Page 122 - IJOCTA-15-3

Amol D. Khandagale et al. / IJOCTA, Vol.15, No.3, pp.493-502 (2025)

d x(t) 1−κ +
= g(t, x(t)), t ∈ J, I + [φ (s) − f (s, φ (s))] (p ) = φ 0 ∈ R,
p
dx f(t, x(t)) (2)
κ = σ + τ − στ,
x(t 0 ) = x 0 ∈ R,
where J = (p, q]. The non-linear terms f :
where J = [t 0 , t 0 + a], for some fixed t 0 ∈ R, a > 0
J × R → R and g : J × R → R be the given con-
and f : J × R → R\{0} and g : J × R → R. Fur- σ,τ
p
thermore, Dhage et al. 31 established the existence tinuous functions. The operator D + is a HFD
operator of order σ and type τ.
results for hybrid differential equations with lin-
ear perturbations. In, 32 Lu et al. formulated the The main contributions of the study are as
theory of hybrid type FDE by using linear per- follows:
turbations of the second kind: (i) Studying the initial value problem involving
the Hilfer fractional derivatives.
q
D [x (t) − f (t, x (t))] = g (t, x (t)) , t ∈ J (ii) Investigating the applicability of the
Krasnoselskii–Dhage fixed point technique to the
x (t 0 ) = x 0 ∈ R,
proposed problem.
where f, g : J × R → R and 0 < q < 1.
(iii) The problem being discussed in this paper
In ref. 33 Akhadkulov et al. utilized this re- is more general than in the literature, such as
vised version of the Krasnoselskii-Dhage type for τ = 1 it reduces to a problem in the Caputo
FPT to obtain the existence results for a hy- fractional derivative, and if τ = 0 it returns to
brid type FDE that incorporates the differential a problem in the Riemann-–Liouville fractional
and integral operators of Riemann–Liouville (RL) derivative.
type: This article is organized as follows: Section
2 presents some essential definitions and lemmas
α
β
D [x (t) − f (t, x (t))] = g t, x (t) , I (x (t)) ,
needed throughout the study. Section 3 investi-
t ∈ J, β > 0 gates the main findings. Section 4 provides an
x (t 0 ) = x 0 , application of the results.
where J = [t 0 , t 0 + a], for some fixed t 0 ∈ R, a > 0 2. Preliminary results
and f : J × R → R and g : J × R × R → R.
+
In ref., 34 Kiataramkul et al. discussed the ex- Definition 1. ( [3]) For σ ∈ R , the RL-
isting results of the following fractional integro- fractional integral of a function h(u) is given as;
differential hybrid boundary value problems for u
1 Z h (s)
σ
differential equations with ψ- Hilfer derivative op- I +h (u) = ds, (u > p) .
erator: p Γ (σ) (u − s) 1−σ
p
n
" !#
β i ,ψ
H α,ρ,ψ x (t) − X I + h i (t, x (t)) Definition 2. ( [3]) For σ ∈ [m − 1, m) , m ∈
D +
a g (t, x (t)) a +
i=1 Z , the RL-fractional derivative of a function
= f (t, x (t)) , h(u) is given as;
m
1 d
x (a) = 0, x (b) = m (x) , D +h (u) =
σ
where t ∈ J = [a, b], 0 < α ≤ 2, 0 ≤ ρ ≤ 1; β i > p Γ (m − σ) du
0, for i = 1, 2, ..., n, g : J × R → R\ {0} , f : Z u h (s)
J × R → R, m : J → R, h i : J × R → R such · σ−m+1 ds, (u > p) .
(u − s)
that h i (a, 0) = 0, i = 1, 2, ..., n. Further, they p
obtained the existence result for inclusion ψ- Hil-
fer fractional integro-differential hybrid boundary Definition 3. ( [1]) The HFD of order σ ∈ (0, 1)
value problems by means of fixed point theorem. and type τ ∈ [0, 1] of a function h(u) is given as;

For some recent development in the field of frac- σ,τ τ(1−σ) (1−τ)(1−σ) h (u) ,
D + h (u) = I +
p p D I +
p
tional differential equations readers are encour-
d
aged to see 35–43 and references therein. where D = du .
Inspired by the aforementioned work, in this Let 0 < p < q < ∞, and let C[p, q] denotes
study, we examine the subsequent initial value a Banach space of all continuous mappings from
problem (IVP) employing the Hilfer fractional de- [p, q] into R with the maximum norm
rivative (HFD) operator: ∥u∥ = max {|u (s)| : s ∈ [p, q]} .
C
σ,τ
D + [φ (s) − f (s, φ (s))] = g (s, φ (s)) , For 0 ≤ κ = σ + τ − στ < 1, we define
p (1)
s ∈ J , 0 < σ < 1, 0 ≤ τ ≤ 1, C 1−κ [p, q], the weighted space of the continuous
494