CUBO A Mathematical Journal

Vol.19, No¯ 01, (89–110). March 2017

Weighted pseudo Almost periodic solutions for fractional

order stochastic impulsive differential equations

Vikram Singh and Dwijendra N Pandey

Department of Mathematics,

Indian Institute of Technology Roorkee

Roorkee-247667, India.

vikramiitr1@gmail.com, dwij.iitk@gmail.com

ABSTRACT

In this paper, we deal with the existence and uniqueness of piecewise square mean

weighted pseudo almost periodic solutions for a class of fractional order stochastic

impulsive differential equations. The working tools are based on fixed point technique,

fractional power operators and stochastic analysis; methods and theory are adopted

from deterministic fractional systems. In addition, an example is given to illustrate the

theory.

RESUMEN

En este art́ıculo estudiamos la existencia y unicidad de soluciones pseudo casi periódicas

con pesos promedio cuadrado a trozos para una clase de ecuaciones diferenciales es-

tocásticas impulsivas de orden fraccional. Las herramientas de trabajo están basadas

en la técnica de punto fijo, operadores de potencia fraccional y análisis estocástico;

los métodos y teoŕıa están adaptados a partir de sistemas fraccionales deterministas.

Adicionalmente, damos un ejemplo para ilustrar la teoŕıa.

Keywords and Phrases: Fractional stochastic impulsive differential equation, Square-mean

piecewise weighted pseudo almost periodicity, Analytic semigroup, Fractional power operator.

2010 AMS Mathematics Subject Classification: 26A33, 34A37, 34C27, 34G20, 35R12,

35R60, 43A60.



90 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

1 Introduction

In recent years, fractional differential equations have been gaining considerable attention of many

scientists and mathematicians because of their demonstrated applications in widespread fields of

science and engineering. Since noises or stochastic perturbations are unavoidable and omnipresent

in nature as well as in man-made systems, so we have to move from deterministic models to

stochastic models. Stochastic differential equations play an important role in formulation and

analysis of fluctuations in stock market prices, asset prices, population modeling, control engi-

neering, and chemical engineering [12, 20] ect. Motivated by these facts many researchers are

showing great interest to establish an appropriate system to investigate qualitative properties such

as existence, uniqueness, controllability and stability of these physical processes with the help of

fractional calculus, stochastic analysis and fixed point theorems. For more details, we refer to

[1, 3, 10, 11, 16, 19, 28, 29] and references therein.

On the other hand, the study of differential equations with impulsive effect constitutes a useful

and important field of research due to a lot of applications. In particular, differential equations with

impulsive effects arise in various deterministic and stochastic processes which appear in chemical

technology, physics, medicine and economics ect. The fractional differential equations involving

impulsive effects came out as a natural description of observed phenomena. For more details see

[5, 13, 14, 21, 22, 24] and the references therein.

The concept of pseudo almost periodic solutions introduced by Zhang [25, 26] is a natural

and good generalization of the classical almost periodic functions. Further, Diagana investigated

weighted pseudo almost periodic solutions in [8]. Moreover, the authors investigated piecewise

almost periodic solutions in [22], piecewise square mean almost periodic solutions in [11], pseudo

almost periodic solutions in [5, 27] for impulsive differential equations. Recently, Zhinan [23] ana-

lyzed piecewise weighted pseudo almost periodic functions, which was more tricky and changeable

than those of the classical functions. Many authors have been made important contributions in

study of almost periodic functions and its generalizations, one can see [6, 11, 13, 14, 22, 23, 24]

and the references therein. However, piecewise square mean weighted pseudo almost periodic mild

solutions for the fractional order stochastic impulsive differential equations, is an untreated topic

in the literature and this fact is the motivation of the present work.

In this paper, we are interested to investigate the existence and uniqueness of piecewise square

mean weighted pseudo almost periodic mild solution for the following fractional order stochastic

impulsive differential system

cDαy(t) + Ay(t) = G(t, y(t)) + F

!

t, y(t),

∫
t

−∞

K(t − s)g(s, y(s))ds

"
dw(t)

dt
, t0 < t ̸= ti, t ∈ R,

(1.1)

y(t+
i
) = y(t−

i
) + Gi(y(ti)), i ∈ Z, (1.2)

y(t0) = y0, (1.3)

where the state y(·) take values in L2(P, H), H is a separable real Hilbert space; cDα,α ∈ (0, 1)



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Weighted pseudo Almost periodic solutions for fractional order . . . 91

symbolizes the Caputo fractional derivative of order α; −A : D(A) ⊂ L2(P, H) → L2(P, H),

is the infinitesimal generator of an analytic semigroup of exponentially bounded linear operator

{S(t)}t≥0; {w(t) : t ≥ 0} is a K-valued Wiener process, K is another separable Hilbert space; G, F, Gi
are some suitable functions will be mention later; δ(·) is Dirac’s delta function and K ∈ L1(R) with

|K(t)| ≤ CKe
−bt, b, CK > 0.

The rest of this paper is organized as follows: In section 2, we define some fundamental results

about the notion of piecewise square mean weighted pseudo almost periodic functions. Section 3 is

devoted to the main results ensuring the existence and uniqueness of mild solutions of (1.1)−(1.3)

via fractional power of operator and fixed point technique. At last, we will provide an example to

show the feasibility of the theory discussed in this paper.

2 Preliminaries

Let L(K, H) denote the collection of all bounded linear operators form K to H. For convenience,

without confusion we will employ the same notation ∥.∥ to denote the norms in H, K and L(K, H)

and ⟨·, ·⟩ for inner product in H and K. Let (Ω, F, {Ft}t≥0, P) be a complete probability space

equipped with a normal filtration {Ft}t≥0 satisfying the usual conditions(i.e right continuous and

{F0} containing all P-null sets). Suppose {w(t) : t ≥ 0} is a K-valued Wiener process with a finite

nuclear covariance operator Q ≥ 0 denote Tr(Q) =
∑

∞

k=1
#λk = #λ < ∞ with Qek = $λkek, where

ek are complete orthonormal basis of K. In fact, w(t) =
∑

∞

k=1

%
#λkwk(t)ek, here {wk(t)}

∞

k=1 are

mutually independent one dimensional standard Wiener process. We consider that Ft = {w(s) :

0 ≤ s ≤ t} is the σ algebra generated by w. Assume that L0
2
= L2(Q

1

2 K, H) represent the space

of all Hilbert Schmidt operators from Q
1

2 K to H with inner product ⟨φ,ψ⟩ = Tr[φQψ∗]. For

more details we refer to the book by Da Prato and Zabczyk [7]. Let the collection of all strongly

measurable, square integrable H valued random variables be denoted by L2(P, H) which a Banach

space endowed with the norm ∥x(·)∥L2 = (E∥x(·)∥
2)

1

2 , where E(·) represents the expectations with

measure P. Moreover L2F0(P, H) denote the collection of all F0 measurable, H valued random

variable y(0).

Let Ω be a subspace of L2(P, H) and E be a compact set of Ω. Assume that R, N, Z, and

C represent the sets of real number, natural number, integers and complex numbers respectively.

For A being a linear operator on L2(P, H), D(A), R(A) and ρ(A) stands for domain, range and

resolvent of A, repectively. Let B = {ti : ti ∈ R, ti < ti+1, i ∈ Z} be the set of all strictly

increasing and unbounded sequences. For {ti : i ∈ Z} ∈ B, let PC(R, L
2(P, H)) denote the space of

all piecewise stochastically continuous processes y : R → L2(P, H) such that y(t) is stochastically

continuous at t for any t /∈ B, y(t−
i
), y(t+

i
) exists and y(t−

i
) = y(ti) for all i ∈ R. In particular,

the space PC(R × Ω, L2(P, H)) consists of all piecewise stochastically continuous processes y :

R × Ω → L2(P, H) such that for any x ∈ Ω,, y(t, ·) ∈ PC(R, L2(P, H)) and for any t ∈ R, y(t, ·) is

stochastically continuous at x ∈ Ω.



92 Vikram Singh and Dwijendra N Pandey CUBO
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2.1 Fractional calculus and fractional power operator

Following [16, 18]) we recall some definitions and basic results of fractional calculus.

Definition 1. The Riemann-Liouville fractional integral of a function g ∈ L1
loc

(R+, R) with the

lower limit zero of order α > 0 is defined by

Jαg(t) =
1

Γ(α)

∫
t

0

(t − ξ)α−1g(ξ)dξ, t > 0,

and J0g(t) := g(t). This fractional integral satisfies the properties Jα ◦ Jb = Jα+b for b > 0.

Definition 2. The Riemann-Liouville fractional derivative of a function g ∈ L1
loc

(R+, R) with the

lower limit zero of order α > 0, n − 1 < α < n, n ∈ N is given by

Dαg(t) =
1

Γ(n − α)

dn

dtn

∫t

0

(t − ξ)n−α−1g(ξ)dξ,

Moreover D0g(t) = g(t) and DαJαg(t) = g(t) for t > 0.

Definition 3. The Caputo fractional derivative of a function g : [0, ∞) → R with the lower limit

0 of order α > 0 is given by

cDαg(t) = Dα
!

g(t) −
n−1∑

k=0

tk

k!
g(k)(0)

"

, t > 0, n − 1 < α < n.

Remark 1. (i) If g(t) ∈ Cn([0, ∞)), then

cDαg(t) =
1

Γ(n − α)

∫t

0

(t − ξ)n−α−1
dn

dξn
g(ξ)dξ,

where n − 1 < α < n, n ∈ N.

(ii) If g is an abstract function with values in H, then integral defined in Definition 1 and 2 are

taken in Bochner’s sense.

If −A generates an analytic semigroup S(t) in L2(P, H) and 0 ∈ ρ(A), then for σ > 0, we can

define fractional power A−σ of the operator A by

A−σ =
1

Γ(σ)

∫
∞

0

tσ−1S(t)dt

where A−σ is bijective, bounded and Aσ = (A−σ)−1, σ > 0 a closed linear operator on

D(Aσ) such that D(Aσ) = R(A−σ). Moreover D(Aσ) is dense in L2(P, H) and the expression

∥y∥σ = ∥A
σy∥, y ∈ D(Aσ) defines a norm on D(Aσ). Let us denote by L2(P, Hσ) the Banach

space D(Aσ) with norm ∥.∥σ. The following properties are well recognized.

Lemma 2.1. [17] Let A be an infinitesimal generator of an analytic semigroup S(t) and 0 ∈ ρ(A).

Then



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Weighted pseudo Almost periodic solutions for fractional order . . . 93

(i) S(t) : L2(P, H) → D(Aσ), for σ ≥ 0, and t > 0.

(ii) For every y ∈ D(Aσ), we have S(t)Aσy = AσS(t)y.

(iii) The operator AσS(t) is bounded and

∥AσS(t)∥ ≤ Mσt
−σe−λt, Mσ, t,λ > 0. (2.1)

(iv) For y ∈ D(Aσ), and 0 < σ ≤ 1, we have

∥S(t)y − y∥ ≤ Cσt
σ∥Aσy∥, Cσ > 0. (2.2)

2.2 Square-mean piecewise weighted pseudo almost periodic function

Now we define square-mean piecewise weighted pseudo almost periodic function and explore its

properties

Definition 4. A stochastic process y : R → L2(P, H) is said to be stochastically continuous for

s ∈ R if limt→s E∥y(t) − y(s)∥
2 = 0.

Definition 5. A stochastically continuous process y : R → L2(P, H) is said to be square mean

almost periodic if for ever ϵ > 0, there exists a l(ϵ) > 0 such that every interval L of length

l(ϵ) > 0 contains a number τ with the property E∥y(t + τ) − y(t)∥2 < ϵ for all t ∈ R.

Definition 6. A sequence zi : Z → L
2(P, H) is said to be square-mean almost periodic sequence

if for ever ϵ > 0, there exists a l(ϵ) > 0 such that every p ∈ Z there is at least one number k in

[p, p+l], with the property E∥zi+k −zi∥
2 < ϵ for all i ∈ Z. We denote the set of all such processes

by AP(Z, L2(P, H)).

Remark 2. Let {zi} ∈ AP(Z, L
2(P, H)), then {zi : i ∈ Z} is stochastically bounded.

Let Wd denote the collection of all functions (weights) ρm : Z → (0, +∞), m ∈ Z. For

ρm ∈ Wd and m ∈ Z
+ = {m ∈ Z, m ≥ 0}, set µ(m,ρ) :=

m∑

k=−m

ρm. Denote Wd,∞ := {ρ ∈ Wd :

lim
m→∞

(m,ρ) = ∞}.

For ρ ∈ Wd,∞, we define

PAPρ(Z, L
2(P, H)) =

{

zm ∈ l
∞(Z, L2(P, H)) : lim

m→∞

1

µ(m,ρ)

m∑

k=−m

E∥zm∥
2ρm = 0

}

(2.3)

Definition 7. Let ρ ∈ Wd,∞. A sequence {zi}i∈Z ∈ l
∞(Z, L2(P, H)) is called square mean dis-

crete weighted pseudo almost periodic if zi = ai + bi, where ai ∈ AP(Z, L
2(P, H)) and bi ∈

PAPρ(Z, L
2(P, H)). The set of all such functions denoted by WPAPρ(Z, L

2(P, H)).

Definition 8. A stochastic process y ∈ PC(R, L2(P, H)) is said to be square-mean piecewise almost

periodic if:



94 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

(i) The set of all sequences {tj
i
: t

j

i
:= ti+j − ti, ti ∈ B, i, j ∈ Z} are equipotentially almost periodic

i.e. for every ϵ > 0 there exists a relatively Dϵ ⊂ R of ϵ periods common for all sequences

{t
j

i
}.

(ii) For any ϵ > 0, there exists a δ > 0 such that if the points s and t are in the same interval of

continuity of y(t) and |t − s| < δ, then E∥y(t) − y(s)∥2 < ϵ.

(iii) For any ϵ > 0, there exists a relatively dense set Rϵ of R such that if τ ∈ Rϵ, then E∥y(t +

τ) − y(t)∥2 < ϵ, with the condition |t − ti| > ϵ, i ∈ Z.

We denote by APp(R, L2(P, H)) the space of all square-mean piecewise almost periodic pro-

cesses. We denote by UPC(R, L2(P, H)) the space of all stochastic processes such that y satisfy

the condition (ii) in Definition 8 and y ∈ PC(R, L2(P, H)).

Definition 9. [6] For {ti} ∈ B, i ∈ Z, the function f(t, y) ∈ PC(R × Ω, L
2(P, H)) is called square-

mean piecewise almost periodic in t ∈ R and uniformly on E ⊆ Ω, {f(·, y) : y ∈ E} is uniformly

bounded, and for every ϵ > 0 there exists a relatively compact set Rϵ of R, such that E∥f(t+τ, y)−

f(t, y)∥2 < ϵ, for all y ∈ E, t ∈ R and τ ∈ Rϵ with |t − ti| > ϵ, i ∈ Z. The set of all such processes

is denoted by APp(R × Ω, L2(P, H)).

Lemma 2.2. [13] Let f ∈ APp(R, L2(P, H)), {zi : i ∈ Z} is square mean almost periodic sequence

in L2(P, H) and {t
j

i
: i, j ∈ Z} is equipotentially almost periodic. Then for each ϵ > 0 there exist

relatively dense sets Rϵ of R and Zϵ of Z such that the following conditions hold:

(i) E∥f(t + τ) − f(t)∥2 < ϵ for all τ ∈ Rϵ, t ∈ R, |t − ti| > ϵ, i ∈ Z.

(ii) E∥zi+p − zi∥
2 < ϵ for all p ∈ Zϵ, and i ∈ Z.

(iii) For any τ ∈ Rϵ there exists at least a number p ∈ Zϵ such that |t
p

i
− τ| < ϵ, i ∈ Z.

Next, we introduce the concept of piecewise square mean weighted pseudo almost periodic

functions and explore its properties.

Let W be the collections of all positive and locally integrable functions ρ : R → (0, ∞). For each

ρ ∈ W and γ > 0, set µ(γ,ρ) :=
∫γ
−γ
ρ(t)dt.

Define

W∞ := {ρ ∈ W : lim
γ→∞

µ(γ,ρ) = ∞},

WB := {ρ ∈ W∞ : ρ is bounded and inf
t∈R

ρ(t) > 0}.

It is clear that WB ⊂ W∞ ⊂ W.

Definition 10. Let ρ1,ρ2 ∈ W∞. ρ1 is said to be equivalent to ρ2 (i.e. ρ1 ∼ ρ2) if
ρ1
ρ2

∈ WB.



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Weighted pseudo Almost periodic solutions for fractional order . . . 95

It is clear that ‘‘ ∼ " binary equivalence relation on W∞. For a given weight ρ ∈ W∞, the

equivalence class is denoted by CL(ρ) := {ρ
∗ ∈ W∞ : ρ ∼ ρ

∗}. Moreover W∞ = ∪ρ∈W∞CL(ρ).

For ρ ∈ W∞, we define

PAAρ(R, L
2(P, H)) :=

{

f ∈ PC(R, L2(P, H)) : lim
γ→∞

1

µ(γ,ρ)

∫
γ

−γ

E∥f(t)∥2ρ(t)dt = 0

}

.

Similarly

PAAρ(R × Ω, L
2(P, H))

:=

{

f ∈ PC(R × Ω, L2(P, H)) : lim
γ→∞

1

µ(γ,ρ)

∫
γ

−γ

E∥f(t, y)∥2ρ(t)dt = 0 uniformly in y ∈ E

}

.

Definition 11. A function f ∈ PC(R, L2(P, H)) is called piecewise square mean weighted pseudo

almost periodic if it has a decomposition of the form f = φ+ ψ, where φ ∈ APp(R, L2(P, H)) and

ψ ∈ PAPρ(R, L
2(P, H)). The set of all such functions denoted by WPAPρ(R, L

2(P, H)).

Definition 12. A function f ∈ PC(R × Ω, L2(P, H)) is called piecewise square mean weighted

pseudo almost periodic if it has a decomposition of the form f = φ + ψ, where φ ∈ APp(R ×

Ω, L2(P, H)) and ψ ∈ PAPρ(R×Ω, L
2(P, H)). The set of all such functions denoted by WPAPρ(R×

Ω, L2(P, H)).

For ρ ∈ W∞ and τ ∈ R define ρ
τ
by ρτ(t) = ρ(t + τ) for all t ∈ R. Define

WT = {ρ ∈ W∞ : ρ ∼ ρ
τ
for each t ∈ R}.

It is clear that WT contains many of weights, such as 1, e
t
and 1 + |t|n with n ∈ N among others.

Remark 3. (i) For ρ ∈ WT , PAPρ(R, L
2(P, H)) is a translation invariant set of PC(R, L2(P, H)).

(ii) It is easy to see that WPAPρ(R, L
2(P, H))(resp., WPAPρ(R × Ω, L

2(P, H))) are Banach

spaces with sup norm.

Similar as the proof of Lemma 2.5 in [9], we have the following result.

Lemma 2.3. Let {fn}n∈N be a sequence of functions in WPAPρ(R, L
2(P, H)). If fn converge

uniformly to f, then f ∈ WPAPρ(R, L
2(P, H)).

Similar as the proof of [14] the following composition theorems hold for piecewise square mean

weighted pseudo almost periodic functions.

Theorem 2.1. Let f(t, y, z) ∈ WPAPρ(R × Ω × Ω, L
2(P, H)),ξ,χ ∈ WPAPρ(R, L

2(P, H)) and

R(ξ) × R(ξ) ⊂ Ω × Ω. Assume that there exists a number Lf > 0 such that

E∥f(t, y1, z1)−f(t, y2, z2)∥
2 ≤ Lf.(E∥y1 −y2∥

2 +E∥z1 −z2∥
2), for all t ∈ R, yi, zi ∈ Ω, i = 1, 2,

then f(·,ξ(·),χ(·)) ∈ WPAPρ(R, L
2(P, H)).



96 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

Theorem 2.2. Let {Ii(y) : i ∈ Z} for any y ∈ Ω be a piecewise square mean weighted pseudo

almost periodic sequence. Assuming that there exists a constant L0 > 0 such that

E∥Ii(x) − Ii(y)∥
2 ≤ L0.E∥x − y∥

2, for all x, y ∈ Ω, i ∈ Z.

If ξ ∈ WPAPρ(R, L
2(P, H)) ∩ UPC(R, L2(P, H)) such that R(ξ) ⊂ Ω, then Ii(ξ(ti)) is piecewise

square mean weighted pseudo almost periodic.

Lemma 2.4. [21] Assume that {t
j

i
: i, j ∈ Z} are equipotentially almost periodic sequences, then

for each p > 0 there exists a positive integer N0 such that each interval of length p has no more

than N0 elements of the sequence {ti} and

n(s, t) ≤ N0(t − s) + N0,

where n(t, s) denotes the number of the points ti in the interval [t, s].

3 Main Results

In this section, we establish piecewise square mean weighted pseudo almost periodic mild solution

to the fractional order stochastic impulsive differential system (1.1)-(1.3).

In formulation of the system (1.1)-(1.3), we consider the following assumptions:

(H1) The collection of sequences {t
j

i
: i, j ∈ Z} is equipotentially almost periodic and there exists

θ > 0 such that infi τ
1
i
= θ.

(H2) −A is the infinitesimal generator of an analytic semigroup S(t), t ≥ 0, on L
2(P, H).

(H3) For ρ ∈ WT , g ∈ WPAPρ(R × L
2(P, Hσ), L

2(P, H)) and there exists a Lg > 0, 0 < η < 1,

such that

E∥g(t1, u1) − g(t2, u2)∥
2 ≤ Lg(|t1 − t2|

η + E∥u1 − u2∥
2
σ
),

for each (ti, ui) ∈ R × L
2(P, Hσ) i = 1, 2.

(H4) For ρ ∈ WT , G ∈ WPAPρ(R × L
2(P, Hσ), L

2(P, H)) and there exists LF > 0, 0 < η < 1,

such that

E∥G(t1, u1) − G(t2, u2)∥
2 ≤ LG(|t1 − t2|

η + E∥u1 − u2∥
2
σ
),

for each (ti, ui) ∈ R × L
2(P, Hσ), i = 1, 2.

(H5) For ρ ∈ WT , F ∈ WPAPρ(R×L
2(P, Hσ)×L

2(P, Hσ), L
2(P, L0

2
)) and there exists LF > 0, 0 <

η < 1, such that

E∥F(t, u1, v1) − F(t, u2, u2)∥
2 ≤ LF(|t1 − t2|

η + E∥u1 − u2∥
2
σ
+ E∥v1 − v2∥

2
σ
),

for each (ti, ui, vi) ∈ R × L
2(P, Hσ) × L

2(P, Hσ) i = 1, 2.



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Weighted pseudo Almost periodic solutions for fractional order . . . 97

(H6) {Gi(y) : k ∈ Z} is piecewise square mean weighted pseudo almost periodic sequence uniformly

y ∈ Ω and that there exists a constant LG > 0 such that

E∥Gi(x) − Gi(y)∥
2 ≤ LG.E∥x − y∥

2
σ, for all x, y ∈ L

2(P, Hσ).

(H7) For any L1, L2 > 0, denote

CF := supt∈R,∥u∥∞<L1,∥v∥∞<L2(E∥F(t, u, v)∥
2)

1

2 < ∞,

CG := sup
t∈R,∥u∥∞<L1

(E∥G(t, u)∥2)
1

2 < ∞, CGi := supk∈Z,∥u∥∞<L1(E∥Gi(u)∥
2)

1

2 < ∞.

Then there exists a constant r0 > 0 such that

3M2σ

&

4C2GiN
2
0

!
1

Mσ
0

+
1

eλ − 1

"2
+ C2G

Γ2(1 − σ)

λ2(1−σ)
+ C2FN0

Γ(1 − 2σ)

λ(2−2σ)

'

≤ r0.

Now, we define the mild solutions for the system (1.1) − (1.3).

Definition 13. A stochastic process y ∈ PC(J, L2(P, H)), J ⊂ R is a mild solution of the system

(1.1) -(1.3), if

(i) y0 ∈ L
2
F0

(P, H).

(ii) y(t) ∈ L2(P, H) has càdlàg path on t ∈ J a.s., and satisfies the following integral equation

y(t) =

⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

I(t − t0)y0 +
∫t
t0
(t − s)α−1J (t − s)G(s, y(s))ds

+
∫
t

t0
(t − s)α−1J (t − s)F(s, y(s),

∫
s

−∞
K(s − ξ)g(ξ, y(ξ))dξ)dw(s), t ∈ [t0, t1];

I(t − t0)y0 + I(t − t1)y1 +
∫t
t0
(t − s)α−1J (t − s)G(s, y(s))ds

+
∫
t

t0
(t − s)α−1J (t − s)F(s, y(s),

∫
s

−∞
K(s − ξ)g(ξ, y(ξ))dξ)dw(s), t ∈ (t1, t2];

...

I(t − t0)y0 +
k∑

i=1

I(t − ti)Gi(y(ti)) +

∫
t

t0

(t − s)α−1J (t − s)G(s, y(s))ds

+
∫
t

t0
(t − s)α−1J (t − s)F(s, y(s),

∫
s

−∞
K(s − ξ)g(ξ, y(ξ))dξ)dw(s), t ∈ (tk, tk+1],

(3.1)

where

I(t) =

∫
∞

0

Nα(θ)S(t
αθ)dθ, J (t) = α

∫
∞

0

θNα(θ)S(t
αθ)dθ,

and for θ ∈ (0, ∞)

Nα(θ) =
1

α
θ−1−

1

α ωα(θ
−

1

α ) ≥ 0, ωα(θ) =
1

π

∞∑

n=1

(−1)n−1θ−nα−1
Γ(nα + 1)

n!
sin(nπα),



98 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

Nα denote the probability density function on (0, ∞) such that

Nα(θ) ≥ 0, θ ∈ (0, ∞) and

∫
∞

0

Nα(θ)dθ = 1.

Noth that when (H2) holds, we observe that if y(t) is stochastically bounded solution of the

system (1.1) − (1.3) on R, then the mild solution (3.1) take the following form as t0 → −∞.

y(t) =
∑

ti<t

I(t − ti)Gi(y(ti)) +

∫t

−∞

(t − s)α−1J (t − s)G(s, y(s))ds

+

∫
t

−∞

(t − s)α−1J (t − s)F

!

s, y(s),

∫
s

−∞

K(s − ξ)g(ξ, y(ξ))dξ

"

dw(s). (3.2)

Lemma 3.1. Assume that (H1) − (H3) hold, if y ∈ WPAPρ(R, L
2(P, Hσ)), then

Θ(A−σy)(t) :=

∫t

−∞

K(t − s)g(s, A−σy(s))ds ∈ WPAPρ(R, L
2(P, H)).

Proof. Since A−σ is bounded, ψ := g(·, A−σy(·)) ∈ WPAPρ(R, L
2(P, H)) by Theorem 2.1. As-

sume that ψ = ψ1 + ψ2 with ψ1 ∈ AP
p(R, L2(P, H)) and ψ2 ∈ PAPρ(R, L

2(P, H)), then

∫
t

−∞

K(t − s)ψ(s)ds =

∫
t

−∞

K(t − s)ψ1(s)ds +

∫
t

−∞

K(t − s)ψ2(s)ds := Θ1(t) + Θ2(t)

where

Θ1(t) =

∫t

−∞

K(t − s)ψ1(s)ds, Θ2(t) =

∫t

−∞

K(t − s)ψ2(s)ds.

It is easy to check that Θ1 ∈ UPC(R, L
2(P, H)). Since ψ1 ∈ AP

p(R, L2(P, H)), for ϵ > 0, there

exists a relatively dense set Rϵ of R formed by ϵ-periods of ψ1. For τ ∈ Rϵ, t ∈ R, |t−ti| > ϵ, i ∈ Z,

we have ∥ψ1(t + τ) − ψ1(t)∥ < ϵ.

Hence for t ∈ R, |t − ti| > ϵ, i ∈ Z, we get

E∥Θ1(t + τ) − Θ1(t)∥
2 =E

(
(
(
(

∫t+τ

−∞

K(t + τ − s)ψ1(s)ds −

∫t

−∞

K(t − s)ψ1(s)ds

(
(
(
(

2

≤E

(
(
(
(

∫
t

−∞

K(t − s)[ψ1(s + τ) − ψ1(s)]ds

(
(
(
(

2

≤C2K

∫
t

−∞

e−2b(t−s)E∥ψ1(s + τ) − ψ1(s)∥
2ds <

C2K
2b
ϵ,

which implies that Θ1 ∈ AP
p
(R, L2(P, H)).



CUBO
19, 1 (2017)

Weighted pseudo Almost periodic solutions for fractional order . . . 99

Next we show that Θ2 ∈ PAPρ(R, L
2(P, H)). In fact for γ > 0, we have

1

µ(γ,ρ)

∫
γ

−γ

E∥Θ2(t)∥
2ρ(t)dt =

1

µ(γ,ρ)

∫
γ

−γ

E

(
(
(
(

∫
t

−∞

K(t − s)ψ2(s)ds

(
(
(
(

2

ρ(t)dt

=
1

µ(γ,ρ)

∫
γ

−γ

E

(
(
(
(

∫
∞

0

K(s)ψ2(t − s)ds

(
(
(
(

2

ρ(t)dt

≤
C2K

µ(γ,ρ)

∫γ

−γ

∫
∞

0

e−2b(s)ρ(t)E∥ψ2(t − s)∥
2dsdt

≤C2K

∫
∞

0

e−2b(s)Λγ(s)ds,

where

Λγ(s) =
1

µ(γ,ρ)

∫
γ

−γ

ρ(t)E∥ψ2(t − s)∥
2dt.

Since ψ2(s) ∈ PAPρ(R, L
2(P, H)), ρ ∈ WT , this implies that ψ2(· − s) ∈ PAPρ(R, L

2(P, H)) for

each s ∈ R by Remark 3. Hence lim
γ→∞

Λγ(s) = 0 for all s ∈ R. Now, by Lebesgue dominated

convergence theorem, we have Θ2 ∈ PAPρ(R, L
2(P, H)).

Lemma 3.2. Assume that (H1) − (H2) hold, if φ(t) ∈ WPAPρ(R, L
2(P, L0

2
)), then

Λφ(t) =

∫t

−∞

Aσ(t − s)α−1J (t − s)φ(s)dw(s) ∈ WPAPρ(R, L
2(P, H)).

Proof. Since φ(t) ∈ WPAPρ(R, L
2(P, L0

2
)) and ∥φ∥∞ := supt∈R(E∥φ(t)∥

2)
1

2 < ∞. Now, using

Ito’s isometry property of stochastic integral and Lemma 2.1 , we get

E∥Λφ(t)∥
2 =E

(
(
(
(

∫t

−∞

Aσ(t − s)α−1J (t − s)φ(s)dw(s)

(
(
(
(

2

≤α2E

(
(
(
(

∫
t

−∞

∫
∞

0

θ(t − s)α−1Nα(θ)A
σS((t − s)αθ)φ(s)dθdw(s)

(
(
(
(

2

≤α2
&∫

t

−∞

∫
∞

0

E∥θ(t − s)α−1Nα(θ)A
σS((t − s)αθ)φ(s)∥2dθds

'

≤α2M2σ

∫t

−∞

∫
∞

0

θ2(1−σ)N 2α(θ)(t − s)
2(α−ασ−1)e2λθ(t−s)

α

E∥φ(s)∥2dθds

≤α2M2
σ
∥φ∥2

∫
∞

0

N 2
α
(θ)

∫
∞

0

θ2(1−σ)ξ2(α−ασ−1)e2λθξ
α

dξdθ.

Since N 2α(θ) ∈ L
1(R+), then by calculating we get (see [11])

α2
∫
∞

0

N 2α(θ)

∫
∞

0

θ2(1−σ)ξ2(α−ασ−1)e2λθξ
α

dξdθ = N0
Γ(1 − 2σ)

λ(2−2σ)
(3.3)

where N0 = supθ≥0 N
2
α
(θ). Then

E∥Λφ(t)∥
2 ≤ M2

σ
∥φ∥2N0

Γ(1 − 2σ)

λ(2−2σ)
,



100 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

This implies that Λφ is well defined. Now let φ = φ1 + φ2, with φ1 ∈ AP
p(R, L2(P, H)) and

φ2 ∈ PAPρ(R, L
2(P, H)), then

Λφ(t) =

∫t

−∞

Aσ(t − s)α−1J (t − s)φ1(s)dw(s) +

∫t

−∞

Aσ(t − s)α−1J (t − s)φ2(s)dw(s)

:=Λφ1(t) + Λφ2(t).

It is easy to check that Λφ1 ∈ UPC(R, L
2(P, H)). Since φ1 ∈ AP

p(R, L2(P, H)), for ϵ > 0, there

exists a relatively dense set Rϵ of R such that E∥φ1(t+τ)−φ1(t)∥
2 < ϵ for τ ∈ Rϵ, t ∈ R, |t−ti| >

ϵ, i ∈ Z,

Note that #w(s) := w(s+τ)−w(s), s ∈ R, is also a Brownian motion with same distribution as

w(s). Now for t ∈ R, |t − ti| > ϵ, i ∈ Z, using Lemma 2.1 and Ito’s isometry property of stochastic

integral, we have

E∥Λφ1(t + τ) − Λφ1(t)∥
2

=E

(
(
(
(

∫
t

−∞

(t − s)α−1AσJα(t − s)[φ1(s + τ) − φ1(s)]d#w(s)

(
(
(
(

2

≤α2E

(
(
(
(

∫
t

−∞

∫
∞

0

θ(t − s)α−1Nα(θ)A
σS((t − s)αθ)[φ1(s + τ) − φ1(s)]dθd#w(s)

(
(
(
(

2

≤α2M2
σ

∫
t

−∞

∫
∞

0

θ2(1−σ)N 2
α
(θ)(t − s)2(α−σα−1)e−2λθ(t−s)

α

E∥[φ1(s + τ) − φ1(s)]∥
2dθds

<ϵα2M2σ

∫t

−∞

∫
∞

0

θ2(1−σ)N 2α(θ)(t − s)
2(α−σα−1)e−2λθ(t−s)

α

dθds.

≤ϵα2M2σ

∫
∞

0

N 2α(θ)

∫
∞

0

θ2(1−σ)ξ2(α−σα−1)e−2λθξ
α

dξdθ

≤M2
σ
N0
Γ(1 − 2σ)

λ(2−2σ)
ϵ,

that is Λφ1 ∈ AP
p(R, L2(P, H)).

Next we show that Λφ2 ∈ PAPρ(R, L
2(P, H)). In fact for γ > 0, we have

1

µ(γ,ρ)

∫
γ

−γ

E∥Λφ2(t)∥
2ρ(t)dt =

1

µ(γ,ρ)

∫
γ

−γ

E

(
(
(
(

∫
t

−∞

Aσ(t − s)α−1J (t − s)φ2(s)dw(s)

(
(
(
(

2

ρ(t)dt

≤
1

µ(γ,ρ)

∫γ

−γ

∫t

−∞

∥Aσ(t − s)α−1J (t − s)∥2E∥φ2(s)∥
2dsρ(t)dt

≤
1

µ(γ,ρ)

∫
γ

−γ

∫
∞

0

∥ξα−1AσJ (ξ)∥2E∥φ2(t − ξ)∥
2dξρ(t)dt.



CUBO
19, 1 (2017)

Weighted pseudo Almost periodic solutions for fractional order . . . 101

Similar as previous calculation, we have

∫
∞

0

∥ξα−1AσJ (ξ)∥2E∥φ2(t − ξ)∥
2dξ

≤α2M2
σ

∫
∞

0

N 2
α
(θ)

∫
∞

0

θ2(1−σ)ξ2(α−σα−1)e−2λθξ
α

E∥φ2(t − ξ)∥
2dξdθ.

Now, we have

1

µ(γ,ρ)

∫γ

−γ

E∥Λ2y(t)∥
2ρ(t)dt ≤α2M2σ

∫
∞

0

N 2α(θ)

∫
∞

0

θ2(1−σ)ξ2(α−σα−1)e−2λθξ
α

Tγ(t)dξdθ.

where

Tγ(t) =
1

µ(γ,ρ)

∫γ

−γ

E∥φ2(t − ξ)∥
2ρ(t)dt. (3.4)

Since φ2(s) ∈ PAPρ(R, L
2(P, H)), ρ ∈ WT and translation invariant, this implies that φ2(· − s) ∈

PAPρ(R, L
2(P, H)) for each s ∈ R by Remark 3. Hence γ→∞Tγ(t) = 0 for all s ∈ R. Now, by

Lebesgue dominated convergence theorem, we have Λφ2 ∈ PAPρ(R, L
2(P, H)).

Theorem 3.1. Assume the conditions (H1) − (H7) are satisfy, if

∆ := 3M2σ

&

4LGN
2
0

!
1

Mσ
0

+
1

eλ − 1

"2
+ LG

Γ2(1 − σ)

λ2(1−σ)
+ LFN0

!

1 +
LgC

2
K

2b

"
Γ(1 − 2σ)

λ(2−2σ)

'

< 1,

then the system (1.1) − (1.3) admits a unique mild solution in WPAPρ(R, L
2(P, H)).

Proof. Let M := {y ∈ WPAPρ(R, L
2(P, H)) with discontinuity of first type at ti, i ∈ Z satisfying

E∥y∥2 ≤ r0, r0 > 0}. Obviously, M is a closet subspace of WPAPρ(R, L
2(P, H)). Define an

operator Q in M by

(Qy)(t) =
∑

ti<t

AσI(t − ti)Gi(A
−σy(ti)) +

∫
t

−∞

(t − s)α−1AσJ (t − s)G(s, A−σy(s)ds

+

∫t

−∞

(t − s)α−1AσJ (t − s)F(s, A−σy(s),Θ(A−σy(s)))dw(s). (3.5)

where Θ(A−σy(s)) =
∫s
−∞

K(s − ξ)g(ξ, A−σy(ξ))dξ.

In fact if y ∈ WPAPρ(R, L
2(P, Hσ)), we have Θ(A

−σy) ∈ WPAPρ(R, L
2(P, H)) by Lemma

3.1, and F(·, A−σy(·),Θ(A−σy(·))) ∈ WPAPρ(R, L
2(P, L0

2
)) by Theorem 2.2. Further by Theorem

3.2 we get
∫t
−∞

(t − s)α−1AσJ (t − s)F(s, A−σy(s),Θ(A−σy(s)))dw(s) ∈ WPAPρ(R, L
2(P, H)).

Similarly by Theorem 2.2 and Theorem 3.2 we come to the conclusion that
∫
t

−∞
(t−s)α−1AσJ (t−

s)G(s, A−σy(s)ds ∈ WPAPρ(R, L
2(P, H)).

Further, we show that

∑

ti<t

AσI(t − ti)Gi(A
−σy(ti)) ∈ WPAPρ(R, L

2(P, H)). (3.6)



102 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

By Theorem 2.2, Gi(A
−σy(ti)) ∈ WPAPρ(Z, L

2(P, H)). Now, let Gi(A
−σy(ti)) = φi +ψi, where

φi ∈ AP(Z, L
2(P, H)) and ψi ∈ PAPρ(Z, L

2(P, H)) for all i ∈ Z, so
∑

ti<t

AσI(t − ti)Gi(A
−σy(ti)) =

∑

ti<t

AσI(t − ti)φi +
∑

ti<t

AσI(t − ti)ψi

:= Υ1(t) + Υ2(t).

Since {t
j

i
: i, j ∈ Z} is equipotentially almost periodic, then by Lemma 2.2, for ϵ > 0 there exists

relative dense set Rϵ of real numbers and Zϵ of integers, such that for ti < t < ti+1, τ ∈ Rϵ,

p ∈ Zϵ, |t − ti| > ϵ, |t − ti+1| > ϵ, i ∈ Z, we have

t + τ > ti + ϵ + τ > ti+p,

ti+p+1 > ti+1 + τ − ϵ > t + τ,

that is, ti+p < t + τ < ti+p+1, then using Cauchy-Schwarz inequality we have

E∥Υ1(t + τ) − Υ1(t)∥
2 ≤E

(
(
(
(

∑

ti<t+τ

AσI(t + τ − ti)φi −
∑

ti<t

AσI(t − ti)φi

(
(
(
(

2

≤E

& ∑

ti<t

∥AσI(t − ti)[φi+p − φi]∥

'2

≤
∑

ti<t

E

&∫
∞

0

Nα(θ)A
σS((t − ti)

αθ)∥φi+p − φi∥dθ

'2

≤
∑

ti<t

E

&∫
∞

0

Nα(θ)Mσ((t − ti)
αθ)−σe−λ(t−ti)

α
θ∥φi+p − φi∥dθ

'2

≤M2σ
∑

ti<t

E

&∫
∞

0

Nα(θ)((t − ti)
αθ)−σe−λ(t−ti)

α
θ∥φk+p − φi∥dθ

'2

<ϵM2σR(θ),

where

R(θ) =

!∫
∞

0

Nα(θ)

& ∑

0<t−ti≤1

((t − ti)
αθ)−σe−λ(t−ti)

α
θ

+

∞∑

l=1

∑

l<t−ti≤l+1

((t − ti)
αθ)−σe−λ(t−ti)

α
θ

'

dθ

"2
. (3.7)

By Lemma 2.4 and (H1), we have

R(θ) ≤

!∫
∞

0

Nα(θ)

&
2N0
Mσ

0

+
2N0
eλ − 1

'

dθ

"2
= 4N2

0

!
1

Mσ
0

+
1

eλ − 1

"2
, (3.8)

where M0 = min{(t − ti)
αθ, 0 < t − ti ≤ 1}. Then

E∥Υ1(t + τ) − Υ1(t)∥
2 < 4ϵM2σN

2
0

!
1

Mσ
0

+
1

eλ − 1

"2
.



CUBO
19, 1 (2017)

Weighted pseudo Almost periodic solutions for fractional order . . . 103

Hence Υ1(t) ∈ AP
p(R, L2(P, H)).

Next, we show that Υ2(t) ∈ PAPρ(R, L
2(P, H)). Define

x(t) = AσI(t − ti)ψi, t ∈ (ti, ti+1], i ∈ Z,

then

lim
t→∞

E∥x(t)∥2 = lim
t→∞

E∥AσI(t − ti)ψi∥
2

≤ lim
t→∞

E

&∫
∞

0

Nα(θ)A
σS((t − ti)

αθ)∥ψi∥dθ

'2

≤ lim
t→∞

M2
σ

∫
∞

0

N 2
α
(θ)((t − ti)

αθ)−2σe−2λ(t−ti)
α
θE∥ψi∥

2dθ

≤ lim
t→∞

M2σE∥ψi∥
2(t − ti)

−2ασ

∫
∞

0

N 2α(θ)θ
−2σe−2λ(t−ti)

α
θdθ = 0,

then x ∈ PAPρ(R, L
2(P, H)). Define xi : R → L

2(P, H) by

xn(t) = A
σI(t − ti−n)ψi−n, t ∈ (ti, ti+1], n ∈ Z

+,

so, xn ∈ PAPρ(R, L
2(P, H)). Moreover

E∥xn(t)∥
2 = E∥AσI(t − ti−n)ψi−n∥

2

≤ E

&∫
∞

0

Nα(θ)A
σS((t − ti−n)

αθ)∥ψi−n∥dθ

'2

≤ M2σ

∫
∞

0

N 2α(θ)((t − ti−n)
αθ)−2σe−2λ(t−ti−n)

α
θE∥ψi−n∥

2dθ

≤ M2
σ
C0

∫
∞

0

N 2
α
(θ)((t − ti + nk)

αθ)−2σe−2λ(t−ti+nk)
α
θdθ

≤ M2σC0

∫
∞

0

N 2α(θ)(t − ti + nk)
−2ασθ−2σe−2λ(t−ti+nk)

α
θdθ

≤
M2σC0

(t − ti + nk)2ασ

∫
∞

0

N 2
α
(θ)θ−2σe−2λ(t−ti+nk)

α
θdθ,

where sup
i∈Z

E∥ψi∥
2 = C0. Therefore by Dirchlet Test the series

∑
∞

n=1 xn converges uniformly

on R. By Lemma 2.3, we have

Υ2(t) =
∑

ti<t

AσI(t − ti)ψi =
∞∑

n=0

xn ∈ WPAPρ(R, L
2(P, H)).

So, (Qy) ∈ WPAPρ(R, L
2(P, H)).



104 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

Moreover

E∥Qy(t)∥2
σ

≤3E∥
∑

ti<t

AσI(t − ti)Gi(A
−σy(ti))∥

2 + 3E∥

∫t

−∞

(t − s)α−1AσJ (t − s)G(s, A−σy(s))ds∥2

+ 3E∥

∫
t

−∞

(t − s)α−1AσJ (t − s)F(s, A−σy(s),Θ(A−σy(s)))dw(s)∥2

≤3M2
σ

∑

ti<t

E∥

∫
∞

0

Nα(θ)((t − ti)
αθ)−σe−λ(t−ti)

α
θGi(A

−σy(ti))dθ∥
2

+ 3α2M2σE

&∫t

−∞

∫
∞

0

θ(1−σ)Nα(θ)(t − s)
(α−σα−1)e−λθ(t−s)

α

∥G(s, A−σy(s))∥dθds

'2

+ 3α2M2σ

∫t

−∞

∫
∞

0

θ2(1−σ)N 2α(θ)(t − s)
2(α−σα−1)e−2λθ(t−s)

α

E∥F(s, A−σy(s),Θ(A−σy(s)))∥2dθds

≤3M2
σ

∑

ti<t

E

&∫
∞

0

Nα(θ)((t − ti)
αθ)−σe−λ(t−ti)

α
θ∥Gi(A

−σy(ti))∥dθ

'2

+ 3α2M2
σ

∫
∞

0

Nα(θ)

∫
t

−∞

θ(1−σ)(t − s)(α−σα−1)e−λθ(t−s)
α

dsdθ

×

∫
∞

0

Nα(θ)

∫t

−∞

θ(1−σ)(t − s)(α−σα−1)e−λθ(t−s)
α

E∥G(s, A−σy(s))∥2dsdθ

+ 3α2M2σ

∫t

−∞

∫
∞

0

θ2(1−σ)N 2α(θ)(t − s)
2(α−σα−1)e−2λθ(t−s)

α

E∥F(s, A−σy(s),Θ(A−σy(s)))∥2dθds

For ξ = t − s, we have

E∥Qy(t)∥2σ ≤3M
2
σC

2
Gi

&∫
∞

0

Nα(θ)
∑

ti<t

((t − ti)
αθ)−σe−λ(t−ti)

α
θdθ

'2

+ 3M2
σ
C2

G

&

α

∫
∞

0

Nα(θ)

∫
∞

0

θ(1−σ)ξ(α−σα−1)e−λθξ
α

dξdθ

'2

+ 3α2M2σC
2
F

∫
∞

0

N 2α(θ)

∫
∞

0

θ2(1−σ)ξ2(α−σα−1)e−2λθξ
α

ξdθ. (3.9)

By standard calculation, we have

&

α

∫
∞

0

Nα(θ)

∫
∞

0

θ(1−σ)ξ(α−σα−1)e−λθξ
α

dξdθ

'2

=

&
1

λ1−σ

∫
∞

0

Nα(θ)

∫
∞

0

(λθξα)−σe−λθξ
α

d(λθξα)dθ

'2

=
Γ2(1 − σ)

λ2(1−σ)
. (3.10)



CUBO
19, 1 (2017)

Weighted pseudo Almost periodic solutions for fractional order . . . 105

Recalling the results (3.3), (3.8) and (3.10) in (3.9), we have

E∥Qy(t)∥2
σ
≤3M2

σ

&

4C2GiN
2
0

!
1

Mσ
0

+
1

eλ − 1

"2
+ C2

G

Γ2(1 − σ)

λ2(1−σ)
+ C2

F
N0
Γ(1 − 2σ)

λ(2−2σ)

'

≤ r0,

this implies that Qy ∈ M. Thus Q is well defined.

Now we show that Q is contraction. For y1, y2 ∈ M, we have

E∥Qy1(t) − Qy2(t)∥
2

≤3E

(
(
(
(
∑

ti<t

AσI(t − ti)[Gi(y1(ti)) − Gi(y2(ti))]

(
(
(
(

2

+ 3E

(
(
(
(

∫
t

−∞

(t − s)α−1AσJ (t − s)[G(s, A−σy1(s)) − G(s, A
−σy2(s))]ds

(
(
(
(

2

+ 3E

(
(
(
(

∫
t

−∞

(t − s)α−1AσJ (t − s)[F(s, A−σy1(s),Θ(A
−σy1(s)))

− F(s, A−σy2(s),Θ(A
−σy2(s)))]dw(s)

(
(
(
(

2

≤3M2σ
∑

ti<t

E

&∫
∞

0

Nα(θ)((t − ti)
αθ)−σe−λ(t−ti)

α
θ∥Gi(y1(ti)) − Gi(y2(ti))∥dθ

'2

+ 3α2M2σ

∫
∞

0

Nα(θ)

∫t

−∞

θ(1−σ)(t − s)(α−σα−1)e−λθ(t−s)
α

dsdθ

×

∫
∞

0

Nα(θ)

∫
t

−∞

θ(1−σ)(t − s)(α−σα−1)e−λθ(t−s)
α

E∥G(s, A−σy1(s)) − G(s, A
−σy2(s))∥

2dsdθ

+ 3α2M2
σ

∫
t

−∞

∫
∞

0

θ2(1−σ)N 2
α
(θ)(t − s)2(α−σα−1)e−2λθ(t−s)

α

E∥F(s, A−σy1(s),Θ(A
−σy1(s)))

− F(s, A−σy2(s),Θ(A
−σy2(s)))∥

2dθds

≤3M2
σ

&

LG

!∫
∞

0

Nα(θ)
∑

ti<t

((t − ti)
αθ)−σe−λ(t−ti)

α
θdθ

"2

+ LG

!

α

∫
∞

0

Nα(θ)

∫
∞

0

θ(1−σ)ξ(α−σα−1)e−λθξ
α

dξdθ

"2

+ LF

!

1 +
LgC

2
K

2b

"∫
∞

0

N 2α(θ)

∫
∞

0

α2θ2(1−σ)ξ2(α−σα−1)e−2λθξ
α

dθdξ

'

E∥y1(t) − y2(t)∥
2.

Recalling the results (3.3), (3.8) and (3.10), we have

E∥Qy1(t) − Qy2(t)∥
2 ≤3M2

σ

&

4LGN
2
0

!
1

Mσ
0

+
1

eλ − 1

"2
+ LG

Γ2(1 − σ)

λ2(1−σ)

+ LFN0

!

1 +
LgC

2
K

2b

"
Γ(1 − 2σ)

λ(2−2σ)

'

sup
t∈R

E∥y1(t) − y2(t)∥
2.



106 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

This implies that ∥Qy1 − Qy2∥∞ ≤
√
∆∥y1 − y2∥∞, where

∆ = 3M2
σ

&

4LGN
2
0

!
1

Mσ
0

+
1

eλ − 1

"2
+ LG

Γ2(1 − σ)

λ2(1−σ)
+ LFN0

!

1 +
LgC

2
K

2b

"
Γ(1 − 2σ)

λ(2−2σ)

'

< 1.

Thus, Q is a contraction. Hence by Banach contraction principle the system (1.1) − (1.3), admits

a unique mild solution #y ∈ M such that Q#y = #y. Moreover, since Aσ is closed, then we have

A−σ#y(t) =
∑

ti<t

I(t − ti)Gi(A
−σ#y(ti)) +

∫t

−∞

(t − s)α−1J (t − s)G(s, A−σ#y(s)ds

+

∫
t

−∞

(t − s)α−1J (t − s)F(s, A−σ#y(s),Θ(A−σ#y(s)))dw(s). (3.11)

such that E∥A−σ#y(t)∥2 < r0, for all t ∈ R. Hence A
−σ#y(t) ∈ M is unique mild solution of the

system (1.1) − (1.3).

Theorem 3.2. Assume that (H1) − (H7) are hold. Then the system (1.1) − (1.3) has a unique

exponentially stable mild solution in WPAPρ(R, L
2(P, H)).

Proof. The existence and uniqueness followed by Theorem 3.1 and adopting the ideas developed

in [11, Theorem 3.3], we come to the conclusion.

Remark 4. Consider the following equation with delay

cDα
t
y(t) + Ay(t) = G(t, y(t − η)) + F

!

t, y(t − η),

∫
t

−∞

K(t − s)g(s, y(s))ds

"
dw(t)

dt
,

t0 < t ̸= ti, t ∈ R, (3.12)

y(t+
i
) = y(t−

i
) + Gi(y(ti)), i ∈ Z, (3.13)

y(t0) = y0, (3.14)

where η is fix and η ∈ R+. Assume that y = y1 + y2 ∈ WPAPρ(R, L
2(P, H)) where

y1 ∈ AP
p(R, L2(P, H)) and y2 ∈ PAPρ(R, L

2(P, H)). For given η ∈ R, it is easy to show that

y(t − η) ∈ APp(R, L2(P, H)).

For γ > 0, we have

1

µ(γ,ρ)

∫γ

−γ

E∥y2(t − η)∥
2ρ(t)dt =

1

µ(γ,ρ)

∫γ−η

−γ−η

E∥y2(t)∥
2ρ(t + η)dt

=
µ(γ + η,ρ)

µ(γ,ρ)
×

1

µ(γ + η,ρ)

∫
γ−η

−γ−η

E∥y2(t)∥
2
ρ(t + η)

ρ(t)
ρ(t)dt.



CUBO
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Weighted pseudo Almost periodic solutions for fractional order . . . 107

Since ρ ∈ WT , then there exists a0 > 0 such that
ρ(t+η)
ρ(t)

≤ a0,
ρ(t−η)
ρ(t)

≤ a0,. For γ > η

µ(γ + η,ρ) =

∫γ−η

−γ−η

ρ(t)dt +

∫γ+η

γ−η

ρ(t)dt

≤

∫γ−η

−γ−η

ρ(t)dt +

∫γ+η

−γ+η

ρ(t)dt

=

∫γ

−γ

ρ(t − η)

ρ(t)
ρ(t)dt +

∫γ

−γ

ρ(t + η)

ρ(t)
ρ(t)dt ≤ 2a0µ(γ,ρ).

Then by y2 ∈ PAPρ(R, L
2(P, H)), we get

1

µ(γ,ρ)

∫γ

−γ

E∥y2(t − η)∥
2ρ(t)dt ≤

2a2
0

µ(γ + η,ρ)

∫γ+η

−γ−η

E∥y2(t)∥
2ρ(t)dt → 0,

as γ → ∞. Hence y(t − η) ∈ WPAPρ(R, L
2(P, H)) for η ∈ R+. Thus the results of Theorem 3.1

holds for the system (3.12) − (3.14).

4 Example

Now,we present an example, which do not aim at generality but indicate how our abstract result

can be applied to concrete problem. Consider the stochastic fractional differential equation with

impulsive effects

∂α

∂tα
z(t, x) −

∂2

∂x2
z(t, x) =G(t, x, z(t, x)) + F

!

t, x, z(t, x),

∫
t

∞

e−2(t−s)g(s, x, z(t, x)ds

"
dw(t)

dt
, t ∈ R,

(4.1)

z(t+
i
, x) =z(t−

i
, x) + λi(z(ti, x)), i ∈ Z, x ∈ (0, 1), (4.2)

z(t, 0) =z(t, 1) = 0, (4.3)

where ti = i +
1

4
| sin 3i + sin

√
3i| and assume that λi ∈ WPAPρ(Z, L

2(P, H)), ρ ∈ WT . Note that

{t
j

i
}, i, j ∈ Z are equipotentially almost periodic and κ = infi∈Z(ti+1 − ti) > 0, for more details see

[14, 21, 24]. Note that w(t) represents a standard Wiener process on a complete probability space

(Ω, F, {Ft}t≥0, P), where {Ft}t≥0 is sigma algebra generated by {w(u) − w(v) : u, v ≤ t}.

Let H = (L2([0, 1], ∥ · ∥L2) be a Hilbert space. Now define the operator

Ay(ξ) := −y′′(ξ), ξ ∈ (0, 1), y ∈ D(A),

where

D(A) := {H2 ∩ H10 : y
′′ ∈ H}.

Then, A is the infinitesimal generator of analytic semigroup S(t) on H. Now, we have zn(t) =

(2)
1

2 sin nπt, n = 1, 2, 3, ..., are the eigenfunction of A corresponding to the eigenvalues nπ. For

σ = 1
4
denote D(A

1

4 ) by L2(P, H 1
4

) is a Banach space equipped with the norm

∥y∥ 1
4

= ∥A
1

4 y∥, y ∈ D(A
1

4 ).



108 Vikram Singh and Dwijendra N Pandey CUBO
19, 1 (2017)

Let y(t)x = z(t, x), t ∈ R, x ∈ [0, 1], then

F

!

t, y(t),

∫
t

−∞

K(t − s)g(s, y(s))ds

"

= F

!

t, x, z(t, x),

∫
t

∞

e−2(t−s)g(s, x, z(t, x)ds

"

.

Now the system (4.1)-(4.3) can be reformulated in the abstract form of the system (1.1)-(1.3).

Since Gi = λi, then (H6) holds with LG = supi∈Z ∥λi∥. We have the following result.

Theorem 4.1. Assume that the following assumptions hold:

(i) For ρ ∈ WT , g ∈ WPAPρ(R × L
2(P, H 1

4

), L2(P, H)) and there exists a Lg > 0, 0 < η < 1 such

that

E∥g(t1, u1) − g(t2, u2)∥
2 ≤ LgE(|t2 − t1|

η + ∥u1 − u2∥
2
1

2

,

for all, (ti, ui) ∈ R × L
2(P, H 1

4

), i = 1, 2.

(ii) For ρ ∈ WT , G ∈ WPAPρ(R × L
2(P, H 1

4

), L2(P, H)) and there exists LG > 0, 0 < η < 1 such

that

E∥G(t1, u1) − G(t2, u2)∥
2 ≤ LG(|t2 − t1|

η + E∥u1 − u2∥
2
1

2

),

for each (ti, ui) ∈ R × L
2(P, H 1

4

), i = 1, 2.

(iii) For ρ ∈ WT , F ∈ WPAPρ(R × L
2(P, H 1

4

) × L2(P, H 1
4

), L2(P, L2
0
)) and there exists a LF > 0

such that

E∥F(t, u1, v1) − f(t, u2, u2)∥
2 ≤ LF(|t2 − t1|

ηE∥u1 − u2∥
2
1

2

+ E∥v1 − v2∥
2
1

2

),

for each (ti, ui, vi) ∈ R × L
2(P, H 1

4

) × L2(P, H 1
4

), i = 1, 2.

Let us choose the constants Mσ = 1,λ = 9, LG = 1, LF = 1, Lg = 1, LG =
1

2
and N0 =

1

4
, then we

have

∆ := 3M2σ

&

4LGN
2
0

!
1

Mσ
0

+
1

eλ − 1

"2
+ LG

Γ2(3
4
)

λ
3

2

+ LFN0

!

1 +
Lg
4

"√
π

λ
3

2

'

= 0.69 < 1,

this implies that the system (4.1)−(4.3) has a unique piecewise square mean weighted pseudo almost

periodic solution.

Acknowledgment

The authors would like to thank the editor and the reviewers for their valuable comments and

suggestions. The work of the first author is supported by the “Ministry of Human Resource and

Development, India under grant number:MHR-02-23-200-44”.



CUBO
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Weighted pseudo Almost periodic solutions for fractional order . . . 109

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