الفهرس 
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Abstract
Mathematical modelling of real-life problems usually results in partial di erential
equations or integral and integro-di erential equations, stochastic equations.
Many mathematical formulation of physical phenomena contain integro-di erential
equations, these equations arises in many elds like
uid dynamics, biological models and
chemical kinetics integro-di erential equations are usually dicult to solve analytically
so it is required to obtain an ecient approximate solution.
Our aim in this thesis are to consider the question of the existence and uniqueness of
solutions of integro- di erential equations.
The thesis consists of three chapters
Chapter 1: Collects the concepts, de nitions and theorems which will be
used in the other chapters.
Chapter 2: This chapter deals with the existence of at least one and
exactly one solution and discuss when this solution is unique of the
nonlocal boundary value problem of Fredholm integro-di erential equation.
Chapter 3: This chapter deals with the existence of at least one and exactly
one solution and discuss when this solution is unique of the nonlocal initial value
problem of Volttera integro-di erential equation.
Chapter 1
Basic concepts and de nitions
1.1 Introduction
In this chapter, we collect together the basic concepts and de nitions which will be needed
in the thesis.
1.2 Preliminaries and Notations
(1) Let C[a; b] be the class of continuous functions on the interval I = [a; b], with the
norm de ned by
kfk = sup
t2[a;b]
jf(t)j:
(2) AC[a,b] denotes the class of absolutely continuous functions de ned on the interval
I=[a,b].
(3) Let L1 = L1[a; b] be the class of Lebesgue integrable functions on the interval [a; b],
0 a < b < 1, with norm de ned by
kfk =
Z b
a
j f(t) j dt ; f 2 L1:
1
CHAPTER 1. BASIC CONCEPTS AND DEFINITIONS 2
(4) The function f be absolutely continuous on [a; b], if given > 0,
9 > 0 such that,
1X
k=1
jf(bk) f(ak)j < ;
8 nite sum of pairwise disjoint subintervals (ak; bk) [a; b]
of total length
1X
k=1
(bk ak) < :
1.3 Fixed point theorems
Fixed.