Kragujevac Journal of Mathematics Volume 40(2) (2016), Pages 224–236. HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS: NEW EXISTENCE AND UNIQUENESS RESULTS LOUIZA TABHARIT 1 AND ZOUBIR DAHMANI 2 Abstract. In this paper, we study a class of high dimensional coupled fractional differential systems using Caputo approach. We investigate the existence of solutions using Schaefer fixed point theorem. Moreover, new existence and uniqueness results are obtained by using the contraction mapping principle. Finally, Some examples are presented to illustrate our main results. 1. Introduction and Preliminaries Fractional differential equations have gained a great interest because of their many applications in modeling of physical and chemical processes and in engineering sci- ences. For the basic theory of fractional differential equations, see [1–10, 12–14, 17, 22]. Moreover, the nonlinear coupled systems involving fractional derivatives are also very important, since they occur in various problems of applied mathematics. In the litera- ture, we can find many papers dealing with the existence and uniqueness of solutions. For more details, we refer the reader to [11,16,18–21] and the references therein. Motivated by cited coupled systems-papers, in this work, we discuss the existence and uniqueness of solutions for the following problem: Key words and phrases. Banach contraction principle, Caputo derivative, fixed point, differential equation, existence, uniqueness. 2010 Mathematics Subject Classification. 30C45, 39B72, 39B82. Received: October 12, 2015. Accepted: December 4, 2015. 224
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Kragujevac Journal of MathematicsVolume 40(2) (2016), Pages 224–236.
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS: NEWEXISTENCE AND UNIQUENESS RESULTS
LOUIZA TABHARIT1 AND ZOUBIR DAHMANI2
Abstract. In this paper, we study a class of high dimensional coupled fractionaldifferential systems using Caputo approach. We investigate the existence of solutionsusing Schaefer fixed point theorem. Moreover, new existence and uniqueness resultsare obtained by using the contraction mapping principle. Finally, Some examplesare presented to illustrate our main results.
1. Introduction and Preliminaries
Fractional differential equations have gained a great interest because of their manyapplications in modeling of physical and chemical processes and in engineering sci-ences. For the basic theory of fractional differential equations, see [1–10,12–14,17,22].Moreover, the nonlinear coupled systems involving fractional derivatives are also veryimportant, since they occur in various problems of applied mathematics. In the litera-ture, we can find many papers dealing with the existence and uniqueness of solutions.For more details, we refer the reader to [11,16,18–21] and the references therein.
Motivated by cited coupled systems-papers, in this work, we discuss the existenceand uniqueness of solutions for the following problem:
Key words and phrases. Banach contraction principle, Caputo derivative, fixed point, differentialequation, existence, uniqueness.
2010 Mathematics Subject Classification. 30C45, 39B72, 39B82.Received: October 12, 2015.Accepted: December 4, 2015.
224
HIGH DIMENSIONAL FRACTIONAL COUPLED SYSTEMS 225
Dα1u1 (t) =l∑
i=1
f 1i (t, u1 (t) , . . . , um (t) , Dγ1u1 (t) , . . . , Dγmum (t)) , t ∈ J,
Dα2u2 (t) =l∑
i=1
f 2i (t, u1 (t) , . . . , um (t) , Dγ1u1 (t) , . . . , Dγmum (t)) , t ∈ J,
where k = 1, 2, . . . ,m. We suppose that n − 1 < αk < n, γk ∈ ]0, n− 1[, k =1, 2, . . . ,m, n ∈ N∗ − 1, m, l ∈ N∗, τ ∈]0, 1[, J := [0, 1]. The derivatives Dαk , Dγk ,k = 1, 2, . . . ,m, are taken in the sense of Caputo. For the functions
(fki)k=1,2,...,m
i=1,...,l:
J × R2m → R, we will specify them later.The paper is organised as follows: We begin by introducing some definitions and
lemmas that will be used in the proof of the main results. Then, in the Main ResultsSection, we prove the existence of solutions theorems. At the last section, someillustrative examples are treated. So, let us now present the basic definitions andlemmas [15].
Definition 1.1. The Riemann-Liouville fractional integral operator of order α > 0,for a continuous function f on [0,∞[ is defined as:
Jαf(t) =
1
Γ (α)
∫ t
0
(t− s)α−1 f (s) ds, α > 0, t ≥ 0,
f(t), α = 0, t ≥ 0,
(1.2)
where Γ (α) :=∫∞0e−xxα−1dx.
Definition 1.2. The Caputo derivative of order α for a function u : [0,∞) → R,which is at least n-times differentiable can be defined as:
Dαu(t) =1
Γ (n− α)
∫ t
0
(t− s)n−α−1 u(n) (s) ds = Jn−αu(n)(t),
for n− 1 < α < n, n ∈ N∗ − 1.
We recall the following lemmas [7, 19].
226 L. TABHARIT AND Z. DAHMANI
Lemma 1.1. For α > 0, the general solution of the fractional differential equationDαu(t) = 0, is given by
u(t) =n−1∑j=0
cjtj,
where cj ∈ R, j = 0, . . . , n− 1, n = [α] + 1.
Lemma 1.2. Let α > 0. Then
JαDαu(t) = u(t) +n−1∑j=0
cjtj,
where cj ∈ R, j = 0, 1, . . . , n− 1, n = [α] + 1.
Lemma 1.3. Let q > p > 0, g ∈ L1 ([a, b]). Then DpJqf (t) = Jq−pf (t), t ∈ [a, b].
Lemma 1.4 (Schaefer fixed point Theorem). Let E be Banach space and T : E → Eis a completely continuous operator. If V = u ∈ E : u = µTu, 0 < µ < 1 is bounded,then T has a fixed point in E.
The proof of the following auxiliary lemma is crucial for the problem (1.1).
Lemma 1.5. Assume that(Qki
)k=1,...,m
i=1,...,l∈ C ([0, 1] ,R), m, l ∈ N∗. And consider the
problem
(1.3) Dαkuk(t) =l∑
i=1
Qki (t), t ∈ J, n− 1 < αk < n, n ∈ N∗ − 1 ,
associated with the conditions:uk (0) = ak0,
u(j)k (0) = 0, j = 1, 2, . . . , n− 2,
u(n−1)k (0) = Jrkuk(τk),
(1.4)
where k = 1, 2, . . . ,m. Then for all k = 1, 2, . . . ,m, we have
The right-hand sides of the above inequalities (2.9) and (2.10) are independent of(u1, u2, . . . , um) and tend to zero as t2 − t1 → 0. Then the operator P , is equi-continuous. Hence, the operator P is a completely continuous.
The functions fki are continuous and bounded on J × R6. Using Theorem 2.2, thesystem (3.2) has at least one solution on J .
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1Department of Mathematics and Informatics, Faculty SEI,University of Mostaganem,AlgeriaE-mail address: [email protected]
2LPAM, Faculty SEI,UMAB of Mostaganem,AlgeriaE-mail address: [email protected]