Classical Mathematical Physics: Dynamical Systems and Field by Walter Thirring, E.M. Harrell

By Walter Thirring, E.M. Harrell

This quantity combines the enlarged and corrected variations of either volumes on classical physics of Thirring's recognized path in mathematical physics. With a variety of examples and comments accompanying the textual content, it really is appropriate as a textbook for college kids in physics, arithmetic, and utilized arithmetic.

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2 Positive Periodic Solutions of the Equation x (t) = a(t)x(t) − τ f (t, x(h(t))) 35 c2 MT , . (1 − λ)(T − λ1 ) λ c4 > max Now for x ∞ K c4 , we have t+T ≥Aτ x≥ = sup τ 0≤t≤T G(t, s) f (s, x(h(s))) ds t t+T ≤ sup τ 0≤t≤T G(t, s)(a(s)x(h(s)) λ1 + M) ds t t+T ≤ sup τ 0≤t≤T ⎡ G(t, s)(a(s)≥x≥ λ1 + M) ds t t+T t+T ≤ τ ⎣ λ1 c4 sup G(t, s) ds ⎦ a(s)G(t, s) ds + sup M 0≤t≤T 0≤t≤T t ⎤ t M T 1−λ ≤ τ λ1 c4 + < c4 . Hence, Aτ : K c4 → K c4 . Next, we define a nonnegative concave continuous functional ψ on K by ψ(x) = min x(t).

A(β, x(β)) dβ − 1 0 In view of the above, we define an operator Aτ by t+T Aτ x = τ G(t, s) f (s, x(s − γ1 (s, x(s))), . . 30) t for every x ∞ X and t ∞ R. Clearly Aτ x(t + T ) = Aτ x(t) and Aτ : X → X . The Green’s kernel G(t, s) satisfies the inequality T c(β) dβ exp 1 ψ= T exp c(β) dβ − 1 0 0 ≤ |G(t, s)| ≤ T exp b(β) dβ − 1 0 for every 0 ≤ t ≤ s ≤ t + T . 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))), .

10 Let τ = 1, f˜0 < 1, and f˜∈ < 1. In addition, assume that there exists ρ > 0 such that f (t, x) > a(t)|x| for μρ < |x| < ρ, where μ = T ex p{− 0 a(s) ds}. 5) has at least two positive T -periodic solutions x1 and x2 such that 0 < ≥x1 ≥ < ρ < ≥x2 ≥. 9. 1 Let f˜0 < T and f˜∈ < T , and assume that there exists a constant c2 > 0 such that either (H1 ) or (H8 ) holds. 5) has at least three positive T -periodic solutions for 1 λ−1 <τ < . 10. 10 it is 1. 1. 11 Let f˜∈ < T . Assume that there are constants 0 < c1 < c2 such that (H1 ) holds and (H9 ) f (t, x) < x for 0 ≤ x ≤ c1 and 0 ≤ t ≤ T .

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