A Study of Fixed-Point and Best Proximity Point - Sumitra Jena*; Akanksha Dubey (Recent Trends in Mathematics, 2026)
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: Fractional Calculus; applied mathematics; fuzzy metric; proximity points; quasi-metric