New Results on Multiple Solutions for th-Order Fuzzy Differential Equations under Generalized Differentiability

Volume 2009, Article ID 395714,13pages doi:10.1155/2009/395714

Research Article

New Results on Multiple Solutions for

N

th-Order Fuzzy Differential Equations under

Generalized Differentiability

A. Khastan,

_{F. Bahrami,}

_{and K. Ivaz}

1_{Department of Applied Mathematics, University of Tabriz, Tabriz 51666 16471, Iran}

2_{Research Center for Industrial Mathematics, University of Tabriz, Tabriz 51666 16471, Iran}

Correspondence should be addressed to A. Khastan,[email protected]

Received 30 April 2009; Accepted 1 July 2009

Recommended by Juan Jos´e Nieto

We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpretNth-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions.

Copyrightq2009 A. Khastan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The term “fuzzy differential equation” was coined in 1987 by Kandel and Byatt 1 and an extended version of this short note was published two years later 2. There are many suggestions to define a fuzzy derivative and in consequence, to study fuzzy differential equation3. One of the earliest was to generalize the Hukuhara derivative of a set-valued function. This generalization was made by Puri and Ralescu4and studied by Kaleva5. It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes by6. Hence, the fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation, H ¨ullermeier

7 interpreted fuzzy differential equation as a family of differential inclusions. The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzy-number-valued function.

The strongly generalized differentiability was introduced in 8 and studied in 9–

generalized derivative is defined for a larger class of fuzzy-number-valued functions than the Hukuhara derivative. Hence, we use this differentiability concept in the present paper. Under this setting, we obtain some new results on existence of several solutions forN th-order fuzzy differential equations. Higher-order fuzzy differential equation with Hukuhara differentiability is considered in 12 and the existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition is proved. Buckley and Feuring13presented two different approaches to the solvability ofNth-order linear fuzzy differential equations.

Here, using the concept of generalized derivative and its extension to higher-order derivatives, we show that we have several possibilities or types to define higher-order derivatives of fuzzy-number-valued functions. Then, we propose a new method to solve higher-order fuzzy differential equations based on the selection of derivative type covering all former solutions. With these ideas, the selection of derivative type in each step of derivation plays a crucial role.

2. Preliminaries

In this section, we give some definitions and introduce the necessary notation which will be used throughout this paper. See, for example,6.

Definition 2.1. LetXbe a nonempty set. A fuzzy setuinXis characterized by its membership functionu:X → 0,1.Thus,uxis interpreted as the degree of membership of an element

xin the fuzzy setufor eachx∈X.

Let us denote byRF the class of fuzzy subsets of the real axisi.e.,u : R → 0,1 satisfying the following properties:

iuis normal, that is, there existss0∈Rsuch thatus0 1,

iiuis convex fuzzy seti.e.,uts1−tr≥min{us, ur},for allt∈0,1, s, r∈R,

iiiuis upper semicontinuous onR,

ivcl{s∈R|us>0}is compact wherecldenotes the closure of a subset.

ThenRF is called the space of fuzzy numbers. Obviously, R ⊂ RF. For 0 < α ≤ 1 denote

uα{s∈R|us≥α}andu0 cl{s∈R|us>0}. Ifubelongs toRF,thenα-level set

uαis a nonempty compact interval for all 0≤α≤1. The notation

uαuα_{, u}α_{, 2.1}

denotes explicitly theα-level set ofu. One refers touanduas the lower and upper branches ofu, respectively. The following remark shows whenuα, uαis a validα-level set.

Remark 2.2see6. The sufficient conditions foruα, uαto define the parametric form of a fuzzy number are as follows:

iuαis a bounded monotonic increasingnondecreasingleft-continuous function on

0,1and right-continuous forα0,

iiuαis a bounded monotonic decreasingnonincreasingleft-continuous function on

0,1and right-continuous forα0,

Foru, v∈RF andλ∈R, the sumuvand the productλ·uare defined byuvα

uα _vα_{, λ·u}α _λuα_{,for allα∈0,1,whereu}α _vα_{means the usual addition of}

two intervalssubsetsofRandλuαmeans the usual product between a scalar and a subset ofR.

The metric structure is given by the Hausdorffdistance:

D:RF×RF −→R∪ {0}, 2.2

by

Du, v sup

α∈0,1max

_uα_−vα_,uα_−vα_{. 2.3}

The following properties are wellknown:

iDuw, vw Du, v,for allu, v, w∈RF,

iiDk·u, k·v |k|Du, v,for allk∈R, u, v∈RF,

iiiDuv, we≤Du, w Dv, e,for allu, v, w, e∈RF,

andRF, Dis a complete metric space.

Definition 2.3. Letx, y ∈RF. If there existsz ∈ RF such thatx yz,then zis called the

H-difference ofx, yand it is denotedxy.

In this paper the sign “” stands always forH-difference and let us remark thatx

y /x −1yin general. Usually we denotex −1y byx−y, whilexystands for the

H-difference.

3. Generalized Fuzzy Derivatives

The concept of the fuzzy derivative was first introduced by Chang and Zadeh14; it was followed up by Dubois and Prade15who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu4, Goetschel and Voxman16, Kandel and Byatt 1, 2. Lakshmikantham and Nieto introduced the concept of fuzzy differential equation in a metric space17. Puri and Ralescu in4introduced H-derivative

differentiability in the sense of Hukuharafor fuzzy mappings and it is based on theH -difference of sets, as follows. Henceforth, we supposeI T1, T2forT1< T2, T1, T2∈R.

Definition 3.1. LetF :I → RF be a fuzzy function. One says,F is differentiable att0 ∈ Iif there exists an elementFt0∈RF such that the limits

lim

h→0

Ft0hFt0

h , hlim→0

Ft0Ft0−h

h 3.1

The above definition is a straightforward generalization of the Hukuhara diff eren-tiability of a set-valued function. From 6, Proposition 4.2.8, it follows that Hukuhara differentiable function has increasing length of support. Note that this definition of derivative is very restrictive; for instance, in9, the authors showed that ifFt c·gt,wherecis a fuzzy number andg : a, b → R is a function withgt < 0, thenFis not differentiable. To avoid this difficulty, the authors9introduced a more general definition of derivative for fuzzy-number-valued function. In this paper, we consider the following definition11.

Definition 3.2. LetF : I → RF and fixt0 ∈ I. One saysF is1-differentiable att0, if there exists an elementFt0 ∈RF such that for allh >0 sufficiently near to 0, there existFt0

hFt0, Ft0Ft0−h,and the limitsin the metricD

lim

h→0

Ft0hFt0

h hlim→0

Ft0Ft0−h

h Ft0. 3.2

Fis2-differentiable if for allh <0 sufficiently near to 0, there existFt0hFt0, Ft0

Ft0−hand the limitsin the metricD

lim

h→0−

Ft0hFt0

h hlim→0−

Ft0Ft0−h

h Ft0. 3.3

IfFisn-differentiable att0, we denote its first derivatives byDn1Ft0, forn1,2.

Example 3.3. Letg : I → R and definef : I → RF byft c·gt,for allt ∈ I. Ifg is differentiable att0 ∈I, thenf is generalized differentiable ont0 ∈Iand we haveft0 c·

gt0. For instance, ifgt0>0,fis1-differentiable. Ifgt0<0,thenfis2-differentiable.

Remark 3.4. In the previous definition,1-differentiability corresponds to the H-derivative introduced in4, so this differentiability concept is a generalization of the H-derivative and obviously more general. For instance, in the previous example, forft c·gtwithgt0< 0,we haveft0 c·gt0.

Remark 3.5. In9, the authors consider four cases for derivatives. Here we only consider the two first cases of9, Definition 5. In the other cases, the derivative is trivial because it is reduced to crisp elementmore precisely,Ft0∈R. For details, see9, Theorem 7.

Theorem 3.6. LetF:I → RFbe fuzzy function, whereFtα fαt, gαtfor eachα∈0,1.

iIf F is (1)-differentiable, then fα and gα are differentiable functions and D₁1Ftα fαt, gαt.

iiIf F is (2)-differentiable, then fα and gα are differentiable functions and D₂1Ftα gαt, fαt.

Proof. See11.

For a given fuzzy functionF, we have two possibilitiesDefinition 3.2to obtain the derivative ofF ott:D₁1FtandD₂1Ft. Then for each of these two derivatives, we have again two possibilities: D₁1D₁1Ft, D₂1D₁1Ft, and D₁1D₂1Ft, D₂1D₂1Ft, respectively.

Definition 3.7. LetF :I → RFandn, m1,2. One says sayFisn, m-differentiable att0∈I, ifDn1F exists on a neighborhood oft0 as a fuzzy function and it ism-differentiable att0. The second derivatives ofFare denoted byDn,m2Ft0forn, m1,2.

Remark 3.8. This definition is consistent. For example, ifF is1,2 and2,1-differentiable simultaneously att0, thenF is1- and2-differentiable aroundt0. By remark in9,Fis a crisp function in a neighborhood oft0.

Theorem 3.9. Let D₁1F : I → RF or D₂1F : I → RF be fuzzy functions, whereFtα fαt, gαt.

iIfD₁1Fis (1)-differentiable, thenfαandgαare differentiable functions andD12,1Ft

α f

αt, gα t.

iiIfD₁1Fis (2)-differentiable, thenf_αandg_αare differentiable functions andD₁2_,2Ftα g

αt, fα t.

iiiIfD₂1Fis (1)-differentiable, thenfαandgαare differentiable functions andD22,1Ft

α g

αt, fα t.

ivIfD₂1Fis (2)-differentiable, thenf_αandg_αare differentiable functions andD₂2_,2Ftα f

αt, gα t.

Proof. We present the details only for the casei, since the other cases are analogous. Ifh >0 andα∈0,1, we have

D1

1 FthD

1

1 Ft

α

f_αth−f_αt, gαth−gαt, 3.4

and multiplying by 1/h,we have

D1

1 FthD

1

1 Ft

α

h

f_αth−f_αt

h ,

g_αth−g_αt h

. 3.5

Similarly, we obtain

D1

1 FtD

1

1 Ft−h

α

h

f_αt−f_αt−h

h ,

g_αt−g_αt−h h

Passing to the limit, we have

D2

1,1Ft

α f

αt, gα t

. 3.7

This completes the proof of the theorem.

Let N be a positive integer number, pursuing the above-cited idea, we write

D_kN

1,...,kNFt0 to denote theNth-derivatives ofF at t0 with ki 1,2 fori 1, . . . , N. Now

we intend to compute the higher derivativesin generalized differentiability sense of the

H-difference of two fuzzy functions and the product of a crisp and a fuzzy function.

Lemma 3.10. Iff, g:I → RF areNth-order generalized differentiable att∈Iin the same case of differentiability, thenfgis generalized differentiable of orderNattandfgNt fNt gN_{t. (The sum of two functions is defined pointwise.)}

Proof. ByDefinition 3.2the statement of the lemma follows easily.

Theorem 3.11. Letf, g : I → RF be second-order generalized differentiable such thatf is (1,1)-differentiable andgis (2,1)-differentiable orfis (1,2)-differentiable andgis (2,2)-differentiable orfis (2,1)-differentiable andgis (1,1)-differentiable orf is (2,2)-differentiable andgis (1,2)-differentiable onI. If theH-differenceftgtexists fort∈I,thenfgis second-order generalized differentiable and

fg t f _{t −1·g t, 3.8}

for allt∈I.

Proof. We prove the first case and other cases are similar. Since f is1-differentiable and

g is2-differentiable onI, by10, Theorem 4,fgtis1-differentiable and we have

fgt ft −1·gt. By differentiation as1-differentiability inDefinition 3.2and usingLemma 3.10, we getfgtis1,1-differentiable and we deduce

fg t ft −1·gt f t −1·g t. 3.9

TheH-difference of two functions is understood pointwise.

Theorem 3.12. Letf : I → Randg : I → RF be two differentiable functions (g is generalized differentiable as inDefinition 3.2).

iIfft·ft>0andgis (1)-differentiable, thenf·gis (1)-differentiable and

iiIfft·ft<0andgis (2)-differentiable, thenf·gis (2)-differentiable and

f·gt ft·gt ft·gt. 3.11

Proof. See10.

Theorem 3.13. Letf : I → Rand g : I → RF be second-order differentiable functions (g is generalized differentiable as inDefinition 3.7).

iIfft·ft>0, ft·f t>0,andgis (1,1)-differentiable thenf·gis (1,1)-differentiable and

f·g t f _{t·gt 2ft·gt ft·g t. 3.12}

iiIfft·ft<0, ft·f t<0andgis (2,2)-differentiable thenf·gis (2,2)-differentiable and

f·g t f _{t·gt 2ft·gt ft·g t. 3.13}

Proof. We provei, and the proof of another case is similar. Ifft·ft > 0 andg is1 -differentiable, then byTheorem 3.12we have

f·gt ft·gt ft·gt. 3.14

Now by differentiation as first case inDefinition 3.2, sincegtis1-differentiable andft·

f _{t>0,then we conclude the result.}

Remark 3.14. By9, Remark 16, letf :I → R, γ∈RFand defineF:I → RFbyFt γ·ft, for allt ∈I. Iffis differentiable onI,thenFis differentiable onI, withFt γ·ft. By

Theorem 3.12, ifft·ft>0,thenFis1-differentiable onI. Also ifft·ft< 0,then

Fis2-differentiable onI. Ifft·ft 0, by9, Theorem 10, we haveFt γ·ft. We can extend this result to second-order differentiability as follows.

Theorem 3.15. Let f : I → Rbe twice differentiable on I,γ ∈ RF and defineF : I → RFby Ft γ·ft, for allt∈I.

iIf ft·ft > 0 and ft·f t > 0,thenFt is (1,1)-differentiable and its second derivative,D₁2_,1F,isF t γ·f t,

iiIf ft·ft > 0 and ft·f t < 0,thenFt is (1,2)-differentiable withD₁2_,2F γ·f _t,

iiiIfft·ft<0andft·f t>0,thenFtis (2,1)-differentiable withD₂2_,1Fγ·f t,

Proof. Casesi andiv follow fromTheorem 3.13. To proveii, sinceft·ft > 0, by

Remark 3.14,Fis1-differentiable and we haveD₁1F γ·ftonI. Also, sinceft·f t< 0, thenD₁1Fis2-differentiable and we conclude the result. Caseiiiis similar to previous one.

Example 3.16. Ifγis a fuzzy number andφ:0,3 → R,where

φt t2_{−3t2 3.15}

is crisp second-order polynomial, then for

Ft γ·φt, 3.16

we have the following

ifor0 < t < 1: φt·φt < 0 and φt·φ t < 0 then by iv, Ft is 2-2 -differentiable and its second derivative,D₂2_,2FisF t 2·γ,

iifor1 < t < 3/2: φt·φt > 0 and φt·φ t < 0 then byii,Ft is1-2 -differentiable withD₁2_,2F 2·γ,

iiifor3/2 < t < 2: φt·φt < 0 andφt·φ t > 0 then byiii,Ft is2-1 -differentiable andD₂2_,1F 2·γ,

ivfor2< t <3: φt·φt>0 andφt·φ t>0 then byi,Ftis1-1-differentiable andD₁2_,1F2·γ,

vfort1,3/2,2: we haveφt·φ t 0, then by9, Theorem 10we haveFt

γ·φt, again by applying this theorem, we getF t 2·γ.

4. Second-Order Fuzzy Differential Equations

In this section, we study the fuzzy initial value problem for a second-order linear fuzzy differential equation:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

y _{t a·yt b·yt σt,} y0 γ0,

y0 γ1,

4.1

wherea, b > 0,γ0, γ1 ∈RF,andσtis a continuous fuzzy function on some intervalI. The intervalI can be0, Afor someA >0 orI 0,∞. In this paper, we supposea, b >0.Our strategy of solving4.1is based on the selection of derivative type in the fuzzy differential equation. We first give the following definition for the solutions of4.1.

Definition 4.1. Lety:I → RFbe a fuzzy function andn, m∈ {1,2}.One saysyis ann, m -solution for problem4.1onI, ifD1

Let y be an n, m-solution for 4.1. To find it, utilizing Theorems 3.6 and 3.9

and considering the initial values, we can translate problem 4.1 to a system of second-order linear ordinary differential equations hereafter, called correspondingn, m-system for problem4.1.

Therefore, four ODEs systems are possible for problem4.1, as follows:

1

,

1

-system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y _{t;α ayt;α byt;α σt;α,}

y t;α ayt;α byt;α σt;α,

y0;α γ0α, y0;α γ0α,

y0;α γ1α, y0;α γ1α,

4.2

1

,

2

-system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y t;α ayt;α byt;α σt;α,

y _{t;α ayt;α byt;α σt;α,}

y0;α γ0α, y0;α γ0α,

y0;α γ1α, y0;α γ1α,

4.3

2

,

1

-system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y t;α ayt;α byt;α σt;α,

y _{t;α ayt;α byt;α σt;α,}

y0;α γ0α, y0;α γ0α,

y0;α γ1, y0;α γ1,

4.4

2

,

2

-system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y _{t;α ayt;α byt;α σt;α,}

y t;α ayt;α byt;α σt;α,

y0;α γ0α, y0;α γ0α,

y0;α γ1α, y0;α γ1α.

Theorem 4.2. Letn, m∈ {1,2}andy y, ybe ann, m-solution for problem4.1onI. Theny andysolve the associatedn, m-systems.

Proof. Supposeyis then, m-solution of problem4.1. According to theDefinition 4.1, then

D1

ny and D2n,myexist and satisfy problem 4.1. By Theorems3.6 and3.9 and substituting y, yand their derivatives in problem4.1, we get then, m-system corresponding ton, m -solution. This completes the proof.

Theorem 4.3. Letn, m ∈ {1,2}and fαtand gαtsolve the n, m-system onI, for everyα ∈ 0,1. Let Ftα fαt, gαt. If F has valid level sets on I and D2n,mF exists, then F is an n, m-solution for the fuzzy initial value problem4.1.

Proof. SinceFtα fαt, gαtisn, m-differentiable fuzzy function, by Theorems 3.6

and3.9we can computeD1

nFandD2n,mFaccording tofα, gα, fα , g α. Due to the fact thatfα, gα

solven, m-system, fromDefinition 4.1, it comes thatFis ann, m-solution for4.1.

The previous theorems illustrate the method to solve problem4.1. We first choose the type of solution and translate problem4.1to a system of ordinary differential equations. Then, we solve the obtained ordinary differential equations system. Finally we find such a domain in which the solution and its derivatives have valid level sets and using Stacking Theorem5we can construct the solution of the fuzzy initial value problem4.1.

Remark 4.4. We see that the solution of fuzzy differential equation4.1depends upon the selection of derivatives. It is clear that in this new procedure, the unicity of the solution is lost, an expected situation in the fuzzy context. Nonetheless, we can consider the existence of four solutions as shown in the following examples.

Example 4.5. Let us consider the following second-order fuzzy initial value problem

y _{t σ}

0, y0 γ0, y0 γ1, t≥0, 4.6

whereσ0 γ0γ1are the triangular fuzzy number havingα-level setsα−1,1−α. Ifyis1,1-solution for the problem, then

ytαyt;α, yt;α, y tαy t;α, y t;α, 4.7

and they satisfy1,1-system associated with4.1. On the other hand, by ordinary differential theory, the corresponding1,1-system has only the following solution:

yt;α α−1

t2 2 t1

, yt;α 1−α

t2 2 t1

. 4.8

We see thatytα yt;α, yt;αare valid level sets fort≥0 and

y α−1,1−α·

t2 2 t1

ByTheorem 3.15,yis1,1-differentiable fort ≥ 0. Therefore,y defines a1,1-solution for

t≥0.

For1,2-solution, we get the following solutions for1,2-system:

yt;α α−1

−t2

2 t1

, yt;α 1−α

−t2

2 t1

, 4.10

whereythas valid level sets fort∈0,1.How ever-alsoytα α−1,1−α·−t2_/2t1 whereyis1,2-differentiable. Thenygives us a1,2-solution on0,1.

2,1-system yields

yt;α α−1

−t2

2 −t1

, yt;α 1−α

−t2

2 −t1

, 4.11

whereythas valid level sets fort∈0,√3−1.We can seeyis a2,1-solution on0,√3−1 Finally,2-2-system gives

yt;α α−1

t2 2 −t1

, yt;α 1−α

t2 2 −t1

, 4.12

whereythas valid level sets for allt∈0,1,and defines a2,2-solution on0,1.

Then we have an example of a second-order fuzzy initial value problem with four different solutions.

Example 4.6. Consider the fuzzy initial value problem:

y _{t yt σ}

0, y0 γ0, y0 γ1 ∀t≥0, 4.13

whereσ0is the fuzzy number havingα-level sets α,2−αandγ0α γ1α α−1,1−α. To find1,1-solution, we have

yt;α α1sint−sint−cost, yt;α 2−α1sint−sint−cost, 4.14

whereythas valid level sets fort≥0 andyt σ0·1sint−sint−cost. FromTheorem 3.15,

y is 1,2-differentiable on 0, π/2, then by Remark 3.8, y is not 1,1-differentiable on

0, π/2. Hence, no1,1-solution exists fort >0. For1,2-solutions we deduce

yt;α α1sinht−sinht−cost,

yt;α 2−α1sinht−sinht−cost,

4.15

For2,1-solutions we get

yt;α α1−sinht sinht−cost,

yt;α 2−α1−sinht sinht−cost,

4.16

we see that the fuzzy functionythas valid level sets fort ∈ 0,ln1√2and define a

2,1-solution for the problem on0,ln1√2. Finally, to find2,2-solution, we find

yt;α α1−sint sint−cost, yt;α 2−α1−sint sint−cost, 4.17

thatythas valid level sets fort≥0 andyis2,2-differentiable on0, π/2.

We then have a linear fuzzy differential equation with initial condition and two solutions.

5. Higher-Order Fuzzy Differential Equations

Selecting different types of derivatives, we get several solutions to fuzzy initial value problem for second-order fuzzy differential equations.Theorem 4.2has a crucial role in our strategy. To extend the results to Nth-order fuzzy differential equation, we can follow the proof of Theorem 4.2 to get the same results for derivatives of higher order. Therefore, we can extend the presented argument for second-order fuzzy differential equation to Nth-order. Under generalized derivatives, we would expect at most 2Nsolutions for anNth-order fuzzy differential equation by choosing the different types of derivatives.

Acknowledgments

We thank Professor J. J. Nieto for his valuable remarks which improved the paper. This research is supported by a grant from University of Tabriz.

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