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POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER GUANG ZHANG AND SUI SUN CHENG Received 5 November 2001 This article investigates the existence of positive periodic solutions for a firstorder functional diﬀerential equations of the form y (t) = −a(t)y(t) + λh(t) f y t − τ(t) , (1) where a = a(t), h = h(t), and τ = τ(t) are continuous T-periodic functions. We will also assume that T > 0, λ > 0, f = f (t) as well as h = h(t) are positive, T 0 a(t) dt > 0. Functional diﬀerential equations with periodic delays appear in a number of ecological models. In particular, our equation can be interpreted as the standard Malthus population model y = −a(t)y subject to perturbation with periodical delay. One important question is whether these equations can support positive periodic solutions. Such questions have been studied extensively by a number of authors (cf. [1, 2, 3, 4, 6, 7] and the references therein). In this paper, we are concerned with the existence and nonexistence of periodic solutions when the parameter λ varies. For this purpose, we call a continuously diﬀerentiable and T-periodic function a periodic solution of (1) associated with λ∗ if it satisfies (1) when λ = λ∗ . We show that there exists λ∗ > 0 such that (1) has at least one positive T-periodic solution for λ ∈ (0, λ∗ ] and does not have any T-periodic positive solutions for λ > λ∗ . Our technique is based on the well-known upper and lower solutions method (cf. [5]). We proceed from (1) and obtain t y(t) exp a(s) ds 0 = λ exp t 0 Copyright © 2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:5 (2002) 279–286 2000 Mathematics Subject Classification: 34B15, 34K13 URL: http://dx.doi.org/10.1155/S1085337502000878 a(s) ds h(t) f y t − τ(t) . (2) 280 Bifurcation in functional diﬀerential equations After integration from t to t + T, we obtain y(t) = λ t+T G(t, s)h(s) f y s − τ(s) ds, (3) t where s G(t, s) = a(u) du exp T exp t . (4) a(u) du − 1 0 Note that the denominator in G(t, s) is not zero since we have assumed that 0 a(t) dt > 0. It is not diﬃcult to check that any T-periodic function y(t) that satisfies (3) is also a T-periodic solution of (1). Note further that T 0 < N ≡ min G(t, s) ≤ G(t, s) ≤ max G(t, s) ≡ M, 0≤s,t ≤T 1≥ 0≤t,s≤T t ≤ s ≤ t + T, min0≤s,t≤T G(t, s) N G(t, s) ≥ = > 0. max0≤s,t≤T G(t, s) max0≤s,t≤T G(t, s) M (5) Now let X be the set of all real T-periodic continuous functions, endowed with the usual linear structure as well as the norm y = sup y(t). (6) 0≤t ≤T Then X is a Banach space with cones Φ = y(t) ∈ X : y(t) ≥ 0 , (7) Ω = y(t) : y(t) ≥ σ y , t ∈ R , where σ = N/M. Define a mapping F : X → X by (F y)(t) = λ t+T G(t, s)h(s) f y s − τ(s) ds. (8) t Then it is easily seen that F is completely continuous on bounded subsets of Ω and for y ∈ Φ, (F y)(t) ≤ λM T h(s) f y s − τ(s) ds (9) ds ≥ σ F y . (10) 0 so that (F y)(t) ≥ λN T 0 h(s) f y s − τ(s) G. Zhang and S. S. Cheng 281 That is, FΦ is contained in Ω. Lemma 1. The mapping F maps Φ into Ω. Lemma 2. Suppose that lim u→+∞ f (u) = +∞. u (11) Let I be a compact subset of (0, +∞). Then there exists a constant bI > 0 such that u < bI for all λ ∈ I and all possible T-periodic positive solutions u of (1) associated with λ. Proof. Suppose to the contrary that there is a sequence {un } of T-periodic positive solutions of (1) associated with {λn } such that λn ∈ I for all n and un → +∞ as n → ∞. Since un ∈ Ω, min un (t) ≥ σ un . (12) 0≤t ≤T By (11), we may choose R f > 0 such that f (u) ≥ ηu for all u ≥ R f , and there exists n0 such that σ un0 ≥ R f , where η satisfies T σηNλn0 h(s) ds > 1. (13) 0 Thus, we have un ≥ un (t) = λn 0 0 0 ≥ σηNλn0 T 0 t+T t G(t, s)h(s) f un0 s − τ(s) ds (14) h(s)un0 ds > un0 . This is a contradiction. The proof is complete. Lemma 3. Suppose that f is nondecreasing on [0, +∞) and f (0) > 0. (15) Let (1) have a T-periodic positive solution y(t) associated with λ > 0. Then (1) also has a positive T-periodic solution associated with λ ∈ (0, λ). Proof. In view of (3) and (15), we have y(t) = λ ≥λ t+T t t+T t G(t, s)h(s) f y s − τ(s) G(t, s)h(s) f y s − τ(s) ds t+T ds, 0<λ (16) G(t, s)h(s) f (0) ds. t 282 Bifurcation in functional diﬀerential equations Let y 0 (t) = y(t), t+T y k+1 (t) = λ G(t, s)h(s) f y k s − τ(s) t ds, k = 0, 1, 2, . . ., (17) k = 0, 1, 2, . . .. (18) y (t) = 0, and 0 y k+1 (t) = λ t+T G(t, s)h(s) f y s − τ(s) k t ds, Clearly, we have y 0 (t) ≥ y 1 (t) ≥ · · · ≥ y k (t) ≥ y (t) ≥ · · · ≥ y (t) ≥ y (t). k 1 (19) 0 If we now let y(t) = limk→∞ y k (t), then y(t) satisfies (3). Clearly, we have y(t) ≥ y (t) = λ 1 t+T G(t, s)h(s) f (0) ds > 0. (20) t This completes our proof. Lemma 4. Suppose that (11) and (15) hold. Then there exists λ∗ > 0 such that (1) has a T-periodic positive solution. Proof. Let β(t) = t+T G(t, s)h(s) ds, t M f = max f β t − τ(t) , 0≤t ≤T λ∗ = 1 . Mf (21) We have β(t) = t+T t G(t, s)h(s) ds ≥ λ∗ t+T t+T 0 < λ∗ G(t, s)h(s) f β s − τ(s) ds, t (22) G(t, s)h(s) f (0) ds. t Let y 0 (t) = β(t), y k+1 (t) = λ∗ t+T t G(t, s)h(s) f y k s − τ(s) ds, k = 0, 1, 2, . . ., (23) k = 0, 1, 2, . . .. (24) y (t) = 0, and 0 y k+1 (t) = λ∗ t+T t G(t, s)h(s) f y s − τ(s) k ds, G. Zhang and S. S. Cheng 283 Clearly, we have y 0 (t) ≥ y 1 (t) ≥ · · · ≥ y k (t) ≥ y (t) ≥ · · · ≥ y (t) ≥ y (t). k 1 0 (25) If we now let y(t) = limk→∞ y k (t), then y(t) satisfies (3). Clearly, we have y(t) ≥ y (t) = λ∗ 1 t+T G(t, s)h(s) f (0) ds > 0. (26) t The proof is complete. ∗ Theorem 5. Suppose that (11) and (15) hold. Then there exists λ > 0 such that (1) has at least one positive T-periodic solution for λ ∈ (0, λ∗ ] and does not have any T-periodic positive solutions for λ > λ∗ . Proof. Suppose to the contrary that there is a sequence {un } of T-periodic positive solutions of (1) associated with {λn } such that limn→∞ λn = ∞. Then either we have un j → +∞ as j → ∞ or there is M̃ > 0 such that un ≤ M̃. Assume the former case holds. Note that un ∈ Ω and thus min un (t) ≥ σ un . 0≤t ≤T (27) By (11), we may choose R f > 0 and η1 > 0 such that f (u) ≥ η1 u when σu ≥ R f . On the other hand, there exist {tn } ⊂ [0, T] such that un j (tn j ) = un j and un j (tn j ) = 0 by the periodicity of {un j (t)}. In view of (1), we have a tn j un j = a tn j u tn j = λn j h tn j f un j tn j − τ tn j ≥ λn j η1 σh tn j un j (28) for all large j. That is, we have λn j ≤ a(tn j )/(η1 σh(tn j )). Note that a(t)/h(t) is bounded. Thus, we obtain a contradiction. Next, suppose that the latter case holds. In view of (15), there exists η2 > 0 such that f (0) ≥ η2 M̃. Then as above, we will obtain a tn un = a tn u tn = λn h tn f un tn − τ tn ≥ λn η2 h tn M̃ ≥ λn η2 h tn un for all n. A contradiction will again be reached. (29) 284 Bifurcation in functional diﬀerential equations Thus, there exists λ∗ > 0 such that (1) has at least one positive T-periodic solution for λ ∈ (0, λ∗ ) and no T-periodic positive solutions for λ > λ∗ . Finally, we assert that (1) has at least one T-periodic positive solution for λ = λ∗ . Indeed, let {λn } satisfy 0 < λ1 < · · · < λk < λ∗ and limk→∞ λk = λ∗ . Since un (t) is T-periodic positive solution of (1) associated with λn and Lemma 2 implies that the set {un (t)} of solutions is uniformly bounded in Ω, the sequence {un (t)} has a subsequence converging to u(t) ∈ Ω. We can now apply the Lebesgue convergence theorem to show that u(t) is a T-periodic positive solution of (1) associated with λ = λ∗ . The proof is complete. Example 6. Consider the equation x (t) + a(t)x(t) = λh(t) xγ t − τ(t) + 1 , γ > 1, (30) where a, h, and τ satisfy the same assumptions stated for (1). In view of Theorem 5, there exists a λ∗ > 0 such that (30) has at least one T-periodic positive solution for λ ∈ (0, λ∗ ] and no T-periodic positive solution for λ > λ∗ . Example 7. Consider the equation y (t) = −ay(t) + λb y 2 (t) + ε , (31) where a, b, ε > 0. Note that the function f (x) = (x2 + ε) satisfies (11) and (15) in Theorem 5. Therefore Theorem 5 may be applied. However, we may give a direct √ proof that, for λ > a/(2b ε), this equation cannot have any positive 2π-periodic solutions associated with λ. Indeed, assume to the contrary that y(t) is such a solution. Then y (ξ) = 0 for some ξ ∈ [0, 2π]. Hence −ay(ξ) + λby 2 (ξ) + λbε = 0. (32) However, since the discriminant of the quadratic equation λbx2 − ax + λbε = 0 (33) a2 − 4λ2 b2 ε < 0, (34) satisfies a contradiction is obtained. We remark that when ε = 0, our equation reduces to the well-known logistic equation. Similarly, we can consider the equation x (t) = a(t)x(t) − λh(t) f x t − τ(t) , (35) G. Zhang and S. S. Cheng 285 where a = a(t), h = h(t), and f = f (t) satisfy the same assumptions stated for (1). By (35), we have x(t) = t+T H(t, s)h(s) f x s − τ(s) ds, (36) t where s exp − a(u) du H(t, s) = t+T a(u) du exp t s T = T 1 − exp − a(u) du exp a(u) du − 1 0 (37) 0 which satisfies M ≥ H(t, s) ≥ N, t ≤ s ≤ t + T, (38) for some M and N > 0, and σ = N/M ≤ 1. Theorem 8. Suppose that (11) and (15) hold. Then there exists λ∗ > 0 such that (35) has at least one positive T-periodic solution for λ ∈ (0, λ∗ ] and no T-periodic positive solution for λ > λ∗ . Acknowledgment This work was supported by the Natural Science Foundation of Shanxi Province and Yanbei Normal College. Part of this work was done during the first author’s visit to the Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences. The first author wishes to express his thanks to Professor Daomin Cao for his kind invitation and nice hospitality. We also thank the referee for his helpful criticisms. References [1] [2] [3] [4] [5] S. S. Cheng and G. Zhang, Existence of positive periodic solutions for non-autonomous functional diﬀerential equations, Electron. J. Diﬀerential Equations 2001 (2001), no. 59, 1–8. M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coeﬃcients, Math. Biosci. 152 (1998), no. 2, 165–177. , Uniform ultimate boundedness and periodic solutions of functionaldiﬀerential equations with infinite delay, J. Systems Sci. Math. Sci. 19 (1999), no. 3, 323–327 (Chinese). D. Q. Jiang and J. J. Wei, Existence of positive periodic solutions for nonautonomous delay diﬀerential equations, Chinese Ann. Math. Ser. A 20 (1999), no. 6, 715–720 (Chinese). Y.-H. Lee, Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Diﬀerential Equations 174 (2001), no. 2, 420–441. 286 [6] [7] Bifurcation in functional diﬀerential equations Y. K. Li, Existence and global attractivity of positive periodic solutions for a class of delay diﬀerential equations, Science in China Series A 28 (1998), no. 2, 108–118 (Chinese). S. N. Zhang, Periodicity in functional-diﬀerential equations, Ann. Diﬀerential Equations 12 (1996), no. 2, 252–257. Guang Zhang: Department of Mathematics, Yanbei Normal College and Datong College, Datong, Shanxi 037000, China E-mail address: [email protected] Sui Sun Cheng: Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, Taiwan E-mail address: [email protected]