Existence and Uniqueness in b-Metric Spaces for Nonlinear Delay Integral Equations

Abstract

The present work establishes a unified theoretical framework for investigating a broad category of nonlinear integral equations that incorporate Fredholm-type operators alongside distributed delay terms. Two complementary fixed-point methodologies are developed and contrasted. Initially, a refined classical approach is presented within exponentially weighted Banach spaces, where innovative delay-regime estimates are derived by distinguishing between the initial transient phase and the later stationary phase. This refinement produces an optimized contraction constant, thereby weakening the sufficient conditions for existence and uniqueness. Subsequently, a novel extension is introduced by embedding the problem into a complete b-metric space, which represents a generalization of standard metric spaces, and applying Agrawal's fixed-point theorem. This alternative pathway offers additional flexibility through a tunable power parameter and an exponential weight, enabling the accommodation of stronger nonlinearities. The critical interplay between the invariance condition, which ensures that the operator maps a suitable ball into itself, and the contractivity condition is examined in detail. Explicit a priori bounds are established, and both approaches are illustrated through carefully constructed examples that demonstrate successful applications as well as inherent limitations. The results provide researchers with versatile analytical tools for handling delay integral equations involving complex nonlinear interactions.