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It is well known fact that fixed point results are very useful for finding the solution of different types of differential equations. In this paper, some new results of multi-valued functions involving rational inequality in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>−</mo></mrow></semantics></math></inline-formula>metric spaces have been obtained. Some of the new concepts are also defined over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>−</mo></mrow></semantics></math></inline-formula>metric spaces that are open set, interior of set, and limit point of set. Moreover, we have presented the application of our main results in the field of Homotopy. Some non-trivial examples are also provided to discuss the validation of the main finding.