A modified extragradient method for inverse-monotone operators in Banach spaces
Let \(X\) be a Banach space with dual space \(X^*\), \(A: X \to X^*\) be a point-to-point operator and \(C\) be a nonempty closed convex subset of \(X\). This paper focuses on the problem of finding \(v \in X\) such that \(Av = 0\) and \(v \in C\). For each \(\alpha\in (0,\infty)\), the authors consider the operator \({A^{g}}_{\alpha}:X\to X\) defined as \({A^{g}}_{\alpha} x= {g^*}'(g'(x)-\alpha Ax)\), where \(g\) is a lower semicontinuous and proper convex function, \({g^*}'\) is the derivative of the Fenchel conjugate of \(g\), and \(g\) and \(g^*\) are Gâteaux differentiable on \(X\) and \(X^*\), respectively. The authors propose an iterative scheme for solving the above problem, which was motivated by the papers [\textit{D.\,Butnariu} and \textit{E.\,Resmerita}, Abstr.\ Appl.\ Anal.\ 84919, 39 (2006; Zbl 1130.47046); \textit{L.-W.\thinspace Li} and \textit{W.\,Song}, Nonlinear Anal., Hybrid Syst.\ 1, No.\,3, 398--413 (2007; Zbl 1117.49011)] for the case when \(g(x)=1/2\|\cdot\|^2\). They prove that their method converges weakly to a solution of the problem under certain assumptions on the operator \(A\) and the space \(X\).
uniformly convex and uniformly smooth Banach space
scientific article; zbMATH DE number 5600133