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An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces. (English) Zbl 0801.47040
Summary: Suppose $$X$$ is an $$s$$-uniformly smooth Banach space $$(s>1)$$. Let $$T:X \to X$$ be a Lipschitzian and strongly accretive map with constant $$k \in (0,1)$$ and Lipschitz constant $$L$$. Define $$S$$: $$X \to X$$ by $$Sx = f - Tx + x$$. For arbitrary $$x_ 0 \in X$$, the sequence $$\{x_ n\}^ \infty_{n = 1}$$ is defined by $x_{n+1} = (1 - \alpha_ n) x_ n + \alpha_ n Sy_ n, \quad y_ n = (1 - \beta_ n) x_ n + \beta_ n Sx_ n,\;n \geq 0,$ where $$\{\alpha_ n\}^ \infty_{n = 0}$$, $$\{\beta_ n \}^ \infty_{n = 0}$$ are two real sequences satisfying:
(i) $$0 \leq \alpha^{p-1}_ n \leq 2^{-1} s(k + k \beta_ n - L^ 2 \beta_ n)$$ $$(w + h)^{-1}$$ for each $$n$$,
(ii) $$0 \leq \beta_ n^{p-1} \leq \min \{k/L^ 2,\;sk/(w + h)\}$$ for each $$n$$,
(iii) $$\sum_ n \alpha_ n = \infty$$,
where $$w = b(1 + L)^ s$$ and $$b$$ is the constant appearing in a characteristic inequality of $$X$$, $$h = \max \{1, s(s-1)/2\}$$, $$p = \min \{2,s\}$$. Then $$\{x_ n\}^ \infty_{n = 1}$$ converges strongly to the unique solution of $$Tx = f$$. Moreover, if $$p = 2$$, $$\alpha_ n = 2^{- 1} s(k + k \beta - L^ 2 \beta)$$ $$(w + h)^{-1}$$, and $$\beta_ n = \beta$$ for each $$n$$ and some $$0 \leq \beta \leq \min \{k/L^ 2,\;sk/(w + h)\}$$, then $\| x_{n + 1} - q \| \leq \rho^{n/s} \| x_ 1 - q \|,$ where $$q$$ denotes the solution of $$Tx=f$$ and $\rho = \bigl( 1 - 4^{-1} s^ 2 (k + k \beta - L^ 2 \beta)^ 2 (w + h)^{-1} \bigr) \in (0,1).$ A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in $$X$$. Suppose $$X$$ is an $$m$$- uniformly convex Banach space $$(m>1)$$ and $$c$$ is the constant appearing in a characteristic inequality of $$X$$, two similar results are showed in the cases of $$L$$ satisfying $$(1-c^ 2) (1+L)^ m <1 + c - cm (1-k)$$ or $$(1-c^ 2) L^ m < 1 + c - cm (1-s)$$.

MSC:
 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H10 Fixed-point theorems 47J25 Iterative procedures involving nonlinear operators
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References:
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