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The present article is devoted to prove some fixed point results for <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>ψ</mi><mo>,</mo><mi>ϕ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-contractions in the framework of metric space equipped with a locally finitely <inline-formula><math display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-transitive relation. The results proved in this article improve and weaken some existing fixed point results available in the literature. Finally, an example is provided for attesting to the credibility of my results.