Hammerstein Equations with Lipschitz and Strongly Monotone Mappings in Classical Banach spaces
Cheikh Talibouya Diop, T. M. M. Sow, Ngalla Djitté, C.E. Chidume · Project Euclid (Cornell University) · 2017
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.