Hardy-Rogers and Jungck Type Fixed Point Theorems in Perturbed Metric Spaces, Stability and Data Dependence

In this paper we establish Hardy-Rogers and Jungck type fixed point theorems in perturbed metric spaces, where the observed distance $D$ is separated from the exact metric $d$ by a nonnegative perturbation $P$. Rather than the uniform absorption of $P$ by $d$ required under domination, we examine a weaker demand, imposed only on the pairs of points that appear with a contractive coefficient. We show that without some such condition, and without a continuity hypothesis on $T$, a perturbed Banach contraction on a complete perturbed metric space may fail to have a fixed point. We further prove Ulam-Hyers stability, well-posedness, and data dependence results in which residuals are measured in the observed distance $D$, with all constants explicit, and we derive a priori error estimates for the Picard and Jungck iterations computable from observed data. The Jungck type theorem is established for weakly compatible pairs.