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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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