我們考慮在一個實Hilbert空間$H$的閉凸子集合$C$上的無窮多個非線性自映射${T_n}$中去找一個共同固定點的問題。即，我們想要找一個具備這個性質的點 $x$ (假設這樣的共同固定點存在):
[
xin igcap_{n=1}^infty ext{Fix}(T_n).
]
我們將使用Krasnoselskii-Mann (KM) 型慣性迭代方法
$$ x_{n+1} = ((1-alpha_n)I+alpha_nT_n)y_n,quad
y_n = x_n + eta_n(x_n-x_{n-1}).eqno(*)$$
我們討論由方法(*)所生成的序列${x_n}$的收斂性。特別地，我們證明當序列${alpha_n}$和${eta_n}$滿足一定條件時，序列${x_n}$會弱收歛到${T_n}$的共同固定點。

We consider the problem of finding a common fixed point of an infinite family ${T_n}$
of nonlinear self-mappings of a closed convex subset $C$ of a real Hilbert space $H$. Namely,
we want to find a point $x$ with the property (assuming such common fixed points exist):
[
xin igcap_{n=1}^infty ext{Fix}(T_n).
]
We will use the Krasnoselskii-Mann (KM) Type inertial iterative algorithms of the form
$$ x_{n+1} = ((1-alpha_n)I+alpha_nT_n)y_n,quad
y_n = x_n + eta_n(x_n-x_{n-1}).eqno(*)$$
We discuss the convergence properties of the sequence ${x_n}$ generated by this algorithm (*).
In particular, we prove that ${x_n}$ converges weakly to a common fixed point of the family
${T_n}$ under certain conditions imposed on the sequences ${alpha_n}$ and ${eta_n}$.

1 Introduction 1
2 Preliminaries 4
3 Inertial KM Algorithms and Their Convergence 11
References 25

[1] H.H. Bauschke, The approximation of xed points of compositions of nonexpansive
mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150159.
[2] M.A. Krasnoselskii, Two remarks on the method of successive approximations,
Usp. Mat. Nauk 10 (1955), 123-127. (In Russian.)
[3] W.R. Mann, 1953 Mean value methods in iteration, Proc. Amer. Math. Soc.
4 (1953), 506-510.
[4] P.E. Mainge, Convergence theorems for inertial KM-type algorithms, Journal
of Computational and Applied Mathematics 219 (2008), 223-236.
[5] J.G. O0Hara, P. Pillay, H.K. Xu, Iterative approaches to !nding nearest common
xed points of nonexpansive mappings in Hilbert spaces, Nonlinear Analysis
54 (2003), 1417-1426.
[6] J.G. O0Hara, P. Pillay, H.K. Xu, Iterative approaches to convex feasibility
problems in Banach spaces, Nonlinear Analysis 64 (2006) 2022-2042.
[7] B.T. Polyak, Introduction to Optimization," Optimization Software, New
York, 1987.
[8] H.K. Xu and R.G. Ori, An implicit iteration process for nonexpansive mappings,
Numer. Funct. Anal. Optimiz. 22 (2001), 767-773.