Higher order convergent iterative algorithm for computing nonlinear Moore–Penrose metric generalized inverses in Banach Spaces and its application to approximate solution problems

Abstract Let X and Y be Banach spaces, and let A : X → Y {A:X\to Y} be a bounded linear operator. Denote by ℛ ⁢ ( A ) {\mathcal{R}(A)} the range and by 𝒩 ⁢ ( A ) {\mathcal{N}(A)} the kernel of A . The operator equation A ⁢ x = b {Ax=b} may be ill-posed in the sense of Hadamard, that is, a solution may fail to exist (e.g., when b ∉ ℛ ⁢ ( A ) {b\notin\mathcal{R}(A)} ) or may not be unique (e.g., when 𝒩 ⁢ ( A ) ≠ { 0 } {\mathcal{N}(A)\neq\{0\}} ). In such cases, it is common to consider approximate solutions in some sense. The nonlinear Moore–Penrose metric generalized inverse has emerged as a valuable tool for addressing the best approximate solution problems in Banach spaces. In recent years, extensive theoretical and applied investigations have been conducted on the Moore–Penrose metric generalized inverse in Banach spaces, including analyses of its continuity, representations, and perturbation properties, among other aspects. Nevertheless, the metric projection associated with it is generally nonlinear, which complicates the computation and leaves its algorithmic aspects largely unexplored. In this paper, we exploit specific geometric properties of Banach spaces to develop an efficient convergent iterative method, along with its higher-order variant, for computing the nonlinear Moore–Penrose metric generalized inverse A M {A^{M}} of a bounded linear operator A between Banach spaces. Convergence conditions and error estimates for the computation of the generalized inverse A M {A^{M}} are derived. As an application, we introduce an iterative method for obtaining the best approximate solution to ill-posed operator equations. Several examples are provided to illustrate the theoretical results and examine the effectiveness of the proposed method.