We prove some fixed point results for mappings satisfying various contractive conditions on Complete G-metric Spaces. Also the Uniqueness of such fixed point are proved, as well as we showed these mappings are G-continuous on such fixed points.

No abstract

The Douglas-Rachford iteration scheme, introduced half a century ago in connection with nonlinear heat flow problems, aims to find a point common to two or more closed constraint sets. Convergence of the scheme is ensured when the sets are convex subsets of a Hilbert space, however, despite the absence of satisfactory theoretical justification, the scheme has been routinely used to successfully solve a diversity of practical problems in which one or more of the constraints involved is non-convex. As a first step toward addressing this deficiency, we provide convergence results for a prototypical non-convex two-set scenario in which one of the sets is the Euclidean sphere.

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. It was Mazur [7] who drew attention to the euclidean space property (I): every bounded closed convex set can be represented as an intersection of closed balls; and he began the investigation to determine those normed linear spaces which possess this property. Phelps [9] continued this investigation, characterising finite dimensional spaces with property (I). Recently, Sullivan [/2] has given a characterisation of smooth spaces with property (I).

No abstract