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Almost Periodic Solutions of Neutral Functional Differential Almost Periodic Solutions of Neutral Functional Equations Syed Abbas 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 Function Almost Periodic Solutions of Neutral Functional Differential Equations Bohr’s early research was mainly concerned with Dirichlet Syed Abbas 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 1924 − 26 in Acta Mathematica, Bohr founded the theory of almost periodic function.

Almost Periodic Solutions of Neutral Functional Differential Equations H. Bohr, ”Zur Theorie der fastperiodischen Funktionen I” Acta Math., 45 (1925) pp. 29-127. Syed Abbas 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.

Definition contd.... Almost Periodic Almost Periodic Function Solutions of Neutral f is said to be almost periodic in the sense of Bohr if to every Functional Differential ǫ > 0 there corresponds a relatively dense set T ( ǫ, f ) (of Equations ǫ -periods) such that Syed Abbas sup � f ( t + τ ) − f ( t ) � ≤ ǫ for each τ ∈ T ( ǫ, f ) . t ∈ R Any such functions can be approximated uniformly on R by a sequence of trigonometric polynomials, N ( n ) � a n , k e i λ n , k t , n = 1 , 2 , ... ; t , λ n , k ∈ R , a n , k ∈ X . P n ( t ) := k =1

√ f ( x ) = cos x + cos 2 x Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas

Properties Almost Periodic Solutions of Neutral Functional Differential Equations Let f and f n , be almost periodic functions with values in a Syed Abbas 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 ; ′ is uniformly continuous, then f ′ is almost periodic; (3) If f (4) If f n → g uniformly, then g is also almost periodic.

Example Almost Assume that z = f t ( p 0 ) is a periodic motion with period Periodic Solutions of T > 0 . Then for any integer k , we have Neutral Functional Differential Equations d ( f t + kT ( p 0 ) , f t ( p 0 )) = 0 , Syed Abbas for all t ∈ R , which implies that d ( f t + kT ( p 0 ) , f t ( p 0 )) < ǫ, for all t ∈ R , for any given constant ǫ > 0 . It follows that kT ∈ E ∗ ( ǫ ) := { τ : d ( f t + τ ( p 0 ) , f t ( p 0 )) < ǫ } . Therefore E ∗ ( ǫ ) is relatively dense with respect to a constant T 1 > T . It follows that the motion z = f t ( p 0 ) is almost periodic.

Example contd.... Almost Periodic Solutions of Neutral Functional Differential Equations Syed Abbas where λ > 0 is a constant. The motion can be described by z = f t ( p 0 ) = ( t + x 0 , λ t + y 0 ) 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 Assume that λ is a irrational number. We claim that the set Periodic Solutions of Neutral Functional E 1 ( ǫ ) := { τ : d ( f t + τ ( p 0 ) , f t ( p 0 )) = | τ | + | λτ | < ǫ, ( mod 2 π ) } Differential Equations Syed Abbas is relatively dense on R . If τ = 2 k π, k ∈ Z , we have | τ | + | λτ | = | 2 k πλ | ( mod 2 π ) . 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 = 2 k π s . t . | 2 k πλ | < ǫ ( mod (2 π )) } is relatively dense in R . It follows that E ∗ ( ǫ ) ⊂ E 1 ( ǫ ) that E 1 ( ǫ ) is relatively dense in R . Thus f t ( p 0 ) is an almost periodic motion on T 2 .

Semigroup Theory Almost Periodic Solutions of Neutral A one parameter family { T ( t ); 0 ≤ t < ∞} of bounded linear Functional Differential operators from X into X is a semigroup of bounded linear Equations operator on X if Syed Abbas (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 Differential Equations, Springer-Verlag, 1983.

Definition contd.... Almost Periodic Solutions of Neutral Functional Differential Equations The infinitesimal generator A of T ( t ) is the linear operator Syed Abbas defined by the formula T ( t ) x − x Ax = lim for x ∈ D ( A ) , , t t → 0 T ( t ) x − x where D ( A ) = { x ∈ X : lim t → 0 exists } denotes the t domain of A .

Evolution semigroup Almost Periodic Solutions of Neutral Functional Differential Equations A family of bounded operators ( U ( t , s )) t , s ∈ R , t ≥ s on a Syed Abbas 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.

Definition contd.... Almost Periodic Solutions of Neutral We say that ( U ( t , s )) t ≥ s solves the Cauchy problem Functional Differential Equations u ( t ) = A ( t ) u ( t ) ˙ for t , s ∈ R , t ≥ s , Syed Abbas 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).

Almost Periodic Solutions of Neutral Bohr and Neugebauer Functional Differential Equations dx Syed Abbas dt = Ax + f ( t ) , (2) A in n th 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 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 Zaidman has shown the existence of almost periodic solution for Syed Abbas 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 Differential equations. Pitman Publising , San Franscisco-London-Melbourne 1979.

Almost Periodic Solutions of Neutral Functional Differential Naito extended these results for Equations dx ( t ) Syed Abbas = Ax ( t ) + L ( t ) x t + f ( t ) , (4) dt 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., Periodic and almost periodic solutions of functional differential equations with finite and infinite delay. Nonlinear Analysis. , 47(2001) 3989-3999.

Neutral Functional Differential Equations Almost Periodic Solutions of Neutral Functional Consider following functional differential equation in a complex Differential Equations Banach space X , Syed Abbas du ( t ) A ( t ) u ( t ) + d = dt F 1 ( t , u ( t − g ( t ))) dt + 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 } .

Kransoselskii’s Theorem Almost Periodic Solutions of Neutral Functional Differential Equations Let M be a nonempty closed convex subset of X . Suppose that Syed Abbas 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 .

Assumptions Almost Periodic Solutions of Neutral The functions F 1 , F 2 are Lipschitz continuous, that is, Functional Differential there exist positive numbers L F 1 , L F 2 such that Equations Syed Abbas � F 1 ( t , φ ) − F 1 ( t , ψ ) � X ≤ L F 1 � 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 F 2 ( � 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;