Weak and strong convergence theorems for maximal monotone operators in a Banach space (Q1771259)

Let \(E\) be a smooth and uniformly convex Banach space with the normalized duality mapping \(J: E \to E^*\) and \(T \subset E \times E^*\) a maximal monotone operator with \(T^{-1} 0 \neq \emptyset\). Let \(J_r=(J+r T)^{-1} J\), \(r>0\), and \(P(x)=\text{ {argmin}}_{y \in T^{-1}0} (\| x\|^2-2 \langle x, Jy\rangle+ \| y\|^2)\). The authors study the iterative process \(x_{n+1}=J^{-1}(\alpha_n J(x_n)+(1-\alpha_n) J(J_{r_n} x_n) )\), \(\alpha_n \in [0,1]\), \(r_n \in (0, \infty)\) \((n=1, 2, \dots)\) and prove that the sequence \(\{ P(x_n) \}\) converges strongly to an element of \(T^{-1} 0\). The results are applied to the convex minimization problem and the variational inequality problem.