LAPLACE TRANSFORM GENERATION THEOREMS AND LOCAL CAUCHY PROBLEMS

We give new criterions to decide if some vector-valued function is a local Laplace transform and apply this to the theory of local Cauchy problems. This leads to an improvement of known results and new Hille-Yosida-type theorems for local convoluted semigroups.

Assume A : D(A) → X is some closed (linear) operator in X.
If w solves (ACP), then the convolution u := w * f solves (ACP f ), since Any function u that solves (ACP f ) is called h-regularized solution of (ACP).Now assume that 0 ∈ supp h, which means that h does not vanish on any interval [0, ε), and which is equivalent to 0 ∈ supp f .By the theorem of Titchmarsh-Foias ¸(see [3]) the convolution operator can be extended to an isometric isomorphism where the space C [ f ] ([0,T],X) of generalized functions is the completion of C([0,T],X) with norm g := S f g ∞ .On this construction see also [9].Thus, if w solves (ACP), then S f w solves (ACP f ).This gives a reason to extend the notion of solutions of (ACP): a generalized function w ∈ C [ f ] ([0,T],X) is called h-generalized solution of (ACP * ) Aw(t) = w (t) if 0 ≤ t ≤ T, w(0) = x if u = S f w solves (ACP f ), that is, if S f w is an h-regularized solution of (ACP).
The notation of generalized solutions was introduced by Cioranescu and Lumer [5,6].
If w is an h-generalized solution of (ACP * ), there is a sequence (v n ) n ⊂ C([0,T], D(A)) with lim v v n = u in C([0,T],X) and with lim n v [1]  n = u in C([0,T],D(A)) (where D(A) denotes the Banach space D(A) with the graph norm).
Thus, lim n v n = u , and lim n (Av [1]  n T],X) converges uniformly and satisfies v [1]  n (t) ∈ D(A) for all t and lim n (Av [1]  n + f (•)x − v n ) = 0 uniformly, then (v n ) n is called h-approximate solution of (ACP).Thus to every h-generalized solution there is an h-approximate solution.
On the other hand, if (v n ) n is an h-approximate solution of (ACP), then w n = S −1 h (v n − v n (0)) converges in C [h] ([0,T],X) to some w, and u = S f w = lim n v [1]  n solves (ACP f ).On approximate solutions see [1,2].Consequently, the notations of generalized solution of (ACP * ), of approximate solutions of (ACP), and (classical) solutions of (ACP f ) are equivalent.That is the reason why it is interesting to study problems of type (ACP f ).
We first clarify the notations.If X is some Banach space, g ∈ L loc 1 ([0,∞),X), and α,τ > 0, we let be the αth integral of g.If there is some h ∈ C([0,τ],X) with g = h [α] , then let g (α) = h be the αth derivative of g.The finite Laplace transform of g on [0, τ] is given by If α > 0 and g is exponentially bounded for large arguments, then g [α] is exponentially bounded for large arguments, and for all large Reλ, which is the Laplace transform of g.
In fact we have, if τ > 0, Claus Müller 71 where (1.8) This follows from (1.9) In trying to get a nice theory how to solve abstract Cauchy problems with continuous inhomogeneity f , it seems to be natural first to consider inhomogeneities of type t β , where β ∈ N or, more generally, β ≥ 0. The reason for this is that if the problem is well posed on [0, τ] with this inhomogeneity, that is, A generates a βintegrated semigroup on [0,τ], then it is well posed with inhomogeneities h [β+1] , where h ∈ L 1 ([0,τ],K), that is, A generates a local h [β+1] -convoluted semigroup on [0,τ].This is shown in Section 2. Section 3 starts with a generalization of the complex representation theorem, see [4,12].This will lead to an improvement of [7,Theorem II].
In Section 4, we generalize a representation theorem of Prüss [12] and develop a new Hille-Yosida-type theorem for integrated semigroups.

Integrated semigroups
In this section, we show that it is worthwhile to study integrated semigroups, that is, abstract Cauchy problems (ACP f ) with inhomogeneities f (t) = t β .
If τ > 0, we say that a subspace The smallest possible Y with property (A) is where 1 (a,b) is the characteristic function on (a,b).
Proof.If x ∈ X and h ∈ Y , let u x,h denote the solution of the abstract Cauchy problem Using the closedness of A and the uniqueness property of the abstract Cauchy problem, it is easy to see that the linear operator has closed graph for all h ∈ Y .Moreover, for all x ∈ X, the linear operator has closed graph.
Corollary 2.2.Let X be a real or complex Banach space, A : D(A) → X a densely defined closed linear operator, τ > 0, and β ≥ 0.

The complex representation theorem and its application
In the following, we generalize the complex representation theorem to the local case.The global version can be found in [4] or [12, Proposition 0.2].
Theorem 3.1 (complex representation theorem).Let X be a complex Banach space, c,a, ω 1 > 0, F(t) := c(e at − 1), and Furthermore, let q : M → X be a holomorphic function satisfying Then for every b > 0 there is some function First, we note that Let C 2 := sup λ∈Γ λq(λ) .We parameterize ∂Γ by the following two functions: Then the sequence of continuous functions converges uniformly on [0, q b] whenever q ∈ (0,a), since if t ∈ [0, q b] and n > m, we obtain e tr e −abr dr e rb(q −a) dr. (3.6) Thus the function Claus Müller 75 is continuous.In the same way we see that the function is bounded on (0, ab).To this end, let R > 0 and consider the three paths We have (3.12) If we let R := 1/t, we obtain the desired result.
Next, we show that, if λ ∈ Γ, To this end, consider a path β R in C consisting of a part of a circle with center ω ∈ C and radius R which connects a point on γ + with a point on γ − .Its parameterization is given by with some ψ ∈ (0,π/2) depending on R.
By Cauchy's formula we have to show that lim Consequently, if ξ ∈ (0,ab) and λ ∈ Γ, (3.17) It remains to show that, for all σ > 0, where if λ is large enough,
Proof.We first show that p is To this end, define q : M → C by q(λ) = λ α−2 p(λ).

Claus Müller 79
Moreover, (3.38) Therefore, by the theory of local convoluted semigroups, the unique solution of the abstract Cauchy problem is given by We now state and prove a lemma which we will need frequently in the sequel.
Furthermore, let h : [0,∞) → [0,∞) be some function with h(t) = o(e (τ/β)t ) as t → ∞ (this condition is always fulfilled if β = 0.In this case Γ is supposed to be some right halfplane in C), and let (3.56) Finally, let X be some real or complex Banach space and let A : D(A) → X be some linear and closed operator which generates a local f -convoluted semigroup on [0,τ].
Theorem 3.13.Let X be a complex Banach space and A : D(A) → X a linear operator.
Then the following assertions are equivalent.

The theorem of Prüss and its application
Next we generalize a result from Prüss [12, Theorem 0.4] to the local case.In its original form, the function q has to be defined on some right half-plane and then is the Laplace transform of some continuous and exponentially bounded function.This will lead to a new Hille-Yosida-type theorem for integrated semigroups.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.

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has a unique solution v x for all x ∈ X;(3) if p can be continuously extended in 0, then A D(A) generates a local p-convoluted semigroup on [0,a).