Journal•ISSN: 0004-9727
Bulletin of The Australian Mathematical Society
Cambridge University Press
About: Bulletin of The Australian Mathematical Society is an academic journal published by Cambridge University Press. The journal publishes majorly in the area(s): Banach space & Mathematics. It has an ISSN identifier of 0004-9727. Over the lifetime, 5813 publications have been published receiving 50552 citations.
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TL;DR: In this article, it was shown that under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces, the convergence of xn to a fixed point is shown to be strong.
Abstract: Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.
677 citations
TL;DR: Delbosco et al. as discussed by the authors established fixed point theorems for selfmaps of complete metric spaces by altering the distances between the points with properties: the use of a function (0 : R -*• R satisfying the following1. cp is continuous and strictly increasing in R ;2. ip(t) = 0 if and only if t = 0 ;3.
Abstract: Let R be the set of nonnegative real numbers and N the set ofpositive integers.Delbosco [/] and Skof [&] have established fixed point theorems forselfmaps of complete metric spaces by altering the distances between thepoints withproperties: the use of a function (0 : R -*• R satisfying the following1. cp is continuous and strictly increasing in R ;2. ip(t) = 0 if and only if t = 0 ;3.
670 citations
TL;DR: A major contribution of the work presented in this thesis is the successful implementation of a system of techniques to solve the graph theoretic model, particularly when applied to large ore-bodies, at a fraction of the time required by current software packages.
Abstract: A fundamental problem in open pit mine planning is that of determining the optimum ultimate pit limits of the mine. These limits are that pit contour which is the result of extracting a volume of material which maximises the difference between the value of extracted ore and the total extraction cost of ore and waste whilst satisfying certain practical operational requirements, such as, safe wall slopes. The determination of the optimum pit contour provides information which is essential in the evaluation of the economic potential of the mineral deposit. A number of optimisation techniques have been proposed for determining the optimum pit contour. Of these techniques, those based on graph theory, linear programming and dynamic programming are mathematically rigorous, but only those based on graph theory are more suited to solving the three-dimensional problem. Unfortunately, direct application of these techniques to large ore-bodies may cause considerable difficulties because of the exceptionally high demand on computer storage and time requirements. Indeed, 25 years of research effort has not satisfactorily resolved these computational problems. A major contribution of the work presented in this thesis is the successful implementation of a system of techniques to solve the graph theoretic model, particularly when applied to large ore-bodies. A measure of this success is the fact that pits, as much as seven times larger, may be designed with a given amount of computer storage, at a fraction of the time required by current software packages. The solution strategy presented involves the application of a modified Dime's Maximum Flow algorithm, together with an efficient 'data reducing' technique. Computational results of these techniques applied on data from gold producing mines in Western Australia are used to demonstrate the success of this strategy. The relationships between the rigorous pit optimisation techniques are also considered in this work. In particular, the Lerchs-Grossman graph-theoretic method is shown
413 citations
TL;DR: In this paper, the Kuhn-Tucker necessary conditions are generalized to a property, called K-invex, of a vector function in relation to a convex cone K. This leads to a new second order sufficient condition for a constrained minimum.
Abstract: If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K-invex, of a vector function in relation to a convex cone K. Necessary conditions and sufficient conditions are obtained for a function f to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.
337 citations
TL;DR: In this article, the authors extended Wittman's result to a class of Banach spaces with a weakly sequentially continuous duality map, and extended it to the class of uniformly smooth spaces.
Abstract: Reich [4] Shioj, i and Takahashi [6 essentiall] y extended Lions and respectively' ,Wittman's result tso the framework of uniformly smooth Banach spaces. Reich [5] alsoextended Wittman's resul to thte class of Banach spaces whic arhe uniformly smoothand have a weakly sequentially continuous duality map. Moreover the contro, l sequence(a
292 citations