Fuzzy Functional Differential Equations under Dissipative-Type Conditions
Fuzzy functional differential equations with continuous right-hand sides are studied. The existence and uniqueness of a solution are proved under dissipative-type conditions. The continuous dependence of the solution on the initial conditions is shown. The existence of the solution on an infinite in...
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irk-123456789-1654922020-02-14T01:26:09Z Fuzzy Functional Differential Equations under Dissipative-Type Conditions Donchev, T. Nosheen, A. Статті Fuzzy functional differential equations with continuous right-hand sides are studied. The existence and uniqueness of a solution are proved under dissipative-type conditions. The continuous dependence of the solution on the initial conditions is shown. The existence of the solution on an infinite interval and its stability are also analyzed. Вивчаються нєчіткі функціонально-диференціальні рівняння з неперервною правою частиною. Доведено існування та єдиність розв'язку за умов дисипативного типу. Встановлено неперервну залежність розв'язку від початкових умов. Також розглянуто питання про існування розв'язку на нескінченному інтервалі та його стійкість. 2013 Article Fuzzy Functional Differential Equations under Dissipative-Type Conditions / T. Donchev, A. Nosheen // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 787–795. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165492 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Donchev, T. Nosheen, A. Fuzzy Functional Differential Equations under Dissipative-Type Conditions Український математичний журнал |
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Fuzzy functional differential equations with continuous right-hand sides are studied. The existence and uniqueness of a solution are proved under dissipative-type conditions. The continuous dependence of the solution on the initial conditions is shown. The existence of the solution on an infinite interval and its stability are also analyzed. |
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Donchev, T. Nosheen, A. |
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Donchev, T. Nosheen, A. |
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Donchev, T. |
| title |
Fuzzy Functional Differential Equations under Dissipative-Type Conditions |
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Fuzzy Functional Differential Equations under Dissipative-Type Conditions |
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Fuzzy Functional Differential Equations under Dissipative-Type Conditions |
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Fuzzy Functional Differential Equations under Dissipative-Type Conditions |
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Fuzzy Functional Differential Equations under Dissipative-Type Conditions |
| title_sort |
fuzzy functional differential equations under dissipative-type conditions |
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Інститут математики НАН України |
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2013 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165492 |
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Fuzzy Functional Differential Equations under Dissipative-Type Conditions / T. Donchev, A. Nosheen // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 787–795. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT donchevt fuzzyfunctionaldifferentialequationsunderdissipativetypeconditions AT nosheena fuzzyfunctionaldifferentialequationsunderdissipativetypeconditions |
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2025-07-14T18:41:32Z |
| last_indexed |
2025-07-14T18:41:32Z |
| _version_ |
1837648832277512192 |
| fulltext |
UDC 517.9
T. Donchev (Univ. Al. I. Cuza, Iaşi, Romania),
A. Nosheen (Abdus Salam School Math. Sci., Pakistan)
FUZZY FUNCTIONAL DIFFERENTIAL EQUATIONS
UNDER DISSIPATIVE-TYPE CONDITIONS*
НЕЧIТКI ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНI РIВНЯННЯ
З УМОВАМИ ДИСИПАТИВНОГО ТИПУ
Fuzzy functional differential equations with continuous right-hand side are studied. Existence and uniqueness of a solution
under dissipative-type conditions are proved. The continuous dependence of the solution on the initial conditions is shown.
The existence of а solution on an infinite interval and its stability are also considered.
Вивчаються нечiткi функцiонально-диференцiальнi рiвняння з неперервною правою частиною. Доведено iснування
та єдинiсть розв’язку за умов дисипативного типу. Встановлено неперервну залежнiсть розв’язку вiд початкових
умов. Також розглянуто питання про iснування розв’язку на нескiнченному iнтервалi та його стiйкiсть.
1. Introduction. The real life models depend not only on the current state of the subject but also
on its prehistory. There are many papers devoted to delayed differential equations. We refer the
readers to [6, 14] where many examples of functional differential equations, representing models in
population dynamics, mathematical biology and medicine are provided. Often the models depend
on uncertainty which requires the usage of a special mathematical approach as stochastic models,
multivalued differential equations or fuzzy set models. The fuzzy differential equations had been
studied first by Kandel and Byatt in [4]. Starting with the work of Kaleva [5], the theory of fuzzy
differential equations was rapidly developed. We refer the books [8, 12] where the fuzzy differential
equations are comprehensively studied (see also [9, 11, 15]).
There are relatively small number of papers devoted to fuzzy functional differential equations
as [1], where the authors prove the existence of solutions using fixed point arguments. In the very
interesting paper [10], the author proves the existence and uniqueness of solution when the right-
hand side satisfied the locally Lipschitz condition (with respect to Kamke function), that paper is
also provided with significant examples. In [15] the authors prove existence and uniqueness of the
solution of fuzzy (ordinary) differential equations using Lyapunov-like functions.
We extend the results of the last two papers to fuzzy functional differential equations satisfying
Razumikhin dissipative-type conditions.
In the next section we give the needed preliminaries from the theory of fuzzy sets. In Section 3,
the system description and a needed preliminary lemma is proved. In Section 4, we prove the local
and global existence as well as uniqueness of the solution under dissipative conditions. Afterward
continuous dependence on the initial conditions and stability of the solution are shown. In the last
section the existence and uniqueness results with the help of Lyapunov-like function are proved.
2. Preliminaries. The space of fuzzy numbers is E = {x : Rn → [0, 1]; x satisfies 1)– 4)}.
1) x is normal, i.e., there exists y0 ∈ Rn such that x(y0) = 1,
* This research is supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI,
project number PN-II-ID-PCE-2011-3-0154. The second author is partially supported by Higher Education Commission of
Pakistan.
c© T. DONCHEV, A. NOSHEEN, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6 787
788 T. DONCHEV, A. NOSHEEN
2) x is fuzzy convex, i.e., x(λy+(1−λ)z) ≥ min{x(y), x(z)} whenever y, z ∈ Rn and λ ∈ [0, 1],
3) x is upper semicontinuous, i.e., for any y0 ∈ Rn and ε > 0 there exists δ(y0, ε) > 0 such that
x(y) < x(y0) + ε whenever |y − y0| < δ, y ∈ Rn,
4) the closure of the set {y ∈ Rn; x(y) > 0} is compact.
The set [x]α = {y ∈ Rn; x(y) ≥ α} is called α-level set of x.
It follows from 1) – 4) that the α-level sets [x]α are in Cc(Rn) for all α ∈ (0, 1], where Cc(Rn)
denotes the compact convex subsets of Rn. The fuzzy zero is defined as
0̂(y) =
{
0 if y 6= 0,
1 if y = 0.
The metric in E is defined by D(x, y) = supα∈(0,1]DH([x]α, [y]α), where DH(·, ·) means the
Hausdorff distance in Cc(Rn).
Some properties of D(x, y) are as follows:
(1) D(x+ z, y + z) = D(x, y) and D(x, y) = D(y, x),
(2) D(λx, λy) = λD(x, y),
(3) D(x, y) ≤ D(x, z) +D(z, y), for all x, y, z ∈ E and λ ∈ R.
We recall some properties of differentiability and integrability for fuzzy functions from [13]. Let
T > 0, further in the paper I = [0, T ].
The map f : I → E is said to be differentiable at t̂ ∈ I = [0, T ] iff for small h > 0 the differences
f(t̂+h)−f(t̂), f(t̂)−f(t̂−h) (in sense of Hukuhara) exist and there exists a fuzzy number A = ḟ(t̂)
such that
lim
h→0+
f(t̂+ h)− f(t̂)
h
= A = lim
h→0+
f(t̂)− f(t̂− h)
h
.
At the end points of I we consider only the one sided (left or right) derivative. It is easy to check
that
D(f(t̂+ h), f(t̂) + hḟ(t̂)) = o(h) where lim
h→0+
o(h)
h
= 0. (1)
The fuzzy valued function F : I → E is said to be (strongly) measurable if for each α ∈ [0, 1], the
set-valued function Fα : I → Cc(Rn) is measurable (see, e.g., [2]).
The integral of F over I, denoted by
∫
I
F (t)dt is defined level-wise as
∫
I
F (t)dt
α =
∫
I
Fα(t)dt =
∫
I
f(t)dt, f(t) ∈ Fα(t) is integrable
for all 0 ≤ α ≤ 1.
The function F : R → E (G : R × E → E) is said to be continuous if it is continuous w.r.t. the
metric D(·, ·). The set of continuous functions from [a, b] into E is denoted by C([a, b],E).
Proposition 1 [5]. If F : I → E is continuous, then it is integrable over I. Moreover, in this
case, the function G(t) =
∫ t
0
F (s)ds is differentiable and Ġ(t) = F (t).
Let m : [−τ,∞) → R+ be continuous. Denote ṁ+(t) = limh→0+
m(t+ h)−m(t)
h
the upper
right Dini derivative.
We need the following known results:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FUZZY FUNCTIONAL DIFFERENTIAL EQUATIONS UNDER DISSIPATIVE-TYPE CONDITIONS 789
Lemma 1 (cf. Lemma 6.1.1 of [7]). Let g : I ×R+ → R+ and m : [−τ, T ]→ R be two contin-
uous functions and ṁ+(t) ≤ g(t,m(t)) for every t ∈ [0, T ]\G (where G is a countable set) with
m(t) = maxs∈[−τ,0]m(t + s). If m(0) ≤ r0, then m(t) ≤ r(t), where r(·) is the maximal solution
of ṙ(t) = g(t, r), r(0) = r0 (provided that r(·) exists on I).
Lemma 2 [7]. Let the assumptions of Lemma 1 hold. If [0, t1] ⊂ [0,∞), then there exists an
ε0 > 0 such that 0 < ε < ε0 and the maximum solution r(t, u0, ε) of
u̇ = g(t, u) + ε, u(t0) = u0 + ε
exists on [0, t1] and limε→0 r(t, t0, u0, ε) = r(t, t0, u0) uniformly on [0, t1].
3. System description. In the paper we consider the following fuzzy functional differential
equation:
ẋ(t) = f(t, xt), x0 = ξ, t ∈ I, (2)
here f : I × E1 → E, ξ ∈ E1 := C([−τ, 0],E) and xt ∈ E1 is defined by xt(s) = x(t + s) for
s ∈ [−τ, 0]. For β, γ ∈ E1 we denote DC(β, γ) = maxs∈[−τ,0]D(β(s), γ(s)). If γ ≡ 0̂ then we will
write DC(β, 0̂).
We refer the reader to [3] for a general theory of functional differential equations.
Notice that the system (2) is very general. It includes for example:
1. Variable times delay system ẋ = f(t, x(t−h(t))), where h(·) is continuous with 0 ≤ h(t) ≤ τ.
2. Distributed time delay ẋ(t) = f
(
t,
∫ 0
−τ
k(s, x(s))ds
)
as in the well know predator–prey
model of Lotka – Wolterra:
ẋ(t) = x(t)
r1 − γ1y(t)−
0∫
−τ
F1(s)y(t+ s)ds
,
ẏ(t) = y(t)
γ2x(t)− r2 +
0∫
−τ
F2(s)x(t+ s)ds
,
where x(·), y(·) represent the population densities of prey and predator at time t ≥ 0, r1 > 0 is the
intrinsic growth rate of the prey, r2 > 0 is the death rate of the predators: γ1 > 0 and γ2 > 0 are the
interaction constants.
The book [14] contains many examples of delayed differential equation from natural sciences.
Some examples of functional differential equations in fuzzy set settings are contained in [10].
The continuous function g : I × R+ → R+ is said to be Kamke function if g(t, 0) ≡ 0, g(t, ·) is
monotone nondecreasing and u ≡ 0 is the only solution of the scalar differential equation
u̇ = g(t, u), u(0) = 0. (3)
We will use the following hypotheses:
(H1) f : I × E1 → E is continuous.
Remark 1. (H1) implies that there exist positive numbers a, b,M such that D(f(t, ϕ), 0̂) ≤M
for every (t, ϕ) ∈ [0, a] × Sb(ξ). Furthermore, one can take a ≤ b
M + 1
. Here Sb(ξ) = {ϕ ∈
∈ E1 : DC(ϕ, ξ) ≤ b}.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
790 T. DONCHEV, A. NOSHEEN
(H2) There exists a Kamke function g : I × R+ → R+ such that
lim
h→0+
[D(β(0) + hf(t, β), γ(0) + hf(t, γ))−D(β(0), γ(0))]
h
≤ g(t,D(β(0), γ(0))),
whenever (β, γ) ∈ Ω. Here Ω is given by Ω = {β, γ ∈ E1 : DC(β, γ) = D(β(0), γ(0))}.
Lemma 3 [5]. Under (H1) the AC function x(·) is a solution of (2) if and only if x0 = ξ and
x(t) = ξ(0) +
t∫
0
f (s, xs) ds. (4)
Our next result is concerned with the construction of approximate solution to the problem (2).
Lemma 4. Let (H1) hold and let a, b,M > 0 be as in Remark 1. Then for every positive integer
n there exists a continuous function xn : [−τ, a]→ E, which is differentiable for every t ∈ [0, a]\Nn
(here Nn ⊂ I is countable set) and satisfies the following conditions:
(i) D(ẋn(t), f(t, xnt )) ≤ 1
n
,
(ii) D(ẋn(t), 0̂) ≤M,
(iii) xnt ∈ Sb(ξ) (recall that xnt ∈ E1).
Proof. If x(·) satisfies (ii) then it is Lipschitz with a constant M. Indeed for 0 ≤ s < t ≤ a, one
has that x(t) = x(s) +
∫ t
s
ẋ(τ)dτ. Hence
D (x(t), x(s)) =
t∫
s
D
(
ẋ(τ), 0̂
)
dτ ≤M(t− s).
Suppose first that the needed xn(·) exists on [0, s], where 0 ≤ s ≤ a. If s = a, then the prove is
complete. Otherwise, for t > s we define
xn(t) = xns (0) + (t− s)f(s, xns ).
The continuity of f(·, ·) implies that there exists t̃ > s such that
D (ẋn(t), f(t, xnt )) = D (f(s, xns ), f(t, xnt )) ≤ 1
n
for all t ∈ [s, t̃],
i.e., (i) is satisfied on [0, t̃]. Since D
(
f(s, xns ), 0̂
)
≤M, we have that (ii) and (iii) also hold on [0, t̃].
Let 0 < t̃ < a and let xn(·) be defined on [0, t̃). Since xn(·) is M -Lipschitz, then
limt→t̃−0 x
n(t) = xn(t̃) exists, i.e., xn(·) is defined on [0, t̃]. By standard application of Zorn’s
lemma, one can conclude that the needed solution xn(·) exists on [0, a].
4. Existence and uniqueness of solution. In this section we prove the main results in the paper.
The first theorem is devoted to the local existence and uniqueness of solution for (2).
Theorem 1. Let the hypothesis (H1) and (H2) hold. Then for each ξ ∈ E1 there exists unique
solution x(·) of the initial value problem (2) defined on [0, a].
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FUZZY FUNCTIONAL DIFFERENTIAL EQUATIONS UNDER DISSIPATIVE-TYPE CONDITIONS 791
Proof. It follows from Lemma 4 that the approximate solution xn(·) exists on [0, a] and
D(f(t, xnt ), 0̂) ≤M.
Let xn(·) and xk(·) be two approximate solutions. Denote m(t) = D(xn(t), xk(t)). By virtue of
the triangle inequality and (1) we have
m(t+ h)−m(t) = D(xn(t+ h), xk(t+ h))−D(xn(t), xk(t)) ≤
≤ D(xn(t) + hẋn(t), xk(t) + hẋk(t)) + o(h)−D(xn(t), xk(t)).
Thus
lim
h→0+
m(t+ h)−m(t)
h
= lim
h→0+
D(xn(t) + hẋn(t), xk(t) + hẋk(t))−D(xn(t), xk(t))
h
≤
≤ lim
h→0+
D(xn(t) + hf(t, xnt ), xk(t) + hf(t, xkt ))−D(xn(t), xk(t))
h
+
1
n
+
1
k
.
By (H2) we have
ṁ(t)+ ≤ g(t,m(t)) +
1
n
+
1
k
for every t ∈ [0, a] with m(t) = maxs∈[t−τ,t]m(s). It follows from Lemma 1 that D(xn(t), xk(t)) ≤
≤ rn,k(t), where rn,k(t) is the maximal solution of
ṙ(t) = g(t, r(t)) +
1
n
+
1
k
, r(0) = 0.
Since g(·, ·) is a Kamke function, one has that xn(·) is a Cauchy sequence in C([0, a],E) and hence
xn(·) → x(·) uniformly on [0, a]. Therefore xnt → xt in E1 uniformly on t ∈ [0, a]. Furthermore,
f(t, ·) is continuous and hence f(t, xnt )→ f(t, xt) uniformly on [0, a]. Therefore
t∫
0
f(s, xns )ds→
t∫
0
f(s, xs)ds.
Consequently x(t) = ξ(0) +
∫ t
0
f(s, xs)ds and hence it is a local solution of (2) thanks to Lemma 3.
Now we prove that the solution is unique.
Let x(t) and y(t) be two solutions, we define m(t) = D(x(t), y(t)) and mt = DC(xt, yt). It is
easy to see that mt = m(t) if and only if (xt, yt) ∈ Ω. In this case triangle inequality and (1) implies
lim
h→0+
m(t+ h)−m(t)
h
= lim
h→0+
[D(x(t+ h), y(t+ h))−D(x(t), y(t))]
h
≤
≤ lim
h→0+
[D(x(t) + hf(t, xt), y(t) + hf(t, yt))−D(x(t), y(t)) + o(h)]
h
.
Since limh→0+ h
−1.o(h) = 0, one has that
lim
h→0+
m(t+ h)−m(t)
h
= lim
h→0+
[D(x(t) + hf(t, xt), y(t) + hf(t, yt))−D(x(t), y(t))]
h
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
792 T. DONCHEV, A. NOSHEEN
Using (H2) we derive
lim
h→0+
m(t+ h)−m(t)
h
≤ g(t,D(x(t), y(t))).
Thus we have ṁ+(t) ≤ g(t,m(t)) for any t for which m(t) = mt. From Lemma 1 and the
last inequality it follows that D(x(t), y(t)) ≤ r(t), where r(·) is the maximal solution of ṙ(t) =
= g(t, r(t)), r(0) = DC(x0, y0). If x0 = y0, then r(t) = 0, i.e., x(t) ≡ y(t) on [0, a].
We are ready to study global existence of solution.
We need the following additional hypothesis:
(H3) There exists a continuous function w : I ×R+ → R+ such that
(1) The maximal solution of µ̇(t) = w(t, µ(t)), µ(0) = DC(ξ, 0̂) exists on I;
(2) for any t ∈ I with DC(xt, 0̂) = D(x(t), 0̂), we have
lim
h→0+
[D(x(t) + hf(t, xt), 0̂)−D(x(t), 0̂)]
h
≤ w(t,D(x(t), 0̂));
(H4) f(·, ·) is bounded on the bounded sets, i.e., for every ϑ > 0 there exists M such that
D(f(t, x(t)), 0̂) ≤M if x ∈ E1, t > 0 and D(x, 0̂) + t ≤ ϑ.
Theorem 2. Under (H1) – (H4) for every ξ ∈ E1 there exists unique solution of (2) defined on
[0, T ].
Proof. By Theorem 1 we know there exists a > 0 such that fuzzy functional differential equation
(2) admits unique solution on [0, a]. Suppose that the maximal interval of existence for x(·) (unique
solution of (2)) is [0, S), where S < T. It follows from (H3) that Ḋ+(x(t), 0̂) ≤ w(t,D(x(t), 0̂))
for t ∈ I with D(x(t), 0̂) = DC(xt, 0̂). Using Lemma 1 we have D(x(t), 0̂) ≤ µ(t) where µ(t) is
the maximal solution of
µ̇(t) = w(t, µ(t)), µ(0) = DC(ξ, 0̂).
Since µ(·) exists on the whole interval [0, T ], one has that there exist a constant M > 0 such that
µ(t) ≤M − T, for all t ∈ [0, T ].
Furthermore f(·, ·) is bounded on the bounded sets and hence x(·) is ϑ-Lipschitz on [0, S) and
hence x(S) = limt→S−0 x(t) exists. Consider the equation (2) on the interval [S, T ]. The initial
condition xS is well defined, because S > 0.
It follows from Theorem 1 that there exists δ > 0 such that (2) has a solution on [S, S+δ], where
S < S + δ ≤ T, a contradiction. Hence x(t) exists on the interval [0, T ], and Zorn’s lemma implies
the existence of unique solution x(·) (as it is shown in Theorem 1) on [0, T ].
The following corollary shows the continuous dependence of the solution of (2) on the initial
condition.
Corollary 1. Let f : I × E1 → E satisfy (H1) – (H3) and let x(·, ϕ), y(·, ψ) be solutions of (2)
with different initial conditions x0 = ϕ and y0 = ψ where ϕ,ψ ∈ E1. Then D(x(t), y(t)) ≤ r(t)
where r(t) is the maximal solution of
ṙ(t) = g(t, r(t)), r(0) = D(ϕ,ψ).
Proof. Denote m(t) = D(x(t, ϕ), y(t, ψ)). Evidently m(0) = D(ϕ,ψ). Using the same argu-
ments as in the proof of Theorem 2 we can show that ṁ+(t) ≤ g(t,m(t)). Since g(·, ·) is a Kamke
function, one has that the map ϕ→ x(·, ϕ) from E1 into C(I,E) is continuous.
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FUZZY FUNCTIONAL DIFFERENTIAL EQUATIONS UNDER DISSIPATIVE-TYPE CONDITIONS 793
Now we will study (2) on [0,∞) = R+. We replace I in (H1) and (H2) by [0,∞), i.e., we set
T =∞.
The hypothesis (H3) in that case has the form:
(H′3) For every S > 0 there exists a continuous function w : [0, S]× R+ → R+such that
(1) the maximal solution of µ̇(t) = w(t, µ(t)), µ(0) = DC(ξ, 0̂) exists on [0, S];
(2) for any t ∈ I with DC(xt, 0̂) = D(x(t), 0̂), we have
lim
h→0+
[D(x(t) + hf(t, xt), 0̂)−D(x(t), 0̂)]
h
≤ w(t,D(x(t), 0̂)).
Theorem 3. Let (H1), (H2) (with I replaced by [0,∞), (H′3) and (H4) hold. Then the differ-
ential equation (2) admits unique solution x(·) defined on [0,∞).
Proof. Let S > 0 be given. From Theorem 2 we know that the initial problem (2) has a solution
on [0, S]. Since S > 0 is arbitrary, by virtue of Zorn lemma the solution x(·) exists on [0,∞).
Definition 1. The solution x(·) (defined on [0,∞)) of (2) is said to be stable if for every
ε > 0 there exists δ such that D(y(t), x(t)) < ε for every t > 0 when DC(y0, x0) < δ. It is called
asymptotically stable if it is stable and there exists ν > 0 such that limt→∞D(x(t), y(t)) = 0 when
DC(y0, x0) < ν.
The following theorem is then valid.
Theorem 4. Under the assumptions of Theorem 3, the unique solution x(·) of (2) is (asymp-
totically) stable if the zero solution of ṙ(t) = w(t, r(t)) is (asymptotically) stable.
Proof. Due to (H2)
lim
h→0+
D(xt(0) + hf(t, xt), yt(0) + hf(t, yt))−D(xt(0), yt(0))
h
≤ g(t,D(xt(0), yt(0)))
when D(xt(0), yt(0)) = DC(xt, yt). Consequently D(x(t), y(t)) ≤ r(t) for every t > 0, where r(·)
is the maximal solution of ṙ(t) = g(t, r(t)) with initial condition r(0) = D(x0, y0).
5. Lyapunov-like function condition. In this section we relax the dissipative condition. Our
results extend the existence results given in [10] and [15]. We assume that f(·, ·) satisfies (H3). Let
M be the constant from the proof of Theorem 2 and denote BM = {x ∈ E : D(x, 0̂) ≤M + 1}.
Definition 2. The continuous map W : (x(0) + BM ) × (x(0) + BM ) → R+ is said to be a
Lyapunov-like function for (2) if it satisfies the following:
1) W (x, x) = 0, W (x, y) > 0 for x 6= y and limn→∞W (xn, yn) = 0 implies
limn→∞D(xn, yn) = 0;
2) there exists a constant L > 0 such that
|W (x1, y1)−W (x2, y2)| ≤ L
(
D(x1, x2) +D(y1, y2)
)
;
3) there exists a Kamke function v(·, ·) such that
lim
h→0+
W (β(0) + hf(t, β), γ(0) + hf(t, γ))−W (β(0), γ(0))
h
≤ v(t,W (β(0), γ(0)))
for any β, γ ∈ E1 with W (β(0), γ(0)) = maxs∈[−τ,0]W (β(s), γ(s)).
Here we also require the following hypothesis to obtain desired result:
(H5) There exists a Lyapunov-like function W (·, ·) for the system (2).
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794 T. DONCHEV, A. NOSHEEN
Theorem 5. Under the assumptions (H1), (H3), (H4) and (H5) the fuzzy functional differential
equation (2) has unique solution defined on I.
Proof. Let a, b, M be as in Remark 1. From Lemma 4 we know that there exists a sequence of
approximate solutions {xn(·)}∞n=1 such that xn0 = ξ and D(ẋn(t), f(t, xnt )) ≤ 1
2n
. Denote m(t) =
= W (xk(t), xn(t)). Notice that xn(t) = xnt (0). Then
m(t+ h)−m(t) = W (xk(t+ h), xn(t+ h))−W (xk(t), xn(t)) ≤
≤W (xk(t) + hf(t, xkt ), x
n(t) + hf(t, xnt ))−W (xk(t), xn(t))+
+L
[
D(xk(t) + hf(t, xkt ), x
k(t+ h)) +D(xn(t) + hf(t, xnt ), xn(t+ h))
]
≤
≤W (xk(t) + hf(t, xkt ), x
n(t) + hf(t, xnt ))−
−W (xk(t), xn(t)) + L.o(h) + L
(
1
2k
+
1
2n
)
.
Consequently
lim
h→0+
m(t+ h)−m(t)
h
≤ v(t,m(t)) + L
(
1
2k
+
1
2n
)
,
i.e.,
m(0) = 0 and ṁ+(t) ≤ v(t,m(t)) + L
(
1
2k
+
1
2n
)
,
for every t ∈ I with m(t) = maxs∈[−τ,0]m(s). It follows from Lemma 1 that m(t) ≤ r(t), where
r(·) is the maximal solution of
ṙ(t) = v(t, r(t)) + L
(
1
2k
+
1
2n
)
, r(0) = 0.
Taking into account Definition 2 we get that the sequence {xn(·)}∞n=1 is a Cauchy sequence in
C(I,E). One can show as in the proof of Theorem 1 that the function x(t) = limn→∞ x
n(t) is the
unique solution of (2).
The proof of the following corollary is omitted, because it follows from the proofs in the previous
section with obvious modifications.
Corollary 2. Let (H1), (H3), (H4), and (H5) hold and let x(·.ϕ), y(·, φ) be solutions of (2)
with different initial conditions x0 = ϕ and y0 = ψ where ϕ,ψ ∈ E1. Then W (x(t), y(t)) ≤ r(t)
where r(t) is the maximal solution of
ṙ(t) = g(t, r(t)), r(0) = max
s∈[−τ,0]
W (ϕ(s), ψ(s)).
Replacing (H3) by (H′3) one can prove the same result on [0,∞).
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Received 02.05.12
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