Nonlinear Rational Displacement Contractions in G-Metric Spaces: Fixed Points, Best Proximity Points, Coupled Fixed Points, and Applications to Fractional Differential Equations

In this paper, a nonlinear rational displacement contraction is introduced in the setting of G-metric spaces. The proposed contractive condition combines the classical three-point G-distance with a nonlinear rational interaction involving the displacement of points under a self-mapping. Unlike several hybrid contractive formulations in which the convergence factor is imposed separately, the present framework derives the effective Picard contraction constant directly from the parameters of the proposed inequality. Under the condition λ+η<1, it is shown that the Picard iteration satisfies a geometric estimate with contraction factor q=λ/(1-η)<1. Consequently, existence and uniqueness of a fixed point are established in a G-complete space, together with convergence and an explicit error estimate for the iterative sequence.The framework is further extended to best proximity points for non-self mappings and to coupled fixed points through an induced mapping on a product space. Finally, an application to a nonlinear Caputo fractional differential equation is presented through an integral operator formulation that actively incorporates the rational displacement structure. The results provide a nonlinear rational alternative for generalized contractive mappings in G-metric spaces and establish a connection between abstract fixed point theory and fractional differential equations.

G-metric space; rational contraction; nonlinear contraction; displacement mapping; fixed point; best proximity point; coupled fixed point; fractional differential equation.

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