In this paper we consider the so called a cone metric type space, which is a generalization of a cone metric space. We prove some common fixed point theorems for four mappings in those spaces. Obtained results extend and generalize well-known comparable results in the literature. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.

a b s t r a c tQuadrature rules with maximal even trigonometric degree of exactness are considered. We give a brief historical survey on such quadrature rules. Special attention is given on an approach given by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (3) (2005) 337-359. Translation in English from Uchenye Zapiski, Vypusk 1 (149). Seria Math. Theory of Functions, Collection of papers, Izdatel'stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, 1959, pp. 31-54]. The main part of the topic is orthogonal trigonometric systems on [0, 2π ) (or on [−π , π )) with respect to some weight functions w(x). We prove that the so-called orthogonal trigonometric polynomials of semi-integer degree satisfy a five-term recurrence relation. In particular, we study some cases with symmetric weight functions. Also, we present a numerical method for constructing the corresponding quadratures of Gaussian type. Finally, we give some numerical examples. Also, we compare our method with other available methods.

No abstract

A method for constructing the generalized Gaussian quadrature rules for Müntz polynomials on (0, 1) is given. Such quadratures possess several properties of the classical Gaussian formulae (for polynomial systems), such as positivity of the weights, rapid convergence, etc. They can be applied to the wide class of functions, including smooth functions, as well as functions with end-point singularities, such as those in boundary-contact value problems, integral equations with singular kernels, complex analysis, potential theory, etc. The constructive method is based on an application of orthogonal Müntz polynomials introduced by Badalyan [Akad. Nauk Armyan. SSR. Armenian summary)] and studied intensively by Borwein, Erdélyi, and Zhang [Trans. Amer. Math. Soc., 342 (1994), pp. 523-542], as well as by Milovanović on a numerical procedure for evaluation of such polynomials with high precision [Müntz orthogonal polynomials and their numerical evaluation, in

The purpose of this article is to generalize common fixed point theorems under contractive condition of Ćirić's type on a cone metric type space. We give basic facts about cone metric type spaces, and we prove common fixed point theorems under contractive condition of Ćirić's type on a cone metric type space without assumption of normality for cone. As special cases we get the corresponding fixed point theorems on a cone metric space with respect to a solid cone. Obtained results in this article extend, generalize, and improve, well-known comparable results in the literature.