Christopher C. Tisdell

Three Point Boundary Value Problems on Time Scales

This work formulates existence theorems for solutions to three-point boundary value problems on time scales. The ideas are based on a relationship between the three point boundary conditions, lower and upper solutions and topological degree theory.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense.

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.

The nonexistence of spurious solutions to discrete, two-point boundary value problems

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which are independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented.

Boundedness and Uniqueness of Solutions to Dynamic Equations on Time Scales

In this work, we investigate the boundedness and uniqueness of solutions to systems of dynamic equations on time scales. We define suitable Lyapunov-type functions and then formulate appropriate inequalities on these functions that guarantee all solutions to first-order initial value problems are uniformly bounded and/or unique. Several examples are given. †cct@maths.unsw.edu.au

Three-point boundary value problems for second-order, ordinary, differential equations

We establish existence results for solutions to three-point boundary value problems for nonlinear, second-order, ordinary differential equations with nonlinear boundary conditions.

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented

A multiplicity result for p-Lapacian boundary value problems via critical points theorem

Abstract In this paper, we deal with the existence of at least three classical solutions for the following two point boundary value problem ( φ p ( u ′ ) ) ′ + λ f ( t , u ) = 0 , α 1 u ( a ) - α 2 u ′ ( a ) = 0 , β 1 u ( b ) + β 2 u ′ ( b ) = 0 , where φ p ( s ) = ∣ s ∣ p - 2 s , p > 1 is a constant, λ is a positive parameter, a , b ∈ R , a b . Our main tool is a recent three critical point theorem of Ricceri. Our emphasis here is constructing an appropriate separable and reflexive real Banach space.

On first-order discrete boundary value problems

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations.

On Exact Controllability of First-Order Impulsive Differential Equations

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work, we present some new results concerning the exact controllability of a nonlinear ordinary differential equation with impulses.