8_devi.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 2, 2012, 187-196

ISSN 1307-5543 – www.ejpam.com

Stability Results for Set Differential Equations Involving Causal

Operators with Memory

J. Vasundhara Devi∗, Ch. Appala Naidu

GVP–Prof. V. Lakshmikantham, Department of Mathematics, Institute for Advanced Studies, GVP

College of Engineering, Visakhapatnam, Andhra Pradesh, India

Abstract. In this paper we study the stability concepts for set differential equations involving causal

operators with memory by considering initial functions as a Hukuhara difference of two functions. This

will enable to obtain results parallel to ordinary differential equations with delay.

2010 Mathematics Subject Classifications: 34A12
Key Words and Phrases: Set differential equations, causal operators, causal operator with memory,

delay, stability, asymptotic stablility.

1. Introduction

It is well recognized and accepted that set differential equations are a generalization of or-

dinary differential equations and vector differential equations in a semilinear metric space and

that they are useful in studying multivalued differential inclusions or multivalued differential

equations.

Also causal operators or Volterra operators or non anticipative operators encompass a

wide range of equations such as ordinary differential equations, integral equations, integro

differential equations, to name a few.

Thus set differential equations involving causal operators with memory include the above

said special cases for various types of equations and such a generalization is interesting as it

gives a comprehensive view of different types of differential equations.

Also it is observed in [1] that solutions for set differential equations contain a lot of unde-

sirable information that needs to be seperated, so that the equations in this setup satisfy the

stability behaviour similar to that of scalar or vector equations. In order to take care of this

situation, the Hukuhara difference in initial values is introduced.

In this paper we extend results in [1] to set differential equations involving causal op-

erators with memory by considering the Hukuhara difference of initial functions. We obtain

stability results using Lyapunov-like functions and the concept of minimal classes.

∗Corresponding author.

Email address: jvdevi�gmail.
om (J. Devi)

http://www.ejpam.com 187 c© 2012 EJPAM All rights reserved.



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 188

2. Preliminaries

We begin with the definitions of Kc(R
n), the semilinear space in which we work. We

next define the Hausdorff Metric, the Hukuhara difference, the Hukuhara derivative and the

Hukuhara Integral. We also state all the important properties that are useful in this paper. We

further define a partial order in Kc(R
n) . We also state all the required results developed in

[4] that will be used in this paper.

Let Kc(R
n) denote the collection of all nonempty, compact and convex subsets of Rn. Define

the Hausdorff metric by

D[A, B] =max[sup
x∈B

d(x ,A), sup
y∈A

d(y, B)], (1)

where d(x ,A) = inf[d(x , y) : y ∈ A], A, B are bounded sets in Rn. We note that Kc(R
n) with

this metric is a complete metric space.

It is known that if the space Kc(R
n) is equipped with the natural algebraic operations

of addition and non-negative scalar multiplication, then Kc(R
n) becomes a semilinear metric

space which can be embedded as a complete cone into a corresponding Banach space.

The Hausdorff metric (1) satisfies the following properties:

D[A+ C , B+ C] = D[A, B] and D[A, B] = D[B,A], (2)

D[λA,λB] = λD[A, B], (3)

D[A, B] ≤ D[A, C] + D[C , B], (4)

for all A, B, C ∈ Kc(R
n) and λ ∈ R+.

Let A, B ∈ Kc(R
n). The set C ∈ Kc(R

n) satisfying A = B + C is known as the Hukuhara

difference of the sets A and B and is denoted by the symbol A− B. We say that the mapping

F : I → Kc(R
n) has a Hukuhara derivative DH F(t0) at a point t0 ∈ I , if

lim
h→0+

F(t0 + h)− F(t0)

h
and lim

h→0+

F(t0)− F(t0 − h)

h

exist in the topology of Kc(R
n) and are equal to DH F(t0). Here I is any interval in R.

With these preliminaries, we consider the set differential equation

DH U = F(t, U), U(t0) = U0 ∈ Kc(R
n), t0 ≥ 0, (5)

where F ∈ C[R+ × Kc(R
n), Kc(R

n)].

The mapping U ∈ C1[J , Kc(R
n)], J = [t0, t0 + a] is said to be a solution of (5) on J if it

satisfies (5) on J .

Since U(t) is continuously differentiable, we have

U(t) = U0 +

∫ t

t0

DH U(s)ds, t ∈ J . (6)



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 189

Hence, we can associate with the IVP (5) the Hukuhara integral

U(t) = U0 +

∫ t

t0

F(s, U(s))ds, t ∈ J . (7)

where the integral is the Hukuhara integral which is defined as,

∫

F(s)ds = {

∫

f (s)ds : f is any continuous selector o f F}

Observe also that U(t) is a solution of (5) on J iff it satisfies (7) on J .

We now proceed to define a partial order in the metric space (Kc(R
n), D). We begin with

the definition of a cone in this set up.

Let K(Ko) be the subfamily of Kc(R
n) consisting of set U ∈ Kc(R

n) such that any u ∈ U is a

non-negative(positive) vector of n components satisfying ui ≥ 0 (ui > 0) for i = 1 . . . n. Then

K is a cone in Kc(R
n) and K0 is the nonempty interior of K .

Definition 1. For any U and V ∈ Kc(R
n), if there exists Z ∈ Kc(R

n) such that Z ∈ K(K0) and

U = V + Z then we say that U ≥ V (U > V ). Similarly we can define U ≤ V (U < V ).

To define the causal operator we introduce the following notation.

Let E = C[[t0, T], Kc(R
n)] and E0 = C[[t0 − h1, T], Kc(R

n)], where U ∈ E0 implies

U(t) = Φ0(t), t0 − h1 ≤ t ≤ t0 and U(t) is any arbitrarily continuous function on [t0, T].

We define a norm on E as follows: for U , V ∈ E

D0[U , V ] = Supt0≤t≤T D[U(t), V (t)]

where D denotes the Hausdorff Metric.

Definition 2. By a causal operator or a Volterra operator or a nonanticipative operator we

mean a mappling Q: E → E satisfying the property that if U(s) = V (s), t0 ≤ s ≤ t < T then

(QU)(s) = (QV )(s), t0 ≤ s ≤ t < T.

By a causal operator with memory we mean a mapping Q:E0→ E such that for U(s) = V (s),

t0 ≤ s ≤ t < T, then Q(U ,Φ0)(s) = Q(V,Φ0)(s), t0 ≤ s ≤ t < T and

Φ0 ∈ C1 = C[[t0 − h1, t0], Kc(R
n)]

3. Comparison Theorems

In this section we give the necessary notations and the comparison theorems required to

prove the stability theorems.

Consider the IVP for set differential equation involving causal operators with memory

given by

DH U(t) = (QU)(t), Ut0
= Φ0 ∈ C1 (8)



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 190

where C1 = C[[t0 − h1, t0], Kc(R
n)] with metric

D1[φ0,ψ0] = sup
t0−h1≤s≤t0

D[φ0(s),ψ0(s)], φ0,ψ0 ∈ C1

Let Ẽ = C[[t0 − h1,∞), Kc(R
n)] with norm

D̃[U ,θ] = sup
t0−h1≤t<∞

D[U(t),θ]

h(t)

where θ is the zero element in Rn, which is regarded as a point set and h : [t0,∞)→ R+ is a

continuous map and (Ẽ, D̃) is a Banach space. In this paper we consider the causal operator

with memory as Q ∈ C[Ẽ, Ẽ] such that Q : Ẽ → Ẽ and U(s) = V (s) for t0 − h1 ≤ s ≤ t implies

(QU)(s) = (QV )(s) for t0 − h1 ≤ s ≤ t.

In order to develop the comparison theorems using Lyapunov-like functions it is useful to

select some class of functions in Kc(R
n) or elements in Ẽ such that the generalized derivative

of the Lyapunov function satisfies certain conditions on these classes. We begin by defining

the following sets.

E0 = {U ∈ Ẽ : L(s, U(s)) ≤ f (L(t, U(t)); t1 ≤ s ≤ t, t1 ≥ t0};

E1 = {U ∈ Ẽ : L(s, U(s)) ≤ L(t, U(t)); t0 ≤ s ≤ t};

Eα = {U ∈ Ẽ : L(s, U(s))α(s) ≤ L(t, U(t))α(t); t0 ≤ s ≤ t};

where

(i) α(t) ≥ 0 is a continuous function on R+,

(ii) f (r) is continuous on R+, non decreasing in r and f (r)≥ r for r ≥ 0.

Now we proceed to state the comparison theorems using the Lyapunov-like functions. As the

proofs are similar to that in [1] we omit them.

Theorem 1. Let L ∈ C[R+ × B,R+], B = {U ∈ Kc(R
n) : D[U ,θ] ≤ ρ} = B(θ ,ρ) and let

L(t,U) be locally Lipschitzian in U (i.e) for U , V ∈ B, t ∈ R+ and K > 0,

|L(t, U)− L(t, V )| ≤ KD̃[U , V ] (9)

(i) Assume that for t ≥ t0 and U ∈ E1

D−L(t, U(t)) ≤ g(t, L(t, U(t))) (10)

where

D−L(t, U(t)) = lim inf
h→0−

1

h
[L(t + h, U(t) + h(QU)(t))− L(t, U(t)]

and g ∈ C[R+ ×R+,R+]



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 191

(ii) Let r(t) = r(t, t0,W0) be the maximal solution of scalar ordinary differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0 (11)

existing on t0 ≤ t <∞.

Let U(t, t0,φ0(t0)) be any solution of the system (8) such that U(t, t0,φ0(t0)) ∈ B for t ∈
[t0, t1] and let L(t0,φ0(t0))≤ w0 then

L(t, U(t, t0 ,φ0(t0))≤ r(t), ∀t ∈ [t0, t1], t ≥ t0.

Theorem 2. Assume that the hypothesis of Theorem 1 holds, except for inequality (10) which is

replaced by

α(t)D−L(t, U(t)) + L(t, U(t))D−α(t) ≤ w(t, L(t, U(t))α(t)), (12)

for t > t0, U ∈ Eα, where α(t) > 0 is continuous on R+ and

D−α(t) = lim inf
h→o−

α(t + h)−α(t)

h

then α(t0)L(t0,φ0)≤ w0 implies α(t)L(t, U(t)) ≤ r(t), t ≥ t0.

Theorem 3. Assume that

(i) L ∈ C[R+ × B,R+] and L(t, U) be locally Lipschitzian in U.

(ii) g0, g ∈ C[R2
+,R] such that g0(t, w) ≤ g(t, w), (t, w) ∈ R+

2, and η(t, T0, v0) is the left

maximal solution of

v′ = g0(t, v), v(T0) = v0 (13)

existing on t0 ≤ t ≤ T0 and r(t, t0, w0) is the right maximal solution of

w′ = g(t, w), w(t0) = w0 (14)

existing on [t0,∞)

(iii)

D−L(t, U(t)) ≤ g(t, L(t, U(t))) (15)

on Ω where Ω = {U ∈ Ẽ : L(s, U(s)) ≤ η(s, t, L(t, U(t))), t0 ≤ s ≤ t} then we have

L(t, U(t, t0 ,φ0(t0))≤ r(t, t0, w0), t ≥ t0 whenever L(t0,φ0(t0))≤ w0.

In the proof of Theorem 3 we use the Lemma 1.4.1 in [2].



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 192

4. Stability Results

In this section we study the stability properties of the solution of (8) in order to do so, we

assume that the solutions of (8) exist for t ≥ t0 and are unique.

It is to be noted that if we have to study the stability properties of any solution U(t, t0,φ0)

of (8). Then we have to find two functions Z and V such that the Hukuhara difference

Z = U − V exists and DH Z = DH U − DH V exists and further Q(0)(t) ≡ 0 for all t. Thus

if all the above conditions are satisfied then studying the stability properties of any solution

U(t, t0,φ0) of (8) reduces to the study of zero solution of (8).

We observe that in the generation of set differential equation involving causal operators

with memory from ordinary differential equations involving causal operators with memory,

certain undesirable elements may enter the solution U(t) of (8). In order that the solutions

of (8) project the behaviour of ordinary differential equations involving causal operator with

memory, from which they can be generated, we introduce the concept of Hukuhara difference

in initial functions. Before we introduce the theory, we consider the following example.

Example 1. Consider the set differential equation with delay on R.

DH U = −U(t −τ), U0 = [φ1,φ2] (16)

where φ1,φ2 are real valued functions and U(t) = [u1(t),u2(t)]

U ′(t) =[u′1(t),u
′
2(t)]

=− U(t −τ)

=− [u1(t −τ),u2(t −τ)]

=[−u2(t −τ),−u1(t −τ)]

therefore

u′1 = −u2(t −τ) and u′2 = −u1(t −τ) (17)

where u1(0) = φ1(0), u2(0) = φ2(0)

u′′1 = u1(t − 2τ) and u′′2 = u2(t − 2τ),−2τ≤ t ≤ 0. (18)

Suppose eλ1 t and e−λ2 t are solutions of the system (18). Let u1(t) = eλ1 t and u2(t) = e−λ2 t

u′1(t) = λ1eλ1 t u′2(t) = −λ2e−λ2 t

u′′1 (t) = λ
2
1eλ1 t u′′2 (t) = λ

2
2e−λ2 t

u1(t − 2τ) = eλ1(t−2τ) u2(t − 2τ) = e−λ2(t−2τ)

since u1(t) and u2(t) are solutions of the system (18) we have to choose λ1,λ2 > 0 such that

λ2
1 = e−2λ1τ,λ2

2 = e2λ2τ, then u1(t) = eλ1 t and u2(t) = e−λ2 t are solutions of the system (18).

As a result we have

u1(t) = C1eλ1 t + C2e−λ2 t and u2(t) = C3eλ1 t + C4e−λ2 t . (19)



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 193

Suppose the initial functions are given by

Φ1(s) =
1

2
[u10 − u20]e

λ1s +
1

2
[u10 + u20]e

−λ2s (20)

Φ2(s) =
1

2
[u20 − u10]e

λ1s +
1

2
[u10 + u20]e

−λ2s (21)

for −2τ ≤ s ≤ 0. By using the relations (17), (19), and (20) we have to compute the values

C1, C2, C3 and C4. Taking t = 0 and t = τ we get

C1 =
1

2
[u10 − u20], C2 =

1

2
[u10 + u20],

C3 =
1

2
[u20 − u10], C4 =

1

2
[u10 + u20].

Therefore

U(t) = [−
1

2
[u20 − u10],

1

2
[u20 − u10]]e

λ1 t + [
1

2
[u10 + u20],

1

2
[u10 + u20]]e

−λ2 t

Choose

ψ0 = [−
1

2
[u20 − u10],

1

2
[u20 + u10]]e

λ1s

and

χ0 = [
1

2
[u10 + u20],

1

2
[u10 + u20]]e

−λ2s

for Φ0 = [Φ1,Φ2] = ψ0 + χ0. Then U(t, t0,χ0) = [
1

2
[u10 + u20],

1

2
[u10 + u20]]e

−λ2s which

implies the stability of the trivial solution of the equation (16).

We suppose that given any solution U(t, t0,Φ0) of (8), we can find a function V such

that the Hukuhara difference U − V exists, satisfies the properties mentioned earlier in this

context so that it is sufficient to study the stability properties of the trivial solution. Other

notations of Lyapunov stability can be formulated in a similar way following the standard

stability definitions in [3]. We now present the stability theorems in this context.

Theorem 4. Assume that there exist functions L(t, U(t)) and g(t, w) satisfying the following

conditions

(i) g ∈ C[R+×R+,R+] and g(t, 0) ≡ 0;

(ii) L ∈ C[R+ × B,R+] where B = B(θ ,ρ) = {U ∈ Kc(R
n) : D[U ,θ] ≤ ρ}; L(t,θ) ≡ 0 and

L(t, U) is positive definite and locally Lipschitzian in U;

(iii) for t > t0 and U ∈ E1

D−L(t, U(t)) ≤ g(t, L(t, U(t)))



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 194

then the stability of the zero solution of

w′ = g(t, w) w(t0) = w0 (22)

implies the stability of the zero solution of

DH U(t) = (QU)(t) Ut0
= χ0 (23)

where Φo =ψ0 +χ0.

Proof. Let 0 < ε < ρ and t0 ∈ R+ be given. Since L(t, U) is positive definite, it follows

that there exists a function b ∈ K such that

b(D̃[U ,θ]) ≤ L(t, U) for (t, U) ∈ R+× B (24)

Suppose that the solution of the system (22) is stable. Then, given b(ε) > 0, t0 ∈ R+, there

exists a δ = δ(t0,ε) > 0 such that when ever w0 < δ, we have w(t, t0 , w0) < b(ε), t ≥ t0,

where w(t, t0, w0) is any solution of the system (22). Choose w0 = L(t0,χ0).

SinceL(t, U) is continuous and L(t,θ) ≡ 0, there exist function δ1 = δ1(t0,ε) > 0 such

that D1[χ0,θ] ≤ δ1 and L(t0,χ0)≤ δ holds simultaneously.

We claim that if D1[χ0,θ] ≤ δ1, then D̃[U(t),θ] < ε for all t ≥ t0.

Suppose this is not true. Then there exists a solution U(t) = U(t, t0,χ0) of the system (23)

satisfying the properties D̃[U(t2),θ] = ε and D̃[U(t),θ] < ε for t0 < t < t2 <∞. From (24)

b(ε)≤ L(t2, U(t2)) (25)

Furthermore, U(t) ∈ B for t ∈ [t0, t2]. Hence, the choice of w0 = L(t0,χ0) and condition (iii)

gives, as a consequence of Theorem 1, the estimate

L(t, U(t)) ≤ r(t), t ∈ [t0, t2], (26)

where r(t) = r(t, t0, w0) is the maximal solution of the comparison equation (22). Now from

equation (25)

b(ε)≤ L(t2, U(t2))≤ r(t2)< b(ε),

which is a contradiction. Hence the zero solution of the system (23) is stable.

The following theorem provides sufficient conditions for asymptotic stability of the system

(23).

Theorem 5. Assume that

(i) there exist functions L(t, U), g(t, w) satisfying the conditions of Theorem 4;

(ii) there exists a function α(t) such that α(t) > 0 is continuous for t ∈ R+ and α(t) −→∞
as t −→∞. Further assume that the relation

α(t)D−L(t, U(t)) + L(t, U(t))D−α(t) ≤ w(t, L(t, U(t))α(t))



J. Devi, Ch. Naidu / Eur. J. Pure Appl. Math, 5 (2012), 187-196 195

for t ≥ t0, U ∈ Eα then, if the zero solution of the system (22) is stable then the zero solution of

the system (23) is asymptotically stable.

Proof. Let 0 < ε < ρ and t0 ∈ R+ be given.Set α0 = mint∈R+α(t), then α0 > 0 follows

from the assumption (ii). Since L(t, U) is positive definite, there exists b ∈ K such that

equation (24) holds.

ε1 = α0 b(ε), (27)

where ε > 0. Then the stability of the zero solution (22) implies that, given ε1 > 0 and

t0 ∈ R+ there exists δ = δ(ε1, t0) such that w0 < δ =⇒ w(t, t0, w0) < ε1, t ≥ t0, where

w(t, t0, w0) is any solution of (23). Choose W0 = L(t0,χ0), then proceeding as in the Theo-

rem 4 with ε1 insted of b(ε), we can prove that the zero solution of (23) is stable.

Let U(t, t0,χ0) be any solution of the system (23) such that

D1[χ0,θ] ≤ δ0

where δ0 = δ(t0, 1

2
ρ). Since the zero solution of the system (23) is stable, it follows that

D̃[U(t),θ] <
1

2
ρ, t ≥ t0.

since α(t) −→∞ as t −→∞, there exists a number T = T (t0,ε)> 0 such that

b(ε)α(t) > ε1, t ≥ t0 + T (28)

Now from the Theorem 2 and the relation (24), we get

α(t)b(D̃[U ,θ]) ≤ α(t)L(t, U) ≤ r(t), t ≥ t0

Where U(t) = U(t, t0,χ0) is any solution of (23) such that D1[χ0,θ] ≤ δ0.

If the zero solution of the system (23) is not asymptotically stable, then there exists a

sequence {tk}, tk ≥ t0 + T and tk −→ ∞ as k → ∞ such that D̃[U(tk),θ] ≥ ε for some

solution U(t) satisfying D1[χ0,θ] ≤ δ0.

Now

b(ε)α(t) ≤b(D̃[U(tk),θ])α(t)

≤α(tk)L(tk, U(tk))

≤r(tk)

<ε1 for tk ≥ t0 + T

which is a contraduction. Hence the solution of IVP (23) is asymptotically stable and the proof

is complete.

The next theorem gives sufficient conditions for the uniformly asymptotic stablity of IVP

(23). As the proofs are similar to that in [1] we omit them.



REFERENCES 196

Theorem 6. Assume that there exists a function L(t, U) satisfies the following properties

(i) L ∈ C[R+ × B,R+] where B = B(θ ,ρ), L(t, U) is positive definite, decrescent and locally

Lipschitzian in U,

(ii)

D−L(t, U(t)) < −c(D̃[U(t),θ]) (29)

for t ≥ t0, U ∈ E0 and c ∈ K then the zero solution of (23) is uniformly assymptotically stable.

Our final stability result is a general result which offers various stability criteria in a single

setup. The proof of this theorem, which can be obtained using the comparison results given

in Theorem 3, is omitted.

Theorem 7. Assume that there exists a function L(t, U) satisfying properties (i), (ii) and (iii)

of the Theorem 3 then the stability properties of the zero solution of IVP (22) implies the corre-

sponding properties of the zero solution of IVP (23).

ACKNOWLEDGEMENTS This work has been done under the project no. SR/S4/MS: 491/07

sanctioned by Department of Science and Technology, Government of India. The authors ac-

knowledge their support.

References

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maps, Dynamic Systems and Applications 1S. 451-464. 2006.

[2] V Lakshmikantham and R Mohan. Theory of Integro Differential equations, Gordan

and Breach Science Publishers, Amsterdam, 1995.

[3] M Lakshmikantham and S Leela. Differential and Integral Inequalities, Vol.I and II ,

Academic press, Newyork, 1969.

[4] V Lakshmikantham, T GnanaBhaskar and J Devi. Theory of Set Differential Equations

in Metric Spaces, Cambridge Scientific Publishers, 2006.