Existence of Solutions: Investigating Fredholm Integral Equations via a Fixed-Point Theorem
Faruk Özger, Merve Temizer Ersoy, Zeynep Ödemiş Özger
Abstract
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 x(s)=1+λx(s)∫01st+sx(t)dt, can be very often encountered in many applications, where x is the function to be determined, λ is a parameter, and t,s∈[0,1]. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form χ(l)=ϱ(l)+χ(l)∫pqk(l,z)(Vχ)(z)dz are investigated in the space Cωp,q, where χ is the unknown function to be determined, V is a given operator, and ϱ,k are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided.