Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Chidume, C.; Zegeye, H.

doi:10.1090/S0002-9939-04-07568-9

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space
HTML articles powered by AMS MathViewer

by C. E. Chidume and H. Zegeye PDF

Proc. Amer. Math. Soc. 133 (2005), 851-858 Request permission

Abstract:

Let $H$ be a real Hilbert space. Let $F:D(F)\subseteq H\rightarrow H$, $K:D(K)\subseteq H\to H$ be bounded monotone mappings with $R(F)\subseteq D(K)$, where $D(F)$ and $D(K)$ are closed convex subsets of $H$ satisfying certain conditions. Suppose the equation $0=u+KFu$ has a solution in $D(F)$. Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on $K$, and the operators $K$ and $F$ need not be defined on compact subsets of $H$.

References

C. E. Chidume and M. O. Osilike, Iterative solution for nonlinear integral equations of Hammerstein type, J. Nigerian Math. Soc. 11 (1992), no. 1, 9–18. Special issue in honour of Professor Chike Obi. MR 1330020
C. E. Chidume and H. Zegeye, Iterative approximation of solutions of nonlinear equations of Hammerstein type, Abstr. Appl. Anal. 6 (2003), 353–365. MR 1982807, DOI 10.1155/S1085337503209052
C. E. Chidume and H. Zegeye, Approximation methods for nonlinear operator equations, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2467–2478. MR 1974645, DOI 10.1090/S0002-9939-02-06769-2
C.E. Chidume, H. Zegeye and S.J. Aneke, Approximation of fixed points of weakly contractive non-self maps in Banach spaces, J. Math. Anal. Appl. 270 (2002), 189-199.
Shiro Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. MR 336469, DOI 10.1090/S0002-9939-1974-0336469-5
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
George J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346. MR 169064
Claudio H. Morales, Surjectivity theorems for multivalued mappings of accretive type, Comment. Math. Univ. Carolin. 26 (1985), no. 2, 397–413. MR 803937
Dan Pascali and Silviu Sburlan, Nonlinear mappings of monotone type, Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978. MR 531036
Simeon Reich, Constructive techniques for accretive and monotone operators, Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978) Academic Press, New York-London, 1979, pp. 335–345. MR 537545
Simeon Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287–292. MR 576291, DOI 10.1016/0022-247X(80)90323-6
W. Takahashi and Pei-Jun Zhang, The closedness property and the pseudo-$A$-properness of accretive operators, J. Math. Anal. Appl. 132 (1988), no. 2, 548–557. MR 943528, DOI 10.1016/0022-247X(88)90083-2
E. H. Zarantonello, Solving functional equations by contractive averaging, Mathematics Research Center Report #160, Mathematics Research Centre, University of Wisconsin, Madison, 1960.