p-th Iterate Contractive Mappings: Extending Rakotch, Edelstein and Bianchini Theorems with Applications to Caputo Fractional Differential Equations

Classical fixed-point theorems for Rakotch, Edelstein, and Bianchini contractions require the contractive condition to hold for the mapping itself at every iteration, which severely limits their applicability to many real-world problems. In this paper, we break this limitation by shifting the contractive requirement to the p-th iterate of the mapping. We introduce three novel classes of p-Rakotch, p-Edelstein, and p-Bianchini contractions and prove that each guarantees the existence of a unique fixed point and global convergence of the Picard sequence from any initial point, under appropriate metric space assumptions (completeness for Rakotch and Bianchini; compactness with continuity for Edelstein). A key feature of our approach is that the original mapping T need not satisfy any contractive condition; only its p-th iterate Tp needs to. This allows us to handle mappings where classical theorems simply do not apply. To validate our theoretical findings, we provide explicit numerical examples for p=3,4,5. More importantly, we demonstrate the practical power of our results through six diverse applications: ordinary differential equations with large coefficients; planar discrete dynamical systems; nonlinear Hammerstein integral equations; Caputo fractional differential equations with large linear terms; fractional equations exploiting the smoothing property; and implicit fractional differential equations. In each application, the classical contractive condition fails, yet our p-iterate approach succeeds. When p=1, all three theorems reduce to their classical counterparts, confirming that our framework is a natural and faithful generalization.