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

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.

Keywords

• Fredholm integral equations
• quadratic Chandrasekhar integral equation
• Hölder space
• tempered modulus of continuity
• the space <i>C_ω</i>(<i>X</i>)
• fixed-point theorem