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In this paper, using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi></mrow></semantics></math></inline-formula>—admissibility and the concept of simulation functions, some near-fixed point results in the setting of metric interval and normed interval spaces are established. The results have been proved using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">Z</mi></semantics></math></inline-formula>-contractions.