## On the stability of fixed points and coincidence points of mappings in the generalized Kantorovich’s theorem.(English)Zbl 1473.54046

One of the very important fixed point theorems for self-mappings on a Banach space is Kantorovich’s fixed point theorem (see [L. V. Kantorovich and G. P. Akilov, Functional analysis. Transl. from the Russian by Howard L. Silcock. 2nd ed. Oxford etc.: Pergamon Press (1982; Zbl 0484.46003), Chapter XVIII, Section 1.2, Theorem 1]). This result was extended to the case of metric spaces in [A. V. Arutyunov et al., Proc. Steklov Inst. Math. 304, 60–73 (2019; Zbl 1439.54025); translation from Tr. Mat. Inst. Steklova 304, 68–82 (2019)].
In this very interesting paper, some stability results for single valued and multi valued fixed point problems of Kantorovich type are given.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems

### Citations:

Zbl 0484.46003; Zbl 1439.54025
Full Text:

### References:

 [1] Kantorovich, L.; Akilov, G., Functional Analysis (1982), Pergamon Press: Pergamon Press Oxford · Zbl 0484.46003 [2] Arutyunov, A.; Zhukovskiy, E.; Zhukovskiy, S., The Kantorovich theorem on fixed points in metric spaces and coincidence points, Proc. Steklov Inst. Math., 304 (2019) · Zbl 1439.54025 [3] Zubelevich, O., Coincidence points of mappings in Banach spaces · Zbl 1419.34065 [4] Zubelevich, O., Coincidence points of mappings in Banach spaces, Fixed Point Theory, 20, 1 (2019) · Zbl 1449.70013 [5] Arutyunov, A., Covering mappings in metric spaces and fixed points, Dokl. Math., 76, 2, 665-668 (2007) · Zbl 1152.54351 [6] Arutyunov, A., Stability of coincidence points and properties of covering mappings, Math. Notes, 86, 1-2, 153-158 (2009) · Zbl 1186.54033 [7] Arutyunov, A.; Vinter, R., A simple “finite approximations” proof of the Pontryagin maximum principle under reduced differentiability hypotheses, Set-Valued Var. Anal., 12, 1-2, 5-24 (2004) · Zbl 1046.49014 [8] Arutyunov, A.; Zhukovskiy, E.; Zhukovskiy, S., On the cardinality of the set of coincidence points of mappings in metric, normed and partially ordered spaces, Sb. Math., 209, 8, 1107-1130 (2018) · Zbl 1408.54014 [9] Arutyunov, A.; Zhukovskiy, E.; Zhukovskiy, S., Coincidence points of set-valued mappings in partially ordered spaces, Dokl. Math., 88, 3, 727-729 (2013) · Zbl 1301.49041 [10] Arutyunov, A.; Greshnov, A., $$( q_1, q_2)$$-quasimetric spaces. covering mappings and coincidence points, Izv. Math., 82, 2, 245-272 (2018) · Zbl 1401.54023 [11] Zhukovskiy, S., On continuous selections of finite-valued set-valued mappings, Eurasian Math. J., 9, 1, 83-87 (2018) [12] Arutyunov, A.; Zhukovskiy, E.; Zhukovskiy, S., On the well-posedness of differential equations unsolved for the derivative, Differ. Equ., 47, 11, 1541-1555 (2011) · Zbl 1252.34009 [13] Arutyunov, A.; Avakov, E.; Zhukovskiy, E., Covering mappings and their applications to differential equations unsolved for the derivative, Differ. Equ., 45, 5, 627-649 (2009) · Zbl 1204.54015 [14] Arutyunov, A.; Zhukovskiy, S., Existence of local solutions in constrained dynamic systems, Appl. Anal., 90, 6, 889-898 (2011) · Zbl 1232.93037 [15] Arutyunov, A., Second-order conditions in extremal problems. The abnormal points, Trans. Am. Math. Soc., 350, 11, 4341-4365 (1998) · Zbl 0922.49017 [16] Arutyunov, A.; Zhukovskiy, S., Variational principles in nonlinear analysis and their generalization, Math. Notes, 103, 6, 1014-1019 (2018) · Zbl 1432.49019 [17] Burlakov, E.; Zhukovskii, E., On well-posedness of generalized neural field equations with impulsive control, Russ. Math., 60, 5, 66-69 (2016) · Zbl 1339.92015 [18] Zhukovskaya, T.; Zhukovskii, E., On iterative methods for solving equations with covering mappings, Numer. Anal. Appl., 9, 4, 277-287 (2016) · Zbl 1375.47043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.