Two step size algorithms for strong convergence for a monotone operator in Banach spaces

For p\geq ۲, let E be a ۲ uniformly smooth and p uniformly convex real Banach spaces and let a mapping \displaystyle \Phi : E \to E^{*} be Lipschitz, and  strongly monotone such that \displaystyle \Phi^{-۱}(۰)\neq \emptyset. For an arbitrary (\{\xi_{۱}\}, \{\psi_{۱}\})\in E, we define the sequences \{\xi_{n}\} and \{\psi_{n}\} by\begin{equation*}    \left\{      \begin{array}{ll}         \psi_{n+۱} = J^{-۱}(J\xi_{n} - \theta_{n}\Phi\xi_{n}), & \hbox{n\geq ۰} \\         \xi_{n+۱} = J^{-۱}(J\psi_{n+۱} - \lambda_{n}\Phi\psi_{n+۱}), & \hbox{n\geq ۰} \\      \end{array}    \right.\end{equation*}where \lambda_{n} and \theta_{n} are positive real number and J is the duality mapping of E. Letting (\lambda_{n}, \theta_{n})\in (۰,\Lambda_{p}) where \Lambda_{p} >۰, then \xi_{n}  and \psi_{n} converges strongly to \xi^{*},   a unique solution of the equation \Phi \xi = ۰.

Keywords:

Lipschitz , Equations , generalized monotone , Bounded