The Douglas–Rachford method is a splitting method frequently employed for finding zeros of sums of maximally monotone operators. When the operators in question are normal cone operators, the iterated process may be used to solve feasibility problems of the following form: Find $x\in \bigcap _{k=1}^{N}S_{k}$. The success of the method in the context of closed, convex, nonempty sets $S_{1},\ldots ,S_{N}$ is well known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less well understood, yet it is surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg and Thibault’s success in applying the method to solving sudoku puzzles, began an investigation of his own. We survey the current body of literature on the subject, and we summarize its history. We especially commemorate Professor Borwein’s celebrated contributions to the area.

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.

A closed convex subset of ${{c}_{0}}$ has the fixed point property (fpp) if every nonexpansive self mapping of it has a fixed point. All nonempty weak compact convex subsets of ${{c}_{0}}$ are known to have the fpp. We show that closed convex subsets with a nonempty interior and nonempty convex subsets which are compact in a topology slightly coarser than the weak topology may fail to have the fpp.

We characterise Opial's condition, the non-strict Opial condition, and the uniform Opial condition for a Banach space X in terms of properties of the duality mapping from X into X*.

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.

We establish the Karlovitz lemma for a nonexpansive self mapping of a nonempty weak* compact convex set in a weak* orthogonal dual Banach lattice.

Rolewicz' drop property is a modification of a concept underlying Daneš' drop theorem. We characterise the drop property by the upper semicontinuity and compact valued property of the duality mapping for the dual. The characterisation suggests that we define a weak drop property which we show characterises the reflexivity of the space.

For an operator on a Hilbert space, points in the closure of its numerical range are characterized as either extreme, non-extreme boundary, or interior in terms of various associated sets of bounded sequences of vectors. These generalize similar results due to Embry, for points in the numerical range.

Stampfli and Embry characterized points in the numerical range which are extreme in terms of the linearity of corresponding sets of vectors. Das and Craven generalized this to include the case of unattained boundary points. We give an alternative proof of this result using a technique of Berberian. This approach appears to be more conceptual in that it enables us to deduce the result from that of Stampfli and Embry. We also illustrate how the same technique may be used to generalize other results of Embry.

An example is given of a Banach space with normal structure which does not satisfy the geometrical conditions commonly expected to be related to normal structure.

We show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

A Banach space is fully 2-rotund if (xn) converges whenever ‖xn + xm‖ converges as m, n → ∞ and locally uniformly rotund if xn → x whenever ‖xn‖ and ‖(xn + x)/2‖ → ‖x‖.

We show that I2 with the equivalent norm

is fully 2-rotund but not locally uniformly rotund, thus answering in the negative a question first raised by Fan and Glicksberg in 1958.

For a Banach space X, Susumu Okada raised the question of whether the unit hall of the dual space X* is weak* separable if X* is weak* separable. The problem occurred in the theory of manifolds modelled on locally convex spaces. We answer the question in the negative but show that it is true for particular types of spaces.

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

For a normed linear space X we investigate conditions for the existence of support maps under which the image of X is a dense subset of the dual. In the case of finite-dimensional spaces a complete answer is given. For more general spaces some sufficient conditions are obtained.

It is known that in a B*-algebra every self-adjoint element is hermitian. We give an elementary proof that this condition characterizes B*-algetras among Banach*-algebras.