In this article, we introduce a new class of non-expansive mappings which is partially more general than Condition \((C_\lambda )\) . We elucidate it by providing a discretized example of general nature. Afterwards, an analog of Condition (E) is provided in the form of a lemma before establishing some existence results. Next, a few convergence results follow in the context of an iteration scheme. Finally, an application to a certain kind of delay differential equations with a finite number of constant delays is discussed by providing the relevant theory and an illustrative example. This application lays a foundation in extending the existing literature on the applications of fixed point theory to delay differential equations by generalizing the result to hold for a finite number of delays instead of a single delay.
