PERIODIC SOLUTIONS OF PERIODIC GENERALIZED FOOD LIMITED MODEL

By using the continuation theorem of coincidence degree theory, the existence of positive periodic solutions for a periodic generalized food limited model with state dependent delays and distributed delays is studied, respectively. 2000 Mathematics Subject Classification. Primary 34K13.


PERIODIC SOLUTIONS OF PERIODIC GENERALIZED FOOD LIMITED MODEL YONGKUN LI
(Received 14 December 1999) Abstract.By using the continuation theorem of coincidence degree theory, the existence of positive periodic solutions for a periodic generalized food limited model with state dependent delays and distributed delays is studied, respectively.
1. Introduction.Gopalsamy, Kulenović, and Ladas [4] studied the existence of periodic solutions of the following periodic food limited model: where K, r , c are continuous positive ω-periodic functions, m is a positive integer.Also, since as it was indicated by Freedman and Wu [1] that it would be of interesting to study the existence of periodic solutions for population models with periodic delays; and since at present, there are only a few papers which have been published on the existence of periodic solutions of state dependent delay differential equations (see [6] and references therein).Our main purpose in this paper is-by using the Mawhin's continuation theorem of coincidence degree theory [2,10]-to establish the existence of positive periodic solutions for the periodic generalized food limited model with state dependent delays and with distributed delays, where K > 0, r > 0, a j ≥ 0, b j ≥ 0 (j = 1, 2,...,n) are continuous ω-periodic functions, τ j ,σ j ∈ C(R 2 ,R) (j = 1, 2,...,n) are ω-periodic with respect to their first arguments, respectively, l j ,h j (j = 1, 2,...,n) are positive constants, η j ,µ j (j = 1, 2,...,n) are nondecreasing with respect to their arguments, respectively, and θ is a positive odd number.As a special case of (1.2), we also obtain the existence of periodic solutions of the following generalized food limited model with periodic delays where K > 0, r > 0, a j ≥ 0, b j ≥ 0, τ j ,σ j (j = 1, 2,...,n) are continuous ω-periodic functions, and θ is a positive odd number.For the ecological sense of (1.4), we refer to [3] and the references therein.For some work concerning the existence of periodic solutions of functional differential equations which was done by using the Mawhin's coincidence degree theory we refer to [5,7,8,9].

Main results.
In this section, by using the Mawhin continuation theorem we show the existence of at least one positive periodic solution of (1.2) and (1.3).To do so, we first make some preparations.
Let X, Y be real Banach spaces, L : Dom L ⊂ X → Y a Fredholm mapping of index zero, and In what follows, we use the notation where u is a continuous ω-periodic function.
We are now in a position to state and prove our first main result.

Theorem 2.2. Equation (1.2) has at least one positive ω-periodic solution.
Proof.Consider the following equation where K, r , a j ,b j ,τ j ,σ j (j = 1, = 2,...,n) and θ are the same as those in (1.2).It is easy to see that if (2.2) has an ω-periodic solution x * (t), then N * (t) = exp(x * (t)) is a positive ω-periodic solution of (1.2).So, to complete the proof, it suffices to show that (2.2) has an ω-periodic solution.
In order to use the continuation theorem of coincidence degree theory to establish the existence of ω-periodic solution of (2.2), we take where Dom L = {x(t) ∈ C 1 (R, R)}, and Define two projectors P and Q as Hence, L is a Fredholm mapping of index zero.Furthermore, through an easy computation, we find that the inverse K P of L P has the form ( Notice that (2.7) We find , λ∈ (0, 1).

.18)
Denote by A = max{| ln |K| 0 |,A 1 + A 2 } + A 3 , here A 3 > 0 is taken sufficiently large such that K(t) − e A − e A n j=1 a j (t) < 0 and K(t) − e −A − e −A n j=1 a j (t) > 0 for t ∈ [0,ω] and take Ω = {x(t) ∈ X : |x| 0 < A}, then it is clear that Ω satisfies condition (i) in Lemma 2.1.When x ∈ ∂Ω ∩ R, x is a constant with |x| = A. Hence, Furthermore, take J = I : ImQ → Ker L, x x, by a straightforward computation, we find (2.20) According to Lemma 2.1, we have completed the proof.
Next, immediately, from Theorem 2.2, we have the following corollary.
Remark 2.4.In (1.4), when b j (t) = 0 for all t ∈ [0,ω] and j = 1, 2,...,n, equation (1.4) becomes the well-known logistic equation with several delays.Finally, similar to the proof of Theorem 2.2, one can prove the following theorem.Proof.The proof is similar to the one of Theorem 2.2, to complete the proof, it suffices to show that the following equation: has an ω-periodic solution, where r ,K,θ,a j ,b j ,l j ,h j ,η j ,µ j , j = 1, 2,...,n are the same as those in (1.3).To this end, set and L, P , Q, X are the same as those in the proof of Theorem 2.2.Corresponding to the operator equation Lx = λx, λ ∈ (0, 1), we have  (2.31) According to this and (2.30), we have (2.32) From this and (2.28) it follows that x(t) 0 < max ln |K| 0 ,B 1 + B 2 . (2.33) The rest of the proof is similar to the one of Theorem 2.2 and will be omitted.

Lemma 2 . 1 .
and X = Ker L Ker P , Y = Im L Im Q. Denote by L P the restriction of L to Dom L Ker P , K P : ImL → Ker P Dom L the inverse (to L P ), and J : ImQ → Ker L an isomorphism of Im Q onto Ker L. For convenience, we introduce the continuation theorem [2, page 40] as follows.Let Ω ⊂ X be an open bounded set and N : X → Y be a continuous operator which is L-compact on Ω (i.e., QN : Ω → Y and K P (I − Q)N : Ω → Y are compact).Assume (i) for each λ ∈ (0, 1), x ∈ ∂Ω Dom L, Lx ≠ λNx; (ii) for each x ∈ ∂Ω Ker L, QNx ≠ 0, and deg{JQN, Ω Ker L, 0} ≠ 0. Then Lx = Nx has at least one solution in Ω Dom L.
n j=1 b j (s) exp x s − σ j s, e x(s) Obviously, QN and K P (I − Q)N are continuous by the Lebesgue theorem and QN( Ω), K P (I − Q)N( Ω) are relatively compact for any open bounded set Ω ⊂ X.Therefore, N is L-compact on Ω for any open bounded set Ω ⊂ X. Corresponding to the operator equation Lx = λNx, λ ∈ (0, 1), we have