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Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Almost Periodic Solutions of Neutral Functional Differential Equations

Syed Abbas

Department of Mathematics and Statistics Indian Institute of Technology Kanpur, Kanpur, 208016, India

June 19th, 2009

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Harald Bohr

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Almost Periodic Function

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

• f the zeta function (now called the Bohr-Landau theorem). In

three papers published in 1924 − 26 in Acta Mathematica, Bohr founded the theory of almost periodic function.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

• H. Bohr, ”Zur Theorie der fastperiodischen Funktionen I” Acta

Math., 45 (1925) pp. 29-127.

• H. Bohr., Zur Theorie der fast periodischen Funktionen. I. Eine

Verallgemeinerung der Theorie der Fourierreihen. Acta math.,

• v. 45, pp. 29-127, 1924.
• H. Bohr., Zur Theorie der fastperiodischen Funktionen. III.

Dirichletentwicklung analytischer Funktionen. Acta math., v. 47, pp. 237-281. 1926.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Definition contd....

Almost Periodic Function f is said to be almost periodic in the sense of Bohr if to every ǫ > 0 there corresponds a relatively dense set T(ǫ, f ) (of ǫ-periods) such that sup

t∈R

f (t + τ) − f (t) ≤ ǫ for each τ ∈ T(ǫ, f ). Any such functions can be approximated uniformly on R by a sequence of trigonometric polynomials, Pn(t) :=

N(n)

• k=1

an,keiλn,kt, n = 1, 2, ...; t, λn,k ∈ R, an,k ∈ X.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

f (x) = cos x + cos √ 2x

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Properties

Let f and fn, 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 fn → g uniformly, then g is also almost periodic.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Example

Assume that z = f t(p0) is a periodic motion with period T > 0. Then for any integer k, we have d(f t+kT(p0), f t(p0)) = 0, for all t ∈ R, which implies that d(f t+kT(p0), f t(p0)) < ǫ, for all t ∈ R, for any given constant ǫ > 0. It follows that kT ∈ E ∗(ǫ) := {τ : d(f t+τ(p0), f t(p0)) < ǫ}. Therefore E ∗(ǫ) is relatively dense with respect to a constant T1 > T. It follows that the motion z = f t(p0) is almost periodic.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Example contd....

where λ > 0 is a constant. The motion can be described by z = f t(p0) = (t + x0, λt + y0) starting from the initial point p0 = (x0, y0) ∈ T2. It is easy to

• bserve that

(1) z = f t(p0) is a periodic motion on T2 if λ is a rational number, (2) z = f t(p0) is not a periodic motion on T2 if λ is an irrational number.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Example contd....

Assume that λ is a irrational number. We claim that the set E1(ǫ) := {τ : d(f t+τ(p0), f t(p0)) = |τ| + |λτ| < ǫ, (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 ∗(ǫ) = {τ ∈ R : τ = 2kπs.t.|2kπλ| < ǫ(mod(2π))} is relatively dense in R. It follows that E ∗(ǫ) ⊂ E1(ǫ) that E1(ǫ) is relatively dense in R. Thus f t(p0) is an almost periodic motion on T2.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Semigroup Theory

A one parameter family {T(t); 0 ≤ t < ∞} of bounded linear

• perators from X into X is a semigroup of bounded linear
• perator on X if

(i) T(0) = I, (ii) T(t + s) = T(t)T(s) for all t, s ≥ 0. C0 semigroup Contraction semigroup Analytic semigroup

• 0A. Pazy, Semigroups of Linear Operators and Applications to Partial

Differential Equations, Springer-Verlag, 1983.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Definition contd....

The infinitesimal generator A of T(t) is the linear operator defined by the formula Ax = lim

t→0

T(t)x − x t , for x ∈ D(A), where D(A) = {x ∈ X : limt→0

T(t)x−x t

exists} denotes the domain of A.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Evolution semigroup

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) ∈ R2 : τ ≥ σ} ∋ (t, s) ֌ U(t, s) is strongly continuous.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Definition contd....

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)

• n a Banach space X, the function t ֌ U(t, s)x is a solution
• f the above problem.

Evolution families are also called evolution systems, evolution

• perators, evolution processes, propagators or fundamental

solution. The Cauchy problem (1) is well posed if and only if there is an evolution family solving (1).

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Bohr and Neugebauer dx dt = Ax + f (t), (2) A in nth order constant matrix and f is almost periodic function from R to Rn. Solution is almost periodic if and only if it is bounded.

• 1H. Bohr and O. Neugebauer, ber lineare Differentialgleichungen mit

konstanten Koeffizienten und Fastperiodischer reder Seite, Nachr. Ges.

• Wiss. Gottingen, Math.-Phys. Klasse, 1926, 8 − 22.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

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 C0 semigroup.

2Zaidman, S., Abstract Differential equations. Pitman Publising, San

Franscisco-London-Melbourne 1979.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Naito extended these results for dx(t) dt = Ax(t) + L(t)xt + 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.

3Naito, T., Nguyen Van Minh., Shin, J. S., Periodic and almost

periodic solutions of functional differential equations with finite and infinite

• delay. Nonlinear Analysis., 47(2001) 3989-3999.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Neutral Functional Differential Equations

Consider following functional differential equation in a complex Banach space X, du(t) dt = A(t)u(t) + d dt F1(t, u(t − g(t))) +F2(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}.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Kransoselskii’s Theorem

Let M be a nonempty closed convex subset of X. Suppose that Λ1 and Λ2 map M into X such that (i) for any x, y ∈ M, Λ1x + Λ2y ∈ M, (ii) Λ1 is a contraction, (iii) Λ2 is continuous and Λ2(M) is contained in a compact set. Then there exists z ∈ M such that z = Λ1z + Λ2z.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Assumptions

The functions F1, F2 are Lipschitz continuous, that is, there exist positive numbers LF1, LF2 such that F1(t, φ) − F1(t, ψ)X ≤ LF1φ − ψAP(X) for all t ∈ R and for each φ, ψ ∈ AP(X) and F2(t, u, φ)−F2(t, v, ψ)X ≤ LF2(u−vX +φ−ψAP(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;

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

U(t, s), t ≥ s, satisfy the condition that, for each ǫ > 0 there exists a number lǫ > 0 such that each interval of length lǫ > 0 contains a number τ with the property that U(t + τ, s + τ) − U(t, s)B(X) < Me− δ

2 (t−s)ǫ.

The functions F1(t, u), F2(t, u, v) are almost periodic for u, v almost periodic. For u, v ∈ AP(X), F1(t, u), F2(t, u, v) is almost periodic, hence it is uniformly bounded. We assume that FiAP(X) ≤ Mi, i = 1, 2. Also, we assume that F

′

2AP(X) ≤ M

′.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Integral form

Define a mapping F by (Fu)(t) = F1(t, u(t−g(t)))+ t

−∞

U(t, s)F2(s, u(s), u(s−g(s)))ds. For u almost periodic, the operator Fu is almost periodic. Consider (Fu)(t) = (Λ1u)(t) + (Λ2u)(t), where Λ1, Λ2 is from AP(X) to AP(X) are given by (Λ1u)(t) = F1(t, u(t − g(t))) and (Λ2u)(t) = t

−∞

U(t, s)F2(s, u(s), u(s − g(s)))ds.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Results

The Operator Λ1 is a contraction provided LF1 < 1. The operator Λ2 is continuous and it’s image is contained in a compact set. Suppose F1, F2 satisfies all the assumption and LF1 < 1. Let Q = {u ∈ AP(X) : uAP(X) ≤ R}. Here R satisfies the inequality RLF1 + b + M δ (2LF2R + a) ≤ R, where F1(·, 0)AP(X) ≤ b. Then equation (5) has a almost periodic solution in Q.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Example

consider the following perturbed Van Der Pol equation for small ǫ1 and ǫ2, u

′′ + (ǫ2u2 + 1)u ′ + u

= ǫ1 d dt (sin(t) + sin( √ 2t))u2(t − g(t)) −ǫ2(cos(t) + cos( √ 2t)), (6) where g(t) is nonnegative, continuous and almost periodic

• function. Using the transformations u = u1 and u

′

1 = u2, we

have U

′ = AU + d

dt F1(t, u(t −g(t)))+F2(t, u(t), u(t −g(t))), (7)

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

where U = u1 u2

• ,

A =

• 1

−1 −1

• and

F1(t, u(t − g(t))) =

• ǫ1(sin(t) + sin(

√ 2t)))u2

1(t − g(t))

• ,

F2(t, u(t), u(t−g(t))) =

• ǫ2(cos(t) + cos(

√ 2t))) − ǫ2u2

1u2)

• .

It is easy to observe that equation (7) has an almost periodic solution which turns out to be the solution of (6).

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

We consider the following non-autonomous model for two competing phytoplankton populations, du(t) dt = u(t)(k1(t) − α1(t)u(t) − β1(t)v(t) − γ1(t)u(t)v(t)), dv(t) dt = v(t)(k2(t) − α2(t)v(t) − β2(t)u(t) − γ2(t)u(t − τ)v(t)), (8) where ki, αi, βi, γi for i = 1, 2 are almost periodic functions and satisfy 0 < ki∗ ≤ ki(t) ≤ k∗

i ,

0 < αi∗ ≤ αi(t) ≤ α∗

i ,

0 < βi∗ ≤ βi(t) ≤ β∗

i ,

0 < γi∗ ≤ γi(t) ≤ γ∗

i

   for t ∈ R.

• 0S. Ahmad, On the nonautonomous VolterraLotka competition

equations, Proceedings of the American Mathematical Society 117, 199-204 (1993).

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

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 {τn}∞

n=1,

there exists a subsequence {τnk}∞

k=1 such that the sequence of

translates {f (t + τnk)}∞

k=1 converges uniformly on R.

Any positive solution {u(t), v(t)} of system (8) satisfies m1 ≤ lim

t→∞ inf u(t) ≤ lim t→∞ sup u(t) ≤ M1,

m2 ≤ lim

t→∞ inf v(t) ≤ lim t→∞ sup v(t) ≤ M2,

whenever α2∗k1∗ > β∗

1k∗ 2 and α1∗k2∗ > β∗ 2k∗ 1.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Suppose that the time dependent coefficients ki, αi, βi, γi (i = 1, 2) are positive and their bounds satisfy α2∗k1∗ > β∗

1k∗ 2,

and α1∗k2∗ > β∗

2k∗ 1,

(9) then the system (8) is permanent.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Suppose that (u1(t), v1(t)) and (u2(t), v2(t)) be two solutions

• f the model system (8) within R2

+ such that

m1 ≤ ui(t) ≤ M1, m2 ≤ vi(t) ≤ M2, t ∈ R, i = 1, 2, and Min{∆1, ∆2} > 0, where ∆1 = (α1∗ − β∗

2 + 2γ1∗m2)m1 − (β∗ 1 + γ∗ 1M1) and

∆2 = 2(α2∗ + γ2∗m1)m2 − (β∗

1 + γ∗ 1M1)M2 − β∗ 2M1 − γ∗ 2M1,

then (u1(t), v1(t)) = (u2(t), v2(t)) ∀ t ∈ R.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

The system of equations (8) has a unique almost periodic solution (¯ u(t), ¯ v(t)) within R2

+ and

m1 ≤ ¯ u(t) ≤ M1, m2 ≤ ¯ v(t) ≤ M2. Moreover as t → ∞, for any solution (u(t), v(t)) of (8) we have u(t) → ¯ u(t) and v(t) → ¯ v(t).

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Consider the following specific example du(t) dt = u(t)((2 + 1 2(sin(.03t) + cos( √ .02t))) −.07u(t) − .05v(t) − .0008u(t)v(t)), dv(t) dt = v(t)((1 + 1 4(cos(.05t) + sin( √ .07t))) −.08v(t) − .015u(t) − .003u(t − τ)v(t)).(10)

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Figure: Almost periodic solution of allelopathic phytoplankton model.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

References

[1] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. [2] Amerio, L. and Prouse, G., Almost periodic functions and functional equations, Van Nostrand- Reinhold, New York, 1971. [3] S. Bochner, A new approach to almost periodicity, Proc. Nat.

• Acad. Sci. U.S.A., 48 (1962), 2039-2043.

[4] S. Bochner, Bei trage zu theorie der Fastperiodischer Funktioner,

• Math. Ann. 96 (1927), 119-147.

[5] L. Amerio and G. Prouse, Almost periodic functions and functional equations, Van Nostrand, New York, 1955.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

[6] G. Sell and R. Sacker, Existence of dichotomies and invariant splittings for linear differential systems. Ill, J. Differential Equations 22 (1976), 497-522. [7] Aulbach, B. and Minh, N.V., Semigroups and Differential equations with almost periodic coefficients, Nonlinear Analysis TMA, Vol.32, No.2 (1998), 287-297. [8] Besicovich, A.S., Almost periodic functions, Dover Publications, New York, 1958. [9] Palmer, J. K., Exponential dichotomies for almost periodic equations Proc. of the Amer. Math. Society, 101,(2) (1987), 293-298. [10] Abbas, S., Bahuguna, D., Almost periodic solutions of neutral functional differential equations, Comp. and math. with appl., 55-11 (2008), 2593-2601.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

THANK YOU

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

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, u(t) = U(t, 0)u(0) + t U(t, ξ)f (ξ)dξ, (11) 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.

0Acquistapace, P., Terreni, B., “Aunified approach to abstract linear

parabolic equations”, Rend. Sem. Math. Uni. Padova, 78 (1987), 47-107.

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

(ATCs): There exist a constant K0 > 0 and a set of real numbers α1, α2, . . . , αk, β1, . . . , βk with 0 ≤ βi < αi ≤ 2, i = 1, 2, . . . , k, such that A(t)(λ−A(t))−1(A(t)−1−A(s)−1)B(X) ≤ K0

k

• i=1

(t−s)αi|λ|βi−1, for t, s ∈ R, λ ∈ Sθ0\{0}, where ρ(A(t)) ⊃ Sθ0 = {λ ∈ C : |argλ| ≤ θ0} ∪ {0}, θ0 ∈ (π 2 , π) and there exists a constant M ≥ 0 such that (λ − A(t))−1B(X) ≤ M 1 + |λ|, λ ∈ Sθ0. 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).

Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Let ∆ = {z : φ1 < argz < φ2, φ1 < 0 < φ2} 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 limz→0,z∈∆ T(z)x = x for every x ∈ X, (iii) T(z1 + z2) = T(z1)T(z2) for z1, z2 ∈ ∆.