Fixed Points of Enriched Mappings with General Real Constants

Building upon classical fixed point theory, the concept of enriched contractions introduces a new class of mappings. For a normed linear space (X,∥·∥), a mapping T:X→X is called an enriched contraction if there exist b∈[0,∞) and θ∈[0,b+1) such that ∥b(x−y)+Tx−Ty∥≤θ∥x−y∥,∀x,y∈X. This class of mappings includes both the well-known Picard–Banach contraction and certain nonexpansive mappings. In this paper, we extend the definition by allowing b∈R\{−1} instead of b∈[0,∞). This extension enables the condition to cover both contraction and certain nonexpansive mappings. We establish results on the existence and uniqueness of fixed points and present the Krasnosel’skii iteration for approximating such points. An example is provided to demonstrate mapping that meets the extended condition but not the original.