Reformulation of Fixed Point Existence: From Banach to Kannan and Chatterjea Contractions

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via Green’s function and establish a sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if the derivative of j is bounded by K and K·(η−ζ)2/8<1/3, then T is a Kannan contraction, ensuring a unique solution. This condition applies even when the Banach contraction principle fails. We also explore the plausibility of applying the Chatterjea contraction, though rigorous verification remains open. Examples illustrate the applicability of the results. This work highlights the utility of generalized contractions in differential equations.