Syed Abbas. June 19th, 2009

Transcription

1 Almost Department of Mathematics and Statistics Indian Institute of Technology Kanpur, Kanpur, , India June 19th, 2009

2 Harald Bohr

3 Function Almost Bohr s early research was mainly concerned with Dirichlet series. Later, he concentrated his efforts on a study of the Riemann zeta function with E. Landau. In 1914, Landau and Bohr formulated a theorem concerning the distribution of zeros of the zeta function (now called the Bohr-Landau theorem). In three papers published in in Acta Mathematica, Bohr founded the theory of almost periodic function.

4 H. Bohr, Zur Theorie der fastperiodischen Funktionen I Acta Math., 45 (1925) pp H. Bohr., Zur Theorie der fast periodischen Funktionen. I. Eine Verallgemeinerung der Theorie der Fourierreihen. Acta math., v. 45, pp , H. Bohr., Zur Theorie der fastperiodischen Funktionen. III. Dirichletentwicklung analytischer Funktionen. Acta math., v. 47, pp

5 Definition contd... Almost Almost Function f is said to be almost periodic in the sense of Bohr if to every ɛ > 0 there corresponds a relatively dense set T (ɛ, f ) (of ɛ-periods) such that sup f (t + τ) f (t) ɛ for each τ T (ɛ, f ). t R Any such functions can be approximated uniformly on R by a sequence of trigonometric polynomials, P n (t) := N(n) k=1 a n,k e iλ n,kt, n = 1, 2,...; t, λ n,k R, a n,k X.

6 f (x) = cos x + cos 2x Almost

7 Properties Almost Let f and f n, be almost periodic functions with values in a Banach space X. Then the followings assertions holds true: (1) f is uniformly continuous on R; (2) The range of f is precompact, i.e., the set {f (t), t R} is a compact subset of X ; (3) If f is uniformly continuous, then f is almost periodic; (4) If f n g uniformly, then g is also almost periodic.

8 Example Almost Assume that z = f t (p 0 ) is a periodic motion with period T > 0. Then for any integer k, we have d(f t+kt (p 0 ), f t (p 0 )) = 0, for all t R, which implies that d(f t+kt (p 0 ), f t (p 0 )) < ɛ, for all t R, for any given constant ɛ > 0. It follows that kt E (ɛ) := {τ : d(f t+τ (p 0 ), f t (p 0 )) < ɛ}. Therefore E (ɛ) is relatively dense with respect to a constant T 1 > T. It follows that the motion z = f t (p 0 ) is almost periodic.

9 starting from the initial point p 0 = (x 0, y 0 ) T 2. It is easy to observe that (1) z = f t (p 0 ) is a periodic motion on T 2 if λ is a rational number, (2) z = f t (p 0 ) is not a periodic motion on T 2 if λ is an irrational number. Example contd... Almost where λ > 0 is a constant. The motion can be described by z = f t (p 0 ) = (t + x 0, λt + y 0 )

10 Example contd... Almost Assume that λ is a irrational number. We claim that the set E 1 (ɛ) := {τ : d(f t+τ (p 0 ), f t (p 0 )) = τ + λτ < ɛ, (mod2π)} is relatively dense on R. If τ = 2kπ, k Z, we have τ + λτ = 2kπλ (mod2π). When λ is an irrational number, the above number set is dense in the neighborhood of 0 S. Hence the number set E (ɛ) = {τ R : τ = 2kπs.t. 2kπλ < ɛ(mod(2π))} is relatively dense in R. It follows that E (ɛ) E 1 (ɛ) that E 1 (ɛ) is relatively dense in R. Thus f t (p 0 ) is an almost periodic motion on T 2.

11 Semigroup Theory Almost A one parameter family {T (t); 0 t < } of bounded linear operators from X into X is a semigroup of bounded linear operator on X if (i) T (0) = I, (ii) T (t + s) = T (t)t (s) for all t, s 0. C 0 semigroup Contraction semigroup Analytic semigroup 0 A. Pazy, Semigroups of Linear Operators and Applications to Partial, Springer-Verlag, 1983.

12 Definition contd... Almost The infinitesimal generator A of T (t) is the linear operator defined by the formula T (t)x x Ax = lim, for x D(A), t 0 t T (t)x x where D(A) = {x X : lim t 0 t domain of A. exists} denotes the

13 Evolution semigroup Almost A family of bounded operators (U(t, s)) t,s R, t s on a Banach space X is called a (strongly continuous) evolution family if (i) U(t, s) = U(t, r)u(r, s) and U(s, s) = I for t r s and t, r, s R, (ii) the mapping {(τ, σ) R 2 : τ σ} (t, s) U(t, s) is strongly continuous.

14 Definition contd... Almost We say that (U(t, s)) t s solves the Cauchy problem u(t) = A(t)u(t) for t, s R, t s, u(s) = x, (1) on a Banach space X, the function t U(t, s)x is a solution of the above problem. Evolution families are also called evolution systems, evolution operators, evolution processes, propagators or fundamental solution. The Cauchy problem (1) is well posed if and only if there is an evolution family solving (1).

15 Bohr and Neugebauer dx dt = Ax + f (t), (2) A in nth order constant matrix and f is almost periodic function from R to R n. Solution is almost periodic if and only if it is bounded. 1 H. Bohr and O. Neugebauer, ber lineare gleichungen mit konstanten Koeffizienten und Fastperiodischer reder Seite, Nachr. Ges. Wiss. Gottingen, Math.-Phys. Klasse, 1926, 8 22.

16 Zaidman has shown the existence of almost periodic solution for du dt = Au + h(t), (3) t R, u AP(X ) and h is an almost periodic function from R to X, A is the infinitesimal generator of a C 0 semigroup. 2 Zaidman, S., Abstract equations. Pitman Publising, San Franscisco-London-Melbourne 1979.

17 Naito extended these results for dx(t) dt = Ax(t) + L(t)x t + f (t), (4) t R, x X and A is the infinitesimal generator of a strongly continuous semigroup, L(t) is a bounded linear operator from a phase space B to X. 3 Naito, T., Nguyen Van Minh., Shin, J. S., and almost periodic solutions of functional differential equations with finite and infinite delay. Nonlinear Analysis., 47(2001)

18 Almost Consider following functional differential equation in a complex Banach space X, du(t) dt = A(t)u(t) + d dt F 1(t, u(t g(t))) +F 2 (t, u(t), u(t g(t))), t R, u AP(X ), (5) where AP(X ) is the set of all almost periodic functions from R to X and the family {A(t) : t R} of operators in X generates an exponentially stable evolution system {U(t, s), t s}.

19 Kransoselskii s Theorem Almost Let M be a nonempty closed convex subset of X. Suppose that Λ 1 and Λ 2 map M into X such that (i) for any x, y M, Λ 1 x + Λ 2 y M, (ii) Λ 1 is a contraction, (iii) Λ 2 is continuous and Λ 2 (M) is contained in a compact set. Then there exists z M such that z = Λ 1 z + Λ 2 z.

20 Assumptions Almost The functions F 1, F 2 are Lipschitz continuous, that is, there exist positive numbers L F1, L F2 such that F 1 (t, φ) F 1 (t, ψ) X L F1 φ ψ AP(X ) for all t R and for each φ, ψ AP(X ) and F 2 (t, u, φ) F 2 (t, v, ψ) X L F2 ( u v X + φ ψ AP(X ) ) for all t R and for each (u, φ), (v, ψ) X AP(X ); A(t), t R, satisfy (ATCs) and A : R B(X ) is almost periodic;

21 U(t, s), t s, satisfy the condition that, for each ɛ > 0 there exists a number l ɛ > 0 such that each interval of length l ɛ > 0 contains a number τ with the property that U(t + τ, s + τ) U(t, s) B(X ) < Me δ 2 (t s) ɛ. The functions F 1 (t, u), F 2 (t, u, v) are almost periodic for u, v almost periodic. For u, v AP(X ), F 1 (t, u), F 2 (t, u, v) is almost periodic, hence it is uniformly bounded. We assume that F i AP(X ) M i, i = 1, 2. Also, we assume that F 2 AP(X ) M.

22 Integral form Almost Define a mapping F by (Fu)(t) = F 1 (t, u(t g(t)))+ t U(t, s)f 2 (s, u(s), u(s g(s)))ds. For u almost periodic, the operator Fu is almost periodic. Consider (Fu)(t) = (Λ 1 u)(t) + (Λ 2 u)(t), where Λ 1, Λ 2 is from AP(X ) to AP(X ) are given by and (Λ 2 u)(t) = (Λ 1 u)(t) = F 1 (t, u(t g(t))) t U(t, s)f 2 (s, u(s), u(s g(s)))ds.

23 Results Almost The Operator Λ 1 is a contraction provided L F1 < 1. The operator Λ 2 is continuous and it s image is contained in a compact set. Suppose F 1, F 2 satisfies all the assumption and L F1 < 1. Let Q = {u AP(X ) : u AP(X ) R}. Here R satisfies the inequality RL F1 + b + M δ (2L F 2 R + a) R, where F 1 (, 0) AP(X ) b. Then equation (5) has a almost periodic solution in Q.

24 Example Almost consider the following perturbed Van Der Pol equation for small ɛ 1 and ɛ 2, u + (ɛ 2 u 2 + 1)u + u = ɛ 1 d dt (sin(t) + sin( 2t))u 2 (t g(t)) ɛ 2 (cos(t) + cos( 2t)), (6) where g(t) is nonnegative, continuous and almost periodic function. Using the transformations u = u 1 and u 1 = u 2, we have U = AU + d dt F 1(t, u(t g(t))) + F 2 (t, u(t), u(t g(t))), (7)

25 where U = ( u1 u 2 ), ( ) 0 1 A = 1 1 and ( 0 F 1 (t, u(t g(t))) = ɛ 1 (sin(t) + sin( 2t)))u1 2 (t g(t)) ( F 2 (t, u(t), u(t g(t))) = ), 0 ɛ 2 (cos(t) + cos( 2t))) ɛ 2 u 2 1 u 2) It is easy to observe that equation (7) has an almost periodic solution which turns out to be the solution of (6). ).

26 We consider the following non-autonomous model for two competing phytoplankton populations, du(t) dt dv(t) dt = u(t)(k 1 (t) α 1 (t)u(t) β 1 (t)v(t) γ 1 (t)u(t)v(t)), = v(t)(k 2 (t) α 2 (t)v(t) β 2 (t)u(t) γ 2 (t)u(t τ)v(t)), (8) where k i, α i, β i, γ i for i = 1, 2 are almost periodic functions and satisfy 0 < k i k i (t) ki, 0 < α i α i (t) αi, for t R. 0 < β i β i (t) βi, 0 < γ i γ i (t) γi 0 S. Ahmad, On the nonautonomous VolterraLotka competition equations, Proceedings of the American Mathematical Society 117, (1993).

27 We use Bochner s criterion of almost periodicity. Such a criterion says that a function g(t), continuous on R is almost periodic if and only if for every sequence of numbers {τ n } n=1, there exists a subsequence {τ nk } k=1 such that the sequence of translates {f (t + τ nk )} k=1 converges uniformly on R. Any positive solution {u(t), v(t)} of system (8) satisfies m 1 lim t inf u(t) lim t sup u(t) M 1, m 2 lim t inf v(t) lim t sup v(t) M 2, whenever α 2 k 1 > β 1 k 2 and α 1 k 2 > β 2 k 1.

28 Suppose that the time dependent coefficients k i, α i, β i, γ i (i = 1, 2) are positive and their bounds satisfy α 2 k 1 > β 1k 2, and α 1 k 2 > β 2k 1, (9) then the system (8) is permanent.

29 Suppose that (u 1 (t), v 1 (t)) and (u 2 (t), v 2 (t)) be two solutions of the model system (8) within R 2 + such that m 1 u i (t) M 1, m 2 v i (t) M 2, t R, i = 1, 2, and Min{ 1, 2 } > 0, where 1 = (α 1 β 2 + 2γ 1 m 2 )m 1 (β 1 + γ 1 M 1) and 2 = 2(α 2 + γ 2 m 1 )m 2 (β 1 + γ 1 M 1)M 2 β 2 M 1 γ 2 M 1, then (u 1 (t), v 1 (t)) = (u 2 (t), v 2 (t)) t R.

30 The system of equations (8) has a unique almost periodic solution (ū(t), v(t)) within R 2 + and m 1 ū(t) M 1, m 2 v(t) M 2. Moreover as t, for any solution (u(t), v(t)) of (8) we have u(t) ū(t) and v(t) v(t).

31 Consider the following specific example du(t) dt dv(t) dt = u(t)(( (sin(.03t) + cos(.02t))).07u(t).05v(t).0008u(t)v(t)), = v(t)(( (cos(.05t) + sin(.07t))).08v(t).015u(t).003u(t τ)v(t)). (10)

32 Figure: Almost periodic solution of allelopathic phytoplankton model.

33 References Almost [1] A. Pazy, Semigroups of Linear Operators and Applications to Partial, Springer-Verlag, [2] Amerio, L. and Prouse, G., Almost periodic functions and functional equations, Van Nostrand- Reinhold, New York, [3] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), [4] S. Bochner, Bei trage zu theorie der Fastperiodischer Funktioner, Math. Ann. 96 (1927), [5] L. Amerio and G. Prouse, Almost periodic functions and functional equations, Van Nostrand, New York, 1955.

34 [6] G. Sell and R. Sacker, Existence of dichotomies and invariant splittings for linear differential systems. Ill, J. 22 (1976), [7] Aulbach, B. and Minh, N.V., Semigroups and equations with almost periodic coefficients, Nonlinear Analysis TMA, Vol.32, No.2 (1998), [8] Besicovich, A.S., Almost periodic functions, Dover Publications, New York, [9] Palmer, J. K., Exponential dichotomies for almost periodic equations Proc. of the Amer. Math. Society, 101,(2) (1987), [10] Abbas, S., Bahuguna, D., Almost periodic solutions of neutral functional differential equations, Comp. and math. with appl., (2008),

35 THANK YOU

36 Acquistapace and Terreni gave conditions on A(t), t R, which ensure the existence of an evolution family {U(t, s), t s > } on X such that, t u(t) = U(t, 0)u(0) + U(t, ξ)f (ξ)dξ, (11) 0 where u(t) satisfies, du(t) dt = A(t)u(t) + f (t), t R. (12) These conditions, now known as the Acquistapace-Terreni conditions (ATCs), are as follows. 0 Acquistapace, P., Terreni, B., Aunified approach to abstract linear parabolic equations, Rend. Sem. Math. Uni. Padova, 78 (1987),

37 (ATCs): There exist a constant K 0 > 0 and a set of real numbers α 1, α 2,..., α k, β 1,..., β k with 0 β i < α i 2, i = 1, 2,..., k, such that k A(t)(λ A(t)) 1 (A(t) 1 A(s) 1 ) B(X ) K 0 (t s) α i λ β i 1, for t, s R, λ S θ0 \{0}, where i=1 ρ(a(t)) S θ0 = {λ C : argλ θ 0 } {0}, θ 0 ( π 2, π) and there exists a constant M 0 such that (λ A(t)) 1 B(X ) M 1 + λ, λ S θ 0. If (ATCs) are satisfied, then from Theorem 2.3 of [?], there exists a unique evolution family {U(t, s), t s > } on X, which governs the linear version of (12).

38 Let = {z : φ 1 < argz < φ 2, φ 1 < 0 < φ 2 } and for z, let T (z) be a bounded linear operator. The family T (z), z is an analytic in if (i) z T (z) is analytic in, (ii) T (0) = I and lim z 0,z T (z)x = x for every x X, (iii) T (z 1 + z 2 ) = T (z 1 )T (z 2 ) for z 1, z 2.