Abstract

This paper introduces and investigates two new classes of contraction mappings—weak enriched F-contractions and weak enriched F′-contractions, in the context of convex metric space (CMS) and convex G-metric space (CGMS). From a given self-mapping, the study constructs a new mapping via different convex combinations, termed the k-fold averaged mapping. The paper establishes that if the underlying space is complete and certain conditions are satisfied, then the k-fold averaged mapping possesses a unique fixed point, and the corresponding iterative scheme converges to this fixed point. It is further shown that the fixed point set of the original mapping is always contained in the fixed point set of k-fold averaged mapping, and under further conditions, both sets of fixed points are equal. These results broaden the scope of fixed point theory in convex metric settings by introducing and exploring these new contraction mappings. Several examples are provided to illustrate the applicability and effectiveness of the theoretical findings.

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