International Journal of Nonlinear Analysis and Applications

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

Document Type : Research Paper

Authors

Mathematics Department, University of The Gambia, Brikama Campus, Gambia

Abstract

For $p\geq 2$, let $E$ be a $2$ 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 $\displaystyle \Phi^{-1}(0)\neq \emptyset$. For an arbitrary $(\{\xi_{1}\}, \{\psi_{1}\})\in E$, we define the sequences $\{\xi_{n}\}$ and $\{\psi_{n}\}$ by
\begin{equation*}
    \left\{
      \begin{array}{ll}
         \psi_{n+1} = J^{-1}(J\xi_{n} - \theta_{n}\Phi\xi_{n}), & \hbox{$n\geq 0$} \\
         \xi_{n+1} = J^{-1}(J\psi_{n+1} - \lambda_{n}\Phi\psi_{n+1}), & \hbox{$n\geq 0$} \\
      \end{array}
    \right.
\end{equation*}
where $\lambda_{n}$ and $\theta_{n}$ are positive real number and $J$ is the duality mapping of $E$. Letting $(\lambda_{n}, \theta_{n})\in (0,\Lambda_{p})$ where $\Lambda_{p} >0$, then $\xi_{n}$  and $\psi_{n}$ converges strongly to $\xi^{*}$,   a unique solution of the equation $\Phi \xi = 0$.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 10
October 2023
Pages 217-225
  • Receive Date: 18 June 2022
  • Accept Date: 27 March 2023