| Summary: | We consider the Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, which is a symmetric Banach space, and prove the existence and approximation of numerical solutions for a broad class of third-order BVPs. Our approach is based on an integral operator that is constructed using Green’s function. The Banach contraction principle (BCP) is applied to guarantee a unique solution to our problem. Moreover, in order to find the value of the numerical solution, this new operator is embedded within the three-step Noor iterative scheme; we named this new iterative scheme the Noor–Green iterative scheme. We provide a convergence theorem for the proposed scheme by employing suitable restrictions on the parameters involved in the problem and in the scheme. The results of the stability of our scheme are also reported. It is worth mentioning that unlike the concept of stability in the classical sense, our result for stability is based on the concept of weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>w</mi><mn>2</mn></msup></semantics></math></inline-formula> stability. In order to support our findings, we carried out various numerical experiments using different third-order BVPs. Finally, we report on the application of our iterative scheme to solve a class of fractional BVPs in the same symmetric Banach space. Our results are essentially new in the present literature and extend several of the results found in the current literature.
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