On a new approach of enriched operators.

We establish the existence and uniqueness of fixed points of generalized contractions in the setting of Banach spaces and prove the convergence of Mann iteration for this general class of mappings. Also, we show the existence of fixed points and the convergence of Mann iteration as well for generalized nonexpansive mappings. Last but not least, we provide two applications, one from the field of numerical analysis of linear systems and another one dealing with functional equations. This new approach significantly extends the classes of enriched contractions and enriched nonexpansive mappings, and allows the use of Mann iteration as opposed to all papers on the subject, which necessarily have to rely on Krasnoselskij iteration.

Review on Optical Methods Used to Characterize the Linear Birefringence of Polymer Materials for Various Applications.

Optical polymers are recognized for their high transparency, raised flexibility, low cost, and good film-forming ability; hence, they introduce a multitude of benefits in a wide range of devices, such as information storage, displays, optical communications, and filters. Among the optical properties, birefringence is an essential parameter in practical cases that demand the control of the state of polarization of light. This review is focused on describing some fundamental and applicative aspects concerning the optical birefringence of the polymer materials. First, elementary notions depicting the phenomenon of light double refraction in macromolecular media are provided. Furthermore, the most relevant optical techniques to determine birefringence are reviewed by highlighting the working principle and mathematical basis for computing this parameter. Then, a series of investigations of optically birefringent polymers are described, summarizing the most utilized approaches to induce light double refraction in such materials. The selected results are analyzed in relation to the pursued applications. In the end, the future of this scientific domain is briefly presented by establishing the research paths that need further exploration. Moreover, the novel directions that could be formulated and might contribute to certain considerable advancements in the materials employed in the modern optical technologies are mentioned.

Birefringence of Thin Uniaxial Polymer Films Estimated Using the Light Polarization Ellipse.

A simple method for determining the linear birefringence of the thin layers based on the determination of the orientation of the polarization ellipse of totally polarized light is proposed and it is applied to PVA thin foils. Theoretical notions and the experimental procedure are described. The linear birefringence of polymer thin foils with different degrees of stretching is determined and the applicability of the method is discussed.

Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces.

In this paper, our aim is to ascertain the distance between two sets iteratively in two simultaneous ways with the help of a multivalued coupling define for this purpose. We define the best proximity points of such couplings that realize the distance between two sets. Our main theorem is deduced in metric spaces. As an application, we obtain the corresponding results in uniformly convex Banach spaces using the geometry of the space. We discuss two examples.