الفهرس 
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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.