Convergence of the steepest descent method for accretive operators

Morales, Claudio; Chidume, Charles

doi:10.1090/S0002-9939-99-04975-8

Convergence of the steepest descent method for accretive operators
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by Claudio H. Morales and Charles E. Chidume PDF

Proc. Amer. Math. Soc. 127 (1999), 3677-3683 Request permission

Abstract:

Let $X$ be a uniformly smooth Banach space and let $A\colon X\to X$ be a bounded demicontinuous mapping, which is also $\alpha$-strongly accretive on $X$. Let $z\in X$ and let $x_0$ be an arbitrary initial value in $X$. Then the approximating scheme \[ x_{n+1}=x_n-c_n(Ax_n-z),\qquad n=0,1,2,\dots ,\] converges strongly to the unique solution of the equation $Ax=z$, provided that the sequence $\{c_n\}$ fulfills suitable conditions.

References

Ronald E. Bruck Jr., The iterative solution of the equation $y\in x+Tx$ for a monotone operator $T$ in Hilbert space, Bull. Amer. Math. Soc. 79 (1973), 1258–1261. MR 328692, DOI 10.1090/S0002-9904-1973-13404-4
C. E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (1987), no. 2, 283–288. MR 870786, DOI 10.1090/S0002-9939-1987-0870786-4
C. E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120 (1994), no. 2, 545–551. MR 1165050, DOI 10.1090/S0002-9939-1994-1165050-6
C. E. Chidume, Steepest descent approximations for accretive operator equations, Nonlinear Anal. 26 (1996), no. 2, 299–311. MR 1359476, DOI 10.1016/0362-546X(94)00282-M
Michael G. Crandall and Amnon Pazy, On the range of accretive operators, Israel J. Math. 27 (1977), no. 3-4, 235–246. MR 442763, DOI 10.1007/BF02756485
Klaus Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365–374. MR 350538, DOI 10.1007/BF01171148
W. G. Dotson Jr., An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space, Math. Comp. 32 (1978), no. 141, 223–225. MR 470779, DOI 10.1090/S0025-5718-1978-0470779-8
Athanassios G. Kartsatos, Zeros of demicontinuous accretive operators in Banach spaces, J. Integral Equations 8 (1985), no. 2, 175–184. MR 777969
Li Shan Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), no. 1, 114–125. MR 1353071, DOI 10.1006/jmaa.1995.1289
R. H. Martin Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399–414. MR 318991, DOI 10.1090/S0002-9947-1973-0318991-4
Claudio Morales, Zeros for accretive operators satisfying certain boundary conditions, J. Math. Anal. Appl. 105 (1985), no. 1, 167–175. MR 773579, DOI 10.1016/0022-247X(85)90103-9
Claudio H. Morales, Surjectivity theorems for multivalued mappings of accretive type, Comment. Math. Univ. Carolin. 26 (1985), no. 2, 397–413. MR 803937
W. V. Petryshyn, On the extension and the solution of nonlinear operator equations, Illinois J. Math. 10 (1966), 255–274. MR 208432
Simeon Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978), no. 1, 85–92. MR 512657, DOI 10.1016/0362-546X(78)90044-5
Kok-Keong Tan and Hong Kun Xu, Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces, J. Math. Anal. Appl. 178 (1993), no. 1, 9–21. MR 1231723, DOI 10.1006/jmaa.1993.1287
Morgan Ward, Note on the general rational solution of the equation $ax^2-by^2=z^3$, Amer. J. Math. 61 (1939), 788–790. MR 23, DOI 10.2307/2371337
Zong Ben Xu and G. F. Roach, A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations, J. Math. Anal. Appl. 167 (1992), no. 2, 340–354. MR 1168593, DOI 10.1016/0022-247X(92)90211-U
Eduardo H. Zarantonello, The closure of the numerical range contains the spectrum, Bull. Amer. Math. Soc. 70 (1964), 781–787. MR 173176, DOI 10.1090/S0002-9904-1964-11237-4