Averaging for fuzzy differential equations

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ISSN_{1842-6298 (electronic), 1843-7265 (print)} Volume 9 (2014), 93 – 104

AVERAGING FOR FUZZY DIFFERENTIAL

EQUATIONS

Mustapha Lakrib, Rahma Guen and Amel Bourada

Abstract. We prove and discuss averaging results for fuzzy differential equations. Our results generalize previous ones.

1 Introduction and preliminaries

In recent years, the fuzzy set theory introduced by Zadeh [20] has emerged as a powerful tool for modeling of uncertainty and for processing of vague or subjective information in mathematical models, whose main directions of development have been diversified and applied in many varied real problems. For such mathematical modeling, using fuzzy differential equations are necessary. For significant results from the theory of fuzzy differential equations and their applications, among many works we refer the interested reader, for instance, to the books [12, 16] and the papers [1,2,4,13,17,18,19] and the references therein.

In the present paper, we establish averaging results for fuzzy differential equations. As in the previous works of the first author on the justification of the method of averaging (see [7, 8, 9, 10, 11]), the conditions we assume on the right-hand sides of the fuzzy differential equations under which our averaging results are stated are more general than those considered in the existing literature as in [5,6,15].

Let Rd _{denotes the} _d_{-dimensional space with the euclidian norm} _{| · |. Conv(}Rd₎ stands for the class of all nonempty compact and convex subsets ofRd_{. In Conv(}Rd₎ the so-called Hausdorff metric is defined by

ρ(A, B) := max

sup a∈A

inf b∈B

|a−b|,sup b∈B

inf a∈A

|a−b|

, A, B∈Conv(Rd₎_.

The metric space (Conv(Rd₎_{, ρ}_{) is complete.}

Denote Ed _{the space of functions} _u _: Rd _→ _[0_,_{1] that satisfy the following} conditions:

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i) u is normal, i.e., there existsx0 ∈Rd such thatu(x0) = 1; ii) u is fuzzy convex, i.e., for anyx, y∈Rd _and_λ_∈_[0_,_{1], one has}

u(λx+ (1−λ)y)≥min{u(x), u(y)};

iii) u is upper semicontinuous, i.e., for any x0 ∈ Rd and ε > 0, there exists

δ = δ(x0, ε) > 0 such that u(x) < u(x0) +ε for all x ∈ Rd that satisfy the condition |x−x0|< δ;

iv) the closure of the set {x∈Rd_:_u₍_x₎_>_0}_{is compact.}

The zero element in Ed _{is defined by ˆ}₀₍_y_{) = 1 if} _y_{= 0 and ˆ}₀₍_y_{) = 0 if} _y_{6= 0.} For α ∈ (0,1], the α-section [u]α of a mapping u ∈ Ed _{is defined as the set} {x∈Rd_:_u₍_x₎_≥_α_{}. The zero section of a mapping}_u_∈Ed_{is defined as the closure} of the set{x∈Rd_:_u₍_x₎_>_0}.

From i)−iv), it follows that [u]α∈Conv(Rd_{) for all} _α_∈_[0_,_1]. For addition and scalar multiplication, we have, forα∈[0,1],

[u+v]α= [u]α+ [v]α, [λu]α =λ[u]α, u, v∈Ed_{, λ}_∈R_. In the space Ed_{, define}_D_:Ed_×Ed_→R₊ _{by setting}

D(u, v) = sup α∈[0,1]

ρ([u]α,[v]α), u, v∈Ed

whereρ is the Hausdorff metric. Dis a metric inEd _{such that:} i) (Ed_{, D}_{) is complete;}

ii) D(x+z, y+z) =D(x, y) for allx, y, z ∈Ed_; iii) D(kx, ky) =|k|D(x, y) for all x, y∈Ed _and _k_∈R_.

The following definitions and theorems are given in [3,14]. Let I be an interval inR_.

Definition 1. A function f :I → Ed _{is called strongly measurable on} _I _{if, for all}

α∈[0,1], the set-valued function fα :I →Conv(Rd) defined by: fα(t) = [f(t)]α, is

Lebesgue-measurable.

Definition 2. A function f :I →Ed_{is called integrally bounded on} _I _{if there exists}

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Definition 3. The integral of a functionf :I →Ed_over _I_{, denoted by}R

If(t)dt, is

defined as an element G∈Ed _{such that, for all} _α_∈₍₀_,_1]_, [G]α =

Z

I

fα(t)dt

=

Z

I

φ(t)dt|φ:I →Rd _{is a measurable selection for} _f_α

.

Theorem 4. If a functionf :I →Ed_{is strongly measurable and integrally bounded,}

thenf is integrable onI.

Theorem 5. If f, g : I → Ed _{are integrable on} _I _and _λ _∈ R_{, then the following}

assertions are true:

i)

Z

I

[f(t) +g(t)]dt=

Z

I

f(t)dt+

Z

I

g(t)dt;

ii)

Z

I

λf(t)dt=λ

Z

I

f(t)dt;

iii) D

Z

I

f(t)dt,

Z

I

g(t)dt

≤

Z

I

D(f(t), g(t))dt.

Definition 6. A function f : I → Ed _{is called continuous at a point} _t₀ _∈ _I _if,

for any ε > 0, there exists δ = δ(t0, ε) > 0 such that D(f(t), f(t0))< ε whenever |t−t0|< δ,t∈I.

A function f :I →Ed_{is called continuous on} _I _{if it is continuous at every point}

t0∈I.

Definition 7. A function f :I×Ed_→Ed _{is called continuous at a point}₍_t₀_{, x}₀₎_∈

I×Ed_{if, for any}_{ε >}₀_{there exists}_δ ₌_δ₍_t₀_{, x}₀_{, ε}₎_>₀_{such that}_D₍_f₍_{t, x}₎_{, f}₍_t₀_{, x}₀₎₎_<

εwhenever |t−t0|< δ and D(x, x0)< δ, t∈I andx∈Ed_.

A function f :I×Ed_→Ed _{is called continuous on}_I _×Ed _{if it is continuous at}

every point(t0, x0)∈I×Ed.

Definition 8. A function f :I×Ed_→Ed _{is called continuous in} _x_∈Ed _uniformly

with respect to t∈I if, for any x0 ∈Ed and ε >0 there existsδ =δ(x0, ε)>0 such

thatD(f(t, x), f(t, x0)< εfor all t∈I whenever D(x, x0)< δ, x∈Ed_.

Definition 9. A function f : I → Ed _{is called differentiable at a point} _t₀ _∈ _I

if, for any α ∈ [0,1], the set-valued function fα : I → Conv(Rd) is Hukuhara

differentiable at the point t0, its derivative is equal to DHfα(t0), and the family of

sets {DHfα(t0) :α ∈[0,1]} defines a function f′(t0) ∈ Ed (which is called a fuzzy

derivative of f(t) at the point t0).

A function f :I → Ed _{is called differentiable on} _I _{if it is differentiable at every}

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Theorem 10. Suppose that a function f : I → Ed _{is differentiable and its fuzzy}

derivative f′

:I →Ed _{is integrable on} _I_{. Then, for any} _{a, t}_∈_I_{, we have}

f(t) =f(a) +

Z t

a

f′ (τ)dτ.

2 Averaging results

Consider the following initial value problem associated to a fuzzy differential equation with a small parameter

˙

x=f

t ε, x

, x(0) =x0, (2.1)

where f : R₊_×U _→ Ed_, U _{an open subset of} Ed_, _x₀ _∈ U _and _{ε >} _{0 is a small} parameter.

Definition 11. A function x : I → U_{, where} _I _{= [0}_{, ω}₎_, ₀ _{< ω} _{≤ ∞}_{, is called a}

solution of problem (2.1) if it is continuous and, for all t∈I, satisfies the integral equation

x(t) =x0+

Z t

0

fτ ε, x(τ)

dτ.

We associate (2.1) with the averaged initial value problem

˙

y=fo(y), y(0) =x0, (2.2)

where the functionfo :U_→Ed _{is such that, for any}_x_∈U lim

T→∞

D

1

T

Z T

0

f(τ, x)dτ, fo(x)

= 0. (2.3)

The main result of this paper establishes nearness of solutions of problems (2.1) and (2.2) on finite time intervals, and reads as follows.

Theorem 12. Suppose that the following conditions are satisfied:

(H1) the functionf :R₊_×U_→Ed _{in (}_2.1_{) is continuous;}

(H2) the continuity off in x∈U _{is uniform with respect to}_t_∈R₊_;

(H3) there exist a locally integrable function m :R₊ _→ R₊ _{and a constant} _{M >}₀

such that

D(f(t, x),ˆ0)≤m(t), ∀t∈R₊_,_∀_x_∈U

with

Z t2

t1

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(H4) for allx∈U_{, the limit (}_2.3_{) exists;}

(H5) for the function fo:U_→ Ed _{in (}_2.2_{), there exists a constant} _{λ >}₀ _{such that}

for continuous functionsu, v:R₊_→U _and_t_∈R₊_,

D

Z t

0

fo(u(τ))dτ,

Z t

0

fo(v(τ))dτ

≤λ

Z t

0

D(u(τ), v(τ))dτ. (2.4)

Let x0 ∈ U. Let xε be a solution of (2.1) and let I = [0, ωε), 0 < ωε ≤ ∞, be

its maximal positive interval of definition. Let y be the solution of (2.2) and let

J = [0, ω0), 0 < ω0 ≤ ∞, be its maximal positive interval of definition. Then, for

any L >0, L ∈ I∩J, and δ > 0, there exists ε0 = ε0(L, δ) > 0 such that, for all

ε∈(0, ε0], we haveD(xε(t), y(t))< δ for allt∈[0, L].

Notice that in condition (H5) we do not require that the functionfo is Lipschitz continuous. On the other hand, the averaging results stated in [5,6,15] are proved under conditions that are stronger compared to the ones above. In particular, the authors assume that the functionf satisfies the Lipschitz condition with respect to the second variable.

We discuss now the result of Theorem 12 when the function f is periodic or more generally almost periodic in t. In those cases some of the conditions in Theorem 12 can be removed. Indeed, if f is periodic in t, from continuity and periodicity properties one can easily deduce condition (H2). Periodicity also implies condition (H4) in an obvious way. The average off is then given, for anyx∈U_{, by}

D

1

P

Z P

0

f(τ, x)dτ, fo(x)

= 0, (2.5)

whereP is the period. If f is almost periodic int, for all x∈U, the limit

lim T→∞D

1

T

Z s+T

s

f(τ, x)dτ, fo(x)

= 0 (2.6)

exists uniformly with respect tos∈ R_{. So, condition (H4) is satisfied when} _s_{= 0.} In a number of cases encountered in applications the function f is a finite sum of periodic functions in t. As in the periodic case above, condition (H2) is then satisfied. Hence we have the following result.

Corollary 13(Periodic and Almost periodic cases). The conclusion of Theorem12

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2.1 Technical lemmas

In what follows we will prove some results we need for the proof of Theorem12.

Lemma 14. Letf :R₊_×U_→Ed_{be a function. Suppose that} _f _{satisfies conditions}

(H2), (H3) and (H4) in Theorem 12. Then the function fo _: _U _→ _Rd _{in (}_2.3_{) is}

continuous and satisfies

D(fo(x),ˆ0)≤M, ∀x∈U

where the constantM is as in condition (H4).

Proof. Continuity of fo_{. Let} _x

0 ∈U. By condition (H2), for anyξ >0 there exists

δ >0 such that, for allx∈U_,_D₍_{x, x}₀₎_≤_δ _{implies that}

D(f(τ, x), f(τ, x0))≤ξ, ∀τ ∈R₊_. _(2.7) Now, by condition (H4), we can easily deduce that, for any η > 0 there exists

T0 =T0(x0, x, η)>0 such that, for allT ≥T0 we have

D(fo(x), fo(x0))≤D

fo(x), 1 T

Z T

0

f(τ, x)dτ

+D

1

T

Z T

0

f(τ, x)dτ, 1 T

Z T

0

f(τ, x0))dτ

+D

fo(x0), 1

T

Z T

0

f(τ, x0)dτ

≤2η+ 1

T

Z T

0

D(f(τ, x), f(τ, x0))dτ ≤2η+ξ.

Since the value of η is arbitrary, in the limit we obtain that D(fo₍_x₎_{, f}o₍_x

0))≤ ξ, which finishes to prove the continuity offo at the point x0.

Boundedness of fo by M. Let x ∈ U_{. By condition (H3), we deduce that, for} any η >0 there exists T0=T0(x, η)>0 such that, for allT ≥T0 we have

D(fo(x),ˆ0) ≤D

fo(x), 1 T

Z T

0

f(τ, x)dτ

+D

1

T

Z T

0

f(τ, x)dτ,ˆ0

≤η+ 1

T

Z T

0

D f(τ, x),ˆ0

dτ ≤η+M.

Since the value of η is arbitrary, in the limit we obtain the desired result.

Lemma 15. Let f :R₊_×U_→Ed _{be a function. Suppose that} _f _{satisfies condition}

(H4) in Theorem 12. Then, for all x∈U_, _{L >}₀ _and _{α >}₀_{, we have}

lim ε→0_t∈sup[0,L]

D ε

α

Z t/ε+α/ε

t/ε

f(τ, x)dτ, fo(x)

!

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Proof. Letx∈U_,_{L >}_{0 and}_{α >}_{0. Let} _t_∈_[0_{, L}_].

Case 1: t= 0. From condition (H4), it follows immediately that

lim ε→0D

ε α

Z α/ε

0

f(τ, x)dτ, fo(x)

!

= 0.

Case 2: t∈(0, L]. We write

ε α

Z t/ε+α/ε

t/ε

f(τ, x)dτ

= 1

α/ε

Z t/ε+α/ε

0

f(τ, x)dτ− 1

α/ε

Z t/ε

0

f(τ, x)dτ

= 1

t/ε+α/ε

t α + 1

Z t/ε+α/ε

0

f(τ, x)dτ

−t α 1 t/ε Z t/ε 0

f(τ, x)dτ

= 1

t/ε+α/ε

Z t/ε+α/ε

0

f(τ, x)dτ

+t

α

"

1

t/ε+α/ε

Z t/ε+α/ε

0

f(τ, x)dτ− 1

t/ε

Z t/ε

0

f(τ, x)dτ

#

.

Then, we obtain

sup t∈(0,L]

D ε

α

Z t/ε+α/ε

t/ε

f(τ, x)dτ, fo(x)

!

≤ sup t∈(0,L]

D 1

t/ε+α/ε

Z t/ε+α/ε

0

f(τ, x)dτ, fo(x)

!

+L

α

"

sup t∈(0,L]

D 1

t/ε+α/ε

Z t/ε+α/ε

0

f(τ, x)dτ, fo(x)

!

+ sup t∈(0,L]

D 1

t/ε

Z t/ε

0

f(τ, x)dτ, fo(x)

!#

.

(2.8)

From condition (H4), we can easily deduce that

lim

ε→0_t_∈sup_(0,L]D

1

t/ε+α/ε

Z t/ε+α/ε

0

f(τ, x)dτ, fo(x)

!

= 0

and

lim

ε→0_t_∈sup_(0,L]D 1

t/ε

Z t/ε

0

f(τ, x)dτ, fo(x)

!

= 0.

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The next corollary follows directly from Lemma 15.

Corollary 16. Suppose that the functionf in (2.1) satisfies conditions (H1), (H3) and (H4) in Theorem12. Letx0 ∈U. Letxεbe a solution of (2.1) and letI = [0, ωε), 0 < ωε ≤ ∞, be its maximal positive interval of definition. Then, for all L > 0,

L∈I, and α >0, we have

lim

ε→0_t_∈sup_[0,L]D

ε α

Z t/ε+α/ε

t/ε

f(τ, xε(t))dτ, fo(xε(t))

!

= 0. (2.9)

Lemma 17. Suppose that the function f in (2.1) satisfies conditions (H1)-(H4) in Theorem 12. Let x0 ∈ U. Let xε be a solution of (2.1) and let I = [0, ωε), 0 < ωε ≤ ∞, be its maximal positive interval of definition. Then, for all L > 0,

L∈I, we have

lim

ε→0_t_∈sup_[0,L]D

Z t

0

fτ ε, xε(τ)

dτ,

Z t

0

fo(xε(τ))dτ

= 0.

Proof. Let L > 0, L ∈ I, and t0 = 0 < t1 < · · · < tn < · · · < tp = L, p ∈ N, a partition of [0, L] with α = α(ε) := tn+1−tn, n = 1, . . . , p and lim

ε→0α = 0. Let

t∈[tm, tm+1] for any m∈ {0,· · ·, p−1}. Then

D

Z t

0

fτ ε, xε(τ)

dτ,

Z t

0

fo(xε(τ))dτ

≤ m−1

X

n=0

D

Z tn+1

tn

fτ ε, xε(τ)

dτ,

Z tn+1

tn

fo(xε(τ))dτ

+D

Z t

tm

fτ ε, xε(τ)

dτ,

Z t

tm

fo(xε(τ))dτ

.

(2.10)

By condition (H3) and Lemma 14we have

D

Z t

tm

fτ ε, xε(τ)

dτ,

Z t

tm

fo(xε(τ))dτ

≤D

Z t

tm

fτ ε, xε(τ)

dτ,ˆ0

+D

Z t

tm

fo(xε(τ))dτ,ˆ0

≤

Z t

tm

Dfτ ε, xε(τ)

,ˆ0dτ+

Z t

tm

D fo(xε(τ)),ˆ0

dτ ≤2M α.

Now, for eachn= 0, . . . , m−1 andτ ∈[tn, tn+1], by condition (H3) we can easily deduce that D(xε(τ), xε(tn))≤ M α so that by conditions (H2) and the continuity offo _(Lemma₁₅_{), it follows, respectively, that}

Dfτ ε, xε(τ)

, fτ

ε, xε(tn)

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and

D(fo(xε(τ)), fo(xε(tn)))≤δn=δn(ε),

with lim

ε→0γn= limε→0δn= 0.

Hence, from (2.10), it follows that

D

Z t

0

fτ ε, xε(τ)

dτ,

Z t

0

fo(xε(τ))dτ

≤ m−1

X

n=0

D

Z tn+1

tn

fτ

ε, xε(tn)

dτ,

Z tn+1

tn

fo(xε(tn))dτ

+ m−1

X

n=0

Z tn+1

tn

(γn+δn)dτ+ 2M α.

(2.11)

For eachn= 0, . . . , m−1, we have

βn :=D

Z tn+1

tn

fτ

ε, xε(tn)

dτ,

Z tn+1

tn

fo(xε(tn))dτ

=αD ε α

Z tn/ε+α/ε

tn/ε

f(τ, xε(tn))dτ, fo(xε(tn))

!

≤α sup t∈[0,L]

D ε

α

Z t/ε+α/ε

t/ε

f(τ, xε(t))dτ, fo(xε(t))

!

:= α̺ (̺=̺(ε)).

Then

m−1

X

n=0

βn≤̺ m−1

X

n=0

α=̺

m−1

X

n=0

(tn+1−tn) =̺t≤̺L,

where, by Corollary 16, lim ε→0̺= 0. On the other hand, we have

m−1

X

n=0

Z tn+1

tn

(γn+δn)dτ ≤η m−1

X

n=0

Z tn+1

tn

dτ =ηt≤ηL,

whereη=η(ε) = max{γn+δn:n= 0, . . . , m−1} and lim ε→0η = 0. Finally, from (2.11) we obtain

sup t∈[0,L]

D

Z t

0

fτ ε, xε(τ)

dτ,

Z t

0

fo(xε(τ))dτ

≤(̺+η)L+ 2M α. (2.12)

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2.2 Proof of Theorem 12

We are now able to prove our main result (there is not much work left). We assume that the assumptions in Theorem 12 are fulfilled.

For t∈[0, L]⊂I∩J, using condition (H5), we obtain

D(y(t), xε(t)) =D

Z t

0

fo(y(τ))dτ,

Z t

0

fτ ε, xε(τ)

dτ

≤D

Z t

0

fo(y(τ))dτ,

Z t

0

fo(xε(τ))dτ

+D

Z t

0

fo(xε(τ))dτ,

Z t

0

fτ ε, xε(τ)

dτ

≤λ

Z t

0

D(y(τ), xε(τ))dτ+σ

(2.13)

where

σ=σ(ε) := sup t∈[0,L]

D

Z t

0

fτ ε, xε(τ)

dτ,

Z t

0

fo(xε(τ))dτ

.

By Lemma17, we have lim ε→0σ= 0.

Now, by Gronwall Lemma, from (2.13) we deduce that

D(y(t), xε(t))≤σ

|eλL−1|

λ ,

which implies that

lim

ε→0_t_∈sup_[0,L]D(xε(t), y(t)) = 0.

The proof is complete.

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Mustapha Lakrib Rahma Guen

Laboratoire de Mathématiques, Laboratoire de Mathématiques, Université Djillali Liabès, Université Djillali Liabès, B.P. 89, 22000 Sidi Bel Abbès, B.P. 89, 22000 Sidi Bel Abbès,

Alg´erie. Alg´erie.

E-mail: m.lakrib@univ-sba.dz E-mail: rah.guen@gmail.com

Amel Bourada

Laboratoire de Mathématiques, Université Djillali Liabès, B.P. 89, 22000 Sidi Bel Abbès, Algérie.