H.B. Thompson

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue

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.

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K -> 2^Y a point-to-set mapping such that for any @g @e K, C(@g) is a pointed, closed, and convex cone in Y and int C(@g) 0. Given a mapping g : K -> K and a vector valued bifunction f : K x K -> Y, we consider the implicit vector equilibrium problem (IVEP) of finding @g^* @e K such that f g(@g^*), y) @? -int C(@g) for all y @e K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.

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.

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.

Extremal points for impulsive Lidstone boundary value problems

The first extremal point for a boundary value problem with impulse for an n^t^h-order linear, ordinary differential equation is characterized by the existence of a nontrivial solution that lies in a cone. Cone theoretic arguments are applied to linear, monotone, compacts maps. To construct such maps, an impulse effect operator is constructed to complement the usual Greens function approach. An application is made to a nonlinear problem.

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

Three-point boundary value problems for second-order discrete equations

We formulate existence results for solutions to discrete equations which approximate three-point boundary value problems for second-order ordinary differential equations

Generalized vector F-variational inequalities and vector F-complementarity problems for point-to-set mappings

In this paper, we study a new generalized vector F-variational inequality and a new generalized vector F-complementarity problem for point-to-set mappings in Hausdorff topological vector spaces. We establish the equivalence between the generalized vector F-variational inequality and the generalized vector F-complementarity problem under certain assumptions. By considering the existence of solutions for the vector F-variational inequality with a single-valued mapping and using the continuous selection theorem, we obtain some new existence theorems of solutions for the generalized vector F-variational inequality and the generalized vector F-complementarity problem, respectively.

Nonlinear multipoint boundary value problems for weakly coupled systems

We establish existence results concerning solutions to multipoint boundary value problems for weakly coupled systems of second order ordinary differential equations with fully nonlinear boundary conditions.