Fixed Point Theory under Generalized Contractive Conditions and Applications to Nonlinear Analysis

Abstract

Fixed point theory provides a fundamental analytical framework for establishing the existence, uniqueness, and approximation of solutions to nonlinear problems. However, the classical Banach contraction principle can be restrictive when a nonlinear mapping's behavior cannot be characterized solely by a global Lipschitz condition. This study develops a generalized max-type contractive framework on a complete metric space, where the contraction condition incorporates direct, displacement, and cross-distance terms. The main theorem shows that a self-mapping satisfying the proposed condition has a unique fixed point and that the corresponding Picard sequence converges to it from any initial value. The proof relies on geometric control of successive iterates, the Cauchy property of the Picard sequence, the completeness of the underlying space, and a uniqueness argument derived from the generalized contractive inequality. The paper illustrates the result's applicability through four mathematical cases: a nonlinear algebraic equation, a nonlinear discrete dynamical system, a Volterra-type integral equation on a space of continuous functions, and a b-metric example involving a nonstandard distance. Numerical Picard iterations and operator estimates further demonstrate the constructive character of the fixed-point approach. The results show that max-type contractive formulations provide a flexible framework for establishing fixed-point existence, uniqueness, and iterative convergence in nonlinear analysis while retaining the constructive advantages of classical contraction methods.