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Abstract DQx(t) The development of Fractional Calculus within the framework of classical functions is now wep-known. One of the main applications of the fractional calculus is to study the existence of solutions of the integral and differential equations of fractional order. The main objects of this thesis are to consider the question of the existence ( and some- times existence and uniqueness) of positive monotonic solutions of initial and boundary value problems of multi-terms retarded functional differential equations of fractional (arbi- trary) orders. The thesis consists of six chapters. Chapter 1 collects some preliminaries, notations and known results which will be used in the other chapters. Also we introduce the main concepts of fractional-order integration, fractional-order differentiation and their properties. Finally we give a survey about the functional differential and integral equations of arbitrary (fractional) orders. Chapter 2 is devoted to prove the existence of at least one monotonic nondecreasing positive solution x E L1 (0, 1] to the initial value problems DQIX(t) = h(t, x(t), x(t - r), D~ix(t - r)), x(t) = 0, t:S 0, and 12(t, x(t), x(t - r), DQIX(t), D~i x(t - r), D~2X(t - r)) Djx(t) = 0, t:S 0, j = 0, l. where fr, fr2 E (1,2]’ a > a2 and a1, ai E (0,1] and r ~ ° is a real number, and the functions !1 and 12 are nonlinear functions satisfy the Caratheodory condition and some monotonicity conditions. Chapter 3 is devoted to prove the existence of at least one monotonic nonincreasing positive solution x E L1(0, 1) of the boundary value problem DQx(t) + h(t, - DQIX(t), - D~i x(t - r)) = 0, x’(t) = 0, t:S 0, x(1) = ° where the function h is a nonlinear functions satisfies the Caratheodory condition and some monotonicity conditions. |