Fixed point iteration for pseudocontractive maps. (English) Zbl 0913.47052

Chidume, C. E.; Moore, Chika

Proc. Am. Math. Soc. 127, No. 4, 1163-1170 (1999).

Summary: Let \(K\) be a compact convex subset of a real Hilbert space \(H\), \(T:K\rightarrow K\) a continuous pseudocontractive map. Let \(\{a_{n}\}\), \(\{b_{n}\}\), \(\{c_{n}\}\), \(\{a_{n}^{'}\}\), \(\{b_{n}^{'}\}\) and \(\{c_{n}^{'}\}\) be real sequences in [0,1] satisfying appropriate conditions. For arbitrary \(x_{1}\in K\), define the sequence \(\{x_{n}\}_{n=1}^{\infty}\) iteratively by \(x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}\); \(y_{n} = a_{n}^{'}x_{n} + b_{n}^{'}Tx_{n} + c_{n}^{'}v_{n}\), \(n\geq 1,\) where \(\{u_{n}\}\), \(\{v_{n}\}\) are arbitrary sequences in \(K\). Then, \(\{x_{n}\}_{n=1}^{\infty}\) converges strongly to a fixed point of \(T\). A related result deals with the convergence of \(\{x_{n}\}_{n=1}^{\infty}\) to a fixed point of \(T\) when \(T\) is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

MSC:

47H10	Fixed-point theorems
47H05	Monotone operators and generalizations
47J05	Equations involving nonlinear operators (general)
47H06	Nonlinear accretive operators, dissipative operators, etc.

Keywords:

fixed point iteration; compact convex subset of a real Hilbert space; continuous pseudocontractive map; continuous hemicontractive nonlinear maps