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Some stability results in complete metric space. (English) Zbl 1199.54232

Let \((E,d)\) be a metric and \(T:E\rightarrow E\) a self map with the set of fixed points nonempty. Consider a fixed point iteration procedure \(\{x_{n}\}_{n=1}^{\infty }\) defined by a general relation of the form \[ x_{n+1}=f(T,x_{n}),\;\;n=0,1,2,\dots, \tag{1} \] where \(x_{0}\in E\) is the initial guess and \(f\) is given.
Let us assume that the sequence \(\{x_{n}\}_{n=1}^{\infty }\) produced by (1) converges to a fixed point \(p\) of \(T.\) Let also \(\{y_{n}\}_{n=0}^{\infty }\) be an arbitrary sequence in \(E\) and set \[ \varepsilon _{n}=d(y_{n+1},\,f(T,y_{n})),\;\;\;\text{for}\;\;\;n=0,1,2,\dots \] The fixed point iteration procedure (1) is said to be \(T\)-stable or stable with respect to \(T\) if and only if \[ \lim_{n\rightarrow \infty }\varepsilon _{n}=0\Leftrightarrow \lim_{n\rightarrow \infty }y_{n}=p. \] In the paper the author obtains stability results for mappings satisfying various general contractive conditions, in the presence of one metric and respectively two metrics on the set \(E\). The stability results in the case of a set endowed with two metrics appear to be given in this paper for the first time.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

[1] Berinde, V.: On the stability of some fixed point procedures. Bul. Stiint. Univ. Baia Mare, Ser. B, Matematica-Informatica 18, 1 (2002), 7-14. · Zbl 1031.47030
[2] Berinde, V.: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare, Romania, 2002. · Zbl 1036.47037
[3] Berinde, V.: A priori and a posteriori error estimates for a class of \(\varphi \)-contractions. Bulletins for Applied Mathematics 90-B (1999), 183-192.
[4] Harder, A. M., Hicks, T. L.: Stability results for fixed point iteration procedures. Math. Japonica 33, 5 (1988), 693-706. · Zbl 0655.47045
[5] Imoru, C. O., Olatinwo, M. O.: On the stability of Picard and Mann iteration processes. Carpathian J. Math. 19, 2 (2003), 155-160. · Zbl 1086.47512
[6] Imoru, C. O., Olatinwo, M. O., Owojori, O. O.: On the stability of Picard and Mann iteration procedures. J. Appl. Func. Diff. Eqns. 1, 1 (2006), 71-80. · Zbl 1388.47002
[7] Jachymski, J. R.: An extension of A. Ostrowski’s theorem on the round-off stability of iterations. Aequationes Math. 53 (1997), 242-253. · Zbl 0885.47023
[8] Osilike, M. O.: Some stability results for fixed point iteration procedures. J. Nigerian Math. Soc. Vol. 14/15 (1995), 17-29.
[9] Osilike, M. O., Udomene, A.: Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings. Indian J. Pure Appl. Math. 30, 12 (1999), 1229-1234. · Zbl 0955.47038
[10] Rhoades, B. E.: Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math. 21, 1 (1990), 1-9. · Zbl 0692.54027
[11] Rhoades, B. E.: Some fixed point iteration procedures. Internat. J. Math. and Math. Sci. 14, 1 (1991), 1-16. · Zbl 0716.47030
[12] Rhoades, B. E.: Fixed point theorems and stability results for fixed point iteration procedures II. Indian J. Pure Appl. Math. 24, 11 (1993), 691-703. · Zbl 0794.54048
[13] Singh, S. L., Bhatnagar, C., Mishra, S. N.: Stability of Jungck-type iterative procedures. Internat. J. Math. & Math. Sc. 19 (2005), 3035-3043. · Zbl 1117.26005
[14] Zeidler, E.: Nonlinear Functional Analysis and its Applications, Fixed-Point Theorems I. Springer-Verlag, New York, 1986. · Zbl 0583.47050
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