| Summary: | In this paper, we define almost <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mi>g</mi></msub></semantics></math></inline-formula>-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mn>2</mn></msub></semantics></math></inline-formula>-metric spaces endowed with binary relations. As consequences of our newly proved results, we deduce some coincidence and common fixed point results for almost <i>g</i>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> Geraghty type contraction mappings in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mn>2</mn></msub></semantics></math></inline-formula>-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mn>2</mn></msub></semantics></math></inline-formula>-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations.
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