International Journal of Analysis and Applications

Volume 17, Number 1 (2019), 26-32

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-26

ON NONCLASSICAL IMPULSIVE ORDINARY DIFFERENTIAL EQUATIONS WITH

NONLOCAL CONDITIONS

S. A. BISHOP∗, M. C. AGARANA AND J. G. OGHONYON

Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria

∗Corresponding author: sheila.bishop@covenantuniversity.edu.ng

Abstract. Results on mild solutions of nonclassical differential equations with impulsive and nonlocal

conditions are extended to a case when the nonlocal conditions are necessarily non Lipschitz and non

compact.

1. Introduction

We study the following quantum stochastic differential equation (QSDE) with impulsive nonlocal condi-

tions introduced in [1];

dz(t) = A(t)z(t) + E(t, (z(t))d∧π (t) + F(t,z(t))dAf (t)

+G(t,z(t))dA+g (t) + H(t,z(t))dt,

almost all t ∈ I,t 6= tk,k = 1, ...,m (1)

∆z(tk) = Jk(z(t
−
k )),k = 1, ...,m

z(t0) = z0 + g(z), t ∈ [0,T]

where

(i) A is a family of semigroup defined in [1]

Received 2018-07-04; accepted 2018-09-12; published 2019-01-04.

2010 Mathematics Subject Classification. 35A24.

Key words and phrases. nonclassical ordinary differential equations (NODEs); non-compact; nonlocal conditions; impulse

effect; non Lipschitz; Stochastic processes.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

26

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-26


Int. J. Anal. Appl. 17 (1) (2019) 27

(ii) E, F, G, H are stochastic processes.

(iii) Jk ∈ C(B̃, B̃),k = 1, 2, ...,m and ∆z(tk) is the difference between z(t+k ) and z(t−k ).

(v) g : B̃ → PC(I,sesq(ID⊗IE)) is a nonlocal condition that is not Lipschitz and not compact.

(vi) z ∈ B̃ is a stochastic process and η,ξ ∈ ID⊗IE is arbitrary.

Problems with nonlocal conditions have been an area of interest, mostly because of the advantage they

have over initial value problems. Existence of solution of nonlocal problems for different types of differential

equations were extensively discussed in the literature by using various methods (See [1, 2, 3-14] and the

references therein). The motivation for this study, is that nonlocal problems occur naturally when modeling

physical problems. In [2], impulsive quantum stochastic differential equations (IQSDE) with initial value

conditions were studied. The multivalued maps are lower respectively upper semicontinuous. In [1], existence

results for Eq.(1) with nonlocal conditions that are completely continuous were established. We showed

that the function g which constitute the nonlocal condition is compact and Lipschitz continuous. Several

interesting results on nonlocal impulsive differential equations satisfying some Lipschitz and compactness

conditions have been established in [6-9]. In this study, existence of solution of Eq. (1) is established

with nonlocal conditions that are not necessarily Lipschitz and compact. We adopt the most suitable fixed

point method to establish this result. Impulsive QSDEs have found applications in quantum continuous

measurements, especially when the mean number of photons up to time ti is momentary giving rise to

impulses on the counting stochastic processes associated with the observables x(ti). See [1, 2] and the

references therein.

2. Preliminaries

The definitions of the following spaces L2loc(B̃)mvs, B̃, PC(I, B̃), PC
′(I, B̃), PC(I,sesq(ID⊗IE)) and

PC′(I,sesq(ID⊗IE)) are adopted from [1, 2]. The spaces B̃ and PC(I,sesq(ID⊗IE)) denote the locally convex

and Banach spaces respectively. The Hausdorff distance, ρ(A,B) is defined as:

ρ(A,B) = max(δ(A,B),δ(B,A)),A,B ∈ clos(C)

and

d(z,B) = Infy∈B|z −y|, δ(A,B) = Supz∈Ad(z,B)

where x ∈ C is as defined in [1] and ρ is a metric.

Definition 1. A stochastic process z ∈ PC(I,Ã) is called a solution of Eq. (1) if it satisfies the integral



Int. J. Anal. Appl. 17 (1) (2019) 28

equation

z(t) = S(t)[z0 + g(z)]

+

∫ t
0

S(t−s)(E(s, (z(s))dΛπ(s) + F(s,z(s))dAf (s)

+G(s,z(s))dA+g (s) + H(s,z(s))ds) +
∑

0<tk<t

S(t− tk)Jk(z(tk)) (2)

for t ∈ [0,T]. To establish a mild solution with non Lipschitz, non compact and nonlocal condition, we must

assume that z(t) near zero does not affect the nonlocal condition g(z). We use the equivalent form of Eq.

(1) given by,

d

dt
〈η,z(t)ξ〉 = P(t,z(t))(η,ξ). (3)

Remark. Equation (3) is also known as nonclassical ordinary differential equation. See [1] and the references

therein. Next, we state the following assumptions:

(H1) The map P in Eq. (3) is continuous and well defined in [1]. Hence, there exists a function K
P
ηξ :

[0,T] → IR+ so that

‖P(t,y) −P(t,z)‖ηξ ≤ K
P
ηξ(t)‖y −z‖ηξ ,

t ∈ [0,T], y,z ∈ B̃.

(H2) Jk ∈ C(B̃, B̃) and T(.) are compact operators

(H3) For each z0 ∈ B̃, we have constants hηξ > 0 and M > 0 so that

‖S(t)‖ηξ ≤ M,t ≥ 0

and

M

(
‖z0‖ηξ + sup

ϕ∈Hh
‖g(ϕ)‖ηξ + K

P
ηξ(t) sup

s,t∈[0,T]
‖P(s,ϕ(s))‖ηξ + sup

ϕ∈Zh

m∑
k=1

‖Jk(ϕ(tk))‖ηξ

)
≤ hηξ

where

Hh :=
{
ϕ ∈ PC([0,T],Ã) : ‖(ϕ(t))‖ηξ ≤ hηξ, t ∈ [0,T]

}
(H4) g : PC([0,T],Ã) →Ã is continuous and constitute the nonlocal condition. Also g : Hh → bd

Let δ depend on hηξ ∈ (0, t1) and g(ϕ) = g(φ), ϕ,φ ∈ Hh where ϕ(s) = φ(s),s ∈ [δ,T] and bd denote a

bounded set.



Int. J. Anal. Appl. 17 (1) (2019) 29

3. Main Result

Theorem 1. Let conditions (H1)-(H4) hold. Then for z0 ∈ B̃, problem (1) has at least a solution.

Proof. Let δ ∈ (0, t1), define H(δ) and Hh(δ) as ;

H(δ) := PC([δ,T], B̃) for functions in PC([0,T], B̃) on [δ,T]

and

Hh(δ) := {ϕ ∈ H(δ) : ‖ϕ(t)‖ηξ ≤ hηξ, t ∈ [δ,T]} .

Let z ∈ Hh(δ) be fixed. Then define a map Γz on Hh by

Γz(ϕ)(t)(η,ξ) = 〈η, [z0 + g(z̃)]ξ〉

+

∫ t
0

S(t−s)P(s, (ϕ(s))(η,ξ)ds

+
∑

0<tk<t

S(t− tk)Jk(u(tk)), t ∈ [0,T], z̃ ∈ Hh(δ)

where

z̃(t) = z(t), if t ∈ [δ,T]

and

z̃(t) = z(δ), if t ∈ [0,δ]

This shows that Γz is a continuous mapping from Hh to Hh. Also, we have

‖Γnz (ϕ)(t) − Γ
n
z (φ)(t)‖ηξ

=

∣∣∣∣
∫ t
0

S(t−s)(P(s,ϕ(s))(η,ξ) −P(s,φ(s))(η,ξ)ds)
∣∣∣∣

≤ M
∫ t
0

‖P(s,ϕ(s)) −P(s,φ(s))‖ηξds

≤ M
∫ t
0

KPηξ(s)‖ϕ(s) −φ(s)‖ηξds (3)

Since the map s → sups∈[0,T] ‖ϕ(s) −φ(s)‖ηξ is continuous, we let

Rηξ = sup
s∈[0,t]

‖ϕ(s) −φ(s)‖ηξ

and

Nηξ(t) =

∫ t
0

KPηξ(s)ds.

Then from (3), we get

‖Γnz (ϕ)(t) − Γ
n
z (φ)(t)‖ηξ ≤

Rηξ(MNηξ(t))
n

n!
,n = 1, 2, .... (4)



Int. J. Anal. Appl. 17 (1) (2019) 30

This can also be proved by induction, see [7] in [1].

So we get

‖Γmz (ϕ)(t) − Γ
m
z (φ)(t)‖ηξ =

∥∥∥∥∥
n∑

m=k+1

(Γmz (ϕ)(t) − Γ
m
z (φ)(t))

∥∥∥∥∥
ηξ

≤
n∑

m=k+1

‖(Γmz (ϕ)(t) − Γ
m
z (φ)(t))‖ηξ

≤
n∑

m=k+1

Rηξ(MNηξ(T))
m

m!

where t ∈ [0,T], ϕ,φ ∈ Hh,m = 1, 2, ...

Now by considering large m, we see that Γmz is a contraction operator on Hh and hence Γz has a unique

fixed point given by

ϕz(t)(η,ξ) = 〈η, [z0 + g(z̃)]ξ〉

+

∫ t
0

S(t−s)P(s,ϕz(s))(η,ξ)ds

+
∑

0<tk<t

S(t− tk)Jk(z(tk)) ∈ Hh, t ∈ [0,T]

From the above, define a map Υ : Hh(δ) → Hh(δ) by

Υ(z)(t)(η,ξ) = ϕz(t)(η,ξ) = 〈η, [z0 + g(z̃)]ξ〉

+

∫ t
0

S(t−s)P(s,ϕz(s))(η,ξ)ds

≤
∑

0<tk<t

S(t− tk)Jk(z(tk)), t ∈ [δ,T] (5)

The next following steps show that the map Υ has a fixed point:

By applying Lebesgue dominated convergence theorem and Arzela-Ascoli theorem we show that; (1) the

operator Υ is continuous;

(2) It maps bd sets into bd sets in PC(I,sesq(ID⊗IE));

(3) Υ maps bd sets into equicontinuous sets in PC(I,sesq(ID⊗IE)) and by applying the Schauder-Tychonov’s

fixed point theorem we get a z∗ ∈ Hh(δ), which is a fixed point.

By replacing Υ(z) with z, g(z̃) with g(z̃∗) in (5), we get

z(t)(η,ξ) = 〈η, [z0 + g(z̃∗)]ξ〉

+

∫ t
0

S(t−s)P(s,z(s))(η,ξ)ds

≤
∑

0<tk<t

S(t− tk)Jk(z∗(tk)), t ∈ [0,T] (6)



Int. J. Anal. Appl. 17 (1) (2019) 31

But

g(z̃∗) = g(z̃), z̃∗(tk) = z̃(tk)

and by the definition of the map Υ,

z̃∗(t) = Υ(z̃∗)(t) = ϕz̃∗ (t) = z̃(t).

So that by (6), we conclude that z(t) is the required solution of Eq. (1) which is the desired result.

Conclusion

Using the equivalent form of QSDE (1) given by Eq. (3) and having satisfied the conditions of the appro-

priate fixed point theorem, we conclude that a fixed point exists and it is a solution of the problem.

Conflict of Interest

The authors declare that there is no conflict of interests.

References

[1] S. A. Bishop, E. O. Ayoola, J. G. Oghonyon, Existence of mild solution of impulsive quantum stochastic differential

equation with nonlocal conditions. Anal. Math. Phys. 7 (3) (2017), 255-265.

[2] S. A. Bishop and P. E. Oguntunde, Existence of Solutions of Impulsive Quantum Stochastic Differential Inclusion, J. Eng.

Appl. Sci. 10 (7) (2015), 181-185.

[3] T. Caraballo, M. A. Hammami, L. Mchiri, Practical exponential stability of impulsive stochastic functional differential

equations. Syst. Control Lett. 109 (2017), 43-48.

[4] T. Cardinali, P. Rubbioni, Nonlinear Anal., Theory Methods Appl. 75 (2) (2012), 871-879.

[5] M. A. Dialloa, K. Ezzinbib, A. Sene, Impulsive integro-differential equations with nonlocal conditions in Banach spaces.

Trans. A. Razmadze Math. Inst. 171 (2017), 304-315.

[6] Q.Dong, G. Li, Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces.

Electron J. Qual. Theory Diff. Equ. 47 (2009), Art. ID 47.

[7] Z.B. Fan, G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct.

Anal. 258 (2010), 1709-1727.

[8] G. Li, Existence results for semilinear differential equations with nonlocal conditions and impulsive conditions. J. Funct.

Anal. 258(2010), 1709-1727.

[9] L. Zhu , Q. Dong and G. Li, Impulsive differential equations with nonlocal conditions in general Banach spaces. Advances

in Diff. Equ. 2012 (2012), Art. ID 10.

[10] A. G. Okeke, S A. Bishop,S. H. Khan Iterative Approximation of Fixed Point of Multivalued ρ-Quasi-Nonexpansive

Mappings in Modular Function Spaces with Applications, J. Funct. Spaces 2018 (2018), Art. ID 1785702.

[11] O. Bolojan-Nica, G. Infante, R. Precupa, Existence results for systems with coupled nonlocal initial conditions. Nonlinear

Anal., Theory Methods Appl. 94 (2014), 231-242

[12] S. Ji and G. Li, A unified approach to nonlocal impulsive differential equations with the measure of noncompactness. Adv.

Diff. Equ. 2012 (2012), Art. ID 182.



Int. J. Anal. Appl. 17 (1) (2019) 32

[13] X.M. Xue, Semilinear nonlocal problems without the assumptions of compactness in Banach spaces. Anal. Appl. 8 (2010),

211-225.

[14] X.M, Xue, Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces.Nonlinear Anal.,

Theory Methods Appl. 70 (2009), 2593-2601.


	1. Introduction
	2. Preliminaries
	3. Main Result
	References