Positive periodic solutions of higher-dimensional nonlinear functional difference equations

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1 J. Math. Anal. Appl. 309 (2005) Positive periodic solutions of higher-dimensional nonlinear functional difference equations Yongkun Li, Linghong Lu Department of Mathematics, Yunnan University Kunming, Yunnan , People s Republic of China Received 21 November 2003 Available online 3 March 2005 Submitted by J. Henderson Abstract By using a well-known fixed point index theorem, we study the existence, multiplicity and nonexistence of positive T -periodic solution(s) to the higher-dimensional nonlinear functional difference equations of the form where x(n + 1) = A(n)x(n) + λh(n)f ( x ( n τ(n) )), n Z, A(n) = diag [ a 1 (n), a 2 (n),..., a m (n) ], h(n)= diag [ h 1 (n), h 2 (n),..., h m (n) ], a j,h j : Z R +, τ : Z Z are T -periodic, j = 1, 2,...,m, T 1; λ>0, x : Z R m, f : R m + R+, where R m + ={(x 1,...,x m ) T R m : x j 0,j = 1, 2,...,m}, R + ={x R: x>0} Elsevier Inc. All rights reserved. Keywords: Functional difference equation; Positive periodic solution; Existence; Fixed point index Project supported by NSFC and Project 2003A0001M supported by the Natural Sciences Foundation of Yunnan Province. * Corresponding author. address: yklie@ynu.edu.cn (Y. Li) X/$ see front matter 2005 Elsevier Inc. All rights reserved. doi: /j.jmaa

2 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) Introduction As pointed in [12], at present, the existence of periodic solutions of functional differential equations has been studied extensively [1,3,4,6 8,10,11]. However, few papers have been published on the same problem for functional difference equations. In [9], Y.N. Raffoul has studied the existence of positive periodic solutions for the following functional difference equation x(n + 1) = a(n)x(n) + λh(n)f ( x ( n τ(n) )), where a(n), h(n) and τ(n) are T -periodic for T is an integer with T 1, λ, a(n), f(x) and h(n) are nonnegative with 0 <a(n)<1 for all n {0, 1,...,T 1}. In [12], by using Krasnosel skii s fixed point theorem and upper and lower solutions method, Zhu and Li have found some sets of positive values λ determining that there exist positive T -periodic solutions to the higher-dimensional functional difference equations of the form where x(n + 1) = A(n)x(n) + λh(n)f ( x ( n τ(n) )), n Z, (1.1) A(n) = diag[a 1 (n), a 2 (n),...,a m (n)], h(n) = diag[h 1 (n), h 2 (n),...,h m (n)], a j,h j : Z R +, τ : Z Z are T -periodic, j = 1, 2,...,m, T 1, λ>0, x : Z R m, function f : R+ m R+ is continuous, where Z is the set of all integers, R+ m ={(x 1,...,x m ) T R m : x j 0,j = 1, 2,...,m}, R + ={x R: x>0}. In the sequel, we say that x is positive whenever x R+ m ; we denote f = (f 1,...,f m ) T, and denote the product of x(n) from n = a to n = b by n=b n=a x(n) with the understanding that n=b n=a x(n) = 1 for all a>b. Also, we denote by N the set of all nonnegative integers and denote [a,b]={a,a + 1,...,b} for a<b Z. The purpose of this paper is to study the existence, multiplicity and nonexistence of positive T -periodic solution(s) of Eq. (1.1) by using a well-known fixed point index theorem. The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, by using a well-known fixed point index theorem, we obtain sufficient conditions of the existence, multiplicity and nonexistence of positive T -periodic solution(s) of Eq. (1.1). 2. Preliminaries Throughout this paper, we always assume that (S 1 ) 0 <a j (n) < 1 for n [0,T 1], j= 1, 2,...,m. The following well-known result of the fixed point index is very useful in the proofs of our main results of this paper.

3 286 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) Lemma 2.1 [2,5]. Let E be a Banach space and K be a cone in E. Forr>0, define K r ={x K: x <r}. Assume that Φ : K r K is completely continuous such that Φx x for x K r ={x K: x =r}. (i) If Φx x for x K r, then i(φ,k r,k)= 0. (ii) If Φx x for x K r, then i(φ,k r,k)= 1. In order to apply Lemma 2.1 to Eq. (1.1), let X be the Banach space X = { x : Z R m : x(n + T)= x(n) } with x = m x j 0, where x j 0 = sup n [0,T 1] x j (n). Lemma 2.2 [12]. x(n) X is a solution of Eq. (1.1) if and only if x(n) = λ G(n, u)h(u)f ( x ( u τ(u) )), where G(n, u) = diag [ G 1 (n, u),..., G m (n, u) ] (2.1) and G j (n, u) = 1 s=u+1 a j (s) 1, u [n, n + T 1], j= 1, 2,...,m. 1 s=n a j (s) It follows from (S 1 ) that the denominator in G j (n, u) is not zero for n [0,T 1]. Note that due to (S 1 ), we have N j G j (n, n) G j (n, u) G j (n, n + T 1) = G j (0,T 1) M j, j = 1, 2,...,m, for u [n, n + T 1], and G j (n, u) 1 G j (n, n + T 1) G j (n, n) G j (n, n + T 1) = N j > 0, M j Let { } Nj γ = min : j = 1, 2,...,m. M j One has γ (0, 1). In what follows, we shall use the following notations: q j = min 0 u T 1 h j (u), p j = max 0 u T 1 h j (u), fj 0 = lim f j (x) x j 0 +, f f j (x) j = lim, x j x j x j j = 1, 2,...,m. j = 1, 2,...,m,

4 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) and q = min j, p= max j, η= min j, N = min j, ɛ = max j, M = max j. Define a cone K by K = { x X: x j (n) γ x j 0,j= 1, 2,...,m }, and for a positive number r, define Ω r by Ω r = { x K: x j 0 <r,j= 1, 2,...,m }. Note that Ω r ={x K: x j 0 = r, j = 1, 2,...,m}. Let the map F : K K be defined by (F x)(n) = λ G(n, u)h(u)f ( x ( u τ(u) )), for x K, n Z, where G(n, u) is defined by (2.1), we denote (F x) = (F 1 x,f 2 x,...,f m x) T. Lemma 2.3. F : K K is well defined. Proof. For each x K, since it is clear that (F x)(n + T)= (F x)(n) for n Z, Fx X. For any x K and n Z, wehave (F j x)(n) = λ G j (n, u)h j (u)f j x u τ(u) Thus F j x 0 = λ G j (0,T 1)h j (u)f j x u τ(u), sup n [0,T 1] (F j x)(n) T 1 λ G j (0,T 1)h j (u)f j x u τ(u), u=0 and for n Z, (F j x)(n) = λ G j (n, u)h j (u)f j x u τ(u) T 1 λn j h j (u)f j x u τ(u) u=0 j = 1, 2,...,m. j = 1, 2,...,m,

5 288 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) T 1 = λn j u=0 G j (0,T 1) M j h j (u)f j ( x ( u τ(u) )) γ F j x 0, j = 1, 2,...,m. Therefore, (F x) K. This completes the proof. Lemma 2.4. F : K K is completely continuous. Proof. The proof is similar to that of [12, Lemma 3.3] and will be omitted. Lemma 2.5. For j = 1, 2,...,m,letη j > 0, ifx K and let f j (x(n)) x j (n)η j for n [0,T 1]. Then Fx λnqηt γ x. Proof. Since min n [0,T 1] x j (n) γ x j 0 for every j = 1, 2,...,mand for all x K,we have that for n [0,T 1], (F j x)(n) = λ G j (n, u)h j (u)f j x u τ(u) λn j q j x j ( u τ(u) ) ηj λn j q j η j Tγ x j 0. Therefore, F j x 0 λn j q j η j Tγ x j 0, j = 1, 2,...,m, and Fx = F j x 0 λn j q j η j Tγ x j 0 λnqηt γ The proof is complete. x j 0 = λnqηt γ x. Lemma 2.6. Let r>0, ifx Ω r and for every j = 1, 2,...,m, there exists an ɛ j > 0 such that f j (x(n)) ɛ j x j (n) for n [0,T 1], then Fx λmpɛt x. Proof. For n [0,T 1],wehave (F j x)(n) = λ G j (n, u)h j (u)f j x u τ(u) λm j p j λm j p j ɛ j T x j 0, x j ( u τ(u) ) ɛj j = 1, 2,...,m.

6 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) Therefore, and F j x 0 λm j p j ɛ j T x j 0, j = 1, 2,...,m, Fx = F j x 0 λm j p j ɛ j T x j 0 λmpɛt The proof is complete. x j 0 = λmpɛt x. Since f is continuous and when x Ω r,forj = 1, 2,...,m, x j 0 r, r x j 0 = sup n [0,T 1] x j (n τ(n)), that is, r x j (n τ(n)) r, for convenience, we can define: (H 1 ) m j (r) = inf f j x u τ(u), 0 u T 1, j= 1, 2,...,m, x Ω r m(r) = min m j (r). (H 2 ) M j (r) = sup f j x u τ(u), 0 u T 1, j= 1, 2,...,m, x Ω r M(r) = max M j (r). Similar to the proofs of Lemmas 2.5 and 2.6, one can show the following two lemmas. Lemma 2.7. If x Ω r, r>0, then Fx λnqmt m(r). Lemma 2.8. If x Ω r, r>0, then Fx λmpmt M(r). 3. Main results We are now in a position to state and prove our first main result of this paper. Theorem 3.1. Suppose (S 1 ) holds. (a) When λ>1/(nqmt m(1)) > 0, if fj 0 = 0 (j = 1, 2,...,m) or f j = 0 (j = 1, 2,...,m), then (1.1) has one positive T -periodic solution and if for every j = 1, 2,...,m, fj 0 = f j = 0, then (1.1) has two positive T -periodic solutions. (b) When 0 <λ<1/(mpmt M(1)), iffj 0 = (j = 1, 2,...,m) or f j = (j = 1, 2,...,m), then (1.1) has one positive T -periodic solution and if for every j = 1, 2,...,m, fj 0 = f j =, then (1.1) has two positive T -periodic solutions. (c) If fj 0 > 0 and f j > 0 (j = 1, 2,...,m),orfj 0 < and f j < (j = 1, 2,...,m), then (1.1) has no positive T -periodic solution for sufficiently large or small λ>0, respectively.

7 290 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) Proof. Part (a). Choose a number r 1 = 1. By Lemma 2.7 we can see that there exists λ 0 = 1/(NqmT m(r 1 )) > 0 such that Fx x for x Ω r1,λ>λ 0. If for every j = 1, 2,...,m, fj 0 2 <r 1 such that f j (x) ɛ j x j for 0 x j r 2, where the constant ɛ j > 0 satisfies ( ) λmp max j T<1. Thus f j (x(n)) ɛ j x j (n) for x Ω r2, n [0,T 1] and j = 1, 2,...,m. It follows from Lemma 2.6 that Fx λmpɛt x < x for x Ω r2. By Lemma 2.1 we have i(f,ω r1,k)= 0, i(f,ω r2,k)= 1. Therefore, i(f,ω r1 \ Ω r2,k) = 1, and F has a fixed point in Ω r1 \ Ω r2, which is a positive T -periodic solution of (1.1) for λ>λ 0. If for every j = 1, 2,...,m, fj = 0, then there is a W>0such that f j (x) ɛ j x j for x j W, where the constant ɛ j > 0 satisfies ( ) λmp max j T<1. Let r 3 = max{2r 1,W/γ}. It follows that x j (n) γ x j 0 W for x Ω r3, n [0,T 1] and j = 1, 2,...,m. Thus f j (x(n)) ɛ j x j (n) for x Ω r3, n [0,T 1] and j = 1, 2,...,m. According to Lemma 2.6, we have Fx λmpɛt x < x for x Ω r3. Again, it follows from Lemma 2.1 that i(f,ω r1,k)= 0, i(f,ω r3,k)= 1. Thus i(f,ω r3 \ Ω r1,k)= 1, and (1.1) has a positive T -periodic solution for λ>λ 0. If fj 0 j = 0, it is easy to see from the above proof that F has a fixed point u 1 in Ω r1 \ Ω r2 and a fixed point u 2 in Ω r3 \ Ω r1 such that r 2 < u 1 <r 1 < u 2 <r 3. Consequently, (1.1) has two positive T -periodic solutions for λ>λ 0. Part (b). Choose a number r 1 = 1. By Lemma 2.8 we know that there exists λ 0 = 1/(MpmT M(r 1 )) > 0, such that Fx < x for x Ω r1, 0 <λ<λ 0. Then for every j = 1, 2,...,m, we obtain the following consequences. If for every j = 1, 2,...,m, fj 0 2 <r 1 such that f j (x) η j x j for 0 x j r 2, where the constant η j > 0 is chosen so that λnq ( min j ) Tγ >1.

8 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) Thus f j (x(n)) η j x j (n) for x Ω r2, n [0,T 1] and j = 1, 2,...,m.Wehaveby Lemma 2.5 that Fx λnqηt γ x > x for x Ω r2. It follows from Lemma 2.1 that i(f,ω r1,k)= 1, i(f,ω r2,k)= 0. Thus i(f,ω r1 \ Ω r2,k)= 1, and F has a fixed point in Ω r1 \ Ω r2, which is a positive T -periodic solution of (1.1) for 0 <λ<λ 0. If for every j = 1, 2,...,m, fj =, then there is a W >0 such that f j (x) η j x j for x j W, where the constant η j > 0 satisfies ( ) λnq max η j Tγ >1. Let r 3 = max{2r 1, W/γ}. It follows that min 0 n T 1 x j (n) γ x j 0 W for x Ω r3, n [0,T 1] and j = 1, 2,...,m. Thus f(x(n)) ηx(n) for x Ω r3 and n [0,T 1]. In view of Lemma 2.5, we have Fx λnqηt γ x > x for x Ω r3. Again, it follows from Lemma 2.1 that i(f,ω r1,k)= 1, i(f,ω r3,k)= 0. Thus i(f,ω r3 \ Ω r1,k)= 1, and (1.1) has a positive T -periodic solution for 0 <λ<λ 0. If for every j = 1, 2,...,m, fj 0 = f j =, then it is easy to see from the above proof that F has a fixed point u 1 in Ω r1 \ Ω r2 and a fixed point u 2 in Ω r3 \ Ω r1 such that r 2 < u 1 <r 1 < u 2 <r 3. Consequently, (1.1) has two positive T -periodic solutions for 0 <λ<λ 0. Part (c). If for every j = 1, 2,...,m, fj 0 > 0 and f j > 0, it is easy to see that there exist positive numbers η j1, η j2, r j1 and r j2 such that r j1 <r j2 and f j (x) η j1 x j for x j [0,r 1 ], f j (x) η j2 x j for x j [r 2, ). Let c j1 = min{η j1,η j2, min r1 x r 2 {f j (x)/x j }} > 0, j = 1, 2,...,m, then we have f j (x) c j1 x j for x j [0, ), j = 1, 2,...,m. Assume ϕ(n) is a positive T -periodic solution of (1.1). We will show that this leads to a contradiction for λ>λ 0, where λ 0 = 1/(Nq(min c j1 )T γ ). Since Fϕ(n)= ϕ(n) for n [0,T 1], it follows from Lemma 2.5 that, for λ>λ 0 ( ) ϕ = Fϕ λnq min j1 Tγ ϕ > ϕ, which is a contradiction. If for every j = 1, 2,...,m, fj 0 j <, it follows that for j = 1, 2,...,m, there exist positive numbers ɛ j1, ɛ j2, r j1 and r j2 such that r j1 <r j2 and

9 292 Y. Li, L. Lu / J. Math. Anal. Appl. 309 (2005) f j (x) ɛ j1 x j for x j [0,r 1 ], f j (x) ɛ j2 x j for x j [r 2, ). Let c j2 = max{ɛ j1,ɛ j2, max r1 x r 2 {f j (x)/x j }} > 0, j= 1, 2,...,m. Then we have f j (x) c j2 x j for x j [0, ). Assume ψ(n) is a positive T -periodic solution of (1.1). We will show that this leads to a contradiction for 0 <λ<λ 0, where λ 0 = 1/(Mp(max c j2 )T ). Since Fψ(n)= ψ(n) for n [0,T 1], it follows from Lemma 2.6 that, for 0 <λ<λ 0, ( ) ψ = Fψ λmp max c j2 T ψ < ψ, which is a contradiction. The proof is complete. Remark. For other results concerning Eq. (1.1) has at least one positive T -periodic solution, we refer the reader to [12]. Finally, it follows from the proof of Theorem 3.1(c) that one can easily show the following statement. Theorem 3.2. Suppose (S 1 ) holds. (a) For every j = 1, 2,...,m, if there is some c j1 such that f j (x) c j1 x j for x [0, ), then there exists λ 0 = 1/(Nq(min c j1 )T γ ) such that for all λ>λ 0, (1.1) has no positive T -periodic solution. (b) For every j = 1, 2,...,m, if there is some c j2 such that f j (x) c j2 x j for x [0, ), then there exists λ 0 = 1/(Mp(max c j2 )T ) such that for all 0 <λ<λ 0, (1.1) has no positive T -periodic solution. References [1] S.-N. Chow, Existence of periodic solutions of autonomous functional differential equations, J. Math. Anal. Appl. 15 (1974) [2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, [3] H.I. Freedman, J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992) [4] D. Jiang, J. Wei, B. Zhang, Positive periodic solutions of functional differential equations and population models, Electron. J. Differential Equations 71 (2002) [5] M. Krasnosel skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, [6] Y. Li, Existence and global attractivity of a positive periodic solution of a class of delay differential equation, Sci. China Ser. A 41 (1998) [7] Y. Li, Y. Kuang, Periodic solutions of periodic delay Lotka Volterra equations and systems, J. Math. Anal. Appl. 255 (2001) [8] P. Liu, Y. Li, Positive periodic solutions of infinite delay functional differential equations depending on a parameter, Appl. Math. Comput. 150 (2004) [9] Y.N. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electron. J. Difference Equations 55 (2002) 1 8.

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