐time existence and uniqueness of the solution to the stochastic problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Input dynamics and nonstandard riccati equations with applications to boundary control of damped wave and plate equations

An optimization problem for a control system governed by an analytic generator with unbounded control actions is considered. The solution to this problem is synthesized in terms of the Riccati operator, arising from a nonstandard Riccati equation. Solvability and uniqueness of the solutions to this Riccati equation are established. This theory is applied to a boundary control problem governed by damped wave and plate equations.Research of this author partially supported by NSF Grant DMS 9204338.     

A note on the dependence on parameters for a nonlinear system via monotonicity theory

Using monotonicity theory, we investigate the continuous dependence on parameters for the discrete BVPs, which can be written in a form of a nonlinear system.     

An identification problem for a conserved phase‐field model with memory

This paper is devoted to recovering a scalar memory kernel in a conserved phase‐field model. For such a problem local in time existence and uniqueness results are proved. The technique used allows to show also the continuous dependence on the kernel of the solution to the direct problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.

We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds.

We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.     

On a class of nonlinear obstacle problems with nonstandard growth

In this paper, we study a class of nonlinear obstacle problems with nonstandard growth. We obtain the L∞ estimate on the solutions and prove the existence and uniqueness of solutions to such problems. Our results are generalizations of the corresponding results in the constant exponent case. Copyright © 2014 John Wiley & Sons, Ltd.     

A short note on the solvability of impulsive fractional differential equations with Caputo derivatives

In this paper, we prove the existence and non-existence of solutions to two impulsive fractional differential equations with strong or weak Caputo derivatives in Euclidean space, respectively.     

Continuous dependence results for a class of nonlinear heat equations in a cylindrical region

In this paper, we derive bounds for the solutions of a quasilinear heat equation in a finite cylindrical region if the far end and the lateral surface are held at zero temperature, and a nonzero temperature is applied at the near end. Some continuous dependence inequalities are also obtained. We also investigate the case in which a given heat flux is prescribed at the near end, instead of a given temperature. Copyright © 2008 John Wiley & Sons, Ltd.