Periodic Solutions of First-Order Functional Differential by Seshadev Padhi, John R. Graef, P. D. N. Srinivasu

By Seshadev Padhi, John R. Graef, P. D. N. Srinivasu

This publication presents state-of-the-art effects at the lifestyles of a number of optimistic periodic recommendations of first-order useful differential equations. It demonstrates how the Leggett-Williams fixed-point theorem could be utilized to review the lifestyles of 2 or 3 confident periodic ideas of sensible differential equations with real-world purposes, really in regards to the Lasota-Wazewska version, the Hematopoiesis version, the Nicholsons Blowflies version, and a few versions with Allee results. Many fascinating enough stipulations are given for the dynamics that come with nonlinear features exhibited by means of inhabitants types. The final bankruptcy presents effects relating to the worldwide charm of strategies to the types thought of within the previous chapters. The strategies utilized in this e-book might be simply understood through someone with a uncomplicated wisdom of study. This e-book bargains a precious reference advisor for college kids and researchers within the box of differential equations with purposes to biology, ecology, and the environment.

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Clearly, f ∈ < ω12 implies that there exist φ ∞ (0, ω12 ) and β > 0 such that f (t, x) < φx for x ≥ β and 0 ≤ t ≤ T . 1. It is easy to show that Aτ : K c4 → K c4 , where c4 > max ω2θ , λc2 . 1. Then for x ∞ K (ψ, c2 , c3 ), from (H5 ) we have ≥Aτ x≥ > c2 . In addition, for x ∞ K c1 , (H6 ) implies T ≥Aτ x≥ ≤ ωτ f (s, x(h(s))) ds 0 T c1 ds ω2 < ωτ < c1 . 1. 1) has at least three positive T -periodic solutions. The theorem is now proved. 4 Let f ∈ < such that (H5 ) holds and (H7 ) f (t, x) < (λ−1)2 c1 λ3 (λ−1)2 λ3 and assume there exist constants 0 < c1 < c2 for x ∞ K , 0 ≤ x ≤ c1 and 0 ≤ t ≤ T .

Xm ) = f (t, x1 , . . , xm ), γi (t + T, x) = γi (t, x) for any x ∞ R+ , t ∞ R, i = 1, . . , m, and T > 0 is a constant. 5 Han and Wang [3] Assume that a1 (t) ≤ a(t, x) ≤ a2 (t) for any (t, x) ∞ R × R+ , where a1 and a2 are nonnegative T -periodic continuous functions T on R and 0 a1 (s) ds > 0. Let lim sup a2 (t) f (t, u 1 , u 2 , . . , u m ) < |u| θ lim sup a2 (t) f (t, u 1 , u 2 , . . 3 Positive Periodic Solutions of the Equation x (t) = a(t)x(t) − τb(t) f (t, x(h(t))) uniformly for t ∞ R, where θ = exp( exp( T 0 T 0 a2 (t)dt)−1 a1 (t)dt)−1 45 .

Let k1 = exp ⎜ T exp 0 c(β) dβ . Then, ψ= =ω ⎜ T 0 b(β) dβ and k2 = k2 k2 (k2 − 1) 1 ω , ω= , k1 ≤ k2 , and λ = = > 1. k2 − 1 k1 − 1 ψ (k1 − 1) For any x ∞ X , we have 48 2 Positive Periodic Solutions of Nonlinear Functional Differential Equations τk2 ≥Aτ x≥ ≤ k1 − 1 T f (s, x(s − γ1 (s, x(s))), . . , x(s − γm (s, x(s)))) ds 0 and τ (Aτ x)(t) ≥ k2 − 1 T f (s, x(s − γ1 (s, x(s))), . . , x(s − γm (s, x(s)))) ds 0 1 (k1 − 1) ≥Aτ x≥ = ≥Aτ x≥. ≥ k2 (k2 − 1) λ Thus, if we define a cone K on X by K = x ∞ X : x(t) ≥ 1 ≥x≥ , λ then Aτ : K → K .

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