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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 dierential 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 dierential equations The thesis consists of three chapters. Chapter 1: Collects some concepts, denitions 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 dierential 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 dierential 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 dened on the Banach space X. In chapter 3 we study the existence of a unique uniformly stable solution of the coupled system of dierential 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 innitesimal generators of the two semigroups fT1(t); t 0g and fT2(t); t 0g respectively and I = [0; T]. Chapter 1 Basic concepts and denitions 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 dene the following. 1. C[0; T] denotes the Banach space of all continous functions f dened on [0; T]. 2. B(X) denotes the space of all bounded linear operators dened on X. 3. C(I;X) denotes the space of all continous functions dened 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 dened by eA = 1X 0 1 k! Ak: Now let A 2 B(X) and t 2 I, then we dene the function etA by: etA = 1X 0 1 k! tkAk: Also the following properties can be easily veried (see[28]) : (1) e0 = I (2) etA:esA = e(s+t)A (3) d dt etA = AetA 1.3 Abstract dierential equations (see[4],[5],[9],[10]) Let X be a Banach space, A 2 B(X). |