الفهرس 
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Abstract
The theory of semigroup of linear operators emphasizing those aspects which are of
importance in applications. Semigroup theory is generally accepted as an integral part
of functional analysis and is included in most standard treatises on functional analysis
which should be consulted for details if necessary. The abstract parts of the theory are in
many ways easier than the specialization to partial di erential equations. Nevertheless
the abstract formulation has the advantage that it provides a direct generalization of nite
dimensional models and makes the transition more transparent,especially in
the application to control problems.
The main object of this thesis is to study the existence of a unique uniformly stable
solution of the coupled systems of di erential equations
The thesis consists of three chapters.
Chapter 1: Collects some concepts, de nitions and auxiliary facts explored in further
chapters.
In chapter 2 we study the existence of a unique uniformly stable solution of each
of the coupled system of di erential equations
du(t)
dt
= A1v(t) + f1(t; v(t)) ; t > 0; and u(0) = u0 2 X (1)
dv(t)
dt
= A2u(t) + f2(t; u(t)) ; t > 0; and v(0) = v0 2 X (2)
v
where A1;A2 2 B(X):
and the coupled system of di erential equations
du(t)
dt
= A1(t)v(t) + f1(t; v(t)) ; t > 0; and u(0) = u0 2 X (3)
dv(t)
dt
= A2(t)u(t) + f2(t; u(t)) ; t > 0; and v(0) = v0 2 X (4)
where fAi(t); t 0g is a family of uniformly bounded linear operators de ned on
the Banach space X.
In chapter 3 we study the existence of a unique uniformly stable solution of
the coupled system of di erential equations
du(t)
dt
= A1u(t) + f1(t; v(t)) ; t > 0; and u(0) = u0 2 X (5)
dv(t)
dt
= A2v(t) + f2(t; u(t)) ; t > 0; and v(0) = v0 2 X (6)
where A1,A2 are in nitesimal generators of the two semigroups
fT1(t); t 0g and fT2(t); t 0g respectively and I = [0; T].
Chapter 1
Basic concepts and de nitions
Introduction
Here we give some preliminaries and known results which will be needed in the thesis.
1.1 Function spaces
Let X be a Banach space, then we can de ne the following.
1. C[0; T] denotes the Banach space of all continous functions f de ned on [0; T].
2. B(X) denotes the space of all bounded linear operators de ned on X.
3. C(I;X) denotes the space of all continous functions de ned on the interval
I = [0; T] with values in X, with the norm
jjfjj
X = sup
t2I
eNt jjf(t)jjX ;N 0; f 2 C(I;X)
which is equivalent to the usual norm
jjfjjX = sup
t2I
jjf(t)jjX; f 2 C(I;X)
and the norm jj:jjX is the norm on X.
4. Y is the Banach space of all 2-column vectors
u
v
!
; u; v 2 X, with the norm
jj
u
v
!
jjY = jjujjX + jjvjjX:
1
CHAPTER 1. BASIC CONCEPTS AND DEFINITIONS 2
5. C(I; Y ) denotes the space of all continous 2-column vectors on the interval I
with values in Y, with the norm
jj
u
v
!
jj
Y = jjujj
X + jjvjj
X:
1.2 Exponential operator (see[3],[19])
Let X be a Banach space, A 2 B(X) then, the exponential operator eA de ned by
eA =
1X
0
1
k!
Ak:
Now let A 2 B(X) and t 2 I, then we de ne the function etA by:
etA =
1X
0
1
k!
tkAk:
Also the following properties can be easily veri ed (see[28]) :
(1) e0 = I
(2) etA:esA = e(s+t)A
(3) d
dt etA = AetA
1.3 Abstract di erential equations (see[4],[5],[9],[10])
Let X be a Banach space, A 2 B(X).