Existence of positive periodic solutions for a periodic logistic equation

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1 Applied Mathematics and Computation 139 (23) Existence of positive periodic solutions for a periodic logistic equation Guihong Fan, Yongkun Li * Department of Mathematics, Yunnan University, Kunming, Yunnan 6591, China Abstract In this paper, we study a more generalized class of periodic logistic equations. By using the method of coincidence degree, a set of sufficient conditions is obtained for the existence of positive periodic solutions of the equations which are under the consideration. Ó 22 Elsevier Science Inc. All rights reserved. Keywords: State dependent delay; Logistic equation; Distributed delay; Periodic solution 1. Introduction Recently Kuang [1] has considered the following model: r _xðtþ ¼xðtÞ f xðt þ sþdlðsþ gðxðtþþ ; ð1þ r where < r < s, f ðxþ and gðxþ are continuously differentiable, lðsþ is nondecreasing and normalized to satisfy lð rþ lð sþ ¼1. Under the assumption of (1) having a unique positive steady state, Kuang has obtained sufficient condition for the positive steady state in (1) to be global attractor regardless of the length of delay. He has also established a global existence result of positive solutions in Eq. (1) when it has only one discrete delay. * Corresponding author. address: yklie@ynu.edu.cn (Y. Li) /2/$ - see front matter Ó 22 Elsevier Science Inc. All rights reserved. PII: S96-33(2)182-

2 312 G. Fan, Y. Li / Appl. Math. Comput. 139 (23) Our aim in this paper is to consider the following more generalized models: " _xðtþ ¼xðtÞ xðt þ sþdlðt; sþ gt; ð xðt sðt; xðtþþþþ ð2þ and " _xðtþ ¼ xðtþ xðt þ sþdlðt; sþ gðt; xðt sðt; xðtþþþþ ; ð3þ here rðtþ, rðtþ 2CðR; ð; 1ÞÞ are x-periodic functions with rðtþ < rðtþ, f ðt; uþ, gðt; uþ, sðt; uþ, lðt; uþ 2CðR R; RÞ are x-periodic with respect to t, lðt; sþ is nondecreasing with respect to s. By using the method of coincidence degree, we obtain sufficient conditions for the existence of at least one positive periodic solution of both Eqs. (2) and (3). 2. Main results In order to establish the existence of x-periodic solutions of both Eqs. (2) and (3), we make some preparations. Let X, Y be normed vector spaces, L: Dom L X be a linear mapping, and N: X be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L ¼ codim Im L < þ1 and Im L is closed in. IfL is a Fredholm mapping of index zero, there exist continuous projectors P: X X and Q: such that Im P ¼ Ker L, Ker Q ¼ Im L ¼ ImðI QÞ. It follows that mapping Lj T Dom L Ker P : ði PÞX ImL is invertible. We denote the inverse of that mapping by K P.IfX is an open bounded subset of X, the mapping N will be called L-compact on X if QNðXÞ is bounded and K P ði QÞN : X X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J: ImQ Ker L. For convenience, we introduce MawhinÕs continuous theorem [2, p. 4], as follows. Lemma 1. Let X 2 X be an open bounded set and let N: X X be a continuous operator which is L-compact on X. Assume (i) X 2 ox T Dom L, Lx 6¼ knx for each k 2ð; 1Þ, (ii) QNx 6¼ and degðjnq; X T Ker L; Þ 6¼ ; for each x 2 ox T Ker L. Then Lx ¼ Nx has at least one solution in X T Dom L.

3 Making the change of variable yðtþ ¼e xðtþ, then Eqs. (2) and (3) are reduced to _xðtþ ¼ G. Fan, Y. Li / Appl. Math. Comput. 139 (23) gðt; e xðt sðt;exðtþþþ Þ; ð4þ and _xðtþ ¼ gðt; e xðt sðt;exðtþ ÞÞ Þ; ð5þ respectively. It is obvious that Eq. (2) has a positive periodic solution if and only if Eq. (4) exists a periodic solution, and the same conclusion also holds for Eqs. (3) and (5). In what follows, we shall use the notation zðtþ ¼ 1 x zðtþ ; where z 2 CðR; RÞ. Our main results in this paper is as follows. Theorem 1. Assume that there exist constants g 1, g 2, c 1 and c 2 with c 2 > 1 > c 1 >, B with B > g 2 such that (i) g 1 < gðt; xþ < g 2 as x P uniformly in t 2 R; (ii) f ðt; xþ > g 2 as x > c 2 uniformly in t 2 R, and f ðt; xþ < g 1 as x < c 1 uniformly in t 2 R; (iii) one of the following two conditions holds (1 ) f ðt; xþ 6 B as x P c 2 uniformly in t 2 R, f ðt; xþ does not change sign as x < c 1 uniformly in t 2 R, (2 ) f ðt; xþ P B as x 6 c 1 uniformly in t 2 R, f ðt; xþ does not change sign as x > c 2 uniformly in t 2 R; (iv) lðt; sþ is strictly increasing with respect to s and min ½lðt; Þ lðt; ÞŠ ¼ a 1 > ; t2½;xš max½lðt; Þ lðt; ÞŠ ¼ a 2 : t2½;xš Then both Eqs. (4) and (5) have at least one x-periodic solution. Proof. We only show that Eq. (4) has at least one x-periodic solution under the hypotheses of the theorem, while the proof of Eq. (5) is similar and will be omitted. Set

4 314 G. Fan, Y. Li / Appl. Math. Comput. 139 (23) X ¼ ¼fx 2 CðR; RÞjxðt þ xþ ¼xðtÞg and kxk ¼max t2½;xš jxðtþj for any x 2 X (or ). Then X and are Banach spaces when they are endowed with the normkk. Let Nx ¼ gðt; e xðt sðt;exðtþþþ Þ; Lx ¼ _x; Px ¼ 1 xðtþ; x 2 X and Qz ¼ 1 zðtþ; z 2 : x x Obviously, Ker L ¼fxjx 2 X ; x ¼ h; h 2 Rg, ImL ¼fzjz2 ; R x zðtþ ¼ g and dim Ker L ¼ n ¼ codim Im L: Since Im L is closed in, L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that Im P ¼ Ker L; Ker and Q ¼ Im L ¼ ImðI QÞ: Furthermore, the generalized inverse (to L) K P :ImL Ker P T Dom L is given by Since K P ðzþ ¼ QN : X x 1 " x x zðsþds 1 x we have K P ði QÞN : X X " x t f u; t rðuþ rðuþ " t 1 f u; x t 1 " x f u; x 2 zðsþ ds: gðt; e xðt sðt;exðtþþþ Þ ; e xðuþsþ dlðu; sþ gðu; e xðu sðu;exðuþþþ Þ du rðuþ rðuþ rðuþ e xðuþsþ dlðu; sþ gðu; e xðu sðu;exðuþþþ Þ du rðuþ e xðuþsþ dlðu; sþ gðu; e xðu sðu;exðuþþþ Þ du: Clearly, QN and K P ði QÞN are continuous. Using the Arcela Ascoli theorem, it is not difficult to show that K P ði QÞNðXÞ is compact for any open bounded set X X. Moreover, QNðXÞ is bounded. Thus, N is L-compact on X

5 with any open bounded set X X. The isomorphism J of Im Q onto Ker L is the identity mapping, since Im Q ¼ Ker L. Now, we reach the position to search for an appropriate open bounded subset X for the application of the continuation theorem. Corresponding to the operator equation Lx ¼ knx; k 2ð; 1Þ, wehave " _xðtþ ¼k k 2ð; 1Þ: g t; e xðt sðt;exðtþ ÞÞ ; Assume that x 2 X is a solution of Eq. (6) for a certain k 2ð; 1Þ. Integrating (6) over the interval ½; xš, we obtain " x g t; e xðt sðt;exðtþ ÞÞ ¼ ; G. Fan, Y. Li / Appl. Math. Comput. 139 (23) ð6þ that is ¼ g t; e xðt sðt;exðtþ ÞÞ : By the mean value theorem, there exists a point n 2½; xš such that gðn; e xðn sðn;exðnþ ÞÞ Þ¼f n; e xðnþsþ dlðn; sþ ð7þ ð8þ where n is dependent of xðtþ. Denote ( D 1 ¼ t 2½; xš > c 2 ); ( D 2 ¼ t 2½; xš ( D 3 ¼ t 2½; xš c 1 < < c 1 ); < c 2 ): Without loss of generality, let us now assume that the condition (1 ) of (iii) holds. It follows from (7) that

6 316 G. Fan, Y. Li / Appl. Math. Comput. 139 (23) ¼ D 2 D 1 D 2 f D 3 t; þ g t; e xðt sðt;exðtþ ÞÞ Bx Cx þ g 1 x; ¼ f D 1 t; f D 3 t; þ g t; e xðt sðt;exðtþ ÞÞ jg 2 jx þ Cx þ g 2 x; where C ¼ max fjf ðt; xþjjt 2½; xš; x 2½c 1 ; c 2 Šg. Taking into account the second half of (1 ), we then deduce that D 2 maxfj Bx Cx þ g 1 xj; jg 2 jx þ Cx þ g 2 xg: From (ii) and the first part of (1 ), we have g 2 < f ðt; xþ < B as x > c 2 uniformly in t 2 R; hence D 1 maxfjkjx; Bxg:

7 Immediately, we have ¼ D 1 þ D 2 þ D 3 maxfjkjx; Bxgþmaxfj Bx Cx þ g 1 xj; jg 2 jx þ Cx þ g 2 xg þ Cx ¼ def M I : By using the assumption of (1 ), we have g t; e xðt sðt;exðtþ ÞÞ maxfjg 1 xj; jg 2 xjg def ¼ M II : Therefore, that is j_xðtþj ¼ G. Fan, Y. Li / Appl. Math. Comput. 139 (23) < k g t; e xðt sðt;exðtþ ÞÞ g t; e xðt sðt;exðtþ ÞÞ þ j_xðtþj < M: < M I þ M II ¼ def M; Under the assumption of the theorem, we claim that there exists a positive constant N 1 which is independent of k, for every periodic solution xðtþ of Eq. (6), there exists a t 2½; xš (which is dependent of xðtþ) such that xðt Þ < N 1. For the sake of contradiction, we assume that for any N >, there exists a periodic solution x N ðtþ of Eq. (6) for a certain k 2ð; 1Þ such that x N ðtþ > N for t 2½; xš: Since x N ðtþ is a periodic solution of Eq. (6) for a certain k 2ð; 1Þ, x N ðtþ satisfies (8), we have

8 318 G. Fan, Y. Li / Appl. Math. Comput. 139 (23) g n; e x N ðn sðn;e x N ðnþ ÞÞ ¼ f n; e x N ðnþsþ dlðn; sþ ; hence f n; e x N ðnþsþ dlðn; sþ < g 2 here n 2½; xš is dependent of x N ðtþ. We let N > lnðc 2 =a 1 Þ (without loss of generality, we think a 1 < c 2 ) then hence f n; e x N ðnþsþ dlðn; sþ > e N ðlðn; Þ lðn; ÞÞ > e N a 1 > c 2 ; e x N ðnþsþ dlðn; sþ > g 2 ; which results in contradiction. Therefore our claim holds. For any periodic solution xðtþ of Eq. (6), according to the claim, there exists a t 2½; xš such that xðt Þ < N 1, integrating _xðtþ from t to t, t 2½; xš, xðtþ ¼xðt Þþ t t _xðsþds N 1 þ j_xðsþjds N 1 þ M ¼ def M 2 : Clearly, M 2 is independent of k. Under the assumption of the theorem, we again claim that there exists a negative constant N 2 which is independent of k, for every periodic solution xðtþ of Eq. (6), there exists a t I 2½; xš (which is dependent of xðtþ) such that xðt I Þ > N 2. For the sake of contradiction, we assume that for any N <, there exists a periodic solution x N ðtþ of Eq. (6) for a certain k 2ð; 1Þ such that x N ðtþ < N for t 2½; xš: Since x N ðtþ is a periodic solution of Eq. (6) for a certain k 2ð; 1Þ, x N ðtþ satisfies (8) g n; e x N ðn sðn;e x N ðnþ ÞÞ ¼ f n; e x N ðnþsþ dlðn; sþ ; hence f n; e x N ðnþsþ dlðn; sþ > g 1 ; here n 2½; xš is dependent of x N ðtþ.

9 We let N < lnðc 1 =a 2 Þ (without loss of generality, we think a 2 > c 1 ) then in view of (ii) f n; e x N ðnþsþ dlðn; sþ < e N ðlðn; Þ lðn; ÞÞ < e N a 2 < c 1 ; e x N ðnþsþ dlðn; sþ < g 1 ; this contradiction establishes our claim. For any periodic solution xðtþ of Eq. (6), according to the claim, there exists a t I 2½; xš such that xðt I Þ > N 2, integrating _xðtþ from t I to t, t 2½; xš, t xðtþ ¼xðt I Þþ _xðsþds N 2 j_xðsþjds N 2 M def ¼ M 1 : t I Clearly, M 1 is independent of k. From (ii), for t 2½; xš there exists c 2 > 1 > c 1 > such that and f ðt; xþ > g 2 for x > c 2 f ðt; xþ < g 1 for x < c 1 : We choose c ¼ max fj lnðc 1 =a 2 Þj; lnðc 2 =a 1 Þg, then Hence and hence e x dlðt; sþ > e c ðlðt; Þ lðt; ÞÞ > e c a 1 c 2 for x > c : e x dlðt; sþ > g 2 for x > c e x dlðt; sþ < e c ðlðt; Þ lðt; ÞÞ < e c a 2 c 1 for x < c ; G. Fan, Y. Li / Appl. Math. Comput. 139 (23) e x dlðt; sþ < g 1 for x < c ;

10 32 G. Fan, Y. Li / Appl. Math. Comput. 139 (23) therefore and > < fðt; e x ðlðt; Þ lðt; rðtþþþþ > gðt; e x Þ for x > c fðt; e x ðlðt; Þ lðt; rðtþþþþ < gðt; e x Þ for x < c g 2 g 1 so x ½ fðt; e x ðlðt; rðtþþ lðt; rðtþþþþ gðt; e x ÞŠ > for jxj > c : Set H ¼jM 1 jþm 2 þ c, let X ¼fx 2 X jkxk < Hg. It is clear that X verifies the requirement (a) in Lemma 1. When x 2 ox T R, x is a constant in R with kxk ¼H. Then QNx ¼ 1 x ¼ " g t; e xðt sðt;exðtþ ÞÞ ½fðt; e x ðlðt; Þ lðt; rðtþþþþ gðt; e x ÞŠ 6¼ : Let F ðx; aþ ¼ax þ 1 a ½fðt; e x ðlðt; rðtþþ lðt; rðtþþþþ 2p gðt; e x ÞŠ ; then for every x 2 ox T Ker L and a 2½; 1Š, we have x T F ðx; aþ ¼ax 2 þ 1 a 2p ½ gðt; e x ÞŠ > : fðt; e x ðlðt; rðtþþ lðt; rðtþþþþ

11 G. Fan, Y. Li / Appl. Math. Comput. 139 (23) Thus F ðx; aþ 6¼, i.e. F ðx; aþ is homotopy transformation. Therefore degfqn; X \ Ker L; g ¼deg ½fðt; e x ðlðt; rðtþþ lðt; rðtþþþþ gðt; e x ÞŠ; X \ R; ¼ degfi; X \ R; g 6¼ : By now we have proved that X verifies all the requirements of Lemma 1. Hence, Eq. (4) has at least one x-periodic solution x I ðtþ in X. Using the same method, we can prove Eq. (5) also has at least one x-periodic solution x I ðtþ in X. The proof is complete. Acknowledgements This work is supported by National Natural Sciences Foundation of PeopleÕs Republic of China and Natural Sciences Foundation of Yunnan Province. References [1] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, [2] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.

Existence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1

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