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In this paper, we present a fixed-point theorem in <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>-complete regular symmetric spaces endowed with a locally <i>T</i>-transitive binary relation <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> using comparison functions that generalizes several relevant existing results. In addition, we adopt an example to substantiate the genuineness of our newly proved result. Finally, as an application of our main result, we establish the existence and uniqueness of a solution of a periodic boundary value problem.