The Sandwich theorem via Pataraia’s fixed point theorem

Positivity - The Sandwich theorem (König in Archiv der Mathematik 23:500–1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and...     

Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems

In this paper, we establish some new generalizations of Darbo’s fixed point theorem for multivalued mappings. Moreover, we prove the existence of solutions for a class of integral equations by Darbo’s fixed point theorem and the existence of solutions for a class of differential inclusions using a generalization of Darbo’s fixed point theorem.     

An extension of Matkowski’s and Wardowski’s fixed point theorems with applications to functional equations

Aequationes mathematicae - In this paper, we prove a fixed point theorem for a system of maps on the finite product of metric spaces. Our result generalizes the result of Matkowski (Bull Acad Pol...     

A novel Ishikawa–Green’s fixed point scheme for the solution of BVPs

In this article, a new numerical approach is introduced for the numerical solution of a wide class of boundary value problems (BVPs). The underlying strategy of the algorithm is based on embedding an integral operator, defined in terms of Green’s function, into Ishikawa fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that confirm the applicability and high efficiency of the method. The absolute error or residual error computations show that the current technique provides highly accurate approximations.     

Counting the order of the group of points of an elliptic curve over a finite field based on Shanks’s algorithm

Shanks’s baby-step-giant-step algorithm to count the order of the group of points of an elliptic curve over a finite prime field is described. The algorithm’s improvements are also discussed. The first one is based on Mestre’s theorem. The second improvement is based on the Sutherland’s probabilistic result allowing to calculate the exponent of an arbitrary finite abelian group.     

Generalizing the Krein–Rutman theorem,measures of noncompactness and the fixed point index

If L : Y → Y is a bounded linear map on a Banach space Y, the “radius of the essential spectrum” or “essential spectral radius” ρ(L) of L is well-defined and there are well-known formulas for ρ(L) in terms of measures of noncompactness. Now let \({C \subset D}\) be complete cones in a normed linear space (X, || · ||) and f : C → C a continuous map which is homogeneous of degree one and preserves the partial ordering induced by D. We prove (see Section 2) that various obvious analogs of the formulas for the essential spectral radius for the case f : C → C have serious defects, even when f is linear on C. We propose (see (3.5)) a definition for ρ _C (f), the “cone essential spectral radius of f,” which avoids these difficulties. If \({{\tilde r}_{C}(f)}\) denotes the (Bonsall) cone spectral radius of f, we conjecture (see Conjecture 4.1) that if \({\rho_{C}(f) < {\tilde r}_{C}(f)}\), then there exists \({u \in C {\backslash} \, \{0\}}\) with f(u) = ru where r ? r _C (f). If f satisfies certain additional conditions (for example, if f is a compact perturbation of a map which is linear on C), we obtain the conclusion of the conjecture; but in general we observe (Remark 4.7) that the conjecture is intimately related to old and difficult conjectures in asymptotic fixed point theory. In Section 5 we briefly discuss extensions of generalized max-plus operators which were our original motivation and for which Conjecture 4.1 is already nontrivial.     

Two applications of Brouwer’s fixed point theorem: in insurance and in biology models

In the first part of the article, a new interesting system of difference equations is introduced. It is developed for re-rating purposes in general insurance. A nonlinear transformation φ of a d-dimensional (d ≥ 2) Euclidean space is introduced that enables us to express the system in the form f^t+1:=φ( f^t), t = 0, 1, 2,. …. Under typical actuarial assumptions, existence of solutions of that system is proven by means of Brouwer’s fixed point theorem in normed spaces. In addition, conditions that guarantee uniqueness of a solution are given. The second, smaller part of the article is about Leslie–Gower’s system of d ≥ 2 difference equations. We focus on the system that satisfies conditions consistent with weak inter-specific competition. We prove existence and uniqueness of the equilibrium of the model under surprisingly simple and very general conditions. Even though the two parts of this article have applications in two different sciences, they are connected with similar mathematics, in particular by our use of Brouwer’s fixed point theorem.     

The analysis of the number of fixed points in the key extending algorithm of RC4

The probabilities of the state transitions of the initial value So in the S table of RC4 are described by a kind of bistochastic matrices, and then a computational formula for such bistochastic matrices is given, by which the mathematical expectation of the number of fixed points in the key extending algorithm of RC4 is obtained. As a result, a statistical weakness of the key extending algorithm of RC4 is presented.     

The role of the isotonizing algorithm in Stein’s covariance matrix estimator

Covariance matrix estimation is central to many applications in statistics and allied fields. A useful estimator in this context was proposed by Stein which regularizes the sample covariance matrix by shrinking its eigenvalues together. This estimator can sometimes yield estimates of the eigenvalues that are negative or differ in order from the observed eigenvalues. In order to rectify this problem, Stein also proposed an ad hoc “isotonizing” procedure which pools together eigenvalue estimates in such a way that the original ordering and positivity of the estimates are enforced. From numerical studies, Stein’s “isotonized” estimator is known to have good risk properties in comparison with the maximum likelihood estimator. However, it remains unclear what role is played by the isotonizing procedure in the remarkable risk reductions achieved by Stein’s estimator. Through two distinct lines of investigations, it is established that Stein’s estimator without the isotonizing algorithm gives only modest risk reductions. In cases where the isotonizing algorithm is frequently used, however, Stein’s estimator can lead to significant risk reductions for certain domains of the parameter. In other cases, Stein’s estimator can even yield risk increases, such as when (1) the theoretical eigenvalues are well separated, and/or (2) when the sample size is moderate to large, leading to over-shrinkage.     

Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

We construct new sequence transformations based on Wynn’s epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn’s epsilon and rho algorithm and of Osada’s generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

     

Strong convergence of a regularization method for Rockafellar’s proximal point algorithm

In this paper, for a monotone operator T, we shall show strong convergence of the regularization method for Rockafellar’s proximal point algorithm under more relaxed conditions on the sequences {r _k } and {t _k }, $$\lim\limits_{k\to\infty}t_k = 0;\quad \sum\limits_{k=0}^{+\infty}t_k = \infty;\quad\ \liminf\limits_{k\to\infty}r_k > 0.$$ Our results unify and improve some existing results.     

Fixed points of n-valued maps,the fixed point property and the case of surfaces—A braid approach

We study the fixed point theory of $n$-valued maps of a space $X$ using the fixed point theory of maps between $X$ and its configuration spaces. We give some general results to decide whether an $n$-valued map can be deformed to a fixed point free $n$-valued map. In the case of surfaces, we provide an algebraic criterion in terms of the braid groups of $X$ to study this problem. If $X$ is either the $k$-dimensional ball or an even-dimensional real or complex projective space, we show that the fixed point property holds for $n$-valued maps for all $n\ge 1$, and we prove the same result for even-dimensional spheres for all $n\ge 2$. If $X$ is the $2$-torus, we classify the homotopy classes of $2$-valued maps in terms of the braid groups of $X$. We do not currently have a complete characterisation of the homotopy classes of split $2$-valued maps of the $2$-torus that contain a fixed point free representative, but we give an infinite family of such homotopy classes.     

Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Aequationes mathematicae - Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are...     

New modified optimal families of King’s and Traub-Ostrowski’s methods

Based on quadratically convergent Schröder’s method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King’s family of fourth-order methods and Traub-Ostrowski’s method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King’s family and Traub-Ostrowski’smethod are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King’s family, Traub-Ostrowski’s method, and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.     

The Bellman function of the dyadic maximal operator related to Kolmogorov’s inequality

We precisely evaluate the Bellman function of two variables of the dyadic maximal operator related to Kolmogorov’s inequality, thus giving an alternative proof of the results in [3]. Additionally, we characterize the sequences of functions that are extremal for this Bellman function. More precisely, we prove that they behave approximately like eigenfunctions of the dyadic maximal operator, for a specific eigenvalue.     

Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems

We define a two-sided analog of the Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on that is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.With 11 PiguresTo the memory of Paul ErdösSupported in part by the INTAS grant 93-0570. The first author was supported by the French foundation PRO MATHEMATICA. The first author expresses his gratitude to l'Institut de Mathématiques de Luminy for support during his stay in Marseille in 1996-97. The second author is grateful to the University of Stony Brook for support during his visit in February–March 1996 and to the Institute for Advanced studies of Hebrew University for support during his being there in 1997