Article ID: MTJPAM-D-20-00008

Title: On Sequences of Certain Contractive Mappings and Their Fixed Points


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00008; Volume 3 / Issue 1 / Year 2021, Pages 70-77

Document Type: Research Paper

Author(s): Mohamed Akkouchi a

aFaculty of Sciences-Semlalia, Cadi Ayyad University, Marrakech, Morocco

Received: 18 April 2020, Accepted: 8 December 2020, Available online: 7 January 2021.

Corresponding Author: Mohamed Akkouchi (Email address: akkm555@yahoo.fr)

Full Text: PDF

Abstract

In 1988, M. Imdad, M.S. Khan and S. Sessa have introduced a general contractive condition of Reich-Hardy-Rogers type in a (complete) metric space (X, d). Let (Tn)n be a sequence of selfmaps of X satisfying this general condition. They proved that each selfmap Tn has a unique fixed point (say) zn in X. Suppose that the sequence (Tn)n converges pointwise on X to a selfmap T. M. Imdad, M.S. Khan and S. Sessa sudied the convergence problem for the sequence (zn)n in X. They established the convergence of this sequence under a regularity assumption, and they raised the question whether this regularity assumption is necessary or not ?
  The aim of this paper is to answer positively to that question. Precisely, we prove that the main results of M. Imdad, M.S. Khan and S. Sessa are still valid without requiring the regularity assumption upon the sequence of fixed points (zn)n of the sequence (Tn)n.

Keywords: Complete metric space, fixed point, sequences of mappings, selfmaps of Reich-Hardy-Rogers type

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