No abstract

In this paper we present a new semilocal convergence theorem from data at one point for the Weierstrass iterative method for the simultaneous computation of polynomial zeros. The main result generalizes and improves all previous ones in this area.

No abstract

The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives. MSC: 34A08; 34A12 Introduction and NotationsFractional Calculus has a long history, but it has attracted considerable attention recently as an important tool for modeling of various real problems, such as viscoelastic systems, diffusion processes, signal and control processing, and seismic processes. Detailed information about the fractional calculus theory and its applications can be found in the monographs [1][2][3][4]. Some results for fractional linear systems with delays are in given in the book [5]. The monograph [6] is devoted to the impulsive differential and functional differential equations with fractional derivatives, as well as to some of their applications.It is well known that the study of linear fractional equations (integral representation, several types of stability, etc.) is an evergreen theme for research. Concerning these fields of fundamental and qualitative investigations for linear fractional ordinary differential equations and systems we refer to [2,4,7] and the references therein. Using the Laplace transform method, several interesting results in this direction are obtained in [8,9] as well. Regarding works concerning fractional differential systems with constant delays, we point out [10][11][12][13]. Concerning the retarded differential systems with variable or distributed delays-fundamental theory and application (stability properties)-we refer to [11,[14][15][16][17][18]. Neutral fractional systems with distributed delays are essentially studied less (see [19][20][21]). Stability properties of retarded fractional systems with derivatives of distributed order are studied in [22]. One of the existing best applications of fractional order equations with delays is modeling human manual control, in which perceptual and neuromuscular delays introduce a delay term. As interesting studies, we refer to [23,24].

In 1891, Weierstrass presented his famous iterative method for finding all the zeros of a polynomial simultaneously. In this paper we establish three new local convergence theorems for the Weierstrass method with a posteriori and a priori error estimates. The main result of the paper generalizes, improves and complements some well known results of Dochev (1962), Kjurkchiev and Markov (1983) and Yakoubsohn (2002). The results are given for polynomials over an arbitrary normed field.