Classification of Lipschitz mappings (Q2868617)

Let \((M,\rho)\) be a metric space; we say that \(T:M\to M\) is a Lipschitz mapping (or it satisfies the Lipschitz condition) if there exists a constant \(k\) such that \(\rho(Tx,Ty)\leq k\rho(x,y)\) for all \(x,y\in M\). The smallest \(k\) for which the inequality holds is called the Lipschitz constant for \(T\) and is denoted by \(k(T)\). This class of mappings plays a great important role in many branches of mathematics.NEWLINENEWLINEThe two constants \(k_\infty(T):=\limsup_nk(T^n)\) and \(k_0(T):=\lim_n\sqrt{[n]{k(T^n)}}\), where \(T^n\) is the \(n\)-th iterate of \(T\), play a special role in fixed point theory. Some basic facts on Banach spaces and on these two constants are given in Chapters 1 and 2.NEWLINENEWLINELet \(C\) be a bounded closed convex subset of a Banach space \((X,\|\cdot\|)\). A mapping \(S:C\to C\) is called nonexpansive if \(\|Sx-Sy\|\leq\|x-y\|\) for all \(x,y\in C\), that is, \(k(S)\leq1\). \textit{K. Goebel} and \textit{M. A. Japón Pineda} [in: Fixed point theory and its applications. Proceedings of the 8th international conference on fixed point theory and its applications (ICFPTA), Chiang Mai, Thailand, July 16--22, 2007. Yokohama: Yokohama Publishers. 71--82 (2008; Zbl 1200.47072)] first introduced the following concept of \(\alpha\)-nonexpansiveness: for \(\alpha:=(\alpha_1,\alpha_2,\dots,\alpha_n)\), where \(n\geq1\), \(\alpha_i\geq0\), \(\alpha_1>0\), \(\alpha_n>0\), and \(\sum_{i=1}^n\alpha_i=1\), we say that a mapping \(T:C\to C\) is \(\alpha\)-nonexpansive if \(\sum_{i=1}^n\alpha_i\|T^ix-T^iy\|\leq\|x-y\|\) for all \(x,y\in C\). Obviously, for \(n=1\), this is the classical nonexpansiveness. The more general situation is naturally to consider this one: for a metric space \((M,\rho)\), a mapping \(T:M\to M\) is called \(\alpha\)-Lipschitzian if there exists a constant \(k\) such that \(\sum_{i=1}^n\alpha_i\rho(T^ix,T^iy)\leq k\rho(x,y)\) for all \(x,y\in M\). This was considered by \textit{K. Goebel} and \textit{B. Sims} [Contemp. Math. 513, 157--167 (2010; Zbl 1218.47075)]. Details on these mappings are presented in Chapter 3.NEWLINENEWLINE In Chapter 4, the author presents many results on bounds for the Lipschitz constants for mean Lipschitzian mappings. For a fixed constant \(p\geq1\), a mapping \(T:M\to M\) is called \((\alpha,p)\)-Lipschitzian if there exists a constant \(k\) such that \(\left(\sum_{i=1}^n\alpha_i\rho(T^ix,T^iy)^p\right)^{1/p}\leq k\rho(x,y)\) for all \(x,y\in M\). This class of mappings was also suggested by Goebel and Japón Pineda [loc. cit.]. A discussion of this class is given in Chapter 5. Chapter 6 deals with mean contractions, that is, \(\alpha\)-Lipschitzian mappings with \(0<k<1\). In Chapter 7, the author presents a brief development of the metric fixed point theory for nonexpansive mappings. More recent results for \(\alpha\)-nonexpansive mappings and \(\alpha\)-Lipschitzian mappings with \(k>1\) are given in Chapters 8 and 9, respectively.NEWLINENEWLINEThe book is self-contained and systematically arranged. Many interesting examples are also presented.