TRICHOTOMY, STABILITY, AND OSCILLATION OF A FUZZY DIFFERENCE EQUATION

Transcription

1 TRICHOTOMY, STABILITY, AND OSCILLATION OF A FUZZY DIFFERENCE EQUATION G. STEFANIDOU AND G. PAPASCHINOPOULOS Received 10 November 2003 We study the trichotomy character, the stability, and the oscillatory behavior of the positive solutionsofa fuzzy difference equation. 1. Introduction Difference equations have already been successfully applied in a number of sciences (for a detailed study of the theory of difference equations and their applications, see [1, 2, 7, 8, 11]. The problem of identifying, modeling, and solving a nonlinear difference equation concerning a real-world phenomenon from experimental input-output data, which is uncertain, incomplete, imprecise, or vague, has been attracting increasing attention in recent years. In addition, nowadays, there is an increasing recognition that for understanding vagueness, a fuzzy approach is required. The effect is the introdution and the study of the fuzzy difference equations (see [3, 4, 13, 14, 15]). In this paper, we study the trichotomy character, the stability, and the oscillatory behavior of the positive solutions of the fuzzy difference equation x n+1 = A + ki=1 c i x n pi mj=1 d j x n q j, (1.1) where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1,2,...,m}, are positive fuzzy numbers, p i, i {1,2,...,k}, q j, j {1,2,...,m}, are positive integers such that p 1 <p 2 < <p k, q 1 <q 2 < <q m, and the initial values x i, i { π, π +1,...,0},where π = max { p k,q m }, (1.2) are positive fuzzy numbers. Studying a fuzzy difference equation results concerning the behavior of a related family of systems of parametric ordinary difference equations is required. Some necessary results Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:4 (2004) Mathematics Subject Classification: 39A10 URL:

2 338 Fuzzy difference equations concerning the corresponding family of systems of ordinary difference equations of (1.1) have been proved in [16] and others are given in this paper. 2. Preliminaries We need the following definitions. For a set B, we denote by B the closure of B. We say that a fuzzy set A,fromR + = (0, ) into the interval [0,1], is a fuzzy number, if A is normal, convex, upper semicontinuous (see [14]), and the support suppa = a (0,1] [A] a = {x : A(x) > 0} is compact. Then from [12, Theorems and 3.1.8], the a-cuts of the fuzzy number A,[A] a ={x R + : A(x) a}, are closed intervals. We say that a fuzzy number A is positive if suppa (0, ). It is obvious that if A is a positive real number, then A is a positive fuzzy number and [A] a = [A,A], a (0,1]. In this case, we say that A is a trivial fuzzy number. We say that x n is a positive solution of (1.1)ifx n is a sequence of positive fuzzy numbers which satisfies (1.1). A positive fuzzy number x is a positive equilibrium for (1.1)if x = A + ki=1 c i x mj=1 d j x. (2.1) Let E, H be fuzzy numbers with [E] a = [E l,a,e r,a ], [H] a = [H l,a,h r,a ], a (0,1]. (2.2) According to [10]and[13, Lemma 2.3], we have that MIN{E,H}=E if E l,a H l,a, E r,a H r,a, a (0,1]. (2.3) Moreover, let c i, f i,d j,g j, i = 1,2,...,k, j = 1,2,...,m, be positive fuzzy numbers such that for a (0,1], [ ci ]a = [ c i,l,a,c i,r,a ], [ fi ] a = [ f i,l,a, f i,r,a ], [ dj ]a = [ d j,l,a,d j,r,a ], [ gj ]a = [ g j,l,a,g j,r,a ], (2.4) E = ki=1 c ki=1 i f i mj=1, H = d mj=1. (2.5) j g j We will say that E is less than H and we will write if E H (2.6) ki=1 sup a (0,1] c ki=1 i,r,a inf a (0,1] f i,l,a mj=1 < inf a (0,1] d mj=1. (2.7) j,l,a sup a (0,1] g j,r,a

3 In addition, we will say that E is equal to H and we will write G. Stefanidou and G. Papaschinopoulos 339 E. = H if E H, H E, (2.8) which means that for i = 1,2,...,k, j = 1,2,...,m,anda (0,1], and so c i,l,a = c i,r,a, f i,l,a = f i,r,a, d j,l,a = d j,r,a, g j,l,a = g j,r,a, (2.9) E l,a = E r,a = H l,a = H r,a, a (0,1], (2.10) which imples that E, H are equal real numbers. For the fuzzy numbers E, H, we give the metric (see [9, 17, 18]) D(E,H) = supmax { E l,a H l,a, E r,a H r,a }, (2.11) where sup is taken for all a (0,1]. The fuzzy analog of boundedness and persistence (see [5, 6]) is given as follows: we say that a sequence of positive fuzzy numbers x n persists (resp., is bounded) if there exists a positive number M (resp., N)suchthat suppx n [M, ) ( resp., suppxn (0,N] ), n = 1,2,... (2.12) In addition, we say that x n is bounded and persists if there exist numbers M,N (0, ) such that suppx n [M,N], n = 1,2,... (2.13) Let x n be a sequence of positive fuzzy numbers such that [ xn ] a = [ L n,a,r n,a ], a (0,1], n = 0,1,..., (2.14) and let x be a positive fuzzy number such that [x] a = [L a,r a ], a (0,1]. (2.15) We say that x n nearly converges to x with respect to D as n if for every δ>0, there exists a measurable set T, T (0,1], of measure less than δ such that where D T ( xn,x ) = limd T ( xn,x ) = 0, as n, (2.16) sup a (0,1] T { max { L n,a L a, R n,a R a }}. (2.17) If T =,wesaythatx n converges to x with respect to D as n.

4 340 Fuzzy difference equations Let X be the set of positive fuzzy numbers. Let E,H X.From[18, Theorem 2.1], we have that E l,a, H l,a (resp., E r,a, H r,a ) are increasing (resp., decreasing) functions on (0,1]. Therefore, using the definition of the fuzzy numbers, there exist the Lebesque integrals E l,a H l,a da, E r,a H r,a da, (2.18) J where J = (0,1]. We define the function D 1 : X X R + such that { } D 1 (E,H) = max E l,a H l,a da, E r,a H r,a da. (2.19) J If D 1 (E,H) = 0, we have that there exists a measurable set T of measure zero such that J J E l,a = H l,a, E r,a = H r,a a (0,1] T. (2.20) We consider, however, two fuzzy numbers E, H to be equivalent if there exists a measurable set T of measure zero such that (2.20) hold and if we do not distinguish between equivalence of fuzzy numbers, then X becomes a metric space with metric D 1. We say that a sequence of positive fuzzy numbers x n converges to a positive fuzzy number x with respect to D 1 as n if limd 1 ( xn,x ) = 0, as n. (2.21) We define the fuzzy analog for periodicity (see [11]) as follows. Asequence{x n } of positive fuzzy numbers x n is said to be periodic of period p if D ( x n+p,x n ) = 0, n = 0,1,... (2.22) Suppose that (1.1) has a unique positive equilibrium x. We say that the positive equilibrium x of (1.1)isstableifforeveryɛ > 0, there exists a δ = δ(ɛ)suchthatforeverypositive solution x n of (1.1) which satisfies D ( x i,x ) δ, i = 0,1,...,π,wehaveD ( x n,x ) ɛ for all n 0. Moreover, we say that the positive equilibrium x of (1.1) is nearlyasymptoticallystable if it is stable and every positive solution of (1.1) nearly tends to the positive equilibrium of (1.1) with respect to D as n. Finally, we give the fuzzy analog of the concept of oscillation (see [11]). Let x n be a sequence of positive fuzzy numbers and let x be a positive fuzzy number. We say that x n oscillates about x if for every n 0 N, there exist s,m N, s,m n 0,suchthat or MIN { x m,x } = x m, MIN { x s,x } = x (2.23) MIN { x m,x } = x, MIN { x s,x } = x s. (2.24)

5 3. Main results G. Stefanidou and G. Papaschinopoulos 341 Arguing as in [13, 14, 15], we can easily prove the following proposition which concerns the existence and the uniqueness of the positive solutions of (1.1). Proposition 3.1. Consider (1.1), where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1, 2,...,m}, are positive fuzzy numbers, and p i, q j, i {1,2,...,k}, j {1,2,...,m}, arepositive integers. Then for any positive fuzzy numbers x π,x π+1,...,x 0, there exists a unique positive solution x n of (1.1) with initial values x π,x π+1,...,x 0. Now, we present conditions so that (1.1)has unboundedsolutions. Proposition 3.2. Consider (1.1), where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1, 2,...,m}, are positive fuzzy numbers, and p i, i {1,2,...,k}, q j, j {1,2,...,m}, are positive integers. If A G, G = ki=1 c i mj=1 d j, (3.1) then (1.1) has unbounded solutions. Proof. Let [A] a = [ A l,a,a r,a ], a (0,1]. (3.2) From (2.4) and(3.2) and since A, c i, d j, i = 1,2,...,k, j = 1,2,...,m, are positive fuzzy numbers, there exist positive real numbers B, C, a i, e i, h j, b j, i = 1,2,...,k, j = 1,2,...,m, such that B = inf a (0,1] A l,a, e i = sup c i,r,a, h j = inf d j,l,a, a (0,1] a (0,1] C = sup A r,a, a i = inf c i,l,a, a (0,1] a (0,1] b j = sup d j,r,a. a (0,1] (3.3) Let x n be a positive solution of (1.1) suchthat(2.14) hold and the initial values x i, i = π, π +1,...,0, are positive fuzzy numbers which satisfy [ xi ] a = [ L i,a,r i,a ], i = π, π +1,...,0, a (0,1] (3.4) and for a fixed ā (0,1], the relations are satisfied, where R i,ā > Z2 W C, L i,ā <W, i = π, π +1,...,0, (3.5) Z = ki=1 e ki=1 i a mj=1, W = i h mj=1. (3.6) j b j

6 342 Fuzzy difference equations Using [15, Lemma 1], we can easily prove that L n,a, R n,a satisfy the family of systems of parametric ordinary difference equations ki=1 c i,l,a L n pi,a L n+1,a = A l,a + mj=1, d j,r,a R n q j,a ki=1 c i,r,a R n pi,a R n+1,a = A r,a + mj=1, d j,l,a L n q j,a n = 0,1,... (3.7) Since (3.1)holds,it is obviousthat A l,ā < ki=1 c i,r,ā mj=1 d j,l,ā. (3.8) Using (3.8) and applying [16, Proposition 1] to the system (3.7)fora = ā,wehavethat lim n L n,ā=a l,ā, limr n,ā =. (3.9) n Therefore, from (3.9), the solution x n of (1.1) which satisfies (3.4) and(3.5) isunbounded. Remark 3.3. From the proof of Proposition 3.2, it is obvious that (1.1) has unbounded solutions if there exists at least one a (0,1] such that (3.8)holds. In the following proposition, we study the boundedness and persistence of the positive solutions of (1.1). Proposition 3.4. Consider (1.1), where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1,2,...,m}, are positive fuzzy numbers, and p i, i {1,2,...,k}, q j, j {1,2,...,m}, are positive integers. If either or holds, then every positive solution of (1.1) is bounded and persists. A. = G (3.10) G A (3.11) Proof. Firstly, suppose that (3.10) is satisfied; then A, c i, d j, i = 1,2,...,k, j = 1,2,...,m, are positive real numbers. Hence, for i = 1,2,...,k, j = 1,2,...,m,weget A = A l,a = A r,a, c i = c i,l,a = c i,r,a, d j = d j,l,a = d j,r,a, a (0,1], (3.12) ki=1 c i A = mj=1. (3.13) d j

7 G. Stefanidou and G. Papaschinopoulos 343 Let x n be a positive solution of (1.1) suchthat(2.14) hold and let x i, i = π, π + 1,...,0, be the positive initial values of x n such that (3.4) hold. Then there exist positive numbers T i, S i, i = π, π +1,...,0, such that T i L i,a,r i,a S i, i = π, π +1,...,0. (3.14) Let (y n,z n ) be the positive solution of the system of ordinary difference equations y n+1 = A + ki=1 c i y ki=1 n pi c i z n pi mj=1, z n+1 = A + mj=1, (3.15) d j z n q j d j y n q j with initial values (y i,z i ), i = π, π +1,...,0, such that y i = T i, z i = S i, i = π, π + 1,...,0. Then from (3.14)and(3.15), we can easily prove that and working inductively, we take y 1 L 1,a, R 1,a z 1, a (0,1], (3.16) y n L n,a, R n,a z n, n = 1,2,..., a (0,1]. (3.17) Since from (3.13) and[16, Proposition 3], (y n, z n ) is bounded and persists, from (3.17), it is obvious that x n is also bounded and persists. Now, suppose that (3.11)holds;then We concider the system of ordinary difference equations B>Z, C>W. (3.18) y n+1 = B + ki=1 a i y ki=1 n pi e i z n pi mj=1, z n+1 = C + mj=1, (3.19) b j z n q j h j y n q j where B,C,a i,e i,b j,h j, i = 1,2,...,k, j = 1,2,...,m,aredefinedin(3.3). Let (y n,z n ) be a solution of (3.19) with initial values (y i,z i ), i = π, π +1,...,0, such that y i = T i, z i = S i, i = π, π +1,...,0, where T i,s i, i = π, π +1,...,0, are defined in (3.14). Arguing as above, we can prove that (3.17) holds.sincefrom(3.18) and[16, Proposition 3], (y n, z n ) is bounded and persists, then from (3.17), it is obvious that, x n is also bounded and persists. This completes the proof of the proposition. In what follows, we need the following lemmas. Lemma 3.5. Let r i,s j, i = 1,2,...,k, j = 1,2,...,m, be positive integers such that ( r1,r 2,...,r k,s 1,s 2,...,s m ) = 1, (3.20) where (r 1,r 2,...,r k,s 1,s 2,...,s m ) is the greatest common divisor of the integers r i,s j, i = 1,2,...,k, j = 1,2,...,m. Then the following statements are true.

8 344 Fuzzy difference equations (I) Thereexistsanevenpositiveintegerw 1 such that for any nonnegative integer p,there exist nonnegative integers α ip, β jp, i = 1,2,...,k, j = 1,2,...,m, such that k m α ip r i + β jp s j = w 1 +2p, p = 0,1,..., (3.21) i=1 j=1 where m j=1 β jp is an even integer. (II) Suppose that all r i, i = 1,2,...,k, are not even and all s j, j = 1,2,...,m, arenotodd integers. Then there exists an odd positive integer w 2 such that for any nonnegative integer p, there exist nonnegative integers γ ip, δ jp, i = 1,2,...,k, j = 1,2,...,m, such that k m γ ip r i + δ jp s j = w 2 +2p, p = 0,1,..., (3.22) i=1 j=1 where m j=1 δ jp is an even integer. (III) Suppose that all r i, i = 1,2,...,k, are not even and all s j, j = 1,2,...,m,arenotodd integers. Then there exists an even positive integer w 3 such that for any nonnegative integer p, there exist nonnegative integers ɛ ip, ξ jp, i = 1,2,...,k, j = 1,2,...,m, such that k m ɛ ip r i + ξ jp s j = w 3 +2p, p = 0,1,..., (3.23) i=1 j=1 where m j=1 ξ jp is an odd integer. (IV) There exists an odd positive integer w 4 such that for any nonnegative integer p,there exist nonnegative integers λ ip, µ jp, i = 1,2,...,k, j = 1,2,...,m, such that k m λ ip r i + µ jp s j = w 4 +2p, p = 0,1,..., (3.24) i=1 where m j=1 µ jp is an odd integer. j=1 Proof. (I) Since(3.20) holds, there exist integers η i, ι j, i = 1,2,...,k, j = 1,2,...,m, such that k m η i r i + ι j s j = 1. (3.25) i=1 j=1 If for any real number a, we denote by [a] the integral part of a, wesetfori = 2,3,...,k, j = 1,2,...,m, k m k m α 1p = 2pη 1 +2 r i +2 s j 2 g ip r i 2 h jp s j, i=2 j=1 i=2 j=1 α ip = 2pη i +2g ip r 1, β jp = 2pι j +2h jp r 1, (3.26)

9 where [ ] pηi g ip = +1, h jp = r 1 [ pιj r 1 G. Stefanidou and G. Papaschinopoulos 345 ] +1, i = 2,3,...,k, j = 1,2,...,m. (3.27) Therefore, from (3.25) and(3.26), we can easily prove that α ip,β jp, i = 1,2,...,k, j = 1,2,...,m, which are defined in (3.26), are positive integers satisfying (3.21)for w 1 = 2r 1 ( k i=2r i + ) m s j j=1 (3.28) and m j=1 β jp is an even number. (II) Firstly, suppose that one of r i, i = 1,2,...,k, is an odd positive integer and without loss of generality, let r 1 be an odd positive integer. Relation (3.22) follows immediately if we set for i = 2,...,k and for j = 1,2,...,m, γ 1p = α 1p +1, γ ip = α ip, δ jp = β jp, w 2 = w 1 + r 1. (3.29) Now, suppose that r i, i = 1,2,...,k, are even positive integers; then from (3.20), one of s j, j = 1,2,...,m, is an odd positive integer and from the hypothesis, one of s j, j = 1,2,...,m, is an even positive integer. Without loss of generality, let s 1 be an odd positive integer and s 2 be an even positive integer. Relation (3.22) follows immediately if we set for i = 1,2,...,k and for j = 3,...,m, γ ip = α ip, δ 1p = β 1p +1, δ 2p = β 2p +1, δ jp = β jp, w 2 = w 1 + s 1 + s 2. (3.30) (III) Firstly, suppose that one of s j, j = 1,2,...,m, is an even positive integer and without loss of generality, let s 1 be an even positive integer. Relation (3.23) follows immediately if we set for i = 1,2,...,k and j = 2,...,m, ɛ ip = α ip, ξ 1p = β 1p +1, ξ jp = β jp, w 3 = w 1 + s 1. (3.31) Now, suppose that s j, j = 1,2,...,m, are odd positive integers; then from the hypothesis, at least one of r i, i = 1,2,...,k, is an odd positive integer, and without loss of generality, let r 1 be an odd integer. Relation (3.23) follows immediately if we set for i = 2,...,k, j = 2,3,...,m, ɛ 1p = α 1p +1, ɛ ip = α ip, δ 1p = β 1p +1, δ jp = β jp, w 3 = w 1 + s 1 + r 1. (3.32) (IV) Firstly, suppose that at least one of s j, j = 1,2,...,m, is an odd positive integer and without loss of generality, let s 1 be an odd positive integer. Relation (3.24) follows immediately if we set for i = 1,2,...,k, j = 2,3,...,m, λ ip = α ip, µ 1p = β 1p +1, µ jp = β jp, w 4 = w 1 + s 1. (3.33)

10 346 Fuzzy difference equations Now, suppose that s j, j = 1,2,...,m, are even positive integers; then from (3.20), at least one of r i, i = 1,2,...,k, is an odd positive integer, and without loss of generality, let r 1 be an odd positive integer. Relation (3.24) follows immediately if we set for i = 2,3,...,k, j = 2,3,...,m, λ 1p = α 1p +1, λ ip = α ip, µ 1p = β 1p + r 1, µ jp = β jp, ( w 4 = w 1 + r 1 s1 +1 ). (3.34) This completes the proof of the lemma. Lemma 3.6. Consider system (3.19), whereb,c arepositive constants such that B = ki=1 e ki=1 i a mj=1, C = i h mj=1. (3.35) j b j Then the following statements are true. (I) Let r be a common divisor of the integers p i +1, q j +1, i = 1,2,...,k, j = 1,2,...,m, such that p i +1= rr i, i = 1,2,...,k, q j +1= rs j, j = 1,2,...,m; (3.36) then system (3.19) has periodic solutions of prime period r. Moreover, if all r i, i = 1,2,...,k, (resp., s j, j = 1,2,...,m) are even (resp., odd) positive integers, then system (3.19)hasperiodic solutions of prime period 2r. (II) Let r be the greatest common divisor of the integers p i +1, q j +1, i = 1,2,...,k, j = 1,2,...,m,suchthat(3.36)hold;thenifallr i, i = 1,2,...,k,(resp.,s j, j = 1,2,...,m)areeven (resp., odd) positive integers, every positive solution of (3.19) tends to a periodic solution of period 2r; otherwise, every positive solution of (3.19) tends to a periodic solution of period r. Proof. (I) From relations (3.35), (3.36), and [16, Proposition 2], system (3.19) hasperiodic solutions of prime period r. Now, we prove that system (3.19) has periodic solutions of prime period 2r, ifallr i, i = 1,2,...,k,(resp.,s j, j = 1,2,...,m) are even (resp., odd) positive integers. Suppose first that p k <q m.let(y n,z n ) be a positive solution of (3.19) with initial values satisfying y rsm+2rλ+ζ = y r+ζ, z rsm+2rλ+ζ = z r+ζ, y rsm+2rν+r+ζ = y 2r+ζ, z rsm+2rν+r+ζ = z 2r+ζ, λ = 0,1,..., s m 1 2, ν = 0,1,..., s m 3, ζ = 1,2,...,r, 2 (3.37) and, in addition, for ζ = 1,2,...,r, y 2r+ζ >B, y r+ζ >B, z r+ζ = Cy 2r+ζ y 2r+ζ B, z 2r+ζ = Cy r+ζ y r+ζ B. (3.38)

11 G. Stefanidou and G. Papaschinopoulos 347 From (3.19), (3.35), (3.36), (3.37), and (3.38), we get for ζ = 1,2,...,r, y ζ = B + C y 2r+ζ z r+ζ = y 2r+ζ, z ζ = C + B z 2r+ζ y r+ζ = z 2r+ζ, y r+ζ = B + C y r+ζ z 2r+ζ = y r+ζ, z r+ζ = C + B z r+ζ y 2r+ζ = z r+ζ. (3.39) Let a v { 2,3,... }. Suppose that for all u = 1,2,...,v 1andζ = 1,2,...,r,wehave y 2ur+ζ = y 2r+ζ, z 2ur+ζ = z 2r+ζ, y 2ur+r+ζ = y r+ζ, z 2ur+r+ζ = z r+ζ. (3.40) Then from (3.19), (3.35) (3.40), we get for ζ = 1,2,...,r, Similarly, we can prove that for ζ = 1,2,...,r, y 2vr+ζ = B + C y 2r+ζ z r+ζ = y 2r+ζ. (3.41) z 2vr+ζ = z 2r+ζ, y 2vr+r+ζ = y r+ζ, z 2vr+r+ζ = z r+ζ. (3.42) Therefore, from (3.39) (3.42), we have that system (3.19) has periodic solutions of period 2r. Now, suppose that q m <p k.let(y n,z n ) be a positive solution of (3.19) suchthatthe initial values satisfy relations (3.38)andforω = 0,1,...,r k /2 1, θ = 1,2,...,2r, y rrk+2rω+θ = y 2r+θ, z rrk+2rω+θ = z 2r+θ. (3.43) Then arguing as above, we can easily prove that (y n,z n ) is a periodic solution of period 2r. This completes the proof of statement (I). (II) Now, we prove that every positive solution of system (3.19) tends to a periodic solution of period κr,where 2 ifr i, i = 1,2,...,k, areeven,s j, j = 1,2,...,m, are odd, κ = (3.44) 1 otherwise. Let (y n,z n ) be an arbitrary positive solution of (3.19). We prove that there exist the lim y κnr+i = ɛ i, i = 0,1,...,κr 1. (3.45) n We fix a τ {0,1,...,κr 1}. Sincefrom[16, Proposition 3], the solution (y n,z n )is bounded and persists, we have liminf κnr+τ = l τ B, liminf κnr+τ = m τ C, n n limsup y κnr+τ = L τ <, limsupz κnr+τ = M τ <. n n Therefore, from relations (3.19), (3.35), and (3.46), we take (3.46) m τ = CL τ L τ B, l τ = BM τ M τ C. (3.47)

12 348 Fuzzy difference equations Weprovethat(3.45)istruefori = τ. Suppose on the contrary that l τ <L τ.thenfrom (3.46), there exists an ɛ > 0suchthat L τ >l τ + ɛ >B+ ɛ. (3.48) In view of (3.46), there exists a sequence n µ, µ = 1,2,...,suchthat lim y κrn µ µ+τ = L τ, lim y r(κnµ r µ i)+τ = T ri,τ L τ, lim µ z r(κn µ s j)+τ = S sj,τ m τ. (3.49) In view of (3.19), (3.35), (3.46), (3.47), and (3.49), we take ki=1 a i T ri,τ L τ = B + mj=1 b j S sj,τ B + CL τ m τ = L τ (3.50) and obviously, we have that T ri,τ = L τ, S sj,τ = m τ, i = 1,2,...,k, j = 1,2,...,m. (3.51) Inaddition,using(3.19), (3.35), (3.46), (3.47), and (3.51), for κ = 2, from statements (I) and (IV) of Lemma 3.5 and arguing as above, we take for γ = 0,1,..., lim y r(2n µ µ w 1 2γ)+τ = L τ, limz r(2nµ w µ 1 s 1 2γ)+τ = m τ, (3.52) and for κ = 1 and from all the statements of Lemma 3.5, lim y r(n µ µ w 1 2γ)+τ = L τ, lim z r(n µ µ w 3 2γ)+τ = m τ, lim y r(nµ w µ 2 2γ)+τ = L τ, limz r(nµ w µ 4 2γ)+τ = m τ, (3.53) w 1,w 2,w 3,w 4 are defined in Lemma 3.5. Let a σ κ {0,1,...,(3 κ)φ}, φ = max { r k,s m }. Suppose first that κ = 2. Then in view of (3.19), there exist positive integers p, q and a continuous function F σ2 : R R R R such that where for i = 0,1,..., p, j = 0,1,...,q, y r(2nµ+2σ 2)+τ = B + F σ2 ( ζnµ,0,...,ζ nµ,p,ξ nµ,0,...,ξ nµ,q), (3.54) ζ nµ,i = y r(2nµ w 1 2i)+τ, ξ nµ,j = z r(2nµ w 1 s 1 2j)+τ. (3.55) If κ = 1, there exist positive integers v 1, v 2, v 3, v 4 and a continuous function G σ1 : R R R R such that y r(nµ+σ 1)+τ = B + G σ1 ( ζnµ,0,...,ζ nµ,v 1, ζ nµ,0,..., ζ nµ,v 2,ξ nµ,0,...,ξ nµ,v 3, ξ nµ,0,..., ξ nµ,v 4 ), (3.56)

13 G. Stefanidou and G. Papaschinopoulos 349 where for i = 0,1,...,v 1, ī = 0,1,...,v 2, j = 0,1,...,v 3,and j = 0,1,...,v 4, ζ nµ,i = y r(nµ w1 2i)+τ, ξ nµ,j = z r(nµ w3 2j)+τ, ζnµ,ī = y r(nµ w 2 2ī)+τ, ξnµ, j = z r(nµ w 4 2 j)+τ. (3.57) Therefore, from (3.47), (3.52), (3.53), (3.54), and (3.56), it follows that lim y r(κn µ µ+κσ κ)+τ = B + CL τ = L τ. (3.58) m τ Using the same argument to prove (3.58) and using (3.19), we can easily prove that for i = 1,2,...,k, j = 1,2,...,m, lim µ y r(κn µ+κσ κ r i)+τ = L τ, limz r(κnµ+κσ µ κ s j)+τ = m τ. (3.59) Therefore, if δ = ɛ(m τ C)/(L τ ɛ B), then in view of (3.19), (3.47), (3.58), and (3.59), there exists a µ 0 {1,2,...} such that for j = 1,2,...,m, z r(κnµ0 +2φ+κ s j)+τ C + B( m τ + δ ) = m τ + δ (3.60) L τ ɛ and so from (3.19), (3.47), (3.48), (3.58), (3.59), and (3.60), we get y r(κnµ0 +2φ+κ)+τ B + C( L τ ɛ ) = L τ ɛ >l τ. (3.61) m τ + δ Using (3.19), (3.47), (3.48), (3.58), (3.59), and (3.61) and working inductively, we can easily prove that y r(κnµ0 +2φ+κω)+τ L τ ɛ >l τ, ω = 2,3,..., (3.62) which is a contradiction since liminf n y κrn+τ = l τ. Therefore, since τ is an arbitrary number such that τ {0,1,...,κr 1},relations(3.45) are satisfied. Moreover, from (3.19) and(3.47), we havethat lim n z κnr+i = ξ i, i = 0,1,...,κr 1. (3.63) This completes the proof of the lemma. In the next proposition, we study the periodicity of the positive solutions of (1.1). Proposition 3.7. Consider (1.1), where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1,2,...,m}, are positive fuzzy numbers, and p i, i {1,2,...,k}, q j, j {1,2,...,m}, are positive integers. If (3.10) holdsandr is a common divisor of the integers p i +1, q j +1, i = 1,2,...,k, j = 1,2,...,m,then(1.1) has periodic solutions of prime period r.moreover,if r i, i = 1,2,...,k,(resp.,s j, j = 1,2,...,m) r i, s j are defined in (3.36) are even (resp., odd) integers, then (1.1) hasperiodicsolutions ofprimeperiod 2r.

14 350 Fuzzy difference equations Proof. From (3.10), we have that A, c i, i = 1,2,...,k, d j, j = 1,2,...,m, are positive real numberssuchthat(3.12) and(3.13) hold. We consider functions L i,a,r i,a, i = π, π + 1,...,0, such that for λ = 0,1,...,φ 1, θ = 1,2,...,r,anda (0,1], L rφ+rλ+θ,a = L r+θ,a, R rφ+rλ+θ,a = R r+θ,a, (3.64) the functions L w,a, w = r +1, r +2,...,0, are increasing, left continuous, and for all w = r +1, r +2,...,0, we have A + ɛ <L w,a < 2A, R w,a = AL w,a L w,a A, (3.65) where ɛ is a positive number such that ɛ <A. Using (3.65) and since the functions L w,a, w = r +1, r +2,...,0, are increasing, if a 1,a 2 (0,1], a 1 a 2,weget AL w,a1 L w,a2 A 2 L w,a1 AL w,a1 L w,a2 A 2 L w,a2 (3.66) which implies that R w,a, w = r +1, r +2,...,0, are decreasing functions. Moreover, from (3.65), we get L w,a R w,a, A + ɛ L w,a,r w,a 2A2 ɛ, (3.67) and so from [18, Theorem 2.1], ( L w,a,r w,a ), w = r +1, r +2,...,0, determine the fuzzy numbers x w, w = r +1, r +2,...,0, such that [x w ] a = [L w,a,r w,a ], w = r +1, r + 2,...,0.Letx n be a positive solution of (1.1) which satisfies (2.14) and let the initial values be positive fuzzy numbers such that (3.4) hold and the functions L i,a,r i,a, i = π, π + 1,...,0, a (0,1], are defined in (3.64), (3.65); L i,a, i = π, π +1,...,0, a (0,1], are increasing and left continuous. Then from [16, Proposition 2], we have that for any a (0,1], the system given by (3.7), (3.12), and (3.13) has periodic solutionsof prime period r, which means that there exists solution ( L n,a,r n,a ), a (0,1], of the system such that L n+r,a = L n,a, R n+r,a = R n,a, a (0,1]. (3.68) Therefore, from (2.22) and(3.68), we have that (1.1) has periodic solutions of prime period r. Now, suppose that r i, i = 1,2,...,k, (resp.,s i, j = 1,2,...,m) are even (resp., odd) integers. We consider the functions L i,a,r i,a, i = π, π +1,...,0, such that analogous relations (3.37), (3.38), and (3.43) hold,l w,a, w = r +1,...,0, are increasing, left continuous functions, and the first relation of (3.65) holds. Arguing as above, the solution x n of (1.1) with initial values x i, i = π, π +1,...,0, satisfying (3.4), where L i,a,r i,a, i = π, π +1,...,0, are defined above, is a periodic solution of prime period 2r. In the following proposition, we study the convergence of the positive solutions of (1.1).

15 G. Stefanidou and G. Papaschinopoulos 351 Proposition 3.8. Consider (1.1), where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1,2,...,m}, are positive fuzzy numbers, and p i, i {1,2,...,k}, q j, j {1,2,...,m}, are positive integers. Then the following statements are true. (i) If (3.11), holds, then (1.1) has a unique positive equilibrium x and every positive solution of (1.1) nearly converges to the unique positive equilibrium x with respect to D as n and converges to x with respect to D 1 as n. (ii) If (3.10)issatisfiedandr is the greatest common divisor of the integers p i +1, q j +1, i = 1,2,...,k, j = 1,2,...,m, suchthat(3.36) holds, then every positive solution of (1.1) nearly converges to a period κr solution of (1.1)with respect to D as n and converges to aperiodκr solution of (1.1)withrespecttoD 1 as n ; κ is defined in (3.44). Proof. (i) Let x n be a positive solution of (1.1) which satisfies (2.14). Since (3.7)and(3.11) hold, we can apply [16, Proposition 4] and we have that for any a (0,1], there exist the lim n L n,a,lim n R n,a,and where lim n L n,a = L a, limr n,a = R a, a (0,1], (3.69) n L a = A l,aa r,a C a D a, R a = A l,aa r,a C a D a, A r,a C a A l,a D a ki=1 c ki=1 i,l,a c C a = mj=1, D a = i,r,a d mj=1. j,r,a d j,l,a (3.70) In addition, from (3.3)and(3.70), we get L a B2 Z 2 C W = λ, R a C2 W 2 = µ, (3.71) B Z where B,C (resp., Z,W)aredefinedin(3.3)(resp.,(3.5)). Then from (3.69), (3.71), and arguing as in [13, 14, 15], we can easily prove that L a,r a determine a fuzzy number x such that [x] a = [L a,r a ]. Finally, using (3.70), we take that x is the unique positive equilibrium of (1.1). Using relations (3.11), (3.69), andarguing as in [15, Proposition2], we can prove that every positive solution of (1.1) nearly converges to the unique positive equilibrium x with respect to D as n and converges to x with respect to D 1 as n. (ii) Suppose that (3.10) holds.letx n be a positive solution of (1.1) suchthat(2.14) holds. Since (L n,a,r n,a ) is a positive solution of the system which is defined by (3.7), (3.12), and (3.13), from Lemma 3.6,wehavethat lim n L κnr+l,a = ɛ l,a, limr κnr+l,a = ξ l,a, a (0,1], l = 0,1,...,κr 1, (3.72) n where κ is defined in (3.44). Using (3.72) andarguingasin[15, Proposition 2], we can prove that everypositive solution of (1.1) nearly converges to a period κr solution of (1.1) with respect to D as n and converges to a period κr solution of (1.1) with respect to D 1 as n. Thus, the proof of the proposition is completed. From Propositions , it is obvious that (1.1) exhibits the trichotomy character described concentratively by the following proposition.

16 352 Fuzzy difference equations Proposition 3.9. Consider the fuzzy difference equation (1.1), wherek,m {1,2,...},and A,c i,d j, i {1,2,...,k}, j {1,2,...,m}, are positive fuzzy numbers. Then (1.1) possesses the following trichotomy. (i) If relation (3.1) is satisfied, then(1.1) has unbounded solutions. (ii) If (3.10) holdsandr is the greatest common divisor of the integers p i +1, q j +1, i = 1,2,...,k, j = 1,2,...,m, suchthat(3.36) holds, then every positive solution of (1.1) nearly converges to a period κr solution of (1.1)with respect to D as n and converges to aperiodκr solution of (1.1)withrespecttoD 1 as n. (iii) If (3.11) holds, then every positive solution of (1.1) nearly converges to the unique positive equilibrium x with respect to D as n and converges to x with respect to D 1 as n. In the next proposition, we study the asymptotic stability of the unique positive equilibrium of (1.1). Proposition Consider the fuzzy difference equation (1.1), where k,m {1,2,...}, A,c i,d j, i {1,2,...,k}, j {1,2,...,m}, are positive fuzzy numbers, and p i, i {1,2,...,k}, q j, j {1,2,...,m}, are positive integers such that (3.11) holds. Suppose that there exists a positive number θ such that θ<b, Z< 2B + C θ (C θ) 2 +4BC, (3.73) 2 where B,C are defined in (3.3) andz is defined in (3.5). Then the unique positive equilibrium x of (1.1) is nearly asymptotically stable. Proof. Since (3.11) holds, from Proposition 3.8, equation (1.1) has a unique positive equilibrium x which satisfies (2.15). Let ɛ be a positive real number. Since (3.18) holds, we can define the positive real number δ as follows: Let x n be a positive solution of (1.1)suchthat From (3.75), we have δ<min{ɛ,λ,θ,b Z}. (3.74) D(x i,x) δ ɛ, i = 0,1,...,π. (3.75) L i,a L a δ, R i,a R a δ, i = 0,1,...,π, a (0,1]. (3.76) In addition, from (3.3), (3.7), (3.74), and (3.76) and since (L a,r a ) satisfies (3.7), we get L 1,a L a = A l,a + ki=1 c i,l,a L ki=1 pi,a c i,l,a (L a + δ) mj=1 L a A l,a + d j,r,a R mj=1 q j,a d j,r,a (R a δ) L a = δ C a A l,a + L a R a δ δ R a (B Z). R a δ (3.77)

17 G. Stefanidou and G. Papaschinopoulos 353 From (3.74)and(3.77), it is obvious that L 1,a L a <δ<ɛ. (3.78) Moreover, arguing as above, we can easily prove that We claim that We fix an a (0,1] and we concider the function R 1,a R a δ D a A r,a + R a. (3.79) L a δ θ<l a R a + A r,a D a, a (0,1]. (3.80) g(h) = A l,aa r,a D a h A r,a h A l,aa r,a D a h A l,a D a + A r,a D a, (3.81) where h is a nonnegative real variable. Moreover, we consider the function f (x, y,z) = x2 (2x + y)z + z 2 x z θ, (3.82) where B x y C and W z Z, B,C (resp., W,Z)aredefinedin(3.3)(resp.,(3.5)). Using (3.82), we can easily prove that the function f is increasing (resp., decreasing) (resp., decreasing) with respect to x (resp., y)(resp.,z)forally,z (resp., x,z)(resp.,x, y) and so from (3.73), f (x, y,z) >f(b,c,z) = B2 (2B + C)Z + Z 2 B Z Therefore, from (3.3), (3.81), (3.82), and (3.83), we have θ>0. (3.83) g(0) = f ( A l,a,a r,a,d a ) + θ>θ. (3.84) In addition, from (3.81), we can prove that g is an increasing function with respect to h andsowehaveg(0) <g(c a ), a (0,1]. Therefore, from (3.70), (3.81), and (3.84), relation (3.80) is true. Hence, from (3.74), (3.79), and (3.80), we get R 1,a R a <δ<ɛ. (3.85) From (3.7), (3.76), (3.78), and (3.85) andworkinginductively, wecaneasilyprovethat L n,a L a ɛ, R n,a R a ɛ, a (0,1], n = 0,1,..., (3.86)

18 354 Fuzzy difference equations and so D ( x n,x ) ɛ, n 0. (3.87) Therefore, the positive equilibrium x is stable. Moreover, from Proposition 3.8, wehave that every positive solution of (1.1) nearly tends to x with respect to D as n.so,x is nearly asymptotically stable. So, the proof of the proposition is completed. Finally, we study the oscillatory behavior of the positive solutions of the fuzzy difference equation x n+1 = A + ks=0 c 2s+1 x n 2s 1 ks=0 d 2s+2 x n 2s, (3.88) where k is a positive integer, and A, c 2s+1, d 2s+2, s {0,1,...,k}, are positive fuzzy numbers. Obviously, (3.88) isaspecialcaseof(1.1). In what follows, we need to study the oscillatory behavior of the positive solutions of the system of ordinary difference equations y n+1 = B + z n+1 = C + ks=0 a 2s+1 y n 2s 1 ks=0 b 2s+2 z n 2s, ks=0 e 2s+1 z n 2s 1 ks=0 h 2s+2 y n 2s, n = 0,1,..., (3.89) where k is a positive integer, B,C,a 2s+1,b 2s+2,e 2s+1,h 2s+2, s {0,1,...,k}, are positive real constants, and the initial values y j, z j, j = 2k 1, 2k,...,0, are positive real numbers. Let (y n, z n ) be a positive solution of (3.89). We say that the solution (y n, z n ) oscillates about (y, z), y,z R +,ifforeveryn 0 N, there exist s, m N, s, m n 0,suchthat ( ys y )( y m y ) 0, ( ys y )( z s z ) 0, ( zs z )( z m z ) 0, ( ym y )( z m z ) 0. (3.90) Lemma Consider system (3.89), where k is a positive integer, B, C, a 2s+1, b 2s+2, e 2s+1, h 2s+2, s {0,1,...,k}, are positive real constants, and the initial values y j, z j, j = 2k 1, 2k,...,0, are positive real numbers. A positive solution ( y n,z n ) of system (3.89)oscillates about the unique positive equilibrium ( x, ȳ ) of system (3.89) if either the relations or the relations Λ max { Λ 1,s,Λ 2,s }, max { 1,s, 2,s }, s = 0,1,...,k, (3.91) Λ min{λ 1,s,Λ 2,s }, min{ 1,s, 2,s }, s = 0,1,...,k, (3.92)

19 z 2 C + λ z ȳ = z. (3.96) G. Stefanidou and G. Papaschinopoulos 355 hold, where for s = 0,1,...,k, Λ = 1,s = 1 [ µ ȳ ( s b 2j+2 z+ a 2s+1 z j=0 2,s = 1 [ ( ȳ s 1 e 2j+1 z + h 2s+2 λ z j=0 Λ 1,s = 1 [ λ z ( s h 2j+2 ȳ+ e 2s+1 ȳ j=0 Λ 2,s = 1 [ ( s 1 z a 2j+1 ȳ+ b 2s+2 µȳ j=0 ks=0 e 2s+1 z ks=0 2s 1 a 2s+1 y 2s 1 ks=0, = h 2s+2 y ks=0, 2s b 2s+2 z 2s k b 2j+2 z 2j+2+2s ) j=s+1 ) k e 2j+1 z 2j+2s j=s k j=s+1 ( s 1 ( s 1 h 2j+2 y 2j+2+2s ) k a 2j+1 y 2j+2s ) j=s a 2j+1 ȳ+ j=0 h 2j+2 ȳ + j=0 ( s 1 ( s 1 e 2j+1 z+ j=0 b 2j+2 z + j=0 k j=s+1 a 2j+1 y 2j+1+2s )] B, )] k h 2j+2 y 2j+1+2s B, j=s+1 k j=s+1 k j=s+1 e 2j+1 z 2j+1+2s )] C, b 2j+2 z 2j+1+2s )] C, λ = ks=0 e ks=0 2s+1 a 2s+1 ks=0, µ = h ks=0. 2s+2 b 2s+2 Proof. Suppose that (3.91)hold.Weprovethatforρ = 0,1,...,k, (3.93) y 2ρ+1 ȳ, z 2ρ+1 z, y 2ρ+2 ȳ, z 2ρ+2 z. (3.94) From (3.89)and(3.91), we have y 1 = B + ks=0 a 2s+1 y 2s 1 ks=0 b 2s+2 z 2s = B + B + 1,k = ȳ, z 1 = C + Λ C + Λ 1,k = z. (3.95) Since from (3.91), Λ Λ 2,0 and 2,0,thenfrom(3.89), we have ks=0 a 2s+1 y 2s y 2 = B + b 2 z 1 + k B + (C + Λ)b 2 + ks=1 b 2s+2 z 1 2s s=1 b 2s+2 z 1 2s b 2 z 1 + µȳ k = B + = ȳ, s=1 b 2s+2 z 1 2s z µȳ z Using (3.89), (3.91), (3.95), and (3.96), relations 1,ρ 1, Λ Λ 1,ρ 1 (resp., 2,ρ, Λ Λ 2,ρ ), ρ = 1,2,...,k, and working inductively, we can easily prove (3.94)forρ = 1,2,...,k: y 2ρ+1 ȳ, z 2ρ+1 z ( resp., y2ρ+2 ȳ, z 2ρ+2 z ). (3.97)

20 356 Fuzzy difference equations Therefore, (3.94) holdforρ = 0,1,...,k. Then since (3.94) holdforρ = 0,1,...,k, using (3.89) and working inductively, we can easily prove that(3.94)holdforanyρ = k +1,k + 2,..., and so if (3.91) hold, the proof of the lemma is completed. Similarly, if (3.92) are satisfied, then we can easily prove that y 2ρ+1 ȳ, z 2ρ+1 z, y 2ρ+2 ȳ, z 2ρ+2 z, ρ = 0,1,... (3.98) This completes the proof of the lemma. Using Lemma 3.11 and arguing as in [13, Proposition 2.4], we can easily prove the following proposition which concerns the oscillatory behavior of the positive solutions of the fuzzy difference equation (3.88). Proposition Consider (3.88), where k is a positive integer, and A, c 2s+1, d 2s+2, s {0, 1,...,k}, are positive fuzzy numbers. Then a positive solution x n of (3.88) satisfying (2.14) oscillates about the positive equilibrium x, which satisfies (2.15)if,foranys = 0,1,...,k and a (0,1], either the relations Λ a max { } Λ 1,s,a, Λ 2,s,a, a max { } 1,s,a, 2,s,a (3.99) or the relations Λ a min { Λ 1,s,a, Λ 2,s,a }, a min { 1,s,a, 2,s,a } (3.100) hold, where Λ a, a, Λ 1,s,a, Λ 2,s,a, 1,s,a, 2,s,a are defined for the analogous system (3.7)inthe same way as Λ,, Λ 1,s,Λ 2,s, 1,s, 2,s were defined in Lemma 3.11 for system (3.89). Using Proposition 3.12, we take the following corollary. Corollary Consider (3.88), where k is a positive integer, and A, c 2s+1, d 2s+2, s {0,1,...,k}, are positive fuzzy numbers. Then a positive solution x n of (3.88) satisfying (2.14) oscillates about the positive equilibrium x, which satisfies (2.15)if,for any p = 0,1,...,k and a (0,1], either the relations or the relations hold. Acknowledgment L 2k 1+2p,a L a, R 2k 1+2p,a R a, L 2k+2p,a L a, R 2k+2p,a R a (3.101) L 2k 1+2p,a L a, R 2k 1+2p,a R a, L 2k+2p,a L a, R 2k+2p,a R a (3.102) This work is a part of the first author Doctoral thesis.

21 G. Stefanidou and G. Papaschinopoulos 357 References [1] R. P. Agarwal, Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics, vol. 155, Marcel Dekker, New York, [2] D. Benest and C. Froeschle, Analysis and Modelling of Discrete Dynamical Systems, Advances in Discrete Mathematics and Applications, vol. 1, Gordon and Breach Science Publishers, Amsterdam, [3] E. Y. Deeba and A. De Korvin, Analysis by fuzzy difference equations of a model of CO 2 level in the blood,appl.math.lett.12 (1999), no. 3, [4] E.Y.Deeba,A.DeKorvin,andE.L.Koh,A fuzzy difference equation with an application, J. Differ. Equations Appl. 2 (1996), no. 4, [5] R. DeVault, G. Ladas, and S. W. Schultz, On the recursive sequence x n+1 = A/xn p + B/xn 1, q Advances in Difference Equations (Veszprém, 1995), Gordon and Breach, Amsterdam, 1997, pp [6], On the recursive sequence x n+1 = A/x n +1/x n 2,Proc.Amer.Math.Soc.126 (1998), no. 11, [7] L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, New York, [8] S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer-Verlag, New York, [9] S. Heilpern, Fuzzymappingsandfixedpointtheorem, J. Math. Anal. Appl. 83 (1981), no. 2, [10] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall PTR, New Jersey, [11] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, [12] H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, CRC Press, Florida, [13] G. Papaschinopoulos and B. K. Papadopoulos, On the fuzzy difference equation x n+1 = A + B/x n, Soft Comput. 6 (2002), [14], On the fuzzy difference equation x n+1 = A + x n /x n m, Fuzzy Sets and Systems 129 (2002), no. 1, [15] G. Papaschinopoulos and G. Stefanidou, Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation, Fuzzy Sets and Systems 140 (2003), no. 3, [16], Trichotomy of a system of two difference equations, J. Math. Anal. Appl. 289 (2004), no. 1, [17] M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983), no. 2, [18] C. Wu and B. Zhang, Embedding problem of noncompact fuzzy number space E (I), Fuzzy Sets and Systems 105 (1999), no. 1, G. Stefanidou: Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Greece address: tfele@yahoo.gr G. Papaschinopoulos: Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Greece address: gpapas@ee.duth.gr

22 Journal of Applied Mathematics and Decision Sciences Special Issue on Intelligent Computational Methods for Financial Engineering Call for Papers As a multidisciplinary field, financial engineering is becoming increasingly important in today s economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions. However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation Authors should follow the Journal of Applied Mathematics and Decision Sciences manuscript format described at the journal site Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Guest Editors Lean Yu, AcademyofMathematicsandSystemsScience, Chinese Academy of Sciences, Beijing , China; Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; yulean@amss.ac.cn Shouyang Wang, AcademyofMathematicsandSystems Science, Chinese Academy of Sciences, Beijing , China; sywang@amss.ac.cn K. K. Lai, Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; mskklai@cityu.edu.hk Hindawi Publishing Corporation