### Abstract:

This dissertation is concerned with the relation between boundary value problems associated with (in general) nonlinear differential operators and variational principles. The Introductory Chapter 1 provides a systematic formulation of the problem. To motivate the discussion, the theory of the simplest problem of the calculus of variations is considered, with special reference to two distinct approaches towards its solution. In the one approach the critical points are determined by differentiation of the functional to obtain the Euler-Lagrange differential equations of the problem. The other approach, that of Direct Methods, computes the solution of the problem by means of certain techniques such as that of Direct Methods, computes the solution of the problem by means of certain techniques such as that of Ritz. The solution found by Direct Methods thus corresponds to the solution of the Euler-Lagrange differential equations. The concern in this study is to express a given boundary value problem as a condition for critical points of a functional. The existence of solutions can then be tackled with the aid of Direct Methods. The basic problem is analogous to the fact that one may regard the zeros of a certain vector field as the critical points of some scalar field in finite dimensional spaces. Chapter 2 is a natural extension of the finite dimensional theory of vector fields to boundary value problems associated (in general) with nonlinear differential operators. The starting point of this chapter is the observation that nonlinear operators can be approximated by linear operators, this being achieved with the aid of differentiation. In this way one is led to consider differentiable functionals which give rise to potential operators. The main part of this chapter is therefore concerned with the characterisation of a given operator as a potential operator. It is also shown that for a potential operator a simple integration provides the associated functional. In the last part of this chapter it is seen that a differentiable operator is a potential operator if a certain system of partial differential equations holds identically. Chapter 3 can be regarded as an extension of the theory contained in Chapter 2. It is concerned with applications of the concept of equivalent integrals of the calculus of variations to a class of boundary value problems whose associated differential operator is a potential operator. The general theory of equivalent integrals is discussed in some detail, after which a concrete application is presented. Concluding remarks are given in the last section of this chapter.