Ian Tweddle

ON MULTIVALUED f-NONEXPANSIVE MAPS

In this paper we prove coincidence and common fixed points results for single-valued maps / and multivalued /-nonexpansive maps with star-shaped weakly compact domains in Banach spaces, which extend the theorems of [3], [6], [8] and others. Moreover weak convergence and strong convergence results for coincidence point sets have also been proved, extending a result in [1].

SOME RESULTS ON COINCIDENCE POINTS

Let (X, d) be a complete metric space, M a nonempty subset of X, and for S = X or5 = M let CB(S) (respectively K(S)) denote the family of all nonempty closed bounded(respectively compact) subsets of 5 endowed with the Hausdorff metric H. A multivaluedmap T of M into CB(X) is called a contraction if there exists a constant h € (0,1) suchthat H\T(x), T{y)j ^ h d(x, y), for al wl x,e hav y € M.e th Ie Lipschitf z constant h = 1,then T is called a nonexpansive mapping. A point x in M is said to be a fixed pointof T if x 6 T(x). Nadler [15] and Markin [12] initiated such a geometric approach tomultivalued maps. In [15] Nadler proved a fixed point result for multivalued contractionmaps of a complete metric space, which is a generalisation of the Banach ContractionPrinciple. Since then various well-known results for single-valued self contraction andnonexpansive mappings have been extended to multivalued analogues. For example, see[4, 5, 9, 11, 17].On the other hand Kaneko [8] has introduced a notion of multivalued /-contractionmap as follows. Let / be a single-valued continuous map of M into X. Then a multivaluedmap T of M into CB(X) is called an J-contraction if there exists a constant he (0,1)such that #(T(z),T(y)) < /id(/(z),/(y)) for all x,y € M. If we have the Lipschitzconstant h = 1, then T is called a f-nonexpansive mapping. A point x in M is said tobe a coincidence point of / and T if f(x) S T(x). We denote by C(f DT) the set ofcoincidence points of / and T. In [8] Kaneko has proved coincidence and common fixedpoint results for self /-contraction maps, extending results of Jungck [7], Nadler [15] andothers. Recently Daffer and Kaneko [2] have studied multivalued /-nonexpansive mapsand extended results of Smithson [19] and Kaneko [8] for such maps of connected metricspaces, using the concept of an /-orbit of the multifunction as a major tool.Geometric fixed point theory in Functional Analysis for such multivalued maps hasbeen extensively developed. One of its developments has led to substantial weakenings

Part Two On the Interpolation of Series

Let there be an arbitrary straight Line PQ which is given in position, above which let any number of ordinates A, B, C, D etc. be erected, which are parallel to each other and separated one from the other by equal intervals: moreover, let these ordinates represent the terms of a regular series, continually increasing or decreasing and having the same sign; and passing through the extremities of them all, there will be exactly one curve, which will in fact be defined by the given equation for the series, that is, from the given equation which expresses in general the relation between any two or more successive ordinates.

A Treatise on Summation & Interpolation of Infinite Series

Just as curves are not determined by some given ordinates no matter how many, but by the general relation between the abscissae and the ordinates, so series are not determined by some given terms no matter how many, but by the relation between successive terms. For any quantities which are finite in number can form terms in different series: in fact the series is unique which has the same initial terms and the same law for forming the remaining terms up to infinity. Therefore in the first place the relations of the terms have to be investigated; then when these have been found they are to be specified by difference equations just as Des Cartes has defined curves by algebraic equations: when these things have been obtained, problems about summation and interpolation and other matters of that type concerning series will be solved by an analysis no less exact than common algebra is.

Appendix Stirling’s Letter to De Moivre Dated 19 June 1729

In this letter Stirling communicated to De Moivre some of his results on the middle-ratio problem. I have translated the letter below using the text which De Moivre reproduced on pp.170–172 of [43] (see also pp.46–49 of [74]). It would appear from the first paragraph that Stirling had already told De Moivre about his solutions and that the letter was written in response to a request for more detailed information which De Moivre wished to include in his Miscellanea Analytica [43]. The main text is very similar to Stirling’s statement and discussion in Proposition 23. The numbers in the second calculation below are not as accurate as those in the corresponding calculation in Proposition 23, although the final result is the same in both places.73

Part One On the Summation of Series

In this first part I have tried to shorten the calculations in the quadrature of curves, and also in more difficult problems, namely by attaining the values of infinite series more readily than by simple addition of terms as is commonly done. For rapidly converging series this indeed amply achieves the purpose, and there is no need for another method: but where they converge slowly, immeasurable work is required for the most part, and it is indeed greater according as the convergence is less; and therefore if they approximate very slowly, they become wholly intractable. For it is very well known that sometimes more than a thousand terms are required in order that the sum may be obtained exact to two or three figures. Therefore we will demonstrate in what follows a method which is easy to apply for transforming those which are slowest converging of all into others which approximate very rapidly; it is clear that from these the sums can be calculated with very little effort to very many places of figures.

Pappus’s Two General Propositions and Euclid’s First Porism

This consists of Simson’s Propositions 7–25 and includes his conjectures for the ten cases of Pappus’s first general Proposition (Propositions 7, 8, 10–16, 19), the second general Proposition (Proposition 21), Euclid’s first Porism (Proposition 23) and various applications and extensions of these results. Pappus’s Lemmas 1–9 for the Porisms are developed in the course of this Part (see Appendix 3).

Lemmas and Restorations

Propositions 26–79 contain the rest of Pappus’s Lemmas for the Porisms (10–38) (see Appendix 3) and Simson’s restorations of several of Euclid’s Porisms, notably the last three of Book 3 as recorded by Pappus (Propositions 50, 53, 57, 58, 61, 62).

Various Porisms: Fermat, Simson and Stewart

In Propositions 80–85 Simson discusses at length four of Fermat’ s Propositions. Some of this material originated with Matthew Stewart, as did the last four Propositions (90–93).

Some results on conic sections in the correspondence between colin MacLaurin and Robert Simson

The relevant correspondence consists of two letters from MacLaurin to Simson which are apparently now lost and Simsons replies to these of 20 January 1735 and 27 July 1736. * MacLaurin had communicated certain results on conic sections to Simson, requesting him to comment on their originality. Obviously impressed by MacLaurins propositions which, with one exception, he neither knew nor was able to locate in books, Simson responded enthusiastically. Unfortunately his observations are too general to allow us at once to identify precisely what these results were; moreover, an enclosure to the first letter which contained detailed proofs is missing. However, in one of Simsons yet unpublished notebooks I have located two sections in which MacLaurins propositions are discussed, proved and generalised.2 MacLaurin had apparently stated these only