On a Fuzzy Logistic Difference Equation

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1 On a Fuzzy Logistic Difference Euation QIANHONG ZHANG Guizhou University of Finance and Economics Guizhou Key Laboratory of Economics System Simulation Guiyang Guizhou CHINA zianhong68@163com JINGZHONG LIU Hunan Institute of Technology Deartment of Mathematics and Physics Hengyang Hunan CHINA hnhyls@126com WENZHUAN ZHANG Guizhou University of Finance and Economics Guizhou Key Laboratory of Economics System Simulation Guiyang Guizhou CHINA zhwenzh97@hotmailcom YUANFU SHAO Guilin University of Technology School of Science Guilin Guangxi CHINA shaoyuanfu@163com Abstract: This aer is concerned with the existence uniueness and asymtotic behavior of the ositive solutions of a fuzzy Logistic difference euation x n+1 = A + Bx n 1 e xn n = 0 1 where (x n ) is a seuence of ositive fuzzy number A B are ositive fuzzy numbers and the initial conditions x 1 x 0 are ositive fuzzy numbers Moreover an illustrative examle is given to demonstrate the effectiveness of the results obtained Key Words: Fuzzy Logistic difference euation Euilibrium oint Bounded Persistence 1 Introduction It is well known that difference euation aears naturally as discrete analogous and as numerical solutions of differential euations and delay differential euation having many alications in economics biology comuter science control engineering The study of asymtotic stability and oscillatory roerties of solutions of difference euations is extremely useful in the behavior of mathematical models of various biological systems and other alications This is due to the fact that difference euations are aroriate models for describing situations where the variable is assumed to take only a discrete set of values and they arise freuently in the study of biological models in the formulation and analysis of discrete time systems etc Recently there has been a lot of work concerning the asymtotic behavior the eriodicity and the boundedness of nonlinear difference euation and system of nonlinear difference euations(see for examle [ ] and the references therein) EI-Metwally et al[7] investigated the asymtotic behavior of oulation model: x n+1 = α + βx n 1 e x n n = 0 1 (1) where α is the immigration rate and β is the oulation growth rate Fuzzy set theory is a owerful tool for modeling uncertainty and for rocessing vague or subjective information in mathematical model Particularly the use of fuzzy difference euations is a natural way to model the dynamical systems with embedded uncertainty Fuzzy difference euation is a difference euation where constants and the initial values are fuzzy numbers and its solutions are seuences of fuzzy numbers Recently there is an increasing interest in the study of fuzzy difference euation (see for examle [ ? 21]) In this aer we study the fuzzy analogs of (1) x n+1 = A + Bx n 1 e x n n = 0 1 (2) where (x n ) is a seuence of ositive fuzzy numbers A B and the initial values x 1 x 0 are ositive fuzzy numbers E-ISSN: Volume

2 For readers convenience we need some definitions: A is said to be a fuzzy number if A : R [0 1] satisfies the below (i)-(iv) (i) A is normal ie there exists an x R such that A(x) = 1; (ii) A is fuzzy convex ie for all t [0 1] and x 1 x 2 R such that A(tx 1 + (1 t)x 2 ) min{a(x 1 ) A(x 2 )}; (iii) A is uer semi-continuous; (iv) The suort of A sua = [A] α = {x : A(x) > 0} is comact The α-cuts of A are denoted by [A] α = {x R : A(x) α} α [0 1] it is clear that the [A] α are closed interval We say that a fuzzy number is ositive if sua (0 ) It is obvious that if A is a ositive real number then A is a fuzzy numbers and [A] α = [A A] α (0 1] Then we say that A is a trivial fuzzy number Let A B be fuzzy numbers with [A] α = [A lα A rα ] [B] α = [B lα B rα ] α (0 1] We define a norm on fuzzy numbers sace as follows: A = We take the following metric : D(A B) = su max{ A lα A rα } su max{ A lα B lα A rα B rα } The fuzzy analog of the boundedness and ersistence (see [4 11]) as follows: we say that a seuence of ositive fuzzy numbers x n ersists (res is bounded) if there exists a ositive real number M (res N) such that sux n [M )(res sux n (0 N]) n N x n is bounded and ersists if there exist ositive real numbers M N > 0 such that sux n [M N] n = 1 2 x n n = 1 2 is an unbounded seuence if the norm x n n = 1 2 is an unbounded seuence x n is a ositive solution of (2) if x n is a seuence of ositive fuzzy numbers which satisfies (2) We say a ositive fuzzy number x is a ositive euilibrium for (2) if x = A + Bxe x Let (x n ) be a seuence of ositive fuzzy numbers and x is a ositive fuzzy number Suose that [x n ] α = [α R nα ] n = α (0 1] (3) and [x] α = [L α R α ] α (0 1] (4) The seuence (x n ) converges to x with resect to D as n if lim D(x n x) = 0 Suose that (2) has a uniue ositive euilibrium x We say that the ositive euilibrium x of (2) is stable if for every ε > 0 there exists a δ = δ(ε) > 0 such that for every ositive solution x n of (2) which satisfies D(x i x) δ i = 0 1 we have D(x n x) ε for all n > 0 Moreover we say that the ositive euilibrium x of (2) is asymtotically stable if it is stable and every ositive solution of (2) tends to the ositive euilibrium of (2) with resect to D as n The urose of this aer is to study existence of the ositive solutions of (2) Furthermore we give some conditions so that every ositive solution of (2) is boundedness and ersistence Finally under some conditions we rove that (2) has a uniue ositive euilibrium x which is asymtotic stable 2 Main results 21 Existence of the ositive solution Firstly we study the existence of the ositive solutions of (2) We need the following lemma Lemma 1 [12] Let f :: R + R + R + R + R + be continuous A B C D are fuzzy numbers Then for α (0 1] [f(a B C D)] α = f([a] α [B] α [C] α [D] α ) (5) Lemma 2 [19] Let u E write [u] α = [u (α) u + (α)] α (0 1] Then u (α) and u + (α) can be regarded as functions on (0 1] which satisfy (i) u (α) is nondecreasing and left continuous; (ii) u + (α) is nonincreasing and left continuous; (iii) u (1) u + (1) Conversely for any functions a(α) and b(α) defined on (0 1] which satisfy (i) (iii) in the above there exists a uniue u E such that [u] α = [a(α) b(α)] for any α (0 1] E-ISSN: Volume

3 Theorem 3 Consider euation (2) where A B are ositive fuzzy numbers Then for any ositive fuzzy numbers x 1 x 0 there exists a uniue ositive solution x n of (2) with initial conditions x 1 x 0 Proof: The roof is similar to Proosition 21 in [12] Suose that there exists a seuence of fuzzy numbers x n satisfying (2) with initial conditions x 1 x 0 Consider the α cuts α (0 1] n = [x n ] α = [α R nα ] [A] α = [A lα A rα ] [B] α = [B lα B rα ] It follows from (2) (6) and Lemma 1 that [x n+1 ] α = [+1α R n+1α ] = [A + Bx n 1 e x n ] α = [A] α + [B] α [x n 1 ] α [e x n ] α = [ A lα + B lα 1α e Rnα A rα + B rα R n 1α e α ] (6) from which we have that for n = α (0 1] +1α = A lα + B lα 1α e Rnα R n+1α = A rα + B rα R n 1α e Lnα (7) Then it is obvious that for any initial condition (L iα R iα ) i = 0 1 α (0 1] there exists a uniue solution (α R nα ) Now we rove that [α R nα ] α (0 1] where (α R nα ) is the solution of system (7) with initial conditions (L iα R iα ) i = 0 1 determines the solution x n of (2) with the initial conditions x i i = 0 1 such that for n = [x n ] α = [α R nα ] α (0 1] (8) From reference [20] and since A B x 1 x 0 are ositive fuzzy numbers for any α 1 α 2 (0 1] α 1 α 2 we have 0 < A lα1 A lα2 A rα2 A rα1 We claim that for n = α1 α2 R nα2 R nα1 (10) We rove it by induction It is obvious from (9) that (10) holds true for n = 1 0 Suose that (10) are true for n k k {1 2 } Then from (7) (9) and (10) for n k it follows that L k+1α1 = A lα1 + B lα1 L k 1α1 e R kα 1 A lα2 + B lα2 L kα2 e R kα 2 = L k+1α2 = A lα2 + B lα2 L kα2 e R kα 2 A rα2 + B rα2 L kα2 e R kα 2 = R k+1α2 = A rα2 + B rα2 L kα2 e R kα 2 A rα1 + B rα1 L kα1 e R kα 1 = R k+1α1 Therefore (10) are satisfied Moreover from (7) we have for α (0 1] L 1α = A lα + B lα L 1α e R 0α (11) R 1α = A rα + B rα R 1α e L 0α Since A B x 1 x 0 are ositive fuzzy numbers then we have that A lα A rα B lα B rα L lα R 1α L 0α R 0α are left continuous So from (11) we have that L 1α R 1α are also left continuous By induction we can get that α R nα n = 1 2 are left continuous Now we rove that the suort of x n sux n = [α R nα ] is comact It is sufficient to rove that [α R nα ] is bounded Let n = 1 since A B x 1 x 0 are ositive fuzzy numbers there exist constants M A > 0 N A > 0 M B > 0 N B > 0 M 1 > 0 N 1 > 0 M 0 > 0 N 0 > 0 such that for all α (0 1] [A lα A rα ] [M A N A ] [B lα B rα ] [M B N B ] [L lα R 1α ] [M 1 N 1 ] [L 0α R 0α ] [M 0 N 0 ] (12) Hence from (11) and (12) we can easily get for α (0 1] [L 1α R 1α ] [ M A + M B M 1 e N 0 N A + N B N 1 e M 0 ] (13) 0 < B lα1 B lα2 B rα2 B rα1 0 < L 1α1 L 1α2 R 1α2 R 1α1 0 < L 0α1 L 0α2 R 0α2 R 0α1 (9) From which it is obvious that [L 1α R 1α ] [ M A +M B M 1 e N 0 N A +N B N 1 e M 0 ] (14) E-ISSN: Volume

4 Therefore (14) imlies that [L 1α R 1α ] is comact and [L 1α R 1α ] (0 ) Deducing inductively we can easily follow that [α R nα ] is comact and [α R nα ] (0 ) n = 1 2 (15) Therefore (10) (15) and since α R nα are left continuous we have that [α R nα ] determines a seuence of ositive fuzzy numbers x n such that (8) holds We rove now that x n is the solution of (2) with initial condition x 1 x 0 Since for all α (0 1] [x n+1 ] α = [+1α R n+1α ] = [ A lα + B lα 1α e R nα A rα + B rα R n 1α e Lnα] = [ A + Bx n 1 e x n] α we have that x n is the solution of (2) with initial condition x 1 x 0 Suose that there exists another solution x n of (2) with initial conditions x 1 x 0 Then from arguing as above we can easily rove thatfor n = [x n ] α = [α R nα ] α (0 1] (16) Then from (8) and (16) we have [x n ] α = [x n ] α α (0 1] n = from which it follows that x n = x n n = 0 1 Thus the roof of the theorem 3 is comleted 22 Boundedness and Permanence In the following we will study the boundedness and ermanence of the fuzzy ositive solution of (2) We need the following lemma Lemma 4 (comarison results) Assume that α (0 ) β [0 ) Let {X n } n= 1 {Y n} n= 1 be seuences of real numbers such that X 1 Y 1 X 0 Y 0 and for n = 0 1 { Xn+1 αx n 1 + β Y n+1 = αy n 1 + β Then X n Y n for all n 1 Lemma 5 Consider the system of the difference euations for n = 0 1 y n+1 = + cy n 1 e z n (17) z n+1 = + dz n 1 e yn where c d are ositive real numbers and the initial values y i z i (i = 0 1) are ositive real numbers If c < e d < e (18) Then system (17) is bounded and ersists Proof: Let {(y n z n )} n= 1 be a ositive solution of system (17) Then it follows from system (17) that for n 0 y n+1 = + cy n 1 e z n > (19) z n+1 = + dz n 1 e y n > Thus y n+1 = + cy n 1 e zn < + ce y n 1 z n+1 = + dz n 1 e yn < + de z n 1 (20) Now we consider the initial value roblem for n = 1 2 Y n+1 = + ce Y n 1 Z n+1 = + de Z n 1 with initial conditions y i Y i z i Z i (i = 0 1) and so it follows from Lemma 4 that Observe that lim Y n = and then lim su y n y n Y n z n Z n n 1 lim 1 ce Z n = lim 1 ce su z n 1 de 1 de Therefore {(y n z n )} n= 1 is bounded and ersists and the roof is comleted Theorem 6 Consider fuzzy difference euation (2) where A B and initial values x 1 x 0 are ositive fuzzy numbers If for all α (0 1] B rα < e A lα (21) Then every ositive solution of (2) is bounded and ersists Proof: Let x n be ositive solution of E(2) such that (8) holds From (7) it is obvious that for n = 1 2 α (0 1] A lα α A rα R nα (22) E-ISSN: Volume

5 From Lemma 4 and (19) it follows that for n = 1 2 α R nα A lα 1 B lα e A rα (23) A rα 1 B rαe A lα Hence from (12) (20) and (21) it is obvious that for n 1 α (0 1] [ ] N A [α R nα ] M A 1 N B e M (24) A From which we get for n 1 [ N A [α R nα ] M A 1 N B e M A and so [α R nα ] [ M A N A 1 N B e M A ] ] ie the ositive solution of (2) is bounded and ersists 23 Dynamic of solution of E(2) Lemma 7 Consider the system of difference euation (17) where c d are ositive real numbers If (18) holds Then there exists a uniue ositive euilibrium (ȳ z) such that < ȳ < 1 ce < z < (25) 1 de Proof: Let (ȳ z) be the solution of the following systems y = + cye z z = + dze y (26) Set Then and f() = de f(z) = + dze 1 ce z z (27) ce 1 > 0 lim f(z) = (28) z + f (z) = de ce z 1 ce z dz (1 ce z ) 2 e 1 ce z 1 < 0 (29) It follows from (26) and (16) that (25) has exactly one solution z > On the other hand set g(y) = + cye 1 de y y (30) Similarly it can easily rove that (28) has exactly one solution ȳ > Noting (18) and (24) it follows that ȳ < z < The roof is comleted 1 ce 1 de Theorem 8 Consider the system (17) where c d are ositive real numbers and the initial values y i z i (i = 0 1) are ositive real numbers Assume that (18) and the following condition hold { ( ) 1 2 max (1 + )ce 1 ce } (1 + )de < 1 (31) 1 de Then the ositive euilibrium (ȳ z) of system (17) is globally asymtotically stable Proof: First we rove that the ositive euilibrium (ȳ z) of (17) is locally asymtotically stable We can easily obtain that the linearized system of (17) about the ositive euilibrium (ȳ z) is where D = Ψ n+1 = DΨ n 0 ce z cȳe z d ze ȳ 0 0 de ȳ Since (18) and (29) hold we can take a ositive number ε such that { ((1 ε = max + ȳ)ce z ) 1 2 (1 + z)de ȳ} Let < 1 (32) We consider the matrix S = diag(1 ε 1 ε 2 ε 3 ) C = (c ij ) = S 1 DS = 0 ce z ε 1 cȳe z ε 2 0 ε d ze ȳ ε de ȳ ε ε 0 with the norm 4 C = max c 1 i 4 ij j=1 Using (30) we can rove that C < 1 Therefore since λ i < C ( where λ i (i = ) are the eigenvalues of D) we have that all eigenvalues of D lie inside the unit disk This imlies that (ȳ z) is locally asymtotically stable E-ISSN: Volume

6 Let now (y n z n ) be a ositive solution of (17) We rove that Using Lemma 4 we have lim y n = ȳ lim z n = z (33) Λ 1 = lim su y n < Λ 2 = lim su z n < λ 1 = lim inf y n > 0 λ 2 = lim inf z n > 0 Then from (17) and (32) we take Λ 1 1 ce λ λ 2 1 Λ 2 1 de λ 1 λ 2 1 ce Λ 2 (34) (35) 1 de Λ 1 From (33) it follows that Λ 1 λ 2 λ 2 1 ce Λ λ 2 2λ 1 Λ 2 1 ce Λ 2 Λ 2 λ 1 λ 1 1 de λ 1 λ 2Λ 1 Λ 1 1 de Λ 1 from which we take Λ 1 1 de λ Λ ce λ 2 Λ 2 1 ce Λ 2 λ 1 1 de λ 1 Now we consider the functions for f(y) = y ( (36) y 1 de y g(z) = z (37) 1 ce z ) ( 1 ce z Then from (35) it follows that From (30) Noting ( y f (y) = (1 de y (1+y)) (1 de y ) 2 ) 1 de g (z) = (1 ce z (1+z)) (1 ce z ) 2 (38) 1 ce ) ( z ) 1 de we get 1 de y (1 + y) > 1 de (1 + ) > 0 1 de 1 ce z (1 + z) > 1 ce (1 + ) > 0 1 ce (39) Therefore from (36) and (37) we obtain for y ( f (y) > 0 g (z) > 0 ) ( 1 ce z ) 1 de Hence f g are increasing functions and this together with (34) imlies that λ 1 = Λ 1 Then from (34) again we see that λ 2 = Λ 2 This imlies that lim y n = ȳ lim z n = z This comletes the roof of the theorem Remark 9 The condition satisfying the uniue ositive euilibrium of (12) is globally asymtotically stable is different from that of [13] Theorem 10 Consider fuzzy difference euation (2) where A B and the initial values x i (i = 0 1) are ositive fuzzy numbers if (19) is satisfied Then the following statements are true (i) The euation (2) has a uniue ositive euilibrium (ii) The every ositive solution x n of (2) converges to the uniue ositive euilibrium x with resect to D as n Proof: (i) We consider the system L α = A lα + B lα L α e Rα R α = A rα + B rα R α e Lα (40) Obviously system (40) has a uniue solution (L α R α ) Let x n be a ositive solution of (2) such that [x n ] α = [α R nα ] α (0 1] n = 0 1 Then alying Theorem 8 to the system (7) we have lim α = L α lim R nα = R α (41) From (22) and (41) we have for 0 < α 1 α < L α1 L α2 R α2 R α1 (42) Since A lα A rα B lα B rα are left continuous it follows from (40) that L α R α are also left continuous From (40) and (12) we get L α A lα M A R α = A rα 1 B rαe Lα N A 1 N B e M A (43) E-ISSN: Volume

7 Therefore (43) imlies that [ N A [L α R α ] M A 1 N B e M A From which it is obvious that [L α R α ] is comact [L α R α ] (0 ) ] (44) So from Lemma 2 relations (40) (42) (44) and since L α R α α (0 1] determines a fuzzy number x such that x = A + Bxe x [x] α = [L α R α ] α (0 1] and 10x 5 05 x 06 B(x) = 5x x 08 (48) We take initial value x 1 x 0 such that 10x 2 02 x 03 x 1 (x) = (49) 10 3 x x 06 and 5x 5 01 x 03 x 0 (x) = 5x x 05 (50) and so x is a ositive euilibrium of (2) Provided that there exists another ositive euilibrium x for (2) then there exist functions L α : (0 1] (0 ) R α : (0 1] (0 ) such that x = A + B xe x [ x] α = [L α R α ] α (0 1] From which we have L α = A lα +B lα L α e R α R α = A rα +B rα R α e L α So L α = L α R α = R α α (0 1] namely x = x This comletes the roof of (i) (ii) From (41) we have lim D(x n x) = lim su {max { α L α R nα R α }} = 0 (45) From which it is obvious that every ositive solution x n of E(2) converges the uniue euilibrium x with resect to D as n 3 Numerical examle To illustrate our results we give some examles in which the conditions of our roositions hold Examle 1 Consider the following fuzzy difference euation x n+1 = A + Bx n 1 e x n n = 0 1 (46) where A B are fuzzy numbers such that 10x 3 03 x 04 A(x) = 5x x 06 (47) From (47) and (48) we get for α (0 1] [A] α = [ α α] And so [B] α = [ α α] [A] α = [03 06] [B] α = [05 08] (51) (52) Moreover from (49) and (50) we get for α (0 1] [x 1 ] α = [ α α] [x 0 ] α = [ α α] It follows that [x 1 ] α = [02 06] (53) [x 0 ] α = [01 05] (54) From (46) it results in a couled system of difference euation with arameter α (0 1] +1α = α + ( α) 1α e Rnα R n+1α = α + ( α) R n 1α e α (55) Therefore B rα < e A lα for any α (0 1] namely the condition (19) is satisfied so from Theorem 6 we have that every ositive solution x n of E(46) is bounded and ersists In addition E(46) has a uniue ositive euilibrium x = ( ) Moreover every ositive solution x n of E(46) converges the uniue euilibrium x with resect to D as n (see Fig1-Fig4) E-ISSN: Volume

8 α=0 α=05 α= R n R n 1 08 &R n n Figure 1: The dynamics of system (55) Figure 3: The solution of system (55) at α = R n R n &R n &R n n n Figure 2: The solution of system (55) at α = 0 4 Conclusion In this work fuzzy Logistic difference x n+1 = A + Bx n 1 e x n n = 0 1 is discussed Firstly the existence of ositive solution to this euation is roved Secondly we find that under condition B rα < e A lα the ositive solutions of fuzzy Logistic difference are bounded and ersistence and there exists uniue ositive euilibrium x such that every ositive solution converges it Finally an examle is given to illustrate our results obtained Acknowledgements: This research was financially suorted by the National Natural Science Foundation of China (Grant No ) and suorted by the Scientific Research Foundation of Guizhou Science and Technology Deartment ([2009]2061[2013]2083) References: [1] D Benest and C Froeschle Analysis and Modelling of Discrete Dynamical Systems Gordon Figure 4: The solution of system (55) at α = 1 and Breach Science Publishers The Netherland 1998 [2] Mukminah Darus and Abd Fatah Wahab Review on fuzzy difference euation Malaysian Journal of Fundamental & Alied Sciences 8 No [3] E Y Deeba and A De Korvin Analysis by fuzzy difference euations of a model of CO 2 level in the blood Al Math Lett [4] E Y Deeba and A De Korvin and E L Koh A fuzzy difference euation with an alication J Difference Euation Al [5] R DeVault G Ladas and S W Schultz Necessary and sufficient conditions the boundedness of x n+1 = A/x n + B/x n 1 JDifference Euations Al [6] R DeVault G Ladas and S W Schultz On the recursive seuence x n+1 = A/x n +1/x n 2 Proc Amer Math Soc [7] E El-Metwally E A Grove G Ladas R Levins and M Radin On the difference eua- E-ISSN: Volume

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