(4.1) D r v(t) ω(t, v(t))


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1 1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution of the differentil inequlity (4.1) D r v(t) ω(t, v(t)) on [, b) if v(t) is continuous nd hs right hnd derivtive on [, b) tht stisfies (4.1). Theorem 4.1. Let ω C r (Ω, lr), r 1, where Ω lr 2 is n open connected set. If u(t) is solution of the eqution (4.2) u = ω(t, u) on [, b] nd v is solution of (4.1) on [, b) with v() u(), then v(t) u(t) for t [, b). Proof. For ny positive integer n, let u n (t) designte the solution of the eqution u = ω(t, u) + 1 n with u n () = u(). From Corollry 3.1 nd Exercise 3.5, there is n n 0 such tht u n, for n n 0, is defined on [, b] nd u n (t) u(t) uniformly on [, b] s n. Suppose tht v(t) is not u(t) for t < b. Then there exist t 1, < t 1 < b, such tht v(t 1 ) > u(t 1 ). Since u n (t) u(t) uniformly on [, b] s n, there is n integer n such tht v(t 1 ) > u n (t 1 ). Thus, there is t 2 < t 1 in (, b) such tht v(t) > u n (t) on t 2 < t t 1, v(t 2 ) = u n (t 2 ). This implies tht D r v(t 2 ) u n (t 2 ) = ω(t 2, u n (t 2 )) + 1 n = ω(t 2, v(t 2 )) + 1 n > ω(t 2, v(t 2 )), which is contrdiction. Consequently, v(t) u(t) for t b. This proves the theorem. Corollry 4.1. Suppose tht ω(t, u) stisfies the conditions of Theorem 4.1 nd, in ddition, is nondecresing in u. If u is solution of (4.2) on [, b] nd v(t) is continuous nd stisfies the integrl inequlity (4.3) v(t) v + ω(s, v(s)) ds, t b, v u(), 1
2 then v(t) u(t), t b. Proof. If V (t) is the right hnd side of (4.3), then v(t) V (t) nd V (t) ω(t, V (t)), V () = v u(). Theorem 4.1 implies tht V (t) u(t) for t < b. Since V (t) is continuous on [, b], we hve V (t) u(t) for t b, which proves the corollry. Remrk 4.1. If it not ssumed tht the function ω(t, u) in Corollry 4.1 is nondecresing in u, then the conclusion in the corollry my not be true. The following exmple ws supplied by X.B. Lin. If ω(t, u) = u nd u(0) = 1, then u(t) = e t. If n 2 is n integer, then v(t) = t 1 for t n nd v(t) = 0 for t > n is solution n of the integrl inequlity (4.3) on [0, ). Corollry 4.2. (The Gronwll Inequlity) If α is rel constnt, β(t) 0 nd ϕ(t) re continuous rel functions for t b which stisfy ϕ(t) α + β(s)ϕ(s) ds, t b, then ϕ(t) αe β(s) ds, t b. Proof. Apply Corollry 4.2 with v = α, ω(t, u) = β(t)u. Corollry 4.3. (Generlized Gronwll Inequlity) If β(t) 0, α(t) nd ϕ(t) re continuous rel functions for t b which stisfy ϕ(t) α(t) + β(s)ϕ(s) ds, t b, then ϕ(t) α(t) + If, in ddition, α(t) is continuous nd α 0, then β(s)α(s)e β(u) du s ds, t b. ϕ(t) α(t)e β(s) ds, t b. Exercise 4.1. Prove Corollry 4.3. Let R(t) = β(s)ϕ(s) ds, obtin differentil inequlity for R nd find solution of the inequlity. If α(t) is continuous, then integrte by prts. Exercise 4.2. Consider the liner system of differentil equtions ẋ = A(t)x + h(t), 2
3 where the d d mtrix A nd the dvector h re continuous on n intervl I, finite or infinite. Prove tht the solution of the initil vlue problem exists on I. Hint: Fix closed intervl Ī I, tke τ Ī, ξ lrd nd let v(t) = x(t). Obtin n integrl inequlity for v nd use the generlized Gronwll inequlity. Differentil inequlities re very convenient for obtining bounds on the solutions of vector systems ẋ = f(t, x). The inequlity is obtined by differentiting sclr vlued functions V (t, x) long the solutions. Exercise 4.3. For x, y lr d, let x y be the inner product of x nd y. Suppose tht f C r (lr lr d, lr d ), r 1, nd there exists continuous function λ C(lR, lr) such tht x f(t, x) λ(t)x x for ll t. For ny τ lr, ξ lr d, show tht the solution of the initil vlue problem exists for ll t nd stisfies the inequlity x(t) e τ λ(s) ds ξ, t τ. Discuss the behvior of the solutions for λ(t) 0. Wht hppens if + τ λ(s) ds = +? Hint: Let V (x) = x x nd find differentil inequlity for V (x(t)) long the solution x(t). Exercise 4.4. Generlize the previous exercise to the cse where x Bf(t, x) λ(t)x x where B is positive definite symmetric mtrix. Hint: Let V (X) = x Bx. Exercise 4.5. Suppose tht f(t, x) λ(t) x for ll t, x nd + λ(s) ds < +. τ Show tht ech solution of ẋ = f(t, x) pproches constnt s t. If, in ddition, f(t, x) f(t, y) λ(t) x y for ll t, x, y, show tht there is onetoone correspondence between the initil positions nd the limit vlues of the solution. Interpret the results for the liner eqution ẋ = A(t)x where the norm of the d d mtrix A(t) is bounded by λ(t). Exercise 4.6. Suppose tht (t) is continuous sclr function, + 0 (s) ds <. As in the previous exercise, show tht the solutions of the eqution ẋ = x + (t)x hve the form x(t) = e t y(t), where y(t) constnt s t nd there is onetoone correspondence between the limits of the solutions nd the initil position. Notice tht you hve shown tht, for ny constnt c, there is function g(t) 0 s t such tht x(t) = e t (c + g(t)) is solution of the differentil eqution. Hint: Find the differentil eqution for y. Exercise 4.7. Consider the eqution ẋ 1 = x 2, ẋ 2 = x 1 + (t)x 1, where is the sme function s in the previous exercise. Show tht the solutions hve the form x 1 (t) = y 1 (t) cos t + y 2 (t) sin t x 2 (t) = y 1 (t) sin t + y 2 (t) cos t 3
4 where y(t) = (y 1 (t), y 2 (t)) constnt s t nd there is onetoone correspondence between the limits of the solutions nd the initil position. Comment bout how this result reltes the solutions to the solutions of the homogeneous eqution ẋ 1 = x 2, ẋ 2 = x 1? 4
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