Topological Existence and Uniqueness of the Mahgoub–Adomian Decomposition Method

Abstract

We develop a rigorous topological theory for the Mahgoub--Adomian Decomposition Method (MADM) in the setting of Banach spaces. The method is formulated as a nonlinear operator equation in the space $C([0,T];H^s(\Omega))$, where the associated Mahgoub--Adomian operator is shown to be continuous and compact. Existence of the MADM solution is established via Schauder’s fixed-point theorem, while uniqueness follows under a strict monotonicity condition on the nonlinear operator. The analysis is carried out independently of contraction assumptions or smallness conditions. The results are applied to both linear and nonlinear Schrödinger equations of the form $i u_t + \Delta u + N(u) = f(x,t), \quad u(x,0)=h(x)$, including the linear problems $N(u)=0$ and nonlinear problems such as $N(u)=\lambda |u|^{p}u$. These results provide a topological well-posedness framework that justifies the convergence and uniqueness of the Mahgoub--Adomian decomposition series for a broad class of Schrödinger-type evolution equations.