Convergence of the steepest descent method for accretive operators. (English) Zbl 0937.47057

Morales, Claudio H.; Chidume, Charles E.

Proc. Am. Math. Soc. 127, No. 12, 3677-3683 (1999).

The authors consider the approximation scheme for the equation \(Ax= z\) in the uniformly smooth Banach space \(X\) and a bounded demicontinuous mapping \(A: X\to X\), which is also \(\alpha\)-strongly accretive on \(X\). For \(z\in X\) and \(x_0\) an arbitrary initial value in \(X\), the approximating scheme \(x_{n+1}= x_n- c_n(Ax_n- z)\), \(n= 0,1,2,\dots\), converges strongly to the unique solution of the equation; provided that the sequence \(\{c_n\}\) fulfils suitable conditions.

Reviewer: U.Kosel (Freiberg)

MSC:

47H10	Fixed-point theorems
47J25	Iterative procedures involving nonlinear operators
65J15	Numerical solutions to equations with nonlinear operators

Keywords:

\(\alpha\)-strongly accretive; approximation scheme; uniformly smooth Banach space