# Article

 Title: Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces (English) Author: Saddeek, A. M. Author: Ahmed, Sayed A. Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 44 Issue: 4 Year: 2008 Pages: 285-293 Summary lang: English . Category: math . Summary: The weak convergence of the iterative generated by $J(u_{n+1}-u_{n})= \tau (Fu_{n}-Ju_{n})$, $n \ge 0$, $\big (0< \tau =\min \big \lbrace 1,\frac{1}{\lambda }\big \rbrace \big )$ to a coincidence point of the mappings $F,J\colon V \rightarrow V^{\star }$ is investigated, where $V$ is a real reflexive Banach space and $V^{\star }$ its dual (assuming that $V^{\star }$ is strictly convex). The basic assumptions are that $J$ is the duality mapping, $J-F$ is demiclosed at $0$, coercive, potential and bounded and that there exists a non-negative real valued function $r(u,\eta )$ such that $\sup _{u,\eta \in V} \lbrace r(u,\eta )\rbrace =\lambda < \infty$ $r(u,\eta )\Vert J(u- \eta ) \Vert _{V^{\star }}\ge \Vert (J -F)(u)-(J-F)(\eta ) \Vert _{V^{\star }}\,, \quad \forall ~ u,\eta \in V\,.$ Furthermore, the case when $V$ is a Hilbert space is given. An application of our results to filtration problems with limit gradient in a domain with semipermeable boundary is also provided. (English) Keyword: iteration Keyword: coincidence point Keyword: demiclosed mappings Keyword: pseudo-monotone mappings Keyword: bounded Lipschitz continuous coercive mappings Keyword: filtration problems MSC: 47H10 MSC: 54H25 idZBL: Zbl 1212.47088 idMR: MR2493425 . Date available: 2009-01-29T09:15:26Z Last updated: 2013-09-19 Stable URL: http://hdl.handle.net/10338.dmlcz/119768 . Reference: [1] Badriev, I. B., Karchevskii, M. M.: On the convergance of the iterative process in Banach spaces.Issledovaniya po prikladnoi matematike (Investigations in Applied Mathematics) 17 (1990), 3–15, in Russian. MR 1127806 Reference: [2] Brezis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle.Boll. Un. Mat. Ital. 6 (1972), 293–300. Zbl 0264.49013, MR 0324498 Reference: [3] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.Akademie Verlag Berlin, 1974. MR 0636412 Reference: [4] Goebel, K., Kirk, W. A.: Topics in metric fixed point theory.Cambridge Stud. Adv. Math. 28 (1990). Zbl 0708.47031, MR 1074005 Reference: [5] Istratescu, V.I.: Fixed Point Theory.Reidel, Dordrecht, 1981. Zbl 0465.47035, MR 0620639 Reference: [6] Karchevskii, M. M., Badriev, I. B.: Nonlinear problems of filtration theory with dis continuous monotone operators.Chislennye Metody Mekh. Sploshnoi Sredy 10 (5) (1979), 63–78, in Russian. MR 0628672 Reference: [7] Lions, J. L.: Quelques Methods de Resolution des Problemes aux Limites Nonlineaires.Dunod and Gauthier-Villars, 1969. MR 0259693 Reference: [8] Lyashko, A. D., Karchevskii, M. M.: On the solution of some nonlinear problems of filtration theory.Izv. Vyssh. Uchebn. Zaved., Matematika 6 (1975), 73–81, in Russian. Reference: [9] Mann, W. R.: Mean value methods in iteration.Proc. Amer. Math. Soc. 4 (1953), 506–510. Zbl 0050.11603, MR 0054846, 10.1090/S0002-9939-1953-0054846-3 Reference: [10] Maruster, S.: The solution by iteration of nonlinear equations in Hilbert spaces.Proc. Amer. Math. Soc. 63 (1) (1977), 69–73. Zbl 0355.47037, MR 0636944 Reference: [11] Zeidler, E.: Nonlinear Functional Analysis and Its Applications.Nonlinear Monotone Operators, vol. II(B), Springer Verlag, Berlin, 1990. Zbl 0684.47029 .

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