A characterization of the generalized projection with the generalized duality mapping and its applications (Q2883958)

Let \((B,\|\cdot\|)\) be a real Banach space with the dual space \(B^*\) and \(\langle\cdot,\cdot\rangle\) denote the duality product. The normalized duality mapping \(I: B\to 2^{B^*}\) is defined by \(I(x) =\{x^*\in B^*:\langle x^*,x\rangle=\| x\|^2\), \(\| x\|=\| x^*\|\}\), \(x\in B\). If \(B\) is a smooth Banach space (and thus \(I\) is single-valued), we can define a function \(\phi: B\times B\to\mathbb{R}\) by \(\phi(x,y)=\| x\|^2- 2\langle I(x),y\rangle+\| y\|^2\), \(x,y\in B\). If \(C\) is a non-empty, closed and convex subset of the smooth Banach space \(B\), \(x\in B\) and \(x_0\in C\), then \(x_0\) is called a generalized projection of \(x\) with the normalized mapping \(I\), denoted by \(x_0\in P^I_C(x)\), if \(\phi(x,x_0)= \text{inf}_{y\in C}\phi(x,y)\). Let \(\psi: [0,\infty)\to [0,\infty)\) be a continuous, strictly increasing function such that \(\psi(t)\to \infty\) as \(t\to\infty\), \(\psi(t)\leq t\) for any \(t\in [0,\infty)\), and \(\psi(0)= 0\). The generalized duality mapping \(I_\psi: B\to 2^{B^*}\) associated with the gauge function \(\psi\) is defined by NEWLINE\[NEWLINEI_\psi(x)= \{x^*\in B^*:\langle x^*,x\rangle=\| x\|\,\psi(\| x\|),\,\| x^*\|= \psi(\| x\|)\}.NEWLINE\]NEWLINE If \(x\in B\) and \(x_0\in C\), then \(x_0\) is called a generalized projection of \(x\) with the generalized duality mapping \(I_\psi\), denoted by \(x_0\in P^{I_\psi}_C(x)\), if \(\phi_\psi(x,x_0)= \text{inf}_{y\in C} \phi_psi(x,y)\), with \(\phi_\psi\) defined analogously to \(\phi\) above. If \(\psi(t)= t\), then \(I_\psi= I\) and thus \(P^{I_\psi}_C= P^I_C\).NEWLINENEWLINE In this paper, the authors define a generalized duality mapping, which is a generalization of the normalized duality mapping and, using this, they extend the notion of a generalized projection and study their properties. Also, they construct an approximating fixed point sequence using the generalized projection with the generalized duality mapping, and prove its strong convergence.