On Cauchy problems for fuzzy differential equations

PII. S0161171204202277 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON CAUCHY PROBLEMS FOR FUZZY DIFFERENTIAL EQUATIONS

D. N. GEORGIOU and I. E. KOUGIAS

Received 25 February 2002

Existence and uniqueness theorems are proved for Cauchy problems of second-order fuzzy differential equations.

2000 Mathematics Subject Classification: 34A30, 54A40, 26E50, 03E72.

1. Introduction. In 1972, Chang and Zadeh [2] first introduced the concept of fuzzy derivative, followed up ten years later by Dubois and Prade [5], who used the extension principle in their approach. In the mean time, Puri and Ralescu [12] used the notion ofH-differentiability to extend the differential of set-valued functions to that of fuzzy functions. This led Seikkala [13] to introduce the notion of fuzzy derivative as an ex-tension of the Hukuhara derivative and the fuzzy integral, which was the same as that proposed by Dubois and Prade [3,4].

Naturally, the investigation of fuzzy differential and integral equations, existence and uniqueness theorems for the solutions of fuzzy initial value problems, drew upon the interest of many researchers of the fuzzy domain. See [1,6,7,8,9,10,11,12,13,14] and the references therein.

In this paper, using the properties of fuzzy integration [8, 9], we examine the ex-istence and uniqueness of initial value problems of second-order fuzzy differential equations:

x(t)=ft, x(t), x(t), xt0

=k1, x

t0

=k2. (1.1)

2. Preliminaries. A nonempty subsetA ofRn _{is calledconvex if and only if (1−} k)x+ky∈Afor everyx, y∈Aandk∈[0,1]. ByPk(Rn), we denote the family of all nonempty compact convex subsets ofRn_.

ForA, B∈Pk(Rn), the Hausdorff metric is defined by

d(A, B)=max

sup a∈A

inf b∈B

a−b,sup b∈B

inf a∈A

a−b

. (2.1)

Afuzzy setinRn_{is a function with domainR}n_{and values in[0,1], that is, an element} of[0,1]Rn(see [15,16]).

Letu, v∈[0,1]Rn. We define the following fuzzy sets (see [15,16]):

(1)u∧v∈[0,1]Rnby(u∧v)(x)=min{u(x), v(x)}, for everyx∈Rn_{(intersection);} (2)u∨v∈[0,1]Rnby(u∨v)(x)=max{u(x), v(x)}, for everyx∈Rn_(union); (3)uc_∈[0,1]Rn

byuc_{(x)=1−u(x), for everyx∈R}n_. Letu∈[0,1]Rn, thea-level setis

[u]a=x∈Rn:u(x)≥a, a∈(0,1],

[u]0_=Clx∈Rn_{:u(x) >0.} (2.2)

ByEn_{, we denote the family of all fuzzy setsu∈[0,1]}Rn

(see, e.g., [12,16]), for which (i) uis normal, that is, there exists an elementx0∈Rnsuch thatu(x0)=1; (ii) uis fuzzy convex, that is, for anyx, y∈Rn_{andk∈[0,1],}

ukx+(1−k)y≥minu(x), u(y); (2.3)

(iii) uis upper-semicontinuous; (iv) [u]0_{is compact.}

Letu∈En_{. Then for eacha∈(0,1], thea-level set[u]}a_{ofuis a nonempty compact} convex subset ofRn_{, that is,u∈P}

k(Rn). Also[u]0∈Pk(Rn). Let

D:En_×En _→[0,∞),

D(u, v)=supd[u]a, [v]a:a∈[0,1], (2.4)

wheredis the Hausdorff metric for nonempty compact convex subsets ofRn_.

It is well known that(En_{, D)is a complete metric space, but it is not locally compact.} Letu, v∈En_{and letcbe a positive number. The additionu+vand (positive) scalar} multiplicationc·uinEn_{are defined in terms of thea-level sets by}

[u+v]a_=[u]a_+[v]a_{, [c·u]}a_=c[u]a_,

(2.5)

for everya∈[0,1].

This defines a linear structure onEn_{such that}

D(u+w, v+w)=D(u, v), D(c·u, c·v)=cD(u, v), (2.6)

for allu, v∈En_{andc >0.}

LetT=[t0, t0+a]witha >0 andx, y∈En.

A mappingf:T→En_{isdifferentiableatt∈T if there exists anf(t)∈E}n_{such that} the limits

lim h→0+

f (t+h)−f (t)

h , hlim→0+

f (t)−f (t−h)

h (2.7)

exist and are equal tof(t).

Letf:T→En_{; the integral off overT, denoted by}

Tf (t)dt, is defined levelwise by the equation[_Tf (t)dt]a ₌

Tfa(t)dt = {

Tf (t)dt:f :T →En is a measurable selection forfa}, for alla∈(0,1].

We say that a mappingf :T →En _{is strongly measurable if, for alla∈[0,1], the} set-valued mappingfa:T→Pk(Rn)is defined byfa(t)=[f (t)]a.

Iff:T→En_{is continuous, then it is integrable (see [8]).}

3. Fuzzy differential equations

Theorem3.1. Lett0∈[a, b]and assume thatf:[a, b]×En×En→Enis continuous. Consider the initial value problem (1.1). A mappingx:[a, b]→En_{is a solution to (1.1)} if and only ifx, xare continuous and satisfy the integral equation

x(t)=k2t−t0+ t

t0 t

t0

fs, x(s), x(s)ds

ds+k1. (3.1)

Proof. Sincef is continuous, by [8, Corollary 4.1], it must be integrable. So, for

x(t)=ft, x(t), x(t), t∈[a, b], (3.2)

we have equivalently

x(t)=

t

t0

fs, x(s), x(s)ds+xt0

(3.3)

(see [9, Lemma 3.1]). Sincex(t0)=k2, we have

x(t)=

t

t0

fs, x(s), x(s)ds+k2. (3.4)

Thus, by [9, Lemma 3.1],

x(t)=

t

t0 t

t0

fs, x(s), x(s)ds+k2

ds+xt0

; (3.5)

equivalently (see [8, Theorem 4.3]),

x(t)=

t

t0 t

t0

fs, x(s), x(s)ds

ds+

t

t0

k2ds+x

t0

; (3.6)

equivalently (see [8, Example 4.1]),

x(t)=

t

t0 t

t0

fs, x(s), x(s)ds

or

x(t)=

t

t0 t

t0

fs, x(s), x(s)ds

ds+k2t−t0+k1. (3.8)

Example3.2. The second-order linear nonhomogeneous fuzzy differential equation

x(t)=q1x(t)+q2x(t)+A(t), (3.9)

wherex,x,A:[a, b]→En_{are continuous functions andq}

1, q2∈R\{0}, with the initial conditions

xt0=k1, xt0=k2, t0∈[a, b], (3.10)

is equivalent to a Volterra-type fuzzy integral equation

x(t)=

t

t0

kt, s, x(s)ds+g(t), (3.11)

wherek:[a, b]×[a, b]×En_→En_andg(t)=k

2(t−t0)+k1.

Indeed, byTheorem 3.1, the second-order linear fuzzy differential equation

x(t)=q1x(t)+q2x(t)+A(t), xt0=k1, xt0=k2, t0∈[a, b], (3.12)

is equivalent to the integral equation

x(t)=

t

t0 t

t0

fs, x(s), x(s)ds

ds+g(t), (3.13)

wheref (t, x(t), x(t))=q1x(t)+q2x(t)+A(t)andg(t)=k2(t−t0)+k1. Letk:[a, b]×[a, b]×En_→En_{be a map such thatk(t, s, x(s))=}t

t0f (s, x(s), x

_(s))ds.

Then the above integral equation is equivalent to the integral equation (3.11).

Thus, all the theorems on existence and uniqueness proved in [6,11] hold true, equiv-alently for the fuzzy differential equation examined here.

Theorem3.3. Letf :[a, b]×En_×En_→En _{be a continuous map and assume that} there existsk >0such that

Dft, x1(t), x2(t)

, ft, y1(t), y2(t)

≤kDx1(t), y1(t)

(3.14)

for allt∈[a, b],x1,x2,y1,y2:[a, b]→En. Then the initial value problem

x(t)=ft, x(t), x(t), xt0=k1, xt0=k2, t0∈[a, b], (3.15)

Proof. Denote byC(J, En_{)the set of all continuous mappings fromJtoE}n_{, where} Jis an interval inR. We metricizeC(J, En_{)by setting}

H(x, y)=supDx(t), y(t):t∈J (3.16)

for allx, y∈C(J, En_{). The pair(C(J, E}n_{), H)is a complete metric space (see [8]).} Now, let(t1, y1(t), y2(t))∈[a, b]×En×Enbe arbitrary and letn >0 be such that

n2_{k <1. We will show that the initial value problem}

x(t)=ft, x(t), x(t), xt1

=y1, x

t1

=y2, t1∈[a, b], (3.17)

has a unique solution onI=[t1, t1+n].

Forx∈C(I, En_{), defineGxonIby the equation}

Gx(t)=y2

t−t1

+

t

t1 t

t1

fs, x(s), x(s)ds

ds+y1. (3.18)

Then by [8, Corollary 4.2],Gx∈C(I, En_{). Furthermore, by [8, Theorem 4.3] and the} Lipschitz condition onf, we have

H(Gx, Gy)=supDGx(t), Gy(t):t∈I

=sup

D t

t1 t

t1

fs, x(s), x(s)ds

ds,

t

t1 t

t1

fs, y(s), y(s)ds

ds

:t∈I

≤

t1+n

t1

t1+n

t1

Dfs, x(s), x(s), fs, y(s), y(s)ds

ds

≤

t1+n

t1

t1+n

t1

kDx(s), y(s)ds

ds

≤kn2_{H(x, y)}

(3.19)

for allx, y∈C(I, En_{). Hence, by Banach’s contraction mapping theorem,Ghas a unique} fixed point, which byTheorem 3.1is the desired solution to problem (1.1).

Express [a, b]as a union of a finite family of intervalsIk with the length of each interval less than n. The preceding paragraph guarantees the existence of a unique solution to problem (1.1) on each intervalIk. Piecing these solutions together gives us the unique solution on the whole interval[a, b].

Example3.4. The second-order linear nonhomogeneous differential equation

x(t)=q1x(t)+A(t), (3.20)

wherex, x, A∈C([a, b], En_)andq

Letf :[a, b]×En_×En_→En _{be such that f (t, x(t), x(t))=q1x(t)+A(t), t∈T.} Clearly, the mapfis continuous and

Dft, x1(t), x2(t), ft, y1(t), y2(t)

=Dq1x1(t)+A(t), q1y1(t)+A(t)

=Dq1x1(t), q1y1(t)

≤q1D

x1(t), y1(t)

(3.21)

for allt∈[a, b].

Thus, byTheorem 3.3, the above problem has a unique solution.

Theorem3.5. Letf:T ×En_×En_→En_{be a continuous map and assume that there} existsk >0such that

dft, x(t), x(t)a,ft, y(t), y(t)a≤kdx(t)a,y(t)a (3.22)

for alla∈[0,1]. Then the initial value problem

x(t)=ft, x(t), x(t), xt0

=k1, x

t0

=k2, t0∈T , (3.23)

has a unique solution.

Proof. Indeed, we have

Dft, x(t), x(t), ft, y(t), y(t)

=supdft, x(t), x(t)a,ft, y(t), y(t)a:a∈[0,1] ≤ksupd[x(t)]a, [y(t)]a:a∈[0,1]

=kDx(t), y(t)

(3.24)

for allt∈T andx, y∈En_.

Thus, byTheorem 3.3, the above problem has a unique solution.

Example3.6. Letx, A:T→En_{be continuous maps andq >0. Then the initial value} problem

x(t)=qx(t)+A(t), xt0=k1, xt0=k2, t0∈T , (3.25)

has a unique solution.

Letf :T ×En _×En_→En _{be such thatf (t, x(t), x(t))=qx(t)+A(t), t∈T, and}

x(t), x(t)∈En_{. Clearly, the mapf is continuous and}

dft, x(t), x(t)a,ft, y(t), y(t)a

≤dqx(t)+A(t)a,qy(t)+A(t)a =dqx(t)a+A(t)a,qy(t)a+A(t)a =dqx(t)a,qy(t)a

≤qdx(t)a,y(t)a

(3.26)

Thus the above problem has a unique solution.

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D. N. Georgiou: Department of Mathematics, Faculty of Sciences, University of Patras, 26500 Patras, Greece

E-mail address:[email protected]

I. E. Kougias: Department of Accounting, School of Business Administration and Economics, Technological Educational Institute of Epirus, 48100 Preveza, Greece