Fixed Point Theory in Partial Ordered Metric Spaces and Their Generalizations: A Systematic Review
Chandni Byapari
*
Dr. C V Raman University, Kargi Road, Kota, Bilaspur, Chhattisgarh, India.
Hiral Raja
Dr. C V Raman University, Kargi Road, Kota, Bilaspur, Chhattisgarh, India.
*Author to whom correspondence should be addressed.
Abstract
Fixed point theory has emerged as a fundamental area of modern functional analysis due to its significant role in establishing the existence, uniqueness, and approximation of solutions to a wide range of linear and nonlinear mathematical problems. Over the past two decades, substantial progress has been made in extending the classical Banach contraction principle through the introduction of generalized contractive conditions, novel metric structures, and auxiliary control functions. This review systematically examines recent developments in fixed point theory with particular emphasis on partial order metric spaces and their important generalizations, including partial metric, partial cone metric, partial b-metric, partial cone b-metric, and partial Aᵦ-metric spaces. Relevant literature published between 2000 and 2025 was identified through a structured literature review approach using major academic databases and predefined inclusion and exclusion criteria. The review is organized into four thematic areas: the role of fixed point theory in functional analysis; advances in generalized metric spaces; fixed point results in partial order metric spaces and their extensions; and recent developments in hybrid, integral-type, and generalized contraction mappings. The synthesis highlights significant theoretical advancements and their applications to differential equations, integral equations, optimization, and dynamic programming. Furthermore, several research gaps are identified, including the lack of a unified framework for partial-type spaces, limited investigations in partial Aᵦ-metric spaces, insufficient development of constructive iterative methods with convergence guarantees, and relatively few applications to fractional and Volterra-type integral equations. The review concludes by outlining promising directions for future research aimed at unifying existing theories and expanding the applicability of fixedpoint techniques to emerging mathematical and applied problems.
Keywords: Fixed point theory, functional analysis, partial order metric spaces, partial metric spaces, contraction mappings, banach contraction principle, partial Aᵦ-metric spaces