Existence Theorems for Some Quadratic Integral Equations

Abstract

Using the theory of measures of noncompactness, we prove a few existence theorems for some quadratic integral equations. The class of quadratic integral equations considered below contains as a special case numerous integral equations encountered in the theories of radiative transfer and neutron transport, and in the kinetic theory of gases. In particular, the well-known Chandrasekhar integral equation also belongs to this class.

On Numerical Treatment for Volterra - Nonlinear QuadraticIntegral Equation in Two-Dimensions

“…ADMis a semi-analytical method, it has been proven as a powerful and reliable scheme for solving a variety of linear and nonlinear problems; integral equations, boundary value problems, ordinary or partial differential equations, algebraic equations, and so on, [16] and [17]. The ADM involves separating the equation into linear and nonlinear portions.…”

On Monotonic Integrable Solutions for Quadratic Functional Integral Equations

“…But first, let us recall that the quadratic equations have numerous applications in the theories of radiative transfer, neutron transport and in the kinetic theory of gases [5,15,22,23]. In order to apply earlier results of the considered type, we have to impose an additional condition that the so-called "characteristic" function ψ is continuous (cf.…”

“…[5,15,22,23]). More general problem (motivated by some practical interests in plasma physics) was investigated in [39].…”

Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra

“…Many authors studied the existence of solutions for several classes of nonlinear quadratic integral equations (see e.g. [2]- [6] and [17]- [18]). However, in most of these papers, the main results are realized with the help of the technique associated with the measure of noncompactness.…”