Fixed Point Results of Banach, Kannan and Wardowski Type in Extended Asymmetric b-Metric Spaces

Abstract

We introduce extended asymmetric b-metric spaces, in which neither symmetry nor a constant in the triangle inequality is required: the constant is replaced by a control function θ(x, z) ≥ 1 depending on the end points. This structure contains metric, asymmetric metric, b-metric, asymmetric b-metric and extended b-metric spaces as special cases. After defining the notions of forward and backward convergence, Cauchy sequences and completeness, we prove a Cauchy criterion for Picard sequences that is adapted to the control function, and use it to establish three fixed point theorems: a Banach-type contraction principle with geometric convergence estimates, a Kannan-type theorem that does not require continuity, and a Wardowski-type F-contraction theorem. We show how the classical results arise as special cases and give examples, including a genuinely extended asymmetric example, a discontinuous Kannan mapping and an F-contraction that is not a Banach contraction. As applications, we treat value iteration in discounted dynamic programming under an asymmetric gauge on the space of value functions, and an iterative feedback law in a scalar control model.