Algorithms for Hammerstein inclusions in certain Banach spaces

Volume 12, Issue 6, pp 387--404 http://dx.doi.org/10.22436/jnsa.012.06.05

Publication Date: February 04, 2019 Submission Date: March 10, 2018 Revision Date: November 19, 2018 Accteptance Date: December 01, 2018

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Authors

Moustapha Sene - Gaston Berger University, Saint Louis, Senegal. Mariama Ndiaye - Gaston Berger University, Saint Louis, Senegal. Ngalla Djitte - Gaston Berger University, Saint Louis, Senegal.

Abstract

Let \(E\) be a reflexive smooth and strictly convex real Banach space. Let \(F: E\rightarrow 2^{E^*}\) and \(K: E^*\rightarrow E\) be bounded maximal monotone mappings such that \(D(F)=E\) and \(R(F)=D(K)=E^*\). Suppose that the Hammerstein inclusion \(0\in u+KFu \) has a solution in \(E\). We present in this paper a new algorithm for approximating solutions of the inclusion \(0\in u+KFu \). Then we prove strong convergence theorems. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings. Furthermore, our technique of proof is of independent interest.

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Keywords

• Hammerstein equation
• monotone
• iterative algorithm

MSC

•  47H04
•  47H06
•  47H17
•  47J25

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