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Abstract Mathematical modelling of real-life problems usually results in partial dierential equations or integral and integro-dierential equations, stochastic equations. Many mathematical formulation of physical phenomena contain integro-dierential equations, these equations arises in many elds like uid dynamics, biological models and chemical kinetics integro-dierential 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- dierential equations. The thesis consists of three chapters Chapter 1: Collects the concepts, denitions 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-dierential 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-dierential equation. Chapter 1 Basic concepts and denitions 1.1 Introduction In this chapter, we collect together the basic concepts and denitions 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 dened by kfk = sup t2[a;b] jf(t)j: (2) AC[a,b] denotes the class of absolutely continuous functions dened 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 dened 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. |