African Diaspora Journal of Mathematics

Hammerstein Equations with Lipschitz and Strongly Monotone Mappings in Classical Banach spaces

C. Diop, T. M. M. Sow, N. Djitte, and C. E. Chidume

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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.

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 20, Number 2 (2017), 1-15.

Dates
First available in Project Euclid: 17 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1494986432

Mathematical Reviews number (MathSciNet)
MR3637472

Zentralblatt MATH identifier
06754627

Subjects
Primary: 47H04: Set-valued operators [See also 28B20, 54C60, 58C06] 47H06: Accretive operators, dissipative operators, etc. 47H15 47H17 47J25: Iterative procedures [See also 65J15]

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

Citation

Diop, C.; Sow, T. M. M.; Djitte, N.; Chidume, C. E. Hammerstein Equations with Lipschitz and Strongly Monotone Mappings in Classical Banach spaces. Afr. Diaspora J. Math. (N.S.) 20 (2017), no. 2, 1--15. https://projecteuclid.org/euclid.adjm/1494986432


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