International Journal of Difference Equations ISSN 0973-6069, Volume 15, Number 2, pp. 209–241 (2020) http://campus.mst.edu/ijde On a Fractional Integro-Differential System Involving Riemann–Liouville and Caputo Derivatives with Coupled Multi-Point Boundary Conditions Bashir Ahmad and Ahmed Alsaedi King Abdulaziz University Nonlinear Analysis and Applied Mathematics Research Group P.O. Box 80203, Jeddah 21589, Saudi Arabia bashirahmad - [email protected][email protected]Sotiris K. Ntouyas University of Ioannina, Department of Mathematics 451 10 Ioannina, Greece [email protected]Ymnah Alruwaily Aljouf University Department of Mathematics, Faculty of Science King Khaled Road, Sakaka 72388, Saudi Arabia [email protected]Abstract We introduce a new class of coupled sequential fractional differential equations involving Riemann–Liouville and Caputo derivatives, integral and nonintegral type nonlinearities and equipped with coupled multi-point boundary conditions. Exis- tence results for the given problem are derived by means of Leray–Schauder non- linear alternative and Krasnosel’ski˘ ı’s fixed point theorem, while the uniqueness of solutions is established via contraction mapping principle. Examples illustrating the main results are presented. AMS Subject Classifications: Riemann–Liouville fractional derivative, Caputo frac- tional derivative, system, nonlocal boundary conditions, existence, fixed point theorem. Keywords: 26A33, 34B15. Received June 9, 2020; Accepted June 22, 2020 Communicated by Paul Eloe
Transcript
International Journal of Difference EquationsISSN 0973-6069, Volume 15, Number 2, pp. 209–241 (2020)http://campus.mst.edu/ijde
On a Fractional Integro-Differential System InvolvingRiemann–Liouville and Caputo Derivatives with
Coupled Multi-Point Boundary Conditions
Bashir Ahmad and Ahmed AlsaediKing Abdulaziz University
Nonlinear Analysis and Applied Mathematics Research GroupP.O. Box 80203, Jeddah 21589, Saudi Arabia
AbstractWe introduce a new class of coupled sequential fractional differential equations
involving Riemann–Liouville and Caputo derivatives, integral and nonintegral typenonlinearities and equipped with coupled multi-point boundary conditions. Exis-tence results for the given problem are derived by means of Leray–Schauder non-linear alternative and Krasnosel’skiı’s fixed point theorem, while the uniqueness ofsolutions is established via contraction mapping principle. Examples illustratingthe main results are presented.
Received June 9, 2020; Accepted June 22, 2020Communicated by Paul Eloe
210 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
1 IntroductionThe tools of fractional calculus are found to be of great utility in improving the mathe-matical modeling of many real world processes. Examples include disease models [8,11], ecological models [16], economic models [23], fractional neural networks [6, 29],chaotic synchronization [27, 28], etc. The interest in fractional calculus owes to thenonlocal nature of fractional-order differential and integral operators, which takes intoaccount the past history of the phenomenon under investigation [10, 17].
Boundary value problems involving different kinds of fractional-order operatorssuch as Riemann–Liouville, Caputo, Hadamard, etc., received much attention in re-cent years. For some recent works on fractional order boundary value problems with avariety of boundary conditions, see [1, 2, 7, 9, 15, 19] and the references cited therein.In a recent paper [4], the authors discussed the existence of solutions for a nonlin-ear fractional integro-differential equation involving two Caputo fractional derivativesof different orders and a Riemann–Liouville integral, equipped with dual anti-periodicboundary conditions. There has also been shown a great interest in the study of cou-pled systems of fractional differential equations in view of their applications in manyphysical situations [3, 12–14, 20, 21, 24–26].
In this paper, we investigate the existence and uniqueness of solutions for a nonlin-ear coupled system of sequential fractional differential equations involving Riemann–Liouville and Caputo derivatives, and integral (in the sense of Riemann–Liouville) andnonintegral type nonlinearities on an arbitrary domain:
RLDq1[(cDp1 + κ1)x(t) + λ1I
γ1h(t, x(t), y(t))]
= φ(t, x(t), y(t)), t ∈ [a, b],
RLDq2[(cDp2 + κ2)y(t) + λ2I
γ2u(t, x(t), y(t))]
= ψ(t, x(t), y(t)), t ∈ [a, b],
(1.1)complemented with coupled (nonconjugate type) multi-point boundary conditions:
x(a) =n−2∑i=1
αiy(ξi), x′(a) = 0, x(b) = 0, x′(b) = 0,
y(a) =n−2∑i=1
βix(ηi), y′(a) = 0, y(b) = 0, y′(b) = 0,
(1.2)
where 1 < p1, q1 ≤ 2, 1 < p2, q2 ≤ 2, cDϑ denotes the Caputo fractional differentialoperator of order ϑ (ϑ = p1, p2), RlD% denotes the Riemann–Liouville fractional dif-ferential operator of order % (% = q1, q2), with p1 + q1 > 3, p2 + q2 > 3, Iγ1 , Iγ2
are Riemann–Liouville fractional integrals of order γ1, γ2 > 1, κi, λi ∈ R, i = 1, 2,h, φ, u, ψ : [a, b] × R2 → R are given continuous functions, a < ξ1 < ξ2 < · · · <ξn−2 < b, a < η1 < η2 < · · · < ηn−2 < b, αj, βj ∈ R, j = 1, 2, · · · , n− 2.
Here we emphasize that the present work is motivated by a recent work [5] in whichexistence and uniqueness results for a mixed fractional order coupled system supple-
Multi-Point Fractional Integro-Differential System 211
mented with nonlocal multi-point and Riemann–Stieltjes integral-multi-strip conditionswere obtained.
We organize the rest of the paper as follows. In Section 2, we recall some basicdefinitions of fractional calculus and present an auxiliary lemma, which plays a key rolein obtaining the desired results. In Section 3, we discuss the existence of solutions forthe given problem while the uniqueness result is presented in the last section.
2 Preliminary MaterialWe begin this section with basic definitions of fractional calculus [10, 17].
Definition 2.1. The Riemann–Liouville fractional integral of order β > 0 for ϕ ∈L1[a, b], existing almost everywhere on [a, b], is defined by
Iβϕ(t) =
∫ t
a
(t− s)β−1
Γ(β)ϕ(s)ds,
where Γ denotes the Euler gamma function.
Definition 2.2. The Riemann–Liouville and Caputo fractional derivatives of order β ∈(n − 1, n], n ∈ N, for ϕ ∈ ACn[a, b], existing almost everywhere on [a, b], are respec-tively defined by
RLDβϕ(t) =dn
dtn
∫ t
a
(t− s)n−β−1
Γ(n− β)ϕ(s)ds,
andcDβϕ(t) =
∫ t
a
(t− s)n−β−1
Γ(n− β)ϕ(n)(s)ds.
Lemma 2.3. The general solution of the fractional differential equation cDβϕ(t) =0, m− 1 < β ≤ m, t ∈ [a, b], is
Lemma 2.4 (See [17]). For β > 0 and ϕ ∈ C[a, b]∩L[a, b], the general solution of theequation (RLDβϕ)(t) = 0 is
ϕ(t) =m∑j=1
σj(t− a)β−j,
212 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
where σj ∈ R, j = 1, 2, . . . ,m. Moreover,
(Iβ RLDβϕ)(t) = ϕ(t) +m∑j=1
σj(t− a)β−j, (RLDβIβϕ)(t) = ϕ(t).
Lemma 2.5. Assume 1 < p1, q1 ≤ 2, and 1 < p2, q2 ≤ 2. For H,Φ, U,Ψ ∈ C[a, b] ∩L[a, b], the solution of the linear system of fractional differential equations:
RLDq1[(cDp1 + κ1)x(t) + λ1I
γ1H(t)]
= Φ(t), t ∈ [a, b],
RLDq2[(cDp2 + κ2)y(t) + λ2I
γ2U(t)]
= Ψ(t), t ∈ [a, b],(2.1)
equipped with the boundary conditions (1.2) is equivalent to the system of integral equa-tions:
x(t) =− κ1∫ t
a
(t− s)p1−1
Γ(p1)x(s)ds− λ1
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)H(s)ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)Φ(s)ds+ f1(t)
[κ1
∫ b
a
(b− s)p1−1
Γ(p1)x(s)ds
+ λ1
∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)H(s)ds−
∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)Φ(s)ds
]+ f6(t)
[κ1
∫ b
a
(b− s)p1−2
Γ(p1 − 1)x(s)ds+ λ1
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)H(s)ds
−∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)Φ(s)ds
]+ f2(t)
[− κ1
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1−1
Γ(p1)x(s)ds
− λ1n−2∑i=1
βi
∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)H(s)ds+
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)Φ(s)ds
]+ f3(t)
[κ2
∫ b
a
(b− s)p2−1
Γ(p2)y(s)ds+ λ2
∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)U(s)ds (2.2)
−∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)Ψ(s)ds
]+ f4(t)
[κ2
∫ b
a
(b− s)p2−2
Γ(p2 − 1)y(s)ds
+ λ2
∫ b
a
(b− s)γ2+p2−2
Γ(γ2 + p2 − 1)U(s)ds−
∫ b
a
(b− s)p2+q2−2
Γ(p2 + q2 − 1)Ψ(s)ds
]+ f5(t)
[− κ2
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2−1
Γ(p2)y(s)ds
− λ2n−2∑i=1
αi
∫ ξi
a
(ξi − s)γ2+p2−1
Γ(γ2 + p2)U(s)ds+
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2+q2−1
Γ(p2 + q2)Ψ(s)ds
],
Multi-Point Fractional Integro-Differential System 213
Multi-Point Fractional Integro-Differential System 215
Proof. Solving the fractional differential equations (2.1) in a standard manner by usingLemmas 2.3 and 2.4, we get
x(t) = −κ1∫ t
a
(t− s)p1−1
Γ(p1)x(s)ds− λ1
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)H(s)ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)Φ(s)ds+ c1
(t− a)p1+q1−1Γ(q1)
Γ(p1 + q1)
+c2(t− a)p1+q1−2Γ(q1 − 1)
Γ(p1 + q1 − 1)+ c3 + c4(t− a), (2.19)
y(t) = −κ2∫ t
a
(t− s)p2−1
Γ(p2)y(s)ds− λ2
∫ t
a
(t− s)γ2+p2−1
Γ(γ2 + p2)U(s)ds
+
∫ t
a
(t− s)p2+q2−1
Γ(p2 + q2)Ψ(s)ds+ b1
(t− a)p2+q2−1Γ(q2)
Γ(p2 + q2)
+b2(t− a)p2+q2−2Γ(q2 − 1)
Γ(p2 + q2 − 1)+ b3 + b4(t− a), (2.20)
where ci, bi ∈ R, i = 1, 2, 3, 4, are unknown arbitrary constants. Using the boundaryconditions (1.2) in equations (2.19)–(2.20), together with notation (2.16) and (2.17), weobtain c4 = 0, b4 = 0, and
where νi, i = 1, . . . , 6, and ωj, j = 1, . . . , 5 are defined by (2.12) and (2.13) respec-tively. Substituting (2.36) in (2.29) and (2.37) in (2.31), we find that
where φi and ψi, i = 1, . . . , 6, are respectively given by (2.8) and (2.9). Insertingc4 = 0, b4 = 0, and the values of ck and bk, k = 1, 2, 3, from (2.36)–(2.41) in (2.19) and(2.20) leads to the solution (2.2) and (2.3). The converse follows by direct computation.This completes the proof.
3 Existence ResultsLetX = x(t)|x(t) ∈ C([a, b],R) denote the Banach space of all continuous functionsfrom [a, b] into R equipped with the norm ‖x‖ = sup
t∈[a,b]|x(t)|. Obviously (X , ‖ · ‖) is
a Banach space and consequently, the product space (X × X , ‖ · ‖) is a Banach spacewith the norm ‖(x, y)‖ = ‖x‖+ ‖y‖ for (x, y) ∈ X × X .
In view of Lemma 2.5, we define an operator T : X × X → X ×X as
T (x, y)(t) := (T1(x, y)(t), T2(x, y)(t)), (3.1)
where
T1(x, y)(t)
218 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
= −κ1∫ t
a
(t− s)p1−1
Γ(p1)x(s)ds− λ1
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds+ f1(t)
[κ1
∫ b
a
(b− s)p1−1
Γ(p1)x(s)ds
+λ1
∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds−
∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+f6(t)
[κ1
∫ b
a
(b− s)p1−2
Γ(p1 − 1)x(s)ds+ λ1
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)h(s, x(s), y(s))ds
−∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)φ(s, x(s), y(s))ds
]+ f2(t)
[− κ1
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1−1
Γ(p1)x(s)ds
−λ1n−2∑i=1
βi
∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+ f3(t)
[κ2
∫ b
a
(b− s)p2−1
Γ(p2)y(s)ds
+λ2
∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds−
∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
]+f4(t)
[κ2
∫ b
a
(b− s)p2−2
Γ(p2 − 1)y(s)ds+ λ2
∫ b
a
(b− s)γ2+p2−2
Γ(γ2 + p2 − 1)u(s, x(s), y(s))ds
−∫ b
a
(b− s)p2+q2−2
Γ(p2 + q2 − 1)ψ(s, x(s), y(s))ds
]+ f5(t)
[− κ2
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2−1
Γ(p2)y(s)ds
−λ2n−2∑i=1
αi
∫ ξi
a
(ξi − s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
], (3.2)
T2(x, y)(t)
= −κ2∫ t
a
(t− s)p2−1
Γ(p2)y(s)ds− λ2
∫ t
a
(t− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+
∫ t
a
(t− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds+ g1(t)
[κ1
∫ b
a
(b− s)p1−1
Γ(p1)x(s)ds
+λ1
∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds−
∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+g2(t)
[κ1
∫ b
a
(b− s)p1−2
Γ(p1 − 1)x(s)ds+ λ1
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)h(s, x(s), y(s))ds
−∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)φ(s, x(s), y(s))ds
]+ g6(t)
[− κ1
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1−1
Γ(p1)x(s)ds
Multi-Point Fractional Integro-Differential System 219
−λ1n−2∑i=1
βi
∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+ g4(t)
[κ2
∫ b
a
(b− s)p2−1
Γ(p2)y(s)ds
+λ2
∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds−
∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
]+g5(t)
[κ2
∫ b
a
(b− s)p2−2
Γ(p2 − 1)y(s)ds+ λ2
∫ b
a
(b− s)γ2+p2−2
Γ(γ2 + p2 − 1)u(s, x(s), y(s))ds
−∫ b
a
(b− s)p2+q2−2
Γ(p2 + q2 − 1)ψ(s, x(s), y(s))ds
]+ g3(t)
[− κ2
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2−1
Γ(p2)y(s)ds
−λ2n−2∑i=1
αi
∫ ξi
a
(ξi − s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
], (3.3)
and fi(t), i = 1, . . . , 6 are given by (2.4), and gj(t), j = 1, . . . , 5 and g6(t) are respec-tively defined by (2.5) and (2.6).
In the forthcoming analysis, we assume that h, φ, u, ψ : [a, b]×R2 → R are contin-uous functions satisfying the following condition:
(H1) ∀ t ∈ [a, b], x, y ∈ R there exist real constants µi, εi, ni, mi ≥ 0, i =1, 2, µ0, ε0, n0,m0 > 0 such that
Now we present our main results. The first existence theorem for the system (1.1)-(1.2) relies on Leray–Schauder alternative.
Lemma 3.1 (Leray–Schauder alternative [22]). Let M : Y −→ Y be a completelycontinuous operator (i.e., a map that restricted to any bounded set in Y is compact).Let G(M) = y ∈ Y : y = λM(y) for some 0 < λ < 1. Then either the set G(M) isunbounded, or M has at lest one fixed point.
Theorem 3.2. Assume that h, φ, u, ψ : [a, b] × R2 → R are continuous functions sat-isfying the assumption (H1). Then the system (1.1)–(1.2) has at least one solution on[a, b] if V < 1, where V is given by (3.9).
Proof. First we show that the operator T : X × X → X × X is completely continu-ous. By continuity of functions h, φ, u and ψ, it is easy to very that the operator T iscontinuous.
Let Br ⊂ X × X where Br = (x, y) ∈ X × X : ‖(x, y)‖ ≤ r. Then there existpositive constants ζi (i = 1, . . . , 4) such that |h(t, x(t), y(t))| ≤ ζ1, |φ(t, x(t), y(t))| ≤ζ2, |u(t, x(t), y(t))| ≤ ζ3, |ψ(t, x(t), y(t))| ≤ ζ4, ∀(x, y) ∈ Br. Then, for any (x, y) ∈Br, we have
|T1(x, y)(t)|
≤ |κ1|∫ t
a
(t− s)p1−1
Γ(p1)|x(s)|ds+ |λ1|
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)ζ1ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)ζ2ds+ |f1(t)|
[|κ1|
∫ b
a
(b− s)p1−1
Γ(p1)|x(s)|ds
+|λ1|∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)ζ1ds+
∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)ζ2ds
]+|f6(t)|
[|κ1|
∫ b
a
(b− s)p1−2
Γ(p1 − 1)|x(s)|ds+ |λ1|
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)ζ1ds
222 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
In consequence, |T1(x, y)(t2) − T1(x, y)(t1)| → 0 independent of x and y as t2 →t1. Analogously, we can obtain that |T2(x, y)(t2) − T2(x, y)(t1)| → 0 independentof x and y as t2 → t1. Therefore, the operator T (x, y) is equicontinuous. In viewof the foregoing steps, it follows by the Arzela–Ascoli theorem that the operator T iscompletely continuous.
Next, it will be verified that the set E = (x, y) ∈ X × X|(x, y) = σT (x, y), 0 ≤σ ≤ 1 is bounded. Let (x, y) ∈ E , then (x, y) = σT (x, y) and for any t ∈ [a, b], wehave x(t) = σT1(x, y)(t), y(t) = σT2(x, y)(t). As before, we can find that
This shows that the set E is bounded. Thus, by Lemma 3.1, we deduce that the operatorT has at least one fixed point. Therefore, the system (1.1)–(1.2) has at least one solutionon [a, b].
Example 3.3. Consider the coupled system of fractional differential equations:RLD3/2
[(cD9/5 +
1
16)x(t) +
2
110I23/5h(t, x(t), y(t))
]= φ(t, x(t), y(t)),
RLD4/3[(cD13/7 +
5
99)x(t) +
3
707I11/3u(t, x(t), y(t))
]= ψ(t, x(t), y(t)),
(3.12)t ∈ [0, 1], equipped with the boundary conditions:
x′(0) = 0, x(1) = 0, x′(1) = 0, x(0) =3∑i=1
αiy(ξi),
y′(0) = 0, y(1) = 0, y′(1) = 0, y(0) =3∑i=1
βix(ξi),
(3.13)
226 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
Moreover, we have µ0 = 3/2, µ1 = 1/15, µ2 = 1/9, ε0 = 1/24, ε1 = 1/30, ε2 =1/9, n0 = 1/2, n1 = 1/10, n2 = 1/16, m0 = 4, m1 = 1/10, and m2 = 1/414 as
|h(t, x(t), y(t))| ≤ 3
2+
1
15‖x‖+
1
9‖y‖, |φ(t, x(t), y(t))| ≤ 1
24+
1
30‖x‖+
1
9‖y‖,
|u(t, x(t), y(t))| ≤ 1
2+
1
10‖x‖+
1
16‖y‖, |ψ(t, x(t), y(t))| ≤ 4 +
1
10‖x‖+
1
414‖y‖.
From (3.7) and (3.8), we get V1 ' 0.164378, V2 ' 0.133751 and V = maxV1, V2 '0.164378 < 1. Therefore, by Theorem 3.2, the problem (3.12)–(3.13) has at least onesolution on [0, 1].
Our next existence result is based on the following version of Krasnosel’skiı’s fixedpoint theorem [18].
Lemma 3.4. Let Y be a closed, bounded, convex and nonempty subset of a Banachspace K. Let J1,J2 be operators mapping Y to K such that
(a) J1y1 + J2y2 ∈ Y where y1, y2 ∈ Y ;
(b) J1 is compact and continuous;
(c) J2 is a contraction mapping.
Then there exists y ∈ Y such that y = J1y + J2y.
In the sequel, it is assumed that h, φ, u, ψ : [a, b]×R2 → R are continuous functionssatisfying the following condition:
(H2) ∀t ∈ [a, b] and xj, yj ∈ R, j = 1, 2, there exist Li, i = 1, . . . , 4 such that
where Ai, Bi (i = 0, . . . , 5) are respectively given by (3.4) and (3.5).
Theorem 3.5. Assume that h, φ, u, ψ : [a, b] × R2 → R are continuous functions sat-isfying the condition (H2). Furthermore, we assume that there exist positive constantsFi (i = 1, . . . , 4) such that ∀ t ∈ [a, b] and x, y ∈ R,
where ∆2, ∆1 and Qi (i = 0, . . . , 5) are defined by (3.17), (3.18) and (3.20) respec-tively.
Proof. Consider a closed ball Mr∗ = (x, y) ∈ X × X : ‖(x, y)‖ ≤ r∗ with
r∗ ≥ maxr1, r2, (3.23)
where
r1 =|λ1|A1F1 +A2F2 + |λ2|A4F3 +A5F4
1− |κ1|A0 − |κ2|A3
,
r2 =|λ1|B1F1 + B2F2 + |λ2|B4F3 + B5F4
1− |κ1|B0 − |κ2|B3.
In order to verify the hypotheses of Lemma 3.4, we decompose the operator T into fouroperators T1,1, T1,2, T2,1 and T2,2 on Mr∗ as follows:
T1,1(x, y)(t) = −κ1∫ t
a
(t− s)p1−1
Γ(p1)x(s)ds− λ1
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds,
T1,2(x, y)(t)
Multi-Point Fractional Integro-Differential System 229
= f1(t)[κ1
∫ b
a
(b− s)p1−1
Γ(p1)x(s)ds+ λ1
∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
−∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+ f6(t)
[κ1
∫ b
a
(b− s)p1−2
Γ(p1 − 1)x(s)ds
+λ1
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)h(s, x(s), y(s))ds−
∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)φ(s, x(s), y(s))ds
]+f2(t)
[− κ1
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1−1
Γ(p1)x(s)ds
−λ1n−2∑i=1
βi
∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+ f3(t)
[κ2
∫ b
a
(b− s)p2−1
Γ(p2)y(s)ds
+λ2
∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds−
∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
]+f4(t)
[κ2
∫ b
a
(b− s)p2−2
Γ(p2 − 1)y(s)ds+ λ2
∫ b
a
(b− s)γ2+p2−2
Γ(γ2 + p2 − 1)u(s, x(s), y(s))ds
−∫ b
a
(b− s)p2+q2−2
Γ(p2 + q2 − 1)ψ(s, x(s), y(s))ds
]+ f5(t)
[− κ2
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2−1
Γ(p2)y(s)ds
−λ2n−2∑i=1
αi
∫ ξi
a
(ξi − s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
],
T2,1(x, y)(t) = −κ2∫ t
a
(t− s)p2−1
Γ(p2)y(s)ds− λ2
∫ t
a
(t− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+
∫ t
a
(t− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds,
T2,2(x, y)(t)
= g1(t)[κ1
∫ b
a
(b− s)p1−1
Γ(p1)x(s)ds+ λ1
∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
−∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+ g2(t)
[κ1
∫ b
a
(b− s)p1−2
Γ(p1 − 1)x(s)ds
+λ1
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)h(s, x(s), y(s))ds−
∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)φ(s, x(s), y(s))ds
]+g6(t)
[− κ1
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1−1
Γ(p1)x(s)ds
230 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
−λ1n−2∑i=1
βi
∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
n−2∑i=1
βi
∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
]+ g4(t)
[κ2
∫ b
a
(b− s)p2−1
Γ(p2)y(s)ds
+λ2
∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds−
∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
]+g5(t)
[κ2
∫ b
a
(b− s)p2−2
Γ(p2 − 1)y(s)ds+ λ2
∫ b
a
(b− s)γ2+p2−2
Γ(γ2 + p2 − 1)u(s, x(s), y(s))ds
−∫ b
a
(b− s)p2+q2−2
Γ(p2 + q2 − 1)ψ(s, x(s), y(s))ds
]+ g3(t)
[− κ2
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2−1
Γ(p2)y(s)ds
−λ2n−2∑i=1
αi
∫ ξi
a
(ξi − s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+
n−2∑i=1
αi
∫ ξi
a
(ξi − s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
].
Notice that T1(x, y)(t) = T1,1(x, y)(t) + T1,2(x, y)(t) and T2(x, y)(t) = T2,1(x, y)(t) +T2,2(x, y)(t) on Mr∗ . To verify condition (a) of Lemma 3.4, we use (3.23) to show thatTMr∗ ⊂Mr∗ . Setting x = (x1, x2), y = (y1, y2), x = (x1, x2) and y = (y1, y2) ∈Mr∗ ,and using condition (3.21), we obtain
|T1,1(x, y)(t) + T1,2(x, y)(t)|
≤ supt∈[a,b]
|κ1|
∫ t
a
(t− s)p1−1
Γ(p1)|x(s)|ds+ |λ1|
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)F1ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)F2ds+ |f1(t)|
[|κ1|
∫ b
a
(b− s)p1−1
Γ(p1)|x(s)|ds
+|λ1|∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)F1ds+
∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)F2ds
]+|f6(t)|
[|κ1|
∫ b
a
(b− s)p1−2
Γ(p1 − 1)|x(s)|ds+ |λ1|
∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)F1ds
+
∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)F2ds
]+ |f2(t)|
[|κ1|
n−2∑i=1
|βi|∫ ηi
a
(ηi − s)p1−1
Γ(p1)|x(s)|ds
+|λ1|n−2∑i=1
|βi|∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)F1ds+
n−2∑i=1
|βi|∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)F2ds
]+|f3(t)|
[|κ2|
∫ b
a
(b− s)p2−1
Γ(p2)|y(s)|ds+ |λ2|
∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)F3ds
+
∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)F4ds
]+ |f4(t)|
[|κ2|
∫ b
a
(b− s)p2−2
Γ(p2 − 1)|y(s)|ds
Multi-Point Fractional Integro-Differential System 231
It clearly follows from the above two inequalities that T1(x, y) + T2(x, y) ∈Mr∗ .Next we show that the operator (T1,1, T2,1) is compact and continuous, which means thatthe condition (b) of Lemma 3.4 is satisfied. Continuity of (T1,1, T2,1) follows from thatof h, φ, u, ψ. For each (x, y) ∈Mr∗ , we have
‖T1,1(x, y)‖ ≤ supt∈[a,b]
|T1,1(x, y)(t)|
≤ supt∈[a,b]
∣∣∣− κ1 ∫ t
a
(t− s)p1−1
Γ(p1)x(s)ds− λ1
∫ t
a
(t− s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
∫ t
a
(t− s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
∣∣∣≤ |κ1|
(b− a)p1r∗
Γ(p1 + 1)+ |λ1|
(b− a)γ1+p1F1
Γ(γ1 + p1 + 1)+
(b− a)p1+q1F2
Γ(p1 + q1 + 1)= S1,
and
‖T2,1(x, y)‖ ≤ supt∈[a,b]
|T2,1(x, y)(t)|
≤ supt∈[a,b]
∣∣∣− κ2 ∫ t
a
(t− s)p2−1
Γ(p2)y(s)ds− λ2
∫ t
a
(t− s)γ2+p2−1
Γ(γ2 + p2)u(s, x(s), y(s))ds
+
∫ t
a
(t− s)p2+q2−1
Γ(p2 + q2)ψ(s, x(s), y(s))ds
∣∣∣≤ |κ2|
(b− a)p2r∗
Γ(p2 + 1)+ |λ2|
(b− a)γ2+p2F3
Γ(γ2 + p2 + 1)+
(b− a)p2+q2F4
Γ(p2 + q2 + 1)= S2,
which leads to‖(T1,1, T2,1)(x, y)‖ ≤ S1 + S2.
Thus the set (T1,1, T2,1)Mr∗ is uniformly bounded. Furthermore, we show that the set(T1,1, T2,1)Mr∗ is equicontinuous. For a ≤ t1 < t2 ≤ b and for any (x, y) ∈ Mr∗ , we
232 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
obtain
|T1,1(x, y)(t2)− T1,1(x, y)(t1)|
≤ supt∈[a,b]
∣∣∣− κ1 ∫ t1
a
(t2 − s)p1−1 − (t1 − s)p1−1
Γ(p1)x(s)ds− κ1
∫ t2
t1
(t2 − s)p1−1
Γ(p1)x(s)ds
− λ1∫ t1
a
(t2 − s)γ1+p1−1 − (t1 − s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
− λ1∫ t2
t1
(t2 − s)γ1+p1−1
Γ(γ1 + p1)h(s, x(s), y(s))ds
+
∫ t1
a
(t2 − s)p1+q1−1 − (t1 − s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
+
∫ t2
t1
(t2 − s)p1+q1−1
Γ(p1 + q1)φ(s, x(s), y(s))ds
∣∣∣≤ |κ1|r
Γ(p1 + 1)
(|(t2 − a)p1 − (t1 − a)p1|+ 2(t2 − t1)p1
)+
|λ1|F1
Γ(γ1 + p1 + 1)
(|(t2 − a)γ1+p1 − (t1 − a)γ1+p1|+ 2(t2 − t1)γ1+p1
)+
F2
Γ(p1 + q1 + 1)
(|(t2 − a)p1+q1 − (t1 − a)p1+q1|+ 2(t2 − t1)p1+q1
).
Similarly, we can get
|T2,1(x, y)(t2)− T2,1(x, y)(t1)|
≤ |κ2|rΓ(p2 + 1)
(|(t2 − a)p2 − (t1 − a)p2|+ 2(t2 − t1)p2
)+
|λ2|F3
Γ(γ2 + p2 + 1)
(|(t2 − a)γ2+p2 − (t1 − a)γ2+p2|+ 2(t2 − t1)γ2+p2
)+
F4
Γ(p2 + q2 + 1)
(|(t2 − a)p2+q2 − (t1 − a)p2+q2|+ 2(t2 − t1)p2+q2
).
Hence, |(T1,1, T2,1)(x, y)(t2)−(T1,1, T2,1)(x, y)(t1)| tends to zero as t1 → t2 independentof (x, y) ∈ Mr∗ . Therefore, the set (T1,1, T2,1)Mr∗ is equicontinuous. Thus, by theArzela–Ascoli theorem, the operator (T1,1, T2,1) is compact on Mr∗ . Next we show thatthe operator (T1,2, T2,2) is a contraction satisfying condition (c) of Lemma 3.4. For(x1, y1), (x2, y2) ∈Mr∗ , t ∈ [a, b], we have
‖T1,2(x1, y1)− T1,2(x2, y2)‖ = supt∈[a,b]
|T1,2(x1, y1)(t)− T1,2(x2, y2)(t)|
≤ supt∈[a,b]
|f1(t)|
[|κ1|
∫ b
a
(b− s)p1−1
Γ(p1)|x1(s)− x2(s)|ds
Multi-Point Fractional Integro-Differential System 233
+|λ1|∫ b
a
(b− s)γ1+p1−1
Γ(γ1 + p1)
∣∣∣h(s, x1(s), y1(s))− h(s, x2(s), y2(s))∣∣∣ds
+
∫ b
a
(b− s)p1+q1−1
Γ(p1 + q1)
∣∣∣φ(s, x1(s), y1(s))− φ(s, x2(s), y2(s))∣∣∣ds]
+|f6(t)|[|κ1|
∫ b
a
(b− s)p1−2
Γ(p1 − 1)|x1(s)− x2(s)|ds
+|λ1|∫ b
a
(b− s)γ1+p1−2
Γ(γ1 + p1 − 1)
∣∣∣h(s, x1(s), y1(s))− h(s, x2(s), y2(s))∣∣∣ds
+
∫ b
a
(b− s)p1+q1−2
Γ(p1 + q1 − 1)
∣∣∣φ(s, x1(s), y1(s))− φ(s, x2(s), y2(s))∣∣∣ds]
+|f2(t)|[|κ1|
n−2∑i=1
|βi|∫ ηi
a
(ηi − s)p1−1
Γ(p1)|x1(s)− x2(s)|ds
+|λ1|n−2∑i=1
|βi|∫ ηi
a
(ηi − s)γ1+p1−1
Γ(γ1 + p1)
∣∣∣h(s, x1(s), y1(s))− h(s, x2(s), y2(s))∣∣∣ds
+n−2∑i=1
|βi|∫ ηi
a
(ηi − s)p1+q1−1
Γ(p1 + q1)
∣∣∣φ(s, x1(s), y1(s))− φ(s, x2(s), y2(s))∣∣∣ds]
+|f3(t)|[|κ2|
∫ b
a
(b− s)p2−1
Γ(p2)|y1(s)− y2(s)|ds
+|λ2|∫ b
a
(b− s)γ2+p2−1
Γ(γ2 + p2)
∣∣∣u(s, x1(s), y1(s))− u(s, x2(s), y2(s))∣∣∣ds
+
∫ b
a
(b− s)p2+q2−1
Γ(p2 + q2)
∣∣∣ψ(s, x1(s), y1(s))− ψ(s, x2(s), y2(s))∣∣∣ds]
+|f4(t)|[|κ2|
∫ b
a
(b− s)p2−2
Γ(p2 − 1)|y1(s)− y2(s)|ds
+|λ2|∫ b
a
(b− s)γ2+p2−2
Γ(γ2 + p2 − 1)
∣∣∣u(s, x1(s), y1(s))− u(s, x2(s), y2(s))∣∣∣ds
+
∫ b
a
(b− s)p2+q2−2
Γ(p2 + q2 − 1)
∣∣∣ψ(s, x1(s), y1(s))− ψ(s, x2(s), y2(s))∣∣∣ds]
+|f5(t)|[|κ2|
n−2∑i=1
|αi|∫ ξi
a
(ξi − s)p2−1
Γ(p2)|y1(s)− y2(s)|ds
+|λ2|n−2∑i=1
|αi|∫ ξi
a
(ξi − s)γ2+p2−1
Γ(γ2 + p2)
∣∣∣u(s, x1(s), y1(s))− u(s, x2(s), y2(s))∣∣∣ds
+n−2∑i=1
|αi|∫ ξi
a
(ξi − s)p2+q2−1
Γ(p2 + q2)
∣∣∣ψ(s, x1(s), y1(s))− ψ(s, x2(s), y2(s))|ds]
234 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
which, in view of (3.22), implies that the operator (T1,2, T2,2) is a contraction. Therefore,the condition (c) of Lemma 3.4 is satisfied. Thus, we deduce by the conclusion ofLemma 3.4 that the system (1.1)–(1.2) has at least one solution on [a, b].
Example 3.6. Consider the problem (3.12)–(3.13) with
h(t, x(t), y(t)) =1√
2500 + t
(2 cosx(t) +
2|y(t)|1 + |y(t)|
),
φ(t, x(t), y(t)) =t2 + 2
270
( (x(t) + 5)2
7 + (x(t) + 5)2+ y(t) + ln 3
),
u(t, x(t), y(t)) =1
8 + t
(tan−1 x(t)
12+y(t) + 13
12
),
ψ(t, x(t), y(t)) =e−t
14√
100 + t6
(sinx(t) + cos y(t) +
7
40
).
(3.26)
Observe that
|h(t, x1, y1)− h(t, x2, y2)| ≤1
25
(|x1 − x2|+ |y1 − y2|
),
Multi-Point Fractional Integro-Differential System 235
|φ(t, x1, y1)− φ(t, x2, y2)| ≤1
90
(|x1 − x2|+ |y1 − y2|
),
|u(t, x1, y1)− u(t, x2, y2)| ≤1
96
(|x1 − x2|+ |y1 − y2|
),
|ψ(t, x1, y1)− ψ(t, x2, y2)| ≤1
140
(|x1 − x2|+ |y1 − y2|
).
With the given data, we find that Q0 ' 0.767776, Q1 ' 0.001045, Q2 ' 0.153414,Q3 ' 0.610335, Q4 ' 0.003852, Q5 ' 0.143089, ∆2 ' 0.037588, ∆1 ' 0.022760,and (|κ1|Q0+|λ1|L1Q1+L2Q2+∆2)+(∆1+|κ2|Q3+|λ2|L3Q4+L4Q5) ' 0.141887 <1. As the hypothesis of Theorem 3.5 holds true, therefore its conclusion applies tothe coupled boundary value problem (3.12)–(3.13) with h(t, x(t), y(t)), φ(t, x(t), y(t)),u(t, x(t), y(t)) and ψ(t, x(t), y(t)) given by (3.26).
4 Uniqueness of SolutionsIn this section we apply Banach’s fixed point theorem to prove the uniqueness of solu-tions for the system (1.1)–(1.2). Before proceeding for this result, let us introduce thefollowing notation:
which implies that the operator T is a contraction by the assumption (4.4)). Hence,by Banach’s fixed point theorem, the operator T has a unique fixed point, which is theunique solution of the system (1.1)–(1.2) on [a, b].
Example 4.2. Consider the system (3.12)–(3.13) with
h(t, x(t), y(t)) =1√
64 + t2
(tan−1 x(t) + ln 7 + y(t)
),
φ(t, x(t), y(t)) =t2
70√
4 + t6
( |x(t)|1 + |x(t)|
+ sin y(t) + 29),
u(t, x(t), y(t)) =e−t + 2
30
(x(t) + cos y(t)
), t ∈ [0, 1],
ψ(t, x(t), y(t)) =t4 + 4√t2 + 3600
(sinx(t) +
tan−1 y(t)√1 + t2
)+
1
4.
(4.7)
It is easy to verify that (H2) holds true with L1 = 1/8, L2 = 1/140, L3 = 1/10,and L4 = 1/12. Using these values together with the data of Example 3.3, we find thatΩ+Ω ' 0.243797 < 1. Thus, all the conditions of Theorem 4.1 are satisfied. Hence, bythe conclusion of Theorem 4.1, the coupled system (3.12)–(3.13) with h(t, x(t), y(t)),φ(t, x(t), y(t)), u(t, x(t), y(t)) and ψ(t, x(t), y(t)) given by (4.7) has a unique solutionon [0, 1].
Remark 4.3. Fixing λ1 = 0 = λ2 in the results of this paper, we obtain the ones for anonlinear Langevin type system involving mixed Riemann-Liouville and Caputo frac-tional derivatives, supplemented with coupled multipoint boundary conditions.
Multi-Point Fractional Integro-Differential System 239
References[1] B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Hadamard-type fractional
differential equations, inclusions and inequalities, Springer, Cham, Switzerland,2017.
[2] B. Ahmad, R. Luca, Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions, Appl.Math. Comput. 339 (2018), 516–534.
[3] B. Ahmad, N. Alghamdi, A. Alsaedi, S.K. Ntouyas, A system of coupled multi-term fractional differential equations with three-point boundary conditions, Fract.Calc. Appl. Anal. 22 (2019), 601–618.
[4] B. Ahmad, Y. Alruwaily, A. Alsaedi, J.J. Nieto, Fractional integro-differentialequations with dual anti-periodic boundary conditions, Differential Integral Equa-tions 33 (2020), 181–206.
[5] B. Ahmad, R.P. Agarwal, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily, Fractional or-der coupled systems for mixed fractional derivatives with nonlocal multi-pointand Riemann–Stieltjes integral-multi-strip conditions, Dynam. Systems Appl. 29(2020), 71–86.
[6] M.S. Ali, G. Narayanan, V. Shekher, A. Alsaedi, B. Ahmad, Global Mittag–Lefflerstability analysis of impulsive fractional-order complex-valued BAM neural net-works with time varying delays, Commun. Nonlinear Sci. Numer. Simul. 83 (2020),105088, 22 pp.
[7] A. Alsaedi, B. Ahmad, M. Alghanmi, Extremal solutions for generalized Caputofractional differential equations with Steiltjes-type fractional integro-initial condi-tions, Appl. Math. Lett. 91 (2019), 113–120.
[8] A. Carvalho, C.M.A. Pinto, A delay fractional order model for the co-infection ofmalaria and HIV/AIDS, Int. J. Dyn. Control, 5:168–186, 2017.
[9] Z. Cen, L.-B. Liu, J. Huang, A posteriori error estimation in maximum norm for atwo-point boundary value problem with a Riemann–Liouville fractional derivative,Appl. Math. Lett. 102 (2020), 106086, 8 pp.
[10] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes inMathematics, Springer Verlag Berlin Heidelberg, 2010.
[11] Y. Ding, Z. Wang, H. Ye, Optimal control of a fractional-order HIV-immune sys-tem with memory, IEEE Trans. Contr. Sys. Techn., 20:763-769, 2012.
240 B. Ahmad, A. Alsaedi, S.K. Ntouyas, Y. Alruwaily
[12] J.A. Gallegos, N. Aguila-Camacho, M.A. Duarte-Mermoud, Smooth solutions tomixed-order fractional differential systems with applications to stability analysis,J. Integral Equations Appl. 31 (2019), 59–84.
[13] J. Henderson, R. Luca, Nonexistence of positive solutions for a system of coupledfractional boundary value problems, Bound. Value Probl. (2015), 2015:138.
[14] S. Hristova, R. Agarwal, D. O’Regan, D. Explicit solutions of initial value prob-lems for systems of linear Riemann-Liouville fractional differential equations withconstant delay, Adv. Difference Equ. (2020), 2020:180.
[15] G. Iskenderoglu, D. Kaya, Symmetry analysis of initial and boundary value prob-lems for fractional differential equations in Caputo sense, Chaos Solitons Fractals134 (2020), 109684.
[16] M. Javidi, B. Ahmad, Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system, Ecological Modelling, 318:8-18, 2015.
[17] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of FractionalDifferential Equations. North-Holland Mathematics Studies, 204. Elsevier ScienceB.V., Amsterdam, 2006.
[18] M. A. Krasnosel’skiı, Two remarks on the method of successive approximations,Usp. Mat. Nauk. 10 (1955),123-127.
[19] S.K. Ntouyas, S. Etemad, On the existence of solutions for fractional differentialinclusions with sum and integral boundary conditions, Appl. Math. Comput. 266(2015), 235–243.
[20] S.K. Ntouyas, H.H. Al-Sulami, A study of coupled systems of mixed order frac-tional differential equations and inclusions with coupled integral fractional bound-ary conditions, Adv. Difference Equ. (2020), 2020:73.
[21] K.M. Owolabi, B. Karaagac, Dynamics of multi-pulse splitting process in one-dimensional Gray-Scott system with fractional order operator, Chaos SolitonsFractals 136 (2020), 109835.
[22] D.R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.
[23] V.V. Tarasova, V.E. Tarasov, Logistic map with memory from economic model,Chaos Solitons Fractals 95 (2017), 84–91.
[24] J. Tariboon, S.K. Ntouyas, S. Asawasamrit, C. Promsakon, Positive solutions forHadamard differential systems with fractional integral conditions on an unboundeddomain, Open Math., 15:645-666, 2017.
Multi-Point Fractional Integro-Differential System 241
[25] H.T. Tuan, A. Czornik, J.J. Nieto, M. Niezabitowski, Global attractivity for someclasses of Riemann-Liouville fractional differential systems, J. Integral EquationsAppl. 31 (2019), 265–282.
[26] C. Wang, Y. Guo, S. Zheng, Y. Chen, Robust stability analysis of LTI systemswith fractional degree generalized frequency variables, Fract. Calc. Appl. Anal.22 (2019), 1655–1674.
[27] Y. Xu, W. Li, Finite-time synchronization of fractional-order complex-valued cou-pled systems, Phys. A 549 (2020), 123903.
[28] F. Zhang, G. Chen C. Li, J. Kurths, Chaos synchronization in fractional differentialsystems, Phil Trans R Soc A, 371:20120155, 2013.
[29] W. Zhang, H. Zhang, J. Cao, H. Zhang, D. Chen, Synchronization of delayedfractional-order complex-valued neural networks with leakage delay, Phys. A 556(2020), 124710.
![Certain Fractional Derivative Formulae Involving · we obtain the fractional derivative formula (2) after simplification, using the theorem 1 and the result ({4], p. Eq. (6)) Theorem](https://static.documents.pub/doc/80x56/5fa6bda43325ba2a6427a287/certain-fractional-derivative-formulae-involving-we-obtain-the-fractional-derivative.jpg)
