Page 125 - IJOCTA-15-3

Investigate the solution of an initial Hilfer fractional value problem
where Let v, w ∈ C 1−κ (J , R), by hypothesis (Q1), we
have
|g (s, v)| ≤ ξ (s) .
1−κ
(s − p) (∆v (s) − ∆w (s))|
Theorem 2. Suppose that the assumptions (Q1)
and (Q2) are fulfilled. If (q − p) 1−κ < 1, then the = (s − p) 1−κ |f (s, v (s)) − f (s, w (s))|
IVP (1)–(2) possesses a solution in C 1−κ (J , R). ≤ (s − p) 1−κ ψ D (|v (s) − w (s)|)

n o 1−κ
Proof. Let K = φ ∈ C 1−κ (J , R) : ∥φ∥ ≤ η , ≤ (s − p) ψ D
C 1−κ

where · (s − p) κ−1 ∥v − w∥ .
σ
|φ 0 | (q−p) β(κ,σ) C 1−κ
Γ(κ) + Λ + Γ(σ) ∥ξ∥ C 1−κ
η ≥ , Applying a maximum over s gives
1 − (q − p) 1−κ
∥∆v − ∆w∥ ≤ ψ D ∥v − w∥ .
(q − p) 1−κ < 1 and Λ = max(s − p) 1−κ f (s, 0). C 1−κ C 1−κ
t∈J
Certainly, K ⊂ C 1−κ (J , R) which is closed, con- Hence, the operator ∆ is a non-linear contraction.
vex, and bounded in nature. Step II: Operator Ω is a continuous and compact
Now, in order to establish the existence result on K into C 1−κ (J , R).
for the IVP (1)–(2), let us consider the general-
First, to show Ω is continuous on K, consider a
ized IVP involving HFD operator of order σ and
sequence {z n } in K converging to z ∈ K. Then,
type τ, for z ∈ K, as follows:
it implies from the Lebesgue-dominated conver-
σ,τ
D + [φ (s) − f (s, φ (s))] = g (s, z (s)) , s ∈ J , gence theorem that
p
(12) lim Ωz n (s)
1−κ
+
I + [φ (s) − f (s, φ (s))] (p ) = φ 0 , (13) n→∞
p
s
where 0 < σ < 1, 0 ≤ τ ≤ 1 and κ = σ + τ − στ, 1 Z σ−1
+ = lim (s − w) g (w, z n (w)) dw
J = [p, q], for some fixed p, q ∈ R and f, g ∈ n→∞ Γ (σ)
C (J × R, R). p
Assume that the hypotheses (Q1) and (Q2) hold Z s
1
for the functions f and g. Using Lemma 9, we = (s − w) σ−1 lim g (w, z n (w)) dw
get the equivalent non-linear VIE to the IVP Γ (σ) n→∞
p
(12)–(13) as s
1 Z
φ 0 κ−1 σ−1
φ (s) = (s − p) + f (s, φ (s)) = (s − w) g (w, z (w)) dw
Γ (κ) Γ (σ)
p
s
1 Z σ−1 = Ωz (s) , ∀s ∈ J .
+ (s − w) g (s, z (w)) dw.
Γ (σ)
p This proves the continuity for Ω. Next, we prove
(14) that ΩK is a uniformly bounded and equicontin-
uous set in K which implies the compactness of
Consider the operators ∆ : C 1−κ (J , R) →
Ω on K.
C 1−κ (J , R) and Ω : K → C 1−κ (J , R) defined
as follows: It follows from the hypothesis (Q2)
φ 0 κ−1 that
(∆φ)(s) = (s − p) + f (s, φ (s)) , s ∈ J ,
Γ (κ)
(15) 1−κ
(s − p) Ωz (s)
and
s
1 Z σ−1
(Ωz)(s) = (s − w) g (w, z (w)) dw, (s − p) 1−κ Z s
Γ (σ) σ−1
p = (s − w) g (w, z (w)) ds
Γ (σ)
(16) p
s ∈ J . (s − p) 1−κ Z s
≤ (s − w) σ−1 |g (w, z (w))| ds
Hence, the equation (14) can be transformed as Γ (σ)
p
φ (s) = ∆φ (s) + Ωz (s) , s ∈ J . (17)
s
1−κ
Z
Next, we will show that the operators ∆ and Ω ≤ (s − p) (s − w) σ−1 ξ (w) ds
satisfy all the conditions of Theorem 1. The proof Γ (σ)
p
is divided into the following steps.
Step I: Operator ∆ is non-linear D−contraction.
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