| Summary: | In this article, we introduce a relatively new concept of multi-valued <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi mathvariant="script">R</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-contractions and utilize the same to prove some fixed point results for a special class of multi-valued mappings in metric spaces endowed with an amorphous binary relation. Illustrative examples are also provided to exhibit the utility of our results proved herein. Finally, we utilize some of our results to investigate the existence and uniqueness of a positive solution for the integral equation of Volterra type.
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