Van Minh Nguyen

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Van Minh Nguyen

• Department of Mathematics
• State University of West GeorgiaE-mail
  vnguyen_at_westga.edu

2
Almost Periodic Solutions of Differential
Equations

• An Approach Via
• Evolution Semi-groups Spectral Theory of
  Functions

3
An Exercise in Calculus

• Given a scalar continuous T-periodic function f
  on R, when is the following function T-periodic
  ?

4
Solution of the Exercise

• The function F can be represented in the form
• F(t) at G(t)
• where G is a T-periodic, continuous function,
  and
• Therefore, F is T-periodic if either a 0, or
  F is bounded.

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Two Classical Results in ODE

• Consider the equation
• where A is a matrix and f is a T-periodic
  function
• Then, the following is classical
• Theorem A. Eq. (F) has a unique T-periodic
  solution for every T-periodic f if and only if
  .
• Theorem B. Eq. (F) has a T-periodic solution if
  and only if it has a bounded solution.

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Theorem As conditions

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History of the Problem

• Theorem B is due to J. Massera (1950), The
  existence of periodic solutions of systems of
  differential equations, Duke Math. J. 17,
  457-475.
• These results can be proved by showing that the
  period maps associated with the equations have
  fixed points.
• (For the proofs, see, e.g., the book J. Liu
  (2003) A first course in the qualitative theory
  of differential equations, Prentice Hall.)

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Extensions of Massera Theorem (Theorem B)

• Consider the delay differential equation
• 
  (1)
• where L, f are continuous and T-periodic,
  r is a positive constant,

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Extensions of Massera Theorem (Theorem B)

• Theorem. (Chow-Hale, 1974) Eq. (1) has a
  T-periodic solution if and only if it has a
  bounded solution.
• Some other extensions were made by Langenhopf
  (1985, Hatvani-Kristin (1992), Hino-Murakami
  (1987), Shin-Naito (1999), Liu (2000),
  Liu-Naito-Minh (2003)
• So far the extensions are concerned with only
  periodic solutions.
• The methods used is to find fixed points of the
  period maps.

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Almost Periodic Functions

• Almost periodic functions (Bochner's criterion)
  A bounded and continuous function f is called
  almost periodic in the sense of Bohr if for every
  numerical sequence there exists a
  subsequence such that the sequence
  of functions converges
  uniformly on R.
• Bohr spectrum of f is the set
• where

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Example.
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Almost periodicity of the primitive of an almost
periodic function

• Consider the integral
• Theorem (Bohr) If f is a scalar and almost
  periodic function, then F is almost periodic iff
  it is bounded on R.
• Theorem (Kadets) If f is an X-valued almost
  periodic function, and the Banach space X does
  not contain any subspaces isomorphic to the space
  of all numerical sequences converging to
  0, then F is almost periodic iff it is bounded
  on R.

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Spectral theory of functions

• The spectrum of a bounded continuous function g
  is the set of all reals such that the
  following Carleman transform of g has no analytic
  extension to any neighborhood of in the
  complex plane
• If g is almost periodic, then
  

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Spectral theory of functions

• Theorem. Let such
  that .
• Then (i)
• (ii)
• (iii) If ,
  then
• (iv) f is T-periodic iff
• (v) If Sp(f) is discrete, then f
  is almost periodic.
• Corollary. If is a closed subset of the
  real line, then
• is a closed subspace of AP(X).

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Extensions of Theorem A for Almost Periodic
Solutions of Evolution Equations

• Consider the evolution equation
• 
  (1)
• where A generates a strongly continuous
  semigroup
• on a Banach space X, f is an
  X-valued almost periodic function in the sense of
  Bohr.

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Almost periodic solutions of evolution equations

• Mild solutions of Eq. (1) on the real line are
  continuous functions u satisfying the following
  for all t gt s
• Problem When does Eq. (1) have almost periodic
  mild solutions on the real line with the same
  structure of spectrum as f ?

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Some principal obstacles in the infinite
dimensional case

• In the infinite dimensional case, the Spectral
  Mapping Theorem fails, i.e., there are no
  equalities of the form
• where A is the generator of the semigroup T(t).
• There are no period maps associated with the
  equations. Therefore, the classical approach to
  the Periodic Solution Problem does not apply.

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Evolution semigroups

• Evolution semigroup associated
  with Eq. (1)
• has the generator G, defined as
• 
  such that u is a mild solution of Eq. (1).
  In this case Gu - f .
• In the finite dimensional case, G -d/dt A .
• Lemma.
• Theorem. Eq. (1) has a mild solution u on R iff
  Gu -f.

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Extension of Theorem A

• Theorem 1. Let the following condition be
  satisfied
• Then, Eq. (1) has a unique almost periodic mild
  solution w with
• Theorem 2. Let A generate an analytic semigroup
  and let the following hold
• Then, Eq. (1) has a unique almost periodic mild
  solution w with

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Nonresonance condition in Theorem 1
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Nonresonance condition in Theorem 2
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Sketch of the proof of Theorem 1

• For Theorem 1 we apply the spectral inclusion of
  strongly continuous semigroups to the evolution
  semigroup
• in where
• To estimate we apply the Weak
  Spectral Mapping Theorem for bounded strongly
  continuous groups
• to translation group S(t) in
  .

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Sketch of the proof of Theorem 2

• For Theorem 2 we apply the method of sums of
  commuting operators. More precisely we use the
  spectral inclusion
• in the function space where
  
• In this function space
• By the assumptions we get the invertibility of
  the operator
• in . This yields the existence
  uniqueness of mild solution of (1) with
  spectrum contained in Sp(f).

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Extensions to other classes of equations

• We can use the evolution semigroup method to
  treat periodic equations
• where A(t) generates a periodic
  evolutionary process.
• The method of sums of commuting operators can be
  applied to higher order evolution equations, e.g.,

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Extensions to other classes of equations

• We can see the functional differential equation
• as an operator equation
• and use the inclusion

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Resonance case

• What happens if one of the conditions in Theorems
  1, 2 fails ?
• Example. The following equation has no bounded
  solution
• In the first example
  the resonance holds, too. The existence of
  periodic solutions is equivalent to the existence
  of bounded solutions.

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Graph of the solution
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Spectral estimate of bounded solution

• Lemma 1. If u is a bounded and uniformly
  continuous mild solution of Eq. (1), then
• where
• Corollary 1. Any bounded and uniformly continuous
  mild solution of (1) is contained in
• where

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Decomposition technique

• Theorem 3. If
• then every bounded, uniformly continuous mild
  solution u can be represented in the form u
  vw, where v is a solution to the homogeneous
  equation, w is a bounded uniformly continuous
  mild solution of (1) with

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Weak resonance condition 1
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Decomposition technique

• Theorem 4. If is bounded, and
• then every bounded, uniformly continuous mild
  solution u can be represented in the form u
  vw, where v is a solution to the homogeneous
  equation, w is a bounded uniformly continuous
  mild solution of (1) with

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Weak resonance condition 2
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Extensions of Theorem B (Massera Theorem)

• Corollary 2. Under the conditions of Theorem 3,
  if
• is countable and X does not
  contain , then Eq. (1) has an almost
  periodic solution w with
• Corollary 3. Under the conditions of Theorem 4,
  if Sp(f) is countable and X does not contain
  , then Eq. (1) has an almost periodic
  solution w with

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Examples

• If the space X is reflexive, then it does not
  contain any subspaces isomorphic to the space
  of numerical sequences converging to 0.
• If A generates a compact semigroup, then weak
  resonance conditions in Theorems 3, 4 hold.
• If A generates an analytic semigroup,
  is bounded. Therefore, Theorem 4 can be applied
  to analytic semigroups.

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Extensions to other classes of evolution equations

• Theorem 3 can be directly extended to periodic
  evolution equations.
• Theorems 3, 4 can be extended to functional
  differential equations.
• Theorem 4 has a version for quasi-periodic
  solutions
• In case the spectrum is finite
  the results can be refined. More precisely, the
  existence of bounded solution implies the
  existence of almost periodic solutions.

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An example

• Consider the partial differential equation
• where
  are scalar functions. Define
• by
  the formula
• with
  are absolutely
• continuous and
• Then, setting we can
  transform the above
• equation into an abstract ODE
• 
  ()

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An application

• By our theory, if f is almost periodic, then
  the following assertions hold
• - Eq. () has an almost periodic mild solution w
  with Sp(w) Sp(f) iff it has a bounded,
  uniformly continuous mild solution on R.
• - If , then () has a
  unique almost periodic mild solution w with
  Sp(w) Sp(f).

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Main results have been published in

• T. Naito, Nguyen Van Minh, Evolutions semigroups
  and spectral criteria for almost periodic
  solutions of periodic evolution equations,
  Journal of Differential Equations 152 (1999),
  358-376.
• T. Naito, Nguyen Van Minh, R. Miyazaki, J.S.
  Shin, A decomposition theorem for bounded
  solutions and the existence of periodic
  solutions to periodic equations, Journal of
  Differential Equations, 160 (2000),263-282.
• S. Murakami, T. Naito, Nguyen Van Minh, Evolution
  semigroups and sums of commuting operators A new
  approach to the admissibility theory of function
  spaces, Journal of Differential Equations, 164
  (2000), 240-285.
• T. Naito, Nguyen Van Minh, J.S. Shin, New
  spectral criteria for almost periodic solutions
  of evolution equations, Studia Mathematica, 145
  (2001), 97-111.

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• Y. Hino, S. Murakami, T. Naito, Nguyen Van Minh,
  A variation of constants formula for abstract
  functional differential equations in the phase
  Space, Journal of Differential Equations, 179
  (2002), 336-355.
• T. Furumochi, T. Naito, Nguyen Van Minh,
  Boundedness and almost periodicity of solutions
  of partial functional differential equations,
  Journal of Differential Equations, 180 (2002),
  125-152.
• Y. Hino, T. Naito, Nguyen Van Minh, J. S. Shin,
  "Almost Periodic Solutions of Differential
  Equations in Banach Spaces", Taylor and Francis
  Group, London-New York, 2002.
• S. Murakami, T. Naito, Nguyen Van Minh, Massera
  Theorem for Almost Periodic Solutions of
  Functional Differential Equations, Journal of the
  Mathematical Society of Japan. To appear.
• T. Naito, Nguyen Van Minh, J. Liu, On the
  bounded solutions of Volterra equations, Appl.
  Anal.. To appear.

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Some open Problems

• Extensions of the above results and methods for
  almost automorphic solutions of well-posed
  evolution equations ?
• Almost automorphic functions have gained a
  resurgence of interest to many mathematicians
  after the appearance of the monograph
• G.M. Nguerekata, Almost automorphic and almost
  periodic functions in abstract spaces. Kluwer
  Academic/Plenum Publishers, New York, 2001.
• The method of sum of commuting operators has been
  considered in
• T. Diagana, G. Nguerekata, Nguyen Van Minh,
  Almost Automorphic Solutions of Evolution
  Equations. Preprint.

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Open Problems

• - Decomposition technique ? periodic equations ?
  functional differential equations ?
• What about the almost periodic solution problem
  for ill-posed equations associated with
  C-semigroups ?
• Some questions have been answered in
• J.-C. Chen, Nguyen Van Minh, S.-Y. Shaw,
  C-Semigroups and Almost Periodic Solutions of
  Evolution Equations. Preprint.
• What about the almost automorphic solution
  problem for ill-posed equations associated with
  C-semigroups ?
• Some obstacles the evolution semigroups
  associated with equations are not strongly
  continuous in AA(X).

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Thank you for your attention