Solvability of Chandrasekhar’s Quadratic Integral Equations in Banach Algebra

Abstract

In this paper, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contain various integral and functional equations that are considered in nonlinear analysis. Our considerations will be discussed in Banach algebra using a fixed point theorem instead of using the technique of measure of noncompactness. An important special case of that functional equation is Chandrasekhar's integral equation which appears in radiative transfer, neutron transport and the kinetic theory of gases [1].

Solvability of a 2 x 2 block operator matrix of chandrasekhar type on a Bananch algebra

“…φ 1 (s) t + s y(s) ds, t ∈ J, y(t) = a 2 (t) + x(t) t φ 2 (s) t + s x(s) ds, t ∈ Jwhich is the same result obtained in[29]. (vii) Letting f 1 (t, y(t)) = a 1 (t), f 2 (t, x(t)) = a 2 (t), then we have the coupled system x(t) = a 1 (t) + t 0 t t + s u 1 (s, y(s)) ds, t ∈ J, y(t) = a 2 (t) + t 0 t t + s u 2 (s, x(s)) ds, t ∈ J.…”

“…which continues the series of publications on the coupled systems ( [4]- [6], [26], [27] and [29]). For example, Su [30] studied a two-point boundary value problems for a coupled system of fractional differential equations.…”

Stabilization of coupled systems of quadratic integral equations of Chandrasekhar type

“…As a particular case of Theorem 2.1 we can obtain an existence theorem for the coupled system of the Chandrasekar's integral equations [29] x(t) = 1 + y(t) where φ i , i = 1, 2, are two functions in L ∞ and if 4λ i ||φ i || L ∞ ≤ 1, i = 1, 2, then the coupled system of Chandrasekar's integral equations (4.1) has at least one solution in C(I ) × C(I ).…”

“…As a particular case of Theorem we can obtain an existence theorem for the coupled system of the Chandrasekar's integral equations where , are two functions in and if , then the coupled system of Chandrasekar's integral equations has at least one solution in . Remark In case of Then and Therefore, the coupled system with (…”