A Fixed Point Theorem for Mappings Satisfying a Contractive Condition of Rational Type on a Partially Ordered Metric Space

The purpose of this paper is to present a fixed point theorem using a contractive condition of rational type in the context of partially ordered metric spaces

New fixed point theorem under R-contractions

In this manuscript we introduce the notions of R-function and R-contractions, and we show an ad hoc fixed point theorem. We prove that this new kind of contractions properly includes the family of all Meir-Keeler contractions and other well-known classes of contractions that have been given very recently (for instance, those using simulation functions and manageable functions). As a consequence, our approach turns out to be appropriate to unify the treatment of different kinds of contractive nonlinear operators.This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author, therefore, acknowledges with thanks DSR for technical and financial support. The authors are grateful to three anonymous referees for their useful suggestions and comments. AF Roldán López de Hierro is grateful to the Department of Quantitative Methods for Economics and Business of the University of Granada. The same author has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE

Just Click Here : A Brief Glance at Absurd Electronic Contracts and the Law Failing to Protect Consumers

As e-commerce explodes around the world, consumers’ rights have been left behind. Before the completion of virtually every transaction on the Internet, the onus is placed on consumers to read and agree to an onslaught of terms and conditions. Often hidden in the middle of this extremely lengthy list of terms are massive exemption and limitation of liability clauses that deny consumers most if not all of their rights as “equal” trading partners. The common law principle that all onerous clauses in a contract need to be brought to the attention of the consumer for them to be binding seems to have been lost with the invention of the “click here to agree” button for signing online contracts. As the courts in Canada have not provided clear guidance on this issue thus far, other means must be pursued in order to protect consumers from the near-tyrannical control of unencumbered electronic standard form contracts in e-commerce. This paper will describe the principle of sufficiency of notice as it applies to paper contracts, and then contrast it with the newer jurisprudence that has refused to apply the principle to electronic contracts. The reasons for the refusal will be explored, followed by an examination of why the principle of sufficiency of notice needs to be applied and strengthened to respond to the increasingly onerous provisions hidden in electronic contracts. Finally, some other options for achieving the goal of consumer protection from hidden onerous clauses will be briefly explored. These other options include introducing stiffer consumer protection legislation domestically, the creation of international treaties, developing voluntary standards of contracting, and relying on Internet self-regulation

On some fixed point theorems under (α,ψ,ϕ) -contractivity conditions in metric spaces endowed with transitive binary relations

After the appearance of Nieto and Rodríguez-López’s theorem, the branch of fixed point theory devoted to the setting of partially ordered metric spaces have attracted much attention in the last years, especially when coupled, tripled, quadrupled and, in general, multidimensional fixed points are studied. Almost all papers in this direction have been forced to present two results assuming two different hypotheses: the involved mapping should be continuous or the metric framework should be regular. Both conditions seem to be different in nature because one of them refers to the mapping and the other one is assumed on the ambient space. In this paper, we unify such different conditions in a unique one. By introducing the notion of continuity of a mapping from a metric space into itself depending on a function α, which is the case that covers the partially ordered setting, we extend some very recent theorems involving control functions that only must be lower/upper semi-continuous from the right. Finally, we use metric spaces endowed with transitive binary relations rather than partial orders.This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. A-F Roldán-López-de-Hierro is grateful to the Department of Quantitative Methods for Economics and Business of the University of Granada. The same author has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE