| Summary: | This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="3.33333pt"></mspace><mo>(</mo><mi>B</mi><mi>C</mi><mrow><mo>(</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub><mo>)</mo></mrow><mo>)</mo><mo>.</mo><mspace width="3.33333pt"></mspace></mrow></semantics></math></inline-formula> The main tool applied in this work is the technique associated with measures of noncompactness in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="3.33333pt"></mspace><mi>B</mi><mi>C</mi><mo>(</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub><mo>)</mo><mspace width="3.33333pt"></mspace></mrow></semantics></math></inline-formula> by a given modulus of continuity. Next, we formulate and prove a sufficient condition for the solvability of that coupled system. We, additionally, provide an example and some particular cases to demonstrate the effectiveness and value of our results.
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