Existence of Solutions: Investigating Fredholm Integral Equations via a Fixed-Point Theorem

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mi>λ</mi><mi>x</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mn>1</mn></msubsup><mfrac><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mfrac><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>,</mo></mrow></semantics></math></inline-formula> can be very often encountered in many applications, where <i>x</i> is the function to be determined, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> is a parameter, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>χ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo>=</mo><mi>ϱ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo>+</mo><mi>χ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><msubsup><mo>∫</mo><mrow><mi>p</mi></mrow><mi>q</mi></msubsup><mi>k</mi><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>V</mi><mi>χ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mi>d</mi><mi>z</mi></mrow></semantics></math></inline-formula> are investigated in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>ω</mi></msub><mfenced separators="" open="[" close="]"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>χ</mi></semantics></math></inline-formula> is the unknown function to be determined, <i>V</i> is a given operator, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϱ</mi><mo>,</mo><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mi>k</mi></mrow></semantics></math></inline-formula> are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided.