We introduce and study a new class of nonlinear operators on metric spaces, called rational
quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence
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We introduce and study a new class of nonlinear operators on metric spaces, called rational
quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence and uniqueness of fixed points, as well as the convergence of the Picard iteration for every initial guess. We show that rational
quasicontractions strictly extend several classical contractive classes, including Hardy-Rogers, Kannan, Chatterjea, and rational contractions, and we provide explicit examples exhibiting the properness of these inclusions. As an application, we consider a nonlocal boundary value problem for a Caputo fractional differential equation of order
with distributed delay and mixed nonlocal boundary conditions. By rewriting the problem as a Hammerstein-Volterra integral equation on a cone, and imposing natural growth and rational Lipschitz conditions on the delayed nonlinearity, we show that the associated Hammerstein operator is a rational
quasicontraction. This yields the existence, uniqueness, and global attractivity of a positive solution. Two model fractional nonlinearities with delayed feedback are discussed in detail, along with a numerical scheme that illustrates the predicted geometric convergence of the discrete Picard iteration in the Caputo fractional setting.
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