Page 41 - IJOCTA-15-3

Modeling and analysis of the dynamics of an excessive gambling problem with modified fractional operator
and we have:
υ
(t 2 − t 1 ) 1 − υ
υ
||I 2 − I 1 || ≤ ||Φ|| + τ (t 2 − t 1 ) υ . ||T[m 2 ](t) − T[m 1 ](t)|| ≤ η Φ ||m 2 − m 1 ||
υ B (υ)
(16) υ
From Equations (14) -(16), we have: υ τ
+ η Φ ||m 2 − m 1 ||
||T[m] (t 2 ) − T[m] (t 1 ) || B (υ) Γ (υ) υ

1 − υ υ
1 − υ = + τ υ η Φ ||m 2 − m 1 ||.
≤ (η Φ (|t 2 − t 1 | + ||m(t 2 − m(t 1 )||)) B (υ) B (υ) Γ (υ + 1)
ψ(υ)
Define the contraction constant:
υ
υ (t 2 − t 1 ) 1 − υ υ
υ
+ ||Φ|| + τ (t 2 − t 1 ) υ k(τ) = + τ υ η Φ < 1.
B (υ) Γ (υ) υ B (υ) B (υ) Γ (υ + 1)
υ This is always possible since ψ(υ) = 1 − υ +
υ
υ
+ Φ 0 |t − t | . υ
2
1
B (υ) Γ (υ + 1) > 1 − υ:
Γ(υ)
For υ ∈ (0, 1) and t 2 > t 1 > 0, we have: 1 − υ
lim k(τ) = η Φ < 1
υ
υ
υ
t = (t 1 + (t 2 − t 1 )) ≤ t + (t 2 − t 1 ) υ τ→0 + B (υ)
2 1
We assume that sup t∈[0, τ] |Φ (t, m (0))| = ϱ,
5
υ
υ
υ
⇒ |t − t | ≤ (t 2 − t 1 ) . S υ = m (t) ∈ C [0, τ] , R : ∥m (t)∥ < υ. For
2 1
m (t) ∈ S υ , and t ∈ [0, τ], we have
Thus,
∥Φ (t, m (t))∥ = ∥Φ (t, m (t)) − Φ 0 + Φ 0 ∥ ≤ η Φ υ+ϱ.
υ
||T[m] (t 2 ) − T[m] (t 1 ) || ≤ c (t 2 − t 1 ) , (17)
5
Furthermore, for m ∈ m ∈ C [0, τ] , R ,
where we have
1 − υ
c = (η Φ + η Φ k)
ψ(υ) 1 − υ
∥m (t)∥ = max ∥Tm (t)∥ ≤ m 0 + (η Φ ∥m (t)∥ + ϱ)
B (υ)
1 υ
υ
+ (2 + υτ ) ||Φ||. + (η Φ ∥m (t)∥ + ϱ) t υ
B (υ) Γ (υ) B (υ) Γ (υ)

1 − υ γ υ υ
+ Φ 0 1 + t
Given ϵ > 0, choose δ = (ϵ/c) 1/υ such that B (υ) Γ (υ + 1)
= π 1 + π 2 ∥m (t) ∥.
|t 2 − t 2 | < δ ⇒ ||T[m] (t 2 ) − T[m] (t 1 ) || < ϵ.
We have that
This holds uniformly for all m, proving equiconti- 1 − υ
nuity. Since T is continuous, compact, and the set π 1 = m 0 + Φ 0 1 + γ υ t υ
m = δT[m] is bounded, there exists at least one B (υ) Γ (υ + 1)

fixed point m = δT[m] which solves the system. 1 − υ υ υ
+ + t ϱ,
B (υ) B (υ) Γ (υ)

1 − υ υ
3.1.1. Existence and uniqueness of the π 2 = η Φ + t υ .
solution B (υ) B (υ) Γ (υ)
Thus,
As shown in Theorem 2, T is continuous, m =
δT[m], δ ∈ (0, 1) is bounded uniformly, and T is π 1
∥m (t) ∥ ≤ . (18)
equicontinuous. By Schaefer’s theorem, T has a 1 − π 2
fixed point ˆm = T[ ˆm], which is a solution. According to Leray-Schauder’s alternative
theorem, system (8) has a solution.
Theorem 3. Assuming that the condition of
Equation (15) is satisfied, there is a solution of 3.1.2. Positivity and boundedness of the result
the mABC fractional system given by Equation
(8). Theorem 4. The set of solutions of the system
(7) with nonnegative initial conditions defined in
5
Proof. Equation (10) confirms the Lipschitz con- the region Ω = {(N, A, M, P, R) ∈ R + : 0 ≤
5
dition of Φ(t, m(t)). For m 1 , m 2 ∈ C([0, τ], R , N(t) + A(t) + M(t) + P(t) + R(t) : T(t) ≤ Λ } is
ψ
we have positively invariant.
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