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Existence and Asymptotic Behavior of Positive Solutions for a Coupled Fractional Differential System Sonia Ben Makhlouf1 · Majda Chaieb1 · Zagharide Zine El Abidine1

© Foundation for Scientific Research and Technological Innovation 2017

Abstract In this paper, we take up the existence and the asymptotic behavior of positive and continuous solutions to the following coupled fractional differential system ⎧ α D u = a(x)u p vr in (0, 1), ⎪ ⎪ ⎨ β D v = b(x)u s v q in (0, 1), u(0) = u(1) = D α−3 u(0) = u  (1) = 0, ⎪ ⎪ ⎩ v(0) = v(1) = D β−3 v(0) = v  (1) = 0, where α, β ∈ (3, 4], p, q ∈ (−1, 1), r, s ∈ R such that (1 − | p|)(1 − |q|) − |r s| > 0, D is the standard Riemann–Liouville differentiation and a, b are nonnegative and continuous functions in (0, 1) allowed to be singular at x = 0 and x = 1 and they are required to satisfy some appropriate conditions related to Karamata regular variation theory. Keywords Riemann–Liouville fractional derivative · Green function · Asymptotic behavior · Karamata function · Schäuder’s fixed point theorem Mathematics Subject Classification 26A33 · 34A08 · 34B27 · 35B40

Introduction In recent years, the subject of fractional differential equations has gained a considerable attention and it has emerged as an interesting and popular field of research. It is mainly due

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Zagharide Zine El Abidine [email protected] Sonia Ben Makhlouf [email protected] Majda Chaieb [email protected]

1

Université de Tunis El Manar, Faculté des Sciences de Tunis, UR11ES22 Potentiels et Probabilités, 2092 Tunis, Tunisie

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Differ Equ Dyn Syst

to the fact that fractional calculus has numerous applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, aerodynamics, wave propagation, etc. An excellent account in the study of fractional differential equations can be found in [14,15,21,22]. Boundary value problems for fractional differential equations have been discussed in [1,5,6,8,12,13,17,28,30]. In [5], Bachar et al. considered the following sublinear fractional two-point boundary value problem 

D α u (x) = a(x)u p (x) , in (0, 1) , u(0) = u(1) = D α−3 u(0) = u(1) = u  (1) = 0,

(1)

where α ∈ (3, 4], p ∈ (−1, 1) and a is a nonnegative continuous function on (0, 1) that may be singular at x = 0 and x = 1 and D α is the fractional derivative in the sense of Riemann–Liouville. In order to state the result proved in [5], we need some notations. We shall use K to denote the set of Karamata functions L defined on (0, η] by  η z(s) ds , L(t) := c exp s t for some η > 1, where c > 0 and z ∈ C ([0, η]) such that z(0) = 0. For two nonnegative functions f and g defined on a set S, the notation f (x) ≈ g(x), x ∈ S, means that there exists a constant c > 0 such that 1c g(x) ≤ f (x) ≤ c g(x), for all x ∈ S. By using a fixed point argument, Bachar et al. studied in [5] problem (1), where the function a verifies (