الفهرس 
Basic concepts and de nitionsOnly 14 pages are availabe for public view
 |  | from | 16 |  |  |
| | | from | 16 | | |
Abstract
Theory of integral equations is one of the most important and useful
branch of mathematical analysis. Integral equations of various types
create the signi cant subject of several mathematical investigations and
appear often in many applications, especially in solving numerous prob-
lem in physics, engineering and economics [32].
The Urysohn integral equation has been studied in several papers and
monographs ([1], [32]). A quite general result has been obtained in [2],
when the solvability of this equation has been studied in the two classes
of integral and monotonic functions on the interval [0; 1].
The coupled system of integral equations are studied in several papers
([15], [16], [17],[18], [19] and [23])
Our aim here is to study the existence of solution of some coupled sys-
tem of functional integral equations and coupled system of functional
integral equations of fractional orders.
In Chapter 1, we collect the concepts, de nitions, theorems and aux-
iliary facts explored in further chapters.
1
In Chapter 2, we study the existence of at least one continuous solu-
tion of nonlinear coupled system of functional integral equations
x(t) = a1(t) +
Z t
0
f1(t; s; y(’1(s)))ds; t 2 [0; T] (1)
y(t) = a2(t) +
Z t
0
f2(t; s; x(’2(s)))ds; t 2 [0; T]: (2)
where the two functions f1 and f2 are continuous in t , where x; y are
continuous on [0; T], and the special case
x(t) = a1(t) +
Z t
0
f1(t; s; x(’1(s)))ds; t 2 [0; T] (3)
will be considered.
The coupled system of Hammerstein functional integral equations
x(t) = a1(t) +
Z t
0
k1(t; s) g1(s; y(’1(s)))ds; t 2 [0; T] (4)
y(t) = a2(t) +
Z t
0
k2(t; s) g2(s; x(’2(s)))ds; t 2 [0; T] (5)
will be considered as an applications.
At the end of this chapter we study the maximal and minimal solution
of the functional integral equation (3).
In Chapter 3, we are concerning with a coupled system of the
nonlinear functional integral equations
x(t) = a1(t) +
Z t
0
f1(t; s; I 1y(s))ds; t 2 [0; T] (6)
y(t) = a2(t) +
Z t
0
f2(t; s; I 2x(s))ds; t 2 [0; T]: (7)