A New Krasnoselskii’s Type Iterative Method for Hammerstein Equations with Monotone Mappings in Certain Banach Spaces

For \(q>1\) , let E be a q-uniformly smooth real Banach space with dual \(E^*\) , and let \(F :E\to E^*\) , \(K: E^*\to E \) be Lipschitz and strongly monotone mappings with \(D(K)=R(F)=E^*\) . Assume that the Hammerstein equation \(u+KFu=0\) has a unique solution \(\bar u\) . We introduce a new iterative algorithm defined as follows: for given \(u_1\in E\) and \(v_1\in E^*\) , let \(\{u_n\}\) and \(\{v_n\}\) be sequences generated iteratively by: \(\displaystyle u_{n+1} = u_n -\lambda J^{-1}(Fu_n-v_n),\,\,\,n\geq 1;\,\,\, v_{n+1} = v_n-\lambda J(Kv_n+u_n),\,\,\,n\geq 1, \) where J is the duality mapping from E into \(E^*\) and \(\lambda \) is a positive real number in \((0,1)\) satisfying suitable conditions. Then we 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.