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2017 Hammerstein Equations with Lipschitz and Strongly Monotone Mappings in Classical Banach spaces
C. Diop, T. M. M. Sow, N. Djitte, C. E. Chidume
Afr. Diaspora J. Math. (N.S.) 20(2): 1-15 (2017).

Abstract

Let $E$ be a Banach space either $l_p$ or $L_p$ or $W^{m,p}$, $1 < p < \infty$, with dual $E^*$, and let $F :E\mapsto E^*$, $K: E^*\mapsto E $ be Lipschitz and strongly monotone mappings with $D(K)=R(F)=E^*$. Assume that the Hammerstein equation $u+KFu=0$ has a unique solution $\bar u$. For given $u_1\in E$ and $v_1\in E^*$, let $\{u_n\}$ and $\{v_n\}$ be sequences generated iteratively by: $u_{n+1} = J^{-1}(Ju_n -\lambda(Fu_n-v_n)),\,\,\,n\geq 1$ and $v_{n+1} = J(J^{-1}v_n-\lambda(Kv_n+u_n)),\,\,\,n\geq 1$, where $J$ is the duality mapping from $E$ into $E^*$ and $\lambda$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that the sequence $\{u_n\}$ converges strongly to $\bar u$, the sequence $\{v_n\}$ converges strongly to $\bar v$, with $\bar{v}= F\bar{u}.$ Furthermore, our technique of proof is of independent interest.

Citation

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C. Diop. T. M. M. Sow. N. Djitte. C. E. Chidume. "Hammerstein Equations with Lipschitz and Strongly Monotone Mappings in Classical Banach spaces." Afr. Diaspora J. Math. (N.S.) 20 (2) 1 - 15, 2017.

Information

Published: 2017
First available in Project Euclid: 17 May 2017

zbMATH: 06754627
MathSciNet: MR3637472

Subjects:
Primary: 47H04 , 47H06 , 47H15 , 47H17 , 47J25

Keywords: duality mappings , Hammerstein Equations , iterative algorithm , Lipschitz maps , Strongly monotone mappings

Rights: Copyright © 2017 Mathematical Research Publishers

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Vol.20 • No. 2 • 2017
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