New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces
- 114 Downloads
Let E be a nonempty closed uniformly convex and 2-uniformly smooth Banach space with dual E∗ and A : E∗ → E be Lipschitz continuous monotone mapping with A− 1(0) ≠ ∅. A new semi-implicit midpoint rule (SIMR) with the general contraction for monotone mappings in Banach spaces is established and proved to converge strongly to x∗ ∈ E, where Jx∗ ∈ A− 1(0). Moreover, applications to convex minimization problems, solution of Hammerstein integral equations, and semi-fixed point of a cluster of semi-pseudo mappings are included.
KeywordsSemi-implicit Monotone mappings Zero point Viscosity approximation
Mathematics Subject Classification (2010)47H09 47H10 47L25
The authors express their deep gratitude to the referees and the editor for their valuable comments and suggestions.
All authors contributed equally to this work. All authors read and approved final manuscript.
This article was funded by the National Science Foundation of China (11471059)and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1706154)and the Research Project of Chongqing Technology and Business University (KFJJ2017069).
Compliance with Ethical Standards
The authors declare that they have no competing interests.
- 5.Duffull, S.B., Hegarty, G.: An Inductive Approximation to the Solution of Systems of Nonlinear Ordinary Differential Equations in Pharmacokinetics-Pharmacodynamics. Journal of Computer Science and Networking pp. 1-14 (2014)Google Scholar
- 6.Khorasani, S., Adibian, A.: Alytical solution of linear ordinary differential equations by differential transfer. Electronic Journal of Differential Equations 79, 1–18 (2003)Google Scholar
- 7.Chidume, C.E., Osilike, M.O.: Iterative solution of nonlinear integral equations of Hammerstein-type. J. Niger. Math. Soc. Appl. Anal. pp. 353-367 (2003)Google Scholar
- 8.Auzinger, W., Frank, R.: Asyptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, vol. 56 (1989)Google Scholar
- 14.Chidume, C.E.: Geometric Properties of Banach Spaces and Nonlinear Iterations Lectures Notes in Mathematics, vol. 1965. Springer, London (2009)Google Scholar
- 19.Reich, S.: A weak convergence theorem for alternating methods with Bergman distance. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accrective and Monotone Type. Lecture Notes in Pure and Appl. Math, vol. 178, pp 313–318. Dekker, New York (1996)Google Scholar
- 20.Chidume, C.E., Djitte, N.: Strong convergence theorems for zeros of bounded maximal monotone nonlinear operators. Abstr. Appl. Anal. 2012 Article ID 681348 (2012)Google Scholar
- 24.Chidume, C.E., Idu, K.O.: Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems. Fixed Point Theory and Applications:97. https://doi.org/10.1186/s13663-016-0582-8 (2016)
- 25.Alber, Y.A.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp 15–50. Marcel Dekker, New York (1996)Google Scholar
- 26.Takahashi, W.: Nolinear Functional Analysis. Yokohama Publishers, Yokohama (2000)Google Scholar
- 33.Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type of Lecture Notes in Pure and Applied Mathematics. In: Kartsatos, A.G. (ed.) , vol. 178, p 15C50. Marcel Dekker, New York (1996)Google Scholar
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.