Page 123 - IJOCTA-15-3

Investigate the solution of an initial Hilfer fractional value problem
functions ϕ as and
n o
κ
κ
C 1−κ [p, q] = {ϕ (u) : [p, q] → R| C 1−κ = ϕ ∈ C 1−κ [p, q] , D +ϕ ∈ C 1−κ [p, q] .
p
o
1−κ
(u − p) ϕ (u) ∈ C [p, q] , (3) It is obvious that
C κ [p, q] ⊂ C σ,τ [p, q] .
where [p, q] is the finite interval. 1−κ 1−κ
Obviously, C 1−κ [p, q] is the weighted Banach
Lemma 6. ( [3, 21]) Let σ, τ > 0 and κ =
space with the norm σ + τ − στ. If f ∈ C κ , then

1−κ
σ,τ
κ
∥ϕ∥ =
(u − p) 1−κ ϕ (u)
. I +D +f = I +D + f,
κ
σ

C 1−κ p p p p
C
At the same time, we define a Banach space σ σ τ(1−σ) f.
D +I +f = D +
p p p
n o
m
C [p, q] := Θ ∈ C m−1 [p, q] : Θ (m) ∈ C κ [p, q] ,
1
κ Lemma 7. ( [3, 21]) Let h ∈ L [p, q] and
with the norm τ(1−σ) h ∈ L [p, q] exists, then
1
D +
p
m−1

X σ,τ σ τ(1−σ) τ(1−σ)

,

+
Θ
∥Θ∥ m =
Θ (k)

(m)
m ∈ N. D + I +h = I + D + h.

C κ p p p p
C C κ
k=0
Lemma 8. ( [14, 21]). Let a real-valued func-
0
Also, C [p, q] := C κ [p, q] . tion ϕ defined on (p, q] × R be such that, for any
κ
Lemma 1. ( [3, 21]) If σ > 0 and τ > 0, there φ ∈ C 1−κ [p, q], ϕ (· , φ (·)) ∈ C 1−κ [p, q]. Then
κ
exists φ ∈ C 1−κ [p, q] is a solution of IVP:
σ,τ
h τ−1 i Γ (τ) τ+σ−1 D + φ (s) = ϕ (s, φ (s)) , 0 < σ < 1, 0 ≤ τ ≤ 1,
σ
I +(t − p) (u) = (u − p) p
p Γ (τ + σ)
1−κ
I + φ p + = φ 0 , κ = σ + τ − στ,
and p
h i iff φ verifies the next integral equation:
σ
D +(t − p) σ−1 (u) = 0, 0 < σ < 1.
p κ−1
φ 0 (s − p) 1
φ (s) = +
Lemma 2. ( [3, 21]) If σ > 0, τ > 0, and Γ (κ) Γ (σ)
1
ϕ ∈ L (p, q), for u ∈ [p, q], there exist the fol- s
Z
lowing properties, · (s − w) σ−1 ϕ (w, u (w)) dw.

σ+τ
σ
τ
I +I +ϕ (u) = I + ϕ (u) p
p
p
p
and Now, the following are some of the concepts,

σ
σ
D +I +ϕ (u) = ϕ (u) . which we will be employed in the main result.
p p
Let C (J × R, R) represents the category of
In particular, if ϕ ∈ C κ [p, q] or ϕ ∈ C [p, q] , continuous functions θ : J × R → R and the kind
then above equalities hold at each u ∈ (p, q] or of mappings ϕ : J × R → R, where the map
u ∈ [p, q], respectively. (1) s 7→ ϕ (s, w) is measurable ∀ w ∈ R,
(2) s 7→ ϕ (s, w) is continuous ∀ w ∈ R.
Lemma 3. ( [3, 21]) Let σ > 0 and 0 ≤ κ < 1.
σ
Then I + is bounded from C κ [p, q] into C κ [p, q]. Moreover, the space C (J × R, R) is referred to be
p
the mappings of Carath´eodory type on J × R,
Lemma 4. ( [3, 21]) Let σ > 0 and 0 ≤ κ < 1. which are Lebesgue integrable functions when
1−σ
1
If ϕ ∈ C κ [p, q] and I + ϕ ∈ C [p, q], then bounded by a Lebesgue integrable function on J .
p κ
1−σ
I + ϕ (p) Definition 4. ( [22–24]) A function Υ : R + →
σ
σ
I +D +ϕ (u) = ϕ (u) − p Γ (σ) (u − p) σ−1 , R + which is upper semi-continuous and non-
p
p
decreasing in nature is said to be a D-function
for all u ∈ (p, q] .
if Υ (0) = 0. Moreover, D represents the category
Lemma 5. ( [3, 21]) If 0 ≤ κ < 1 and ϕ ∈ of each D-functions on R + .
C κ [p, q], then
Definition 5. ( [22–24]) For an operator ∆ :
σ
σ
I +ϕ (p) := lim I +ϕ (u) = 0, 0 ≤ κ < σ. C 1−κ (J , R) → C 1−κ (J , R), if there is a D-
p p
u→p +
function Υ ∆ ∈ D, where
Now, we present the following weighted spaces ∥∆v − ∆w∥ ≤ Υ ∆ (∥v − w∥) ,
which are required in our main result.
for all v, w ∈ C 1−κ (J , R), where 0 < Υ ∆ (α) < α
for all α > 0, then the operator ∆ is referred as a
n o
σ,τ
C σ,τ = ϕ ∈ C 1−κ [p, q] , D + ϕ ∈ C 1−κ [p, q]
1−κ p nonlinear D-contraction on C 1−κ (J , R).
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