| Summary: | The aim of this article is to establish the existence of the solution of non-linear functional integral equations <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mi>U</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>l</mi> </msubsup> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>h</mi> </msubsup> <mi>P</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>r</mi> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> </mfenced> </mfenced> <mo>×</mo> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>a</mi> </msubsup> <mi>Q</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mfenced> <mi>d</mi> <mi>r</mi> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics> </math> </inline-formula> of two variables, which is of the form of two operators in the setting of Banach algebra <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> <mo>×</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> </mfenced> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> <mo>×</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> </mfenced> </mrow> </semantics> </math> </inline-formula> and a fixed point theorem, which is a generalization of Darbo’s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example.
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