Hyers--Ulam stability, well-posedness, and data dependence for Fisher-type contractive mappings
Nội dung chính của bài viết
Tóm tắt
In this paper, we investigate Hyers--Ulam stability, well-posedness, and data dependence for mappings satisfying a Fisher-type inequality.
Our stability results are proved in an arbitrary metric space under the sole assumption that a fixed point exists.
A basic ingredient is a structural consequence of the inequality: if $z$ is a fixed point of $T$, then $T^{-1}(\{z\})=\{z\}$.
This unique-preimage property is used to obtain stability estimates with an explicit constant.
As applications, we combine these general results with fixed point existence theorems from \cite{Hoc2022} to derive stability conclusions for boundedly compact, $T$-orbitally compact, and complete metric spaces.
A discontinuous example is included to illustrate the scope of the theory.
Từ khóa
Fixed point, Fisher-type contraction, well-posedness, Hyers–Ulam stability, data dependence.
Chi tiết bài viết
Tài liệu tham khảo
[2] V. Berinde (2007), Iterative Approximation of Fixed Points, Springer, Berlin.
[3] S. K. Chatterjea (1972), Fixed-point theorems, C. R. Acad. Bulgare Sci., 25, 727-730.
[4] Lj. B. Ciric (1974), A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45(2), 267-273.
[5] R. E. Edwards (1965), Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York.
[6] B. Fisher (1976), A fixed point theorem for compact metric spaces, Publ. Inst. Math. (Beograd) (N.S.), 25(39), 193-194.
[7] H. Garai, L. K. Dey and T. Senapati (2018), On Kannan-type contractive mappings, Numer. Funct. Anal. Optim., 39(13), 1466-1476.
[8] N. H. Hoc, H. K. Anh and L. T. Thu (2022), Some fixed point theorems for Fisher-type contractive mappings, Hong Duc Univ. J. Sci., 12, 23-30.
[9] D. H. Hyers (1941), On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27(4), 222-224.
[10] R. Kannan (1968), Some results on fixed points, Bull. Calcutta Math. Soc., 60, 71-76.
[11] A. Petrusel and I. A. Rus (2007), Well-posedness of the fixed point problem for multivalued operators, in: Applied Analysis and Differential Equations, World Scientific, 295-306.
[12] A. Petrusel, I. A. Rus and J.-C. Yao (2007), Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11(3), 903-914.
[13] S. Reich and A. J. Zaslavski (2001), Well-posedness of fixed point problems, Far East J. Math. Sci., Special Volume(Part III), 393-401.
[14] I. A. Rus (2001), Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca.
[15] I. A. Rus (2009), Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10(2), 305-320.
[16] S. M. Ulam (1960), Problems in Modern Mathematics, John Wiley & Sons, New York.