Journal of Advances in Mathematics and Computer Science

In this manuscript we develop a self-contained theory of rational-type contractive operators acting on partial metric spaces. A new inequality is proposed in which the contraction bound is assembled from three quantities - the ambient partial distance and two ratio terms built from forward and mixed distances - and is then passed through a continuous control map followed by a comparison map. The resulting class is shown to be broad enough to absorb the classical Banach operator, the Dass-Gupta rational operator, and a number of ordered variants that have appeared recently. Under completeness of the underlying partial metric space and continuity of the operator, we prove that exactly one fixed point exists, that Picard iterates converge to it, and that its self-distance vanishes. The argument is broken into four transparent stages and is accompanied by a schematic flow diagram. A concrete numerical model on the interval [0, 2] is examined, and two- and three-dimensional graphical evidence is supplied to confirm the contractive inequality pointwise. We then establish that the fixed point problem is well posed in the sense of limit-solution stability, and we show that small perturbations of the operator produce small perturbations of the fixed point. Finally, the abstract theorem is applied to a nonlinear Fredholm integral equation of the second kind; sufficient conditions on the kernel are derived, and a numerical table illustrates the contraction constant for several parameter choices. The paper closes with a discussion of open directions.