A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS

Transcription

1 IJMMS 28: ) PII. S hp://ijmms.hindawi.com Hindawi Publishing Corp. A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS ELIAS DEEBA and ANDRE DE KORVIN Received 29 January 2001) Absrac. We define he concep of a larges and a smalles soluion o an underlying equaion and hen show ha a fuzzy soluion of he corresponding fuzzy equaion is bounded above and below by hese soluions, respecively Mahemaics Subjec Classificaion. 03E72, 39A Inroducion. The sudy of funcional equaions wih is wide range of applicaions is an ineresing subjec ha daes back o he work of Abel. Mahemaical models of numerous applied problems resul in eiher ordinary differenial, parial differenial, difference or algebraic equaions. I is sandard o assume ha he sough variable in hese equaions is deerminisic and can be obained using numerical or analyical mehods. However, in modeling applied problems only parial informaion may be known or here may be a degree of uncerainy in he parameers used in he model or some measuremens may be imprecise. Because of such feaures, we are emped o consider he sudy of funcional equaions in he fuzzy seing. Indeed, in [1, 2, 3, 4, 5] he auhors did iniiae his sudy. The mehods presened for solving he fuzzy funcional equaions depended on having a soluion o he classical equaions for end poins of α-cus. In his paper, we presen a raher differen approach ha will provide an upper and a lower bound of he soluion. In Secion 2, we will discuss he basic background needed for he manuscrip and in Secion 3, we will presen he main resuls of he paper along wih examples. 2. Preliminaries. We presen, for he sake of compleeness, some background maerial needed in he sequel. For a deailed sudy, we refer he reader o [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Le X be any nonempy se. The se A is a fuzzy se on X if A is a funcion from X ino he inerval [0,1]. The value Ax) is someimes referred o as he membership of x in A. The se A is said o be convex if every [0,1], Ax )x 2 ) minax 1 ), Ax 2 )}. A is said o be normalized if here exiss an x such ha Ax) = 1. By he α-level of a fuzzy subse A, denoed by [A] α, we mean [A] α =x X Ax) α}. 2.1) I is well known ha he α-levels of a fuzzy subse A deermine A. Also, i can be easily verified ha for any fuzzy subse A of X: C1) A is convex if and only if [A] α is convex and

2 734 E. DEEBA AND A. DE KORVIN C2) if A is coninuous, hen A is convex if and only if [A] α is a closed inerval. In his seing by a fuzzy number N, we mean a fuzzy subse of X, he se of posiive reals, which is coninuous, convex, normalized and vanishing a infiniy. Define a funcion g : X X. The exension principle, in one dimension see [12, 13, 14, 15]) saes ha g can be exended o fuzzy subses of X as follows: ga)y) = sup A y 1 )), 2.2) where he sup is aken over all y 1 such ha gy 1 ) = y and A is a fuzzy subse of X. Finally, we will need a basic resul of Nguyen [10] which addresses he following quesion. When does he relaion [ ga) ]α = g [A] α ) 2.3) hold for any coninuous funcion g and for any fuzzy subse A? I was shown in [10] ha 2.3) holds if and only if he sup in 2.2) is aained. Tha is, here exiss x such ha ga)z) = A x ), 2.4) where z = gx ). 3. Main resuls. Before we discuss he bounds on he fuzzy soluion o a funcional equaion we need he following lemmas. Firs, we will define he noion of he larges soluion. We consider he equaion } sup x) A) = λ, 3.1) where A) is a given funcion of, λ is a known parameer, 0 λ 1 and x) is an unknown funcion. If here is a soluion x), 0 x) 1, hen we claim ha he larges soluion o 3.1) is x ) = 1, A) λ, λ, A) > λ. 3.2) Firs we show ha x is a soluion. Indeed, x ) A) λ, if A) λ, x ) A) λ, if A) > λ. 3.3) Thus sup x ) A)} λ and 0 x ) 1. Moreover, if A) exceeds or assumes he value of λ for some value, hen sup x ) A)} =λ. We now show ha x is he larges soluion. If x) is any oher soluion saisfying 3.1), hen for A) > λ, x) λ = x ) and for A) λ, x) 1 = x ). We have, herefore, shown he following lemma. Lemma 3.1. The funcion x is he larges soluion o 3.1) among all soluions x, 0 x) 1, provided A) λ for some.

3 A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS 735 Nex, we define he noion of smalles soluion. To his end, we consider he equaion } inf x) B) = µ, 3.4) where B) is a given funcion of, µ is a known parameer, 0 µ 1 and x) is an unknown funcion. If here is a soluion x), 0 x) 1, hen we claim ha he smalles soluion o 3.4) is x ) = 0, B) µ, µ, B)<µ. 3.5) We firs show ha x ) is a soluion. Indeed, x ) B) µ, if B) µ, x ) B) µ, if B)<µ. 3.6) Thus inf x ) B)} µ and 0 x ) 1. Moreover, if B) is less han or assumes he value of µ for some value, hen inf x ) B)} =µ. We now show ha x is he smalles soluion. If x) is any oher soluion saisfying 3.4), hen for B)<µ, µ = x ) x) and for B) µ, 0= x ) x). Thus we have shown he following lemma. Lemma 3.2. The funcion x is he smalles soluion of 3.4) among all soluions x), 0 x) 1, provided B) µ for some. Remark 3.3. The compuaion of λ and µ for a specified choice of A) and B) is cenral for deermining he bounds on he soluion. We now look a he equaion x n+1 = f x n ) 3.7) wih given iniial condiion x 0. We assume x 0 is a fuzzy subse of R and we exend f o a fuzzy subses of R using he exension principle. Then he difference equaion reads x n+1 ) = f x n ) ) = sup xn 1 )), 3.8) where he sup is aken over all 1 for which f 1 ) =. However, for simpliciy of noaion we wrie 3.8) as x n+1 ) = f ) x n ). 3.9) We assume ha 1) f : R R is a one-o-one funcion. Thus f n is also a one-o-one funcion for every nf 1 denoes he inverse of f), 2) lim n f n ) is eiher or a singleon for all. If i is a poin, we denoe his by f ). We may define he fuzzy soluion as x ) = x 0 [ f ) ]. 3.10)

4 736 E. DEEBA AND A. DE KORVIN The moivaion for his definiion comes from he exension principle where x n ) = sup f n ) x 0 ) 3.11) and he assumpion ha f is one-o-one x n ) = x 0 f n )). Then as n, we ge he above definiion. The problem we consider is as follows: if we pu cerain bounds on he iniial fuzzy se x 0, wha bounds do we have on he fuzzy soluion x? Theorem 3.4. Le f : R R be a one-o-one funcion such ha lim n f n ) = f ) for all. Assume here exis funcions A) and B) wih values in [0,1] such ha sup x0 f ) ) A) } = λ, inf x0 f ) ) B) } = µ, 3.12) and A) λ and B) µ for some. Then he larges soluion, x, and smalles soluion, x, are respecively he upper and lower bounds of he soluion o 3.9) wih fuzzy iniial condiion x 0. Proof. The equaion x n+1 = fx n ) wih iniial condiion x 0 has formally he soluion x n+1 = f n+1 x 0 ). The fuzzy exension of his is x n+1 ) = sup f n+1) ) x 0 ). 3.13) Since f 1 is one-o-one and since f ) = lim n f n+1) ) exiss, we can wrie he above expression as x n+1 ) = x 0 f n+1) ) ) 3.14) or wrie he soluion as x ) = x 0 f ) ). 3.15) Thus sup x ) A) } = λ, inf x ) B) } = µ. 3.16) The proof hen follows from Lemmas 3.1 and 3.2. Example 3.5. Consider he sandard equaion x n+1 = αx n 3.17) wih specified iniial condiion x 0 and known α. The soluion o his equaion is given by 0, α<1, x = lim α n x 0 = x 0, α= 1, 3.18) n, α>1. In his example f)= α. We consider each case separaely.

5 A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS 737 Case 1. Consider ha α<1. In his case f, 0, ) = 0, = ) Assume ha A) = χ a ), he characerisic funcion for a given a R. We will look for he larges λ = sup x0 f ) ) A) } = χ x0 0) χ a 0). 3.20) If x 0 0ora 0, hen λ = 0. In his case x ) = 1, a, 0, = a. 3.21) If x 0 = a = 0, hen λ = 1 and x ) = 1 for all. In eiher case, x x. Case 2. Consider α = 1. Noe ha f 1 ) =. In his case, λ = sup χx0 ) χ a ) } 0, x 0 a, = 1, x 0 = a. 3.22) Thus when x 0 a, x ) = 1 and x x 0) = 1; when x 0 = a, x ) = 1 for all. In eiher case, x x. Case 3. Consider α>1. In his case, x = means ha no is a soluion o he given equaion. In his case, λ = sup χx0 0) χ a ) } = 0, x 0 0, x ) = 1, x a, 0, x = a. 3.23) Again, in his case x x. We now look for he smalles soluion. When α<1 and x 0 0 µ = inf χx0 0) χ b ) }. 3.24) If x 0 0 and we pick b, µ = 0. Thus x ) = 0 for all. Ifx 0 0, we have when α = 1orα 1, µ = 0. Thus, x ) = 0 for all. Hence x x. The previous echniques may be applied o soluions where much less informaion is given on x 0 and f. The nex example illusraes his siuaion. Example 3.6. Assume ha f is an increasing funcion on [0,1] wih f > 0on [0, ] and f < 0on[,1] where [0,1] is such ha f ) =. Then f ) = for 0, 1 and f 0) = 0 and f 1) = 1. Assume ha x 0, A, and B are fuzzy subses of [0,1]. Then λ = [ x 0 ) A )] [ x 0 0) A0) ] [ x 0 1) A1) ], µ = [ x 0 ) B )] [ x 0 0) B0) ] [ x 0 1) B1) ] 3.25) and he previous bounds x and x are readily compued.

6 738 E. DEEBA AND A. DE KORVIN In he previous examples f was assumed o be one-o-one funcion, wha happens when f is no one-o-one? In he compuaion of λ and µ, x 0 [f )] mus be replaced by sup f ) x 0 ). The nex example illusraed his siuaion. Example 3.7. Consider he equaion x n = 4x n 1 1 xn 1 ) 3.26) wih fuzzy iniial condiion x 0. In his example fx) = 4x1 x). Assume, for simpliciy sake, ha x 0, A, and B are all fuzzy subses of [0,1]. Then f ) =1 1 )/2,1+ 1 )/2}. I is easy o see ha f ) =0,1/2,1} for all. Ifsup A) = 1, hen 1 λ = x 0 0) x 0 2 ) 1 µ = x 0 0) x 0 2 ) x 0 1), ) x 0 1) inf B) 3.27) and he bound esimaes are readily compued. Noe 1. The soluion o 3.9) wih iniial condiion x 0 may be viewed as an inervalvalued fuzzy se. The bounds previously obained involve, 0, 1, λ, and µ. Can we ge more precise esimaes? If we consider λ and µ as given by 3.1) and 3.4), we se λ = poss[x,a], µ = nec [ x,b c], 3.28) where poss[x,a] is he larges inersecion of x and A and nec[x,b c ] is he weakes implicaion no B x. I herefore makes sense ha we could sharpen he bounds if we had a family of known possibiliies and known necessiies. Again assume f is oneo-one and f n ) f ) for all. We define an implicaion operaor on [0,1] by 1, a b, aφb = b, a>b. 3.29) We assume ha for each y Y, poss[x 0 f,a y ] is known. We se py) = poss [ x 0 f ],A y 3.30) and le ux) = inf y A y x)φpy)}. Theorem 3.8. Assume ha for every y Y, here exiss x y such ha A y x y )> py). Then u as defined above is an upper bound for he soluion of 3.30). Proof. Consider poss[x 0 f,a y ] = py) for y Y.ByTheorem 3.4, for each fixed y, he larges soluion is given by x,y ) = 1, A y ) py), py), A y ) > py). 3.31)

7 A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS 739 Therefore, by definiion of he implicaion operaor, φ, x,y ) = A y)φpy). 3.32) Since his is an upper bound for all y Y, he sharpes upper bound generaed is u) = inf y x,y ) = inf Ay )φpy) }. 3.33) y We now apply a similar approach o esablish a lower bound for he soluion of he difference equaion 3.30) wih fuzzy iniial condiion x 0. The operaor analogous o φ is defined as b, a<b, aβb = 0, a b. We assume nec[x 0 f,by c ] is known for a family of y Y.Wese 3.34) nec [ x 0 f,by] c = Ny), 3.35a) L) = sup By )βny) }. 3.35b) y Theorem 3.9. Assume ha for every y Y, here exiss x y such ha B y x y ) Ny). Then L as defined in 3.35b) is a lower bound of he soluion o 3.35a). Proof. Le nec[x 0 f,by c ] = Ny).ByTheorem 3.8, for each y, he smalles soluion is given by x,y ) = 0, B y ) Ny), Ny), B y ) < Ny). 3.36) Thus, by definiion of β, x,y ) = B y)βny). This is a lower bound for all y so he sharpes lower bound generaed is Lx) = supx,y ) = sup By )βny) }. 3.37) y The previous resul can be applied when x 0 is no compleely deermined. This is demonsraed in he nex example. Example Consider a funcion f such ha we may find a pariion P i such ha f P i ) = i,1 i l. We also assume ha f n converges o a poin for all which we denoe by f ). Also assume given wo fuzzy ses: A y1 and A y2. Then ) sup x0 i Ay1 ) } = p 1 = p ) y 1, i ) sup x0 i Ay2 ) } = p 2 = p ) 3.38) y 2. i

8 740 E. DEEBA AND A. DE KORVIN The lef-hand sides of he above wo equaions are poss[x,a y1 ] and poss[x,a y2 ] where x is a soluion o 3.38) wih iniial fuzzy condiion. Now for k = 1,2 1, A yk ) p k A yk )φp k = p k, A yk )>p k. 3.39) So if [A y1 ] p 1 [A y2 ] p 2 p1 >p 1, p2 >p 2 ), hen x ) p 1 p 2. We noe ha x 0 i ) sup P A k ) can be reinerpreed as follows: le  k,i be he resricion of A k o P i, hen x 0 i ) sup i A y k ) = poss[p i,â k,i ] x 0 i ), where he se P i is idenified wih is characerisic funcion. The possibiliy of he soluion and A k is hen given by ) [ ] )} sup x0 i poss pi,â k,i x0 i 1 i l 3.40) no knowing he precise value of x 0 i ) bu knowing he above expression enables us o assign an upper bound o he soluion. A similar example could be consruced for a lower bound. The approach we have aken here is o define he iniial condiion, x 0 ), as a funcion of wih values in [0,1]. This indicaes he degree o which he number is he iniial condiion. The bounds obained on x 0 along wih he assumpions on f led o a soluion o he fuzzy difference equaion. The novely of his approach over he one used in [1, 2, 3, 4, 5] is ha in his approach we do no assume ha he classical equaion mus have an explici soluion. We raher obain bounds on he soluion. The approach can be applied o a variey of problems. References [1] E. Deeba and A. de Korvin, On a fuzzy difference equaion, IEEE Transacion on Fuzzy Sysems ), no. 4, [2] E. Deeba, A. de Korvin, and E. L. Koh, A fuzzy difference equaion wih an applicaion, J. Differ. Equaions Appl ), no. 4, MR 99b: Zbl [3], On a fuzzy logisic difference equaion, Differenial Equaions Dynam. Sysems ), no. 2, MR 99g: Zbl [4] E. Deeba, A. de Korvin, and S. Xie, Pexider funcional equaions heir fuzzy analogs, In. J. Mah. Mah. Sci ), no. 3, MR 97h: Zbl [5], Techniques and applicaions of fuzzy se heory o difference and funcional equaions and heir uilizaion in modeling diverse sysems, Fuzzy Theory Sysems: Techniques and Applicaions, vol. 1 4, Academic Press, California, 1999, pp CMP [6] D. Dubois and H. Prade, Operaions on fuzzy numbers, Inerna. J. Sysems Sci ), no. 6, MR 58# Zbl [7], Fuzzy Ses and Sysems. Theory and Applicaions, Mahemaics in Science and Engineering, vol. 144, Academic Press, New York, MR 82c: Zbl [8] H. M. Hersch and A. Camarazza, A fuzzy-se approach o modifiers and vagueness in naural languages, J. Exp. Psychol. General ), [9] O. Kaleva, Fuzzy performance of a coheren sysem, J. Mah. Anal. Appl ), no. 1, MR 87g: Zbl [10] H. T. Nguyen, A noe on he exension principle for fuzzy ses, J. Mah. Anal. Appl ), no. 2, MR 58#243. Zbl

9 A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS 741 [11] G. C. Oden, Inegraion of fuzzy logical informaion, Journal of Experimenal Psychology: Human Percepion and Performance ), no. 4, [12] L. A. Zadeh, Fuzzy ses, Informaion and Conrol ), MR 36#2509. Zbl [13], The concep of a linguisic variable and is applicaion o approximae reasoning. I, Informaion Sci ), MR 52#7225a. Zbl [14], The concep of a linguisic variable and is applicaion o approximae reasoning. II, Informaion Sci ), MR 52#7225b. Zbl [15], The concep of a linguisic variable and is applicaion o approximae reasoning. III, Informaion Sci ), no. 1, MR 52#7225c. Zbl [16] H. J. Zimmermann, Resuls of empirical sudies in fuzzy se heory, Applied General Sysems Research G. J. Klir, ed.), Plenum Press, New York, 1978, pp Zbl Elias Deeba: Deparmen of Compuer and Mahemaical Sciences, Universiy of Houson-Downown, One Main Sree, Houson, TX 77002, USA address: deebae@d.uh.edu Andre de Korvin: Deparmen of Compuer and Mahemaical Sciences, Universiy of Houson-Downown, One Main Sree, Houson, TX 77002, USA address: dekorvin@d.uh.edu

10 Boundary Value Problems Special Issue on Singular Boundary Value Problems for Ordinary Differenial Equaions Call for Papers The purpose of his special issue is o sudy singular boundary value problems arising in differenial equaions and dynamical sysems. Survey aricles dealing wih ineracions beween differen fields, applicaions, and approaches of boundary value problems and singular problems are welcome. This Special Issue will focus on any ype of singulariies ha appear in he sudy of boundary value problems. I includes: Theory and mehods Mahemaical Models Engineering applicaions Biological applicaions Medical Applicaions Finance applicaions Numerical and simulaion applicaions Composela, Saniago de Composela 15782, Spain; juanjose.nieo.roig@usc.es Gues Edior Donal O Regan, Deparmen of Mahemaics, Naional Universiy of Ireland, Galway, Ireland; donal.oregan@nuigalway.ie Before submission auhors should carefully read over he journal s Auhor Guidelines, which are locaed a hp:// Auhors should follow he Boundary Value Problems manuscrip forma described a he journal sie hp:// Aricles published in his Special Issue shall be subjec o a reduced Aricle Processing Charge of C200 per aricle. Prospecive auhors should submi an elecronic copy of heir complee manuscrip hrough he journal Manuscrip Tracking Sysem a hp://ms.hindawi.com/ according o he following imeable: Manuscrip Due May 1, 2009 Firs Round of Reviews Augus 1, 2009 Publicaion Dae November 1, 2009 Lead Gues Edior Juan J. Nieo, Deparameno de Análisis Maemáico, Faculad de Maemáicas, Universidad de Saniago de Hindawi Publishing Corporaion hp://