Daniel C. Biles

An application of time scales to economics

Economics is a discipline in which there appears to be many opportunities for applications of time scales. The time scales approach will not only unify the standard discrete and continuous models in economics, but also, for example, allows for payments which arrive at unequally spaced points in time. We present a dynamic optimization problem from economics, construct a time scales model, and apply calculus of variations to derive a solution. Time scale calculus would allow exploration of a variety of situations in economics.

A generalization of the Lane–Emden equation

Abstract We consider a class of second-order initial-value problems, which includes the well-known Lane–Emden equation from astrophysics as a special case. Local existence, global existence and uniqueness of solutions are proven.

Numerical approximation for singular second order differential equations

We consider numerical approximation of solutions of singular second order differential equations. In particular, we study the backward (or implicit) Euler method. We prove results concerning consistency, global error and stability. We show that the global error is linear with respect to the step size. Numerical results are also given, which demonstrate the linear convergence and we compare the numerical results with known approximations.

First-Order Singular and Discontinuous Differential Equations

We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and singular nonlinearities.

A utility maximisation problem on multiple time scales

In this paper, we present a technique for a dynamic optimisation problem in which the value function and constraints can be on different time scales. Using this technique we solve a perfect-foresight utility maximisation problem for a finitely-lived consumer while making very general assumptions about timing of consumption and investing decisions.

Fixed point approaches to the solution of integral inclusions

Solutions to generalizations of the Volterra and Hammerstein integral inclusions are found by using the fixed point theorems of Covitz-Nadler and Bohnenblust-Karlin. Several illustrative examples are presented. Some conditions are given which also allow Lipschitz solutions to be obtained.

Lower bounds for the first zero for nonlinear second order differential equations

We consider establishing lower bounds for the first zero of the solution of the nonlinear second order initial value problem (p(x)y′(x))′ + f (x,y(x)) = 0, x 0 y(0) = a > 0, y′(0) = 0. Using the linear case as a starting point, we prove several of these theorems, comparing them by considering several examples. Mathematics subject classification (2010): 34C10, 34A34, 34A36.

Results on Uniqueness of Solutions for a Generalized Second Order Differential Equation

ABSTRACT We present two theorems concerning uniqueness of solutions for second order ordinary differential equations that have a generalized left-hand side with initial conditions. Despite the fact that the hypotheses are weak, the theorems cover a large number of cases and the proofs are not especially complicated.

Generalized Differential and Integral Equations

The theory of differential equations is always and necessarily “under construction,” because more accurate mathematical models usually demand new theoretical developments. For instance, discontinuities or singularities often occur in applications and they are often removed from models just for technical limitations. On the other hand, the last few years have witnessed an increasing interest in different types of differential and integral equations as mathematical models for real life situations. Besides the classical examples of impulsive equations or equations with deviating arguments, many other types of revisions of the classical concepts of differential or integral equations are being intensively studied: set-valued equations, stochastic equations, fractional equations, fuzzy equations, and many more. This diversity notwithstanding, it appears that generalized Stieltjes integration provides an unified framework for many of the above types of equations, thus simplifying and improving the theory at the same time. The special issue succeeded in bringing together a number of papers on many different branches of the theory of differential equations which clearly deserve the adjective “generalized.” The editors in charge of this special issue did not expect such a variety when they first proposed a special issue mainly focused on the following topics: