Literature Review: Fixed Point Generalizations of the Banach Contraction Principle in Classical Metric Spaces

Abstract

The Banach contraction principle (1922) holds a central position in fixed point theory on metric spaces. Over time, various generalizations have emerged to weaken the contraction condition while retaining the guarantee of a fixed point. However, simple reviews that map the logical relationships among these generalizations are still limited, especially in the Indonesian language. This article presents a literature review of six main classes of generalizations of the Banach contraction principle in classical metric spaces, namely Boyd-Wong (1969), Meir-Keeler (1969), Ciric quasi-contraction (1974), Reich (1971), weak -contraction (Berinde, 2004), and orbital contraction (Rus-Hicks-Rhoades). The selection is restricted to single-valued mappings on complete metric spaces. Each class is described by its definition, fixed point theorem, a note on when the condition reduces to the original Banach contraction, and a brief example (or a reference to the original literature for more complex cases). Based on a comparative analysis, an implication table is constructed, showing that the Banach class is the strongest (it implies all other classes), Ciric implies Reich but not conversely, and the Boyd-Wong, Meir-Keeler, weak -contraction, and orbital classes are mutually independent. This review concludes that the visual implication map, the simplified language, and the explicit reduction notes to the Banach case are three main contributions that distinguish it from previous surveys. Five directions for further research are also proposed, including extensions to non-complete metric spaces or to b-metric spaces