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.
Citation: Jie Shi. A fixed-point theorem for generalized strictly nonexpansive mappings on bounded sets in complete metric spaces[J]. AIMS Mathematics, 2026, 11(3): 7143-7154. doi: 10.3934/math.2026294
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.
| [1] |
M. Abbas, T. Nazir, S. Radenović, Common fixed point of power contraction mappings satisfying (E.A) property in generalized metric spaces, Appl. Math. Comput., 219 (2013), 7663–7670. https://doi.org/10.1016/j.amc.2012.12.090 doi: 10.1016/j.amc.2012.12.090
|
| [2] | A. Mortaza, Fixed point theorems for meir-keeler type contractions in metric spaces, preprint paper, 2016. https://doi.org/10.48550/arXiv.1604.01296 |
| [3] |
E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc., 13 (1962), 459–465. https://doi.org/10.1090/S0002-9939-1962-0148046-1 doi: 10.1090/S0002-9939-1962-0148046-1
|
| [4] |
D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–464. https://doi.org/10.2307/2035677 doi: 10.2307/2035677
|
| [5] |
W. A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl., 277 (2003), 645–650. https://doi.org/10.1016/S0022-247X(02)00612-1 doi: 10.1016/S0022-247X(02)00612-1
|
| [6] |
T. Lindstrøm, D. A. Ross, A nonstandard approach to asymptotic fixed point theorems, J. Fixed Point Theory Appl., 25 (2023), 35. https://doi.org/10.1007/s11784-022-01028-6 doi: 10.1007/s11784-022-01028-6
|
| [7] | L. E. J. Brouwer, Über abbildung von mannigfaltigkeiten, Math. Ann., 71 (1912), 97–115. |
| [8] | J. Schauder, Der fixpunktsatz in funktionalräumen, Stud. Math., 2 (1930), 171–180. |
| [9] | F. E. Browder, D. G. de Figueiredo, Nonexpansive nonlinear operators in a Banach space, In: Proceedings of the National Academy of Sciences of the United States of America, 54 (1965), 1041–1044. |
| [10] | A. Robinson, Non-Standard Analysis, Princeton: Princeton University Press, 1996. |
| [11] | P. A. Loeb, M. P. H. Wolff, Nonstandard Analysis for the Working Mathematician, 2 Eds., Berlin: Springer, 2015. |
Jie Shi. A fixed-point theorem for generalized strictly nonexpansive mappings on bounded sets in complete metric spaces[J]. AIMS Mathematics, 2026, 11(3): 7143-7154. doi: 10.3934/math.2026294
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