A Study of Fixed-Point and Best Proximity Point

Year : 2026 | Volume : 03 | Issue : 01 | Page : 28 39
    By

    Sumitra Jena*,

  • Akanksha Dubey,

  1. Research Center, Department of Mathematics, Faculty of Science, Shri Rawatpura Sarkar University, Raipur, Chhattisgarh, India
  2. Assistant Professor and HOD I/C, Department of Mathematics, Shri Rawatpura Sarkar University, Raipur, Chhattisgarh, India

Abstract

Fixed-point theory plays a fundamental role in nonlinear analysis and has significant applications in optimization, differential equations, and applied mathematics. This study investigates the existence and properties of fixed points and best proximity points for various classes of mappings defined on metric and normed spaces. While fixed-point results guarantee the existence of a point that remains invariant under a given mapping, such points may not exist when the mapping is defined between disjoint subsets. In such cases, the concept of best proximity points provides an optimal approximate solution by minimizing the distance between the point and its image. The paper presents generalized conditions under which fixed points and best proximity points exist, focusing on contractive-type mappings and cyclic mappings. Several theoretical results are established using completeness, compactness, and continuity assumptions in metric spaces. The relationship between fixed-point theorems and best proximity point theorems is also examined, showing that fixed-point results can be obtained as special cases of best proximity point principles when the involved sets intersect. Illustrative examples are provided to demonstrate the applicability of the obtained results. The findings contribute to the development of nonlinear analysis and provide a unified framework for studying fixed points and best proximity points in different mathematical settings.

Keywords: proximity points, applied mathematics, fuzzy metric, quasi-metric, fractional calculus

[This article belongs to Recent Trends in Mathematics ]

How to cite this article:
Sumitra Jena*, Akanksha Dubey. A Study of Fixed-Point and Best Proximity Point. Recent Trends in Mathematics. 2026; 03(01):28-39.
How to cite this URL:
Sumitra Jena*, Akanksha Dubey. A Study of Fixed-Point and Best Proximity Point. Recent Trends in Mathematics. 2026; 03(01):28-39. Available from: https://journals.stmjournals.com/rtm/article=2026/view=239178


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Regular Issue Subscription Review Article
Volume 03
Issue 01
Received 09/02/2026
Accepted 26/02/2026
Published 10/03/2026
Publication Time 29 Days
52

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