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The chaos of the solution semigroup for some partial differential equations in weighted Banach spaces
Advances in Difference Equations volume 2014, Article number: 123 (2014)
In this paper we deal with the solution semigroup of some partial differential equations in a weighted Banach space on the real axis. We aim at showing the connection between complex-analytic approach and chaotic theory. With the approach of Carleman’s formula and Joel H Shapiro’s construction, we could construct dense systems of functions from which the chaos of the solution semigroup follows. The novelty of our paper is the usage of the complex-analytic approach in investigation on chaos of some partial differential equations. As far as we know, our manuscript is the first paper in this direction.
with an initial condition
where or with a a positive constant and is a continuous bounded function on . In case of , where consists of all complex-valued functions on satisfying with the norm , both the hypercyclicity and the chaos of the solution semigroup of (1) and (2) in the form
It is natural to ask the following question:
Does the hypercyclicity or chaos of the solution semigroup of (1) and (2) still hold when is in other Banach spaces?
In the present paper we are concerned with the above question. Our study will be focused on the weighted Banach space . Let be a nonnegative continuous function defined on , henceforth called a weight, satisfying
Given a weight , the weighted Banach space consists of complex continuous functions f defined on the half real axis with vanishing at infinity, and is normed by
In , the normal family in an open set in the complex plane which is integer translates of an entire function is characterized. The hypercyclicity of bounded translation operators on Hilbert spaces of entire functions which have slow growth is characterized in . The translation operators are engaged in  to get the intriguing and beautiful chaotic characterizations of simple connectivity. For the reader’s convenience, we shall recall some basic facts on the concept of chaos.
In the last decade it has been observed that chaotic behavior in the sense of Devaney  can occur in some infinite-dimensional space for a linear operator. Recall a continuous linear operator T on a topological vector space X is called hypercyclic if there exists a vector whose orbit is dense in X. A periodic point for T is a vector such that for some . T is said to be chaotic if it is hypercyclic and its set of periodic points is dense in X.
Lemma 1.1 (The hypercyclicity criterion)
Suppose T is an operator on a Fréchet space X. Suppose further that there are dense subsets and of X, and a mapping , such that:
pointwise on ,
pointwise on ,
TS is the identity map on .
Then T is hypercyclic on X.
In this paper we shall show that the solution semigroup of (1) which is defined in (3) is chaotic in some . Our proof is based on constructing function system which is dense and periodic under the acting of the solution semigroup, which is a totally complex-analytic approach. In Section 2, we introduce some basic results from complex analysis. Our theorem on chaos of the solution semigroup of (1) will be proved in Section 3.
2 Preliminary lemmas
From now on, A denotes positive constants and it may be different at each occurrence.
Let us recall Carleman’s formula, which connects the zeros of a holomorphic function with its behavior on the boundary of a circle.
With a sequence of numbers , , we associate the function
Lemma 2.1 Let be a function analytic on , then
where is the function associated with the zeros of in S defined by (5) and is the corresponding sequence of arguments. Furthermore, is a function of R depends on f, satisfying
Let be a positive continuous function on the half-axis, if for every fixed , the quantity
is finite. It is called the Legendre transform or the Young dual function for α (see ).
Lemma 2.2 Let be a nonnegative continuous function satisfying (4), let be a sequence of complex numbers satisfying , furthermore, let be the function associated to Λ defined by (5), and be defined by (6) and (7) separately. If
then the system is dense in .
Proof We use the Hahn-Banach theorem. Suppose ϕ is a continuous linear functional on that annihilates each exponential function for . By Hahn-Banach we will be done if we can show that on .
The Riesz representation theorem provides a complex measure μ satisfying
for . In particular,
for each . Now the last equation shows that the function defined on the complex plane by
is holomorphic in the closed right half plane , satisfying
for each .
By the definition of μ, we have
Thanks to Lemma 2.1, we will see that (8) verifies Lemma 2.2. Consider in the closed half circle . Without loss of generality, we can suppose that . Application of Carleman’s formula in Lemma 2.1 yields
Recall the function defined in Lemma 2.1 remains bounded as . This forces
which gives , proving Lemma 2.2. □
For a function regular in the right half plane , the indicator function of is defined by (see )
We also need the following uniqueness theorem on holomorphic functions of exponential type growth on the right half plane characterized by the indicator functions (see ).
Lemma 2.3 (Carleson’s theorem)
Suppose that is regular and exponential type in and ; then if and only if , .
3 The chaotic theorem
With the useful criteria for the density in Lemma 2.2 and Lemma 2.3 in hand, we are able to prove Theorem 3.1, which is the main result of this paper.
Theorem 3.1 Let be the solution semigroup of (1) defined in (3) with , where and is a bounded continuous function. If
holds for some positive constant a, then the discrete semigroup is chaotic in .
Proof It is obvious that the discrete semigroup is very close to the translation operators . We shall follow the proof for in . We will proceed with the proof in two steps.
Step 1: The solution semigroup is hypercyclic in .
Our business is to find the dense subspaces and and the inverting operator S required by the hypercyclicity criterion in Lemma 1.1.
Let us define
By the density Lemma 2.2, it is obvious that is dense in . The case of can be done in a similar fashion, that is, applying Carleman’s formula in Lemma 2.1 to the closed half circle on the left half plane . This argument works as well for Lemma 2.1 and Lemma 2.2. Thus, we have obtained the dense subspaces of .
Let us verify the hypercyclicity criterion for the dense subset :
For every , . Note that can be written as , therefore, .
Let be the operator translation by −a, i.e. . For every , . Note that can be written as , therefore, .
Finally, we have on where I is the identity operator.
Step 2: admits a dense periodic points subset.
Since the obvious periodic points for , where q is a (real) rational number, no longer span a dense subspace of , it requires a bit more work.
Denote by . Since satisfies (4), we know that E is a subset of . We are going to show that E is also dense in . We proceed with the proof in Lemma 2.2 word by word. Denote by the function induced by the bounded linear functional ϕ, then
Denote by , then for fixed n, we have
where A is some positive constant depend on n. Thus is regular and exponential type in the closed right half plane , satisfying . By Lemma 2.3, we can deduce from for all . Thus E is dense in .
Now we proceed to construct dense periodic subset of with the help of E. We claim that for each point λ of the unit circle, the series
converges in the norm of to an eigenvector f of corresponding to the eigenvalue λ. Since , for fixed , we have
Thus the absolute series
Let denote the collection of all these eigenvectors where λ is a root of unity, it is obvious that is a set of periodic points of . As aforementioned, the set
is another collection of periodic points for . Thus the linear span of also consists entirely of periodic points. To prove Theorem 3.1, it remains to show that is dense in . We will see that it reduces to show that is dense in .
Recall that we have proved the density of the systems E. So it is just a job for the Hahn-Banach theorem to show that the closure of the span of contains E. Suppose ϕ is a nontrivial bounded linear functional that annihilates every function in . By the Hahn-Banach theorem it is enough to prove that ϕ also annihilates every function in . That is, we are assuming that for all and we want to prove for .
Denote by and write
for which we will prove are zero. Since ϕ is continuous and the series in (10) converges to f in the norm of , it follows that
We will show that the left-hand side of (11) has square-summable coefficients. Thus we can deduce all of the coefficients must be zero by the uniqueness theorem from the theory of Fourier series.
By the definition of the continuity of a linear functional, there exists some positive constant A such that
In particularly, (4) yields
for all , where . Since , we have , which show that the right-hand side of (11) belongs to of the unit circle. Thus by (11) all the coefficients must be zero, proving Theorem 3.1. □
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The author gratefully acknowledges the help of the referees and the editors, which lead to the improvement of the original manuscript. His thanks also goes to Prof. Hui Fang who introduced the work of Fukiko Takeo to the author. This work was supported by National Natural Science Foundation of China (Grant No. 11261024).
The author declares that they have no competing interests.
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Cite this article
Yang, X. The chaos of the solution semigroup for some partial differential equations in weighted Banach spaces. Adv Differ Equ 2014, 123 (2014). https://doi.org/10.1186/1687-1847-2014-123
- partial differential equation
- weighted Banach spaces
- Carleman’s formula