<p>This paper investigates the approximation of fixed points for quasi-nonexpansive mappings in Banach spaces using the general Picard-Mann (GPM) algorithm. Under mild conditions such as the demiclosedness of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(I - \digamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <mi>ϝ</mi> </mrow> </math></EquationSource> </InlineEquation> at zero and the Opial property, we derive weak and strong convergence theorems for the iterative sequences generated by the GPM method. We establish several stability results for the GPM scheme, including summably almost stability property for quasi-contractive mappings. The theoretical findings are applied to the classical relaxation method for solving systems of linear inequalities, demonstrating the practical relevance of our approach. Numerical examples in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}^4\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ell ^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> are provided to illustrate the efficiency and convergence behavior of the proposed algorithm. The results presented herein extend and complement existing work in fixed point theory and iterative approximation methods.</p>