Regularization methods for nonlinear ill-posed equations involving \(m\)-accretive mappings in Banach spaces
The topic of this paper is the regularization of nonlinear ill-posed equations involving \( m \)-accretive operators in smooth Banach spaces. We present some details in the sequel, and for this let \( E \) be a real, reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, and let \( j : E \to E^* \) be a (single-valued) normalized duality mapping, where \( E^* \) denotes the dual space of \( E \). In addition, let \( A: E \supset \mathcal{D}(A) \to E \) be a single-valued nonlinear operator which is \( m \)-accretive, i.e., the following two conditions are satisfied: (i) for each \( x,y \in \mathcal{D}(A) \), we have \( \langle A(x) - A(y),j(x-y) \rangle \geq 0 \), and (ii) for each \( \lambda > 0 \), the nonlinear mapping \( A + \lambda I \) is onto, i.e., there holds \( \mathcal{R}(A + \lambda I) = E \). Here, \( \langle x, x^* \rangle \in \mathbb{R} \) means evaluation of \( x^* \in E^* \) at an element \( x \in E \), and \( I : E \to E \) denotes the identity mapping. The considered equation \( A(x) = f \) with \( f \in E \) may be ill-posed, and it is assumed that the set \( S \subset E \) of solutions to this equation is nonempty. Two regularization schemes are considered in this paper. The first scheme is \( A(x) + \alpha(x-x^+) = f_\delta \), where \( f_\delta \in E \) satisfies \( \| f_\delta -f \| \leq \delta \), and \( x^+ \in E \) is a given initial guess. It is shown that this regularized equation has for each \( \alpha > 0 \) a unique solution \( x = x_\alpha^\delta \in \mathcal{D}(A) \). In addition, \( x_\alpha^\delta \) converges strongly to a solution \( p_* \in S \) satisfying \( \langle p_* - x^+, j(p_*-p) \rangle \leq 0 \) for each \( p \in S \), where \( \alpha = \alpha_\delta \) satisfies \( \alpha \to 0 \) and \( \delta/\alpha \to 0 \). The second considered regularization method is a regularized version of an inertial proximal point algorithm: \( c_k( A(x_{k+1}) + \alpha_k x_{k+1} -f_k) +x_{k+1} - x_{k} = \gamma_k(x_k-x_{k-1}) \) for \( k = 1,2,\dots \), where \( x_0, x_1 \in E \) are given, and \( f_k \in E \) satisfies \( \| f_k - f \| \leq \delta_k \to 0 \) as \( k \to \infty \). It is shown that the sequence \(\{ x_k\} \) converges strongly to a solution \( p_* \in S \) as \( k \to \infty \) if the parameters \( c_k > c_0 > 0, \alpha_k > 0 \) and \( \gamma_k \geq 0 \) satisfy some additional conditions. This paper extends similar results obtained earlier under stronger assumptions.
scientific article; zbMATH DE number 6152680