Functionals strictly subjected to convergent series and search for singularities of mappings (Q480901)

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 {\parindent=0.5cm \begin{itemize}\item[(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 \(\rho (x,x^\prime) \leq t,\) \( t^\prime \leq c_k\); \item[(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 \(\rho (x,x^\prime) \leq t,\) \( t^\prime \leq \frac{t}{c_k}c_{k+1}.\) \end{itemize}} 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. \(\downarrow\)