Page 114 - IJOCTA-15-1

D. Kasinathan et.al. / IJOCTA, Vol.15, No.1, pp.103-122 (2025)


), ς ∈ (−∞, 0];
 φ(ς) + ˜p(x ς 1 , x ς 2 , . . . , x ς n


 ) ς Z
 S α (ς)[φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n
 p p

 ), 0)] E∥ [σ i (ς, ϑ, x) − σ i (ς, ϑ, y)]dϑ∥ ≤ M σ i E∥x − y∥ ;
 +g(0, φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n B
ς


 R 0
 +T α (ς)[x 1 + η] − g(ς, x ς , σ 1 (ς, τ, x τ )dτ)


 ς Z
0


ς ϑ σ i (ς, 0, 0)dϑ = 0.


 R R
 + T α (ς − ϑ) f(ϑ, x ϑ , σ 2 (ϑ, τ, x τ )dτ) dϑ 0



 0 0

 ς (H 3 ) The nonlinear continuous function f :
 R
 + T α (ς − ϑ)G(ς, τ, x τ )dw(τ)


 J × B × H → H and ∃ L f > 0 ∋ ∀ ς ∈
 0

 ς J, x 1 , x 2 ∈ B, y 1 , y 2 ∈ L p (Ω, H),

R
H
 + T α (ς − ϑ)σ(ϑ)dB (ϑ), ς ∈ [0, ς 1 ]


 Q
 0
 p p
 . E∥f(ς, x 1 , y 1 ) − f(ς, x 2 , y 2 )∥ ≤ L f (E∥x 1 − x 2 ∥ B

 .

 .
 p
 ) +E∥y 1 − y 2 ∥ ); f(ς, 0, 0) = 0.
B
x(ς) = S α (ς)[φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n
), 0)]
 +g(0, φ + ˜p(x ς 1 , x ς 2 , . . . , x ς n

ς


R
 +T α (ς)[x 1 + η] − g(ς, x ς , σ 1 (ς, τ, x τ )dτ) (H 4 ) The nonlinear function G : J × J × B →


 0
0
 L (K, H) is continuous and there exist
 Q

ς ϑ
¯

 R R L G , L G > 0 ∋ ∀ ς, ϑ ∈ J and x, y ∈ B,
 + T α (ς − ϑ) f(ϑ, x ϑ , σ 2 (ϑ, τ, x τ )dτ) dϑ


 − G(ς, ϑ, y)∥ p ≤
 0 0 (i) E∥G(ς, ϑ, x)

 ς p

R
 + T α (ς − ϑ)G(ς, τ, x τ )dw(τ)
 L G (E∥x − y∥ ).

 p ¯ p
0
 (ii) E∥G(ς, ϑ, x)∥ ≤ L G E∥x∥ .

 ς

R
 + T α (ς − ϑ)σ(ϑ)dB (ϑ)
 H

Q n
 (H 5 ) The function ˜p : B
 → B is continuous
 0
 P 1 ¯
 + ) and there exist L ˜p , L ˜p > 0 ∋ ∀ ς ∈ J and

S α (ς − ς k )I (x ς k
 k
0<ς k <ς


 x, y ∈ B,
P 2
 + T α (ς − ς k )I (x ς k ), ∀ ς ∈ (ς k , ς k+1 ], p


 k (i) E∥˜p(x ς 1 , . . . , x ς n )(ς)−˜p(y ς 1 , . . . , y ς n )(ς)∥ ≤
 0<ς k <ς
 p p

k = 1, m,
 L ˜p (E∥x ς 1 −y ς 1 ∥ , . . . , E∥x ς n −y ς n ∥ ).
)(ς)∥ p ≤
(ii) E∥˜p(x ς 1 , x ς 2 , . . . , x ς n
¯ ∥ .
p
ς L ˜p E∥x ς 1 , . . . , x ς n
R
where η = d [g(ς, x ς , σ 1 (ς, τ, x τ )dτ)] ς=0 .
dς i
0 (H 6 ) The impulsive functions I : B → H, i =
k
In order to prove the existence result, we demon- 1, 2, k = 1, m s.t x, y ∈ B and there exist
i ¯ i
strate the following primary hypotheses hold: L , L > 0,
k k
i
i
i
(i) E∥I (ς, x) − I (ς, y)∥ p ≤ L [E∥x −
k
k
k
(H 1 ) A is the infinitesimal generator of a y∥ ]
p
strongly continuous cosine family of (ii) E∥I (ς, x)∥ ≤ L [E∥x∥ ]
p
i
p
¯ i
bounded linear operators {S α (ς)} ς≥0 and k k
{T α (ς)} ς≥0 on H, there exist +ve con- (H 7 ) The function σ : J → L (K, H) satisfies
0
Q
stants M ϑ , M ς s.t ∀ ς, ϑ ∈ J
ς
Z

p ˆ
E
σ(ϑ)dϑ
≤ L.
p
p
∥S α (ς)∥ ≤ M ϑ , ∥T α (ς)∥ ≤ M ς .
0
(H 2 ) The function g : J×B×H → H is continu- p−1 n p p p p p
¯
ous on J×J and there exist +ve constants (H 8 ) 32 M g (1 + M σ 1 ) + M ς ς [L f (1 +
¯
M g , M g s.t ∀ ς ∈ J, x 1 , x 2 ∈ B, y 1 , y 2 ∈ p )] + M ς C p ς 2 ¯ p ∞ 1 p q
p
p
˜
P
L p (Ω, H), M σ 2 L G + M ϑ L k
k=1
∞ p o
p P 2 q p
p +M ς L k l < 1.
E∥g(ς, x 1 , y 1 ) − g(ς, x 2 , y 2 )∥ ≤ M g k=1
p p
× (E∥x 1 − x 2 ∥ + E∥y 1 − y 2 ∥ )
B
B
p
¯
p
E∥g(ς, x, y)∥ ≤ M g E∥x − y∥ ; g(ς, 0, 0) = 0. 4. Existence result
B
(ii) For each (ς, ϑ) ∈ J × J, the func- In this section, we present the existence result for
tion σ i : J × J × B → H, i = 1, 2 mild solution of Cauchy problem (1). To develop
is continuous and ∃ M σ i > 0 s.t our results, first we need to derive the following
∀ ς, ϑ ∈ J, x, y ∈ B, existence result of mild solution.
108