Fixed point theory and iteration procedures (Q1105228)

The following two main problems are discussed: Let \(T: X\to X\), where (X,d) is a complete metric space. (1) When does T have a fixed point? (2) When does a sequence \(\{x_ n\}\), given by some iteration process, converge to a fixed point of T? Suppose \(A=[a_{ij}]\) is an infinite matrix which satisfies: (1) \(a_{ij}\geq 0\) for every i and j. (2) \(a_{ij}=0\) for \(j>i\), (3) \(\sum^{i}_{j=1}a_{ij}=1\) for every i. (4) \(\lim_{i\to \infty}a_{ij}=0\) for every j. Suppose C is a convex subset of a linear space X and \(T: C\to C\). Choose \(x_ 1\) in C. Then the Mann iteration process \(M(x_ 1,A,T)\) is defined \(v_ n=\sum^{n}_{k=1}a_ nkx_ k\) and \(x_{n+1}=Tv_ n\) for \(n=1,2,... \). Mann proved that if \((X,\|.\|)\) is a Banach space, C is closed as well as convex, and T is continuous, then the convergence of either \(\{x_ n\}^{\infty}_{n=1}\) or \(\{v_ n\}^{\infty}_{n=1}\) to y implies the convergence of the other to y, and their common limit y is a fixed point of T. If \(\sum^{\infty}_{n=1}\| x_ n-x_{n+1}\| <\infty,\{x_ n\}\) is a Cauchy sequence and Mann's result implies that \(x_ n\to y\), \(v_ n\to y\), and \(Ty=y:\) A weaker condition when A is the Cesaro matrix is taken to prove the following main result. Let \((X,\|.\|)\) be a Banach space. Let C be a closed, convex subset of X. Suppose \(T: C\to C\), \(x_ 1\in C\), and \(\{x_ n\}^{\infty}_{n=1}\) and \(\{v_ n\}^{\infty}_{n=1}\) are the sequences in the Mann iteration process \(M(x_ 1,A,T)\) where A is the Cesaro matrix. That is, \(A=[a_{ij}]\) is the infinite matrix such that \(a_{ij}=0\) if \(j>i\) and \(a_{ij}=1/i\) if \(j\leq i\). If \(\sum^{\infty}_{n=1}1/n\| v_ n-Tv_ n\|\) converges, then there exists x * in C such that \(v_ n\to x\) * as \(n\to \infty\). Furthermore, if T is continuous, then \(x_ n\to x\) * as \(n\to \infty\) and Tx \(*=x\) *.