Optimal pair of fixed points: existence, uniqueness and approximation

In this paper, we introduce a novel class of mappings, referred to as noncyclic generalized θ-contractions. By employing the geometric concept of $WUC$ property in metric spaces, we establish new existence and convergence theorems for the fixed points associated with these mappings. The results presented herein generalize and improve several existing fixed point theorems related to generalized φ-contractions. Furthermore, we address the issue of error estimation and derive both a priori and a posteriori error bounds for the fixed points obtained via the Picard iterative process applied to a noncyclic generalized θ-contraction mapping defined on a uniformly convex Banach space. A distinctive feature of our analysis lies in avoiding the use of geometric progression techniques. Consequently, the resulting error estimates hold unconditionally in uniformly convex Banach spaces, thereby removing the need for any restrictive power-type condition on the modulus of convexity. We then present a comprehensive example to illustrate and validate the applicability and robustness of the main theoretical results. Finally, we apply the existence and convergence results for optimal pairs of fixed points to a system of differential equations.