الفهرس 
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Abstract
Quadratic integral equations are often applicable in the theory of radiative
transfer, kinetic theory of gases, in the theory of neutron transport and in
the traffic theory. Especially, the so-called quadratic integral equation of
chandraskher type can be very often encountered in many applications(see
[4], [8], [16], [14] and [15]).
Our aim in this thesis is (to generalization Caratheodory Theorem (see[12])
to prove the existence of at least one continuous solution for the quadratic
integral equation.
t(t-S)O:-l t(t-S),6-1
x(t) = a(t) + 10 f(a) f(s, x(s)) ds . 10 ,,10\ g(s, x(s)) ds
= a(t) + I” f(t, x(t)) . J,6g(t, x(t)); a, j3 E (0,1) (1)
The thesis consists of three chapters.
Chapter 1 collects the main concepts of fractional-order integration, fractional-
order differentiation and their properties. Also some preliminaries and
known results which will be used in the other chapters.
Chapter 2 collects the quadratic integral equations from different papers.
fixed-point theorems which use it in the prove existence of a solution: and
their assumptions.
Chapter 3 is devoted to prove Caratheodory Theorem (see[12]) for the
retarded with existence of a solution for the quadratic integral equation
(1). We use Tychonov’s fixed-point Theorem to prove the global exis-
tence at least one solution of the quadratic integral equation (1), where
f(t, x), g(t, x) satisfies Caratheodory condition and a(t) is a continuous
function on [0, T].
Also; is devoted to prove the existence of the maximal and the minimal solu-
tions of the quadratic integral equation when f (t, x), g( t, x) are monotonic
nondecreasing in x for each t E [0, T], and some applications.