Abstract
Let X be a uniformly convex and uniformly smooth real Banach space with dual X*. Let F : X → X* and K : X* → X be continuous monotone operators. Suppose that the Hammerstein equation u + KFu = 0 has a solution in X. It is proved that a hybrid-type approximation sequence converges strongly to u*, where u* is a solution of the equation u + KFu = 0. In our theorems, the operator K or F need not be defined on a compact subset of X and no invertibility assumption is imposed on K.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alber, Y.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Leture Notes in Pure and Applied Mathematics, vol. 178, pp. 15–50. Dekker, New York (1996)
Brez̀is H., Browder F.E.: Nonlinear integral equations and systems of Hammerstein type. Bull. Am. Math. Soc. 82, 115–147 (1976)
Chang S.S., Cho Y.J., Zhou Y.Y.: Iterative sequences with mixed errors for asymptotically quasi-nonexpansive type mappings in Banach spaces. Acta Math. Hung. 100, 147–155 (2003)
Chidume C.E., Zegeye H.: Iterative approximation of solutions of nonlinear equations of Hammerstein type. Abstract Appl. Anal. 6, 353–365 (2003)
Dominquez Benavides T., Lopez Acedo G., Xu H.K.: Iterative solutions for zeros of accretive operators. Math. Nachr. 248/249, 62–71 (2003)
Hammerstein A.: Nichtlineare integralgleichungen nebst anwendungen. Acta Math. Soc. 54, 117–176 (1930)
Ishikawa S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)
Jung J.S., Morales C.H.: The Mann process for perturbed m-accretive operators in Banach spaces. Nonlinear Anal. 46(2), 231–243 (2001)
Kacurovski R.I.: On monotone operators and convex functionals. Uspekhi Mat. Nauk. 15, 213–215 (1960)
Mainge P.E.: Strong convergence of projected subgradient methods for nonsmooth and non- strictly convex minimization. Set Valued Anal. 16, 899–912 (2008)
Mann W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Minty G.J.: Monotone operators in Hilbert spaces. Duke Math. J. 29, 341–346 (1962)
Sburlan P.D.: Nonlinear mappings of monotone type. Editura Academiae, Bucaresti (1978)
Reich, S.: A weak convergence theorem for the alternating method with Bergman distance. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Leture Notes in Pure and Applied Mathematics, vol. 178, pp. 313–318. Dekker, New York (1996)
Reich, S.: Constructive techniques for accretive and monotone operators. In: Applied Nonlinear Analysis, pp. 335–345. Academic Press, New York (1979)
Rhoades B.E., Soltuz S.M.: The equivalence between the convergence of Ishikawa and Mann iterations for an asymptotically pseudocontractive map. J. Math. Anal. Appl. 283, 681–688 (2003)
Kamimura S., Takahashi W.: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2002)
Ofoedu E.U., Malonza D.M.: Hybride approximation of nonlinear operator equations and applications to equation of Hammerstein. Appl. Math. Comput. 217, 6019–6030 (2010)
Takahashi W.: Nonlinear Functional Analysis (Japanese). Kindikagaku, Tokyo (1988)
Vainberg M.M., Kacurovskii R.I.: On the variational theory of nonlinear operators and equations. Dokl. Akad. Nauk. 129, 1199–1202 (1959)
Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Zarantonello, E.H.: Solving functional equations by contractive averaging. Mathematics Research Center Report #160, Mathematics Research Center, University of Wisconsin, Madison (1960)
Zegeye H., Ofoedu E.U., Shahzad N.: Convergence theorems for equilibrium problem, variotional inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 216, 3439–3449 (2010)
Zegeye, H.; Shahzad, N.: Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces. Optim. Lett. doi:10.1007/s11590-010-0235-5
Zeidler E.: Nonlinear Functional Analysis its Applications II/B. Springer, New York (1990)
Zhou H.Y., Chang S.S., Agarwal R.P., Cho Y.J.: Stability results for the Ishikawa iteration procedures. Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal. 9, 477–486 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
This article is published under license to BioMed Central Ltd. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Zegeye, H., Malonza, D.M. Hybrid approximation of solutions of integral equations of the Hammerstein type. Arab. J. Math. 2, 221–232 (2013). https://doi.org/10.1007/s40065-012-0060-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-012-0060-z