A fixed-point theorem for generalized strictly nonexpansive mappings on bounded sets in complete metric spaces

This paper offers substantial advances in the theory of fixed points for generalized strictly nonexpansive mappings. We develop a novel proof technique based on nonstandard analysis to establish a new fixed-point theorem. The core result demonstrates that, in a complete metric space, every continuous generalized strictly nonexpansive mapping with a bounded orbit possesses a unique fixed point to which all iterative sequences converge. The significance of this theorem lies in its substantial relaxation of the classical framework: It entirely dispenses with compactness and convexity requirements, which are typically indispensable in the study of nonexpansive mappings (such as in the Browder–Göhde theorem), replacing them solely with a boundedness condition.