A modular extension of Brøndsted's fixed point theorem in ℓ^p(⋅)

We establish a fixed point theorem for inward mappings in the variable exponent sequence space ℓ^p(⋅), extending the classical Brøndsted fixed point theorem to the setting of modular spaces. Our approach is based on a modular analogue of the Bishop–Phelps partial order and exploits modular uniform convexity. The main results provide a unified framework that recovers both Brøndsted's original theorem and its recent refinement by Zubelevich. Although ℓ^p(⋅) serves as the primary setting for our investigation, the techniques developed are applicable to a broader class of modular spaces.