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Functionals strictly subjected to convergent series and search for singularities of mappings. (English) Zbl 1310.54046

Let \((X,\rho)\) be a metric space; a multivalued functional \(\varphi: X \to P(\mathbb{R}_+)\) is called strictly subjected to the series \(\sum_{j=1}^\infty c_j < \infty, \,\, 0 <c_{n+1} < c_n, \, n \geq 1\) provided that
(1)
if \(t = \varphi(x)\) is such that \(t > c_1\), then there exist \(t^\prime = \varphi(x^\prime)\) and \(k \geq 1\) such that \(\rho (x,x^\prime) \leq t,\) \( t^\prime \leq c_k\);
(2)
if \(t = \varphi(x)\) is such that \(t \leq c_k\) for some \(k \geq 1\), then there exist \(t^\prime = \varphi(x^\prime)\) and \(k \geq 1\) such that \(\rho (x,x^\prime) \leq t,\) \( t^\prime \leq \frac{t}{c_k}c_{k+1}.\)
Some examples of such functionals are presented and their comparison with the classes of functionals considered earlier by the author (see, e.g., [Math. Notes 86, No. 1, 107–120 (2009); translation from Mat. Zametki 86, No. 1, 110–125 (2009; Zbl 1196.54071)]; ibid. 86, No. 2, 276–281 (2009); translation from Mat. Zametki 86, No. 2, 304–309 (2009; Zbl 1197.54062)]; ibid. 93, No. 1, 172–186 (2013); translation from Mat. Zametki 93, No. 1, 127–143 (2013; Zbl 1273.54053)] is made. The cascade search principle for zeros of such functionals is suggested and new iteration schemes are developed. Among applications, the author considers the existence and approximation problems of preimages of a closed subspace under the action of a given multimap, fixed and coincidence points and other problems. \(\downarrow\)

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47J25 Iterative procedures involving nonlinear operators
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References:

[1] Fomenko T.N.: Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces. Math. Notes 86, 107–120 (2009) · Zbl 1196.54071 · doi:10.1134/S0001434609070104
[2] Fomenko T.N.: Cascade search for preimages and coincidences: Global and local versions. Math. Notes 93, 172–186 (2013) · Zbl 1273.54053 · doi:10.1134/S0001434613010173
[3] Fomenko T.N.: Cascade search of the coincidence set of collections of multivalued mappings. Math. Notes 86, 276–281 (2009) · Zbl 1197.54062 · doi:10.1134/S0001434609070293
[4] Fomenko T. N.: Cascade search principle and its applications to the coincidence problem of n one-valued or multi-valued mappings. Topology Appl. 157, 760–773 (2010) · Zbl 1191.54037 · doi:10.1016/j.topol.2009.08.006
[5] Fomenko T.N.: Cascade search: Stability of reachable limit points. Moscow Univ. Math. Bull. 65, 179–185 (2010) · Zbl 1304.54078 · doi:10.3103/S0027132210050013
[6] T. N. Fomenko, Remarks on the local cascade search for roots and preimages. In: Progress in Analysis (Proceedings of the 8th Congress of the ISAAC, Moscow, 2011), Volume 2, Peoples’ Friendship University of Russia, Moscow, 2012, 165– 172. · Zbl 1303.39012
[7] Fomenko T.N.: Stability of cascade search. Izv. Math. 74, 1051–1068 (2010) · Zbl 1210.54052 · doi:10.1070/IM2010v074n05ABEH002515
[8] A. Arutyunov, E. Avakov, B. Gelman, A. Dmitruk and V. Obukhovskii, Locally covering maps in metric spaces and coincidence points. J. Fixed Points Theory Appl. 5 (2009), 105–127 · Zbl 1182.54050
[9] Nguyen Thi Hong Van and B. A. Pasynkov, On the continuous sections of metric mappings. Moscow Teachers’ Training State University, Moscow, 2012. Deposited in All-Russia Institute of Scientific and Technical Information of Russian Academy of Sciences, 26.11.2012, No.435-B2012.
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