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Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>D</mi><mrow><mo>*</mo><mo>*</mo></mrow></msup></semantics></math></inline-formula> iteration technique, for approximating fixed points in generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>D</mi><mrow><mo>*</mo><mo>*</mo></mrow></msup></semantics></math></inline-formula> iteration approach to the fixed points of generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results.