| Summary: | In this manuscript, we introduce two notions, Pata−Suzuki <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula>-contraction and Pata <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">Z</mi> </semantics> </math> </inline-formula>-contraction for the pair of self-mapping <inline-formula> <math display="inline"> <semantics> <mrow> <mi>g</mi> <mo>,</mo> <mi>f</mi> </mrow> </semantics> </math> </inline-formula> in the context of metric spaces. For such types of contractions, both the existence and uniqueness of a common fixed point are examined. We provide examples to illustrate the validity of the given results. Further, we consider ordinary differential equations to apply our obtained results.
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