ON A FUNCTIONAL EQUATION ASSOCIATED WITH ( a, k ) -REGULARIZED RESOLVENT FAMILIES

. Let a ∈ L 1 loc ( R + ) and k ∈ C ( R + ) be given. In this paper, we study the functional equation


Introduction
Functional equations arise in most parts of mathematics.Well known examples are Cauchy's equation, the functional equations for the Riemann zeta function, the equation for entropy and numerous equations in combinatorics.Still other examples arise in probability theory, geometry and operator theory [1].
The theory of functional equations for bounded operators, emerged after the book of Hille and Phillips [11] in 1957.A strongly continuous semigroup T (t) of bounded and linear operators on a Banach space X, is defined by means of Abel's functional equation: T (t)T (s) = T (t + s), t ≥ 0, T (0) = I, which, in turn, characterizes the well posedness of the abstract Cauchy problem of first order: u (t) = Au(t), t ≥ 0; u(0) = u 0 , where Ax = lim t→0 + T (t)x − x t is defined on the domain D(A) := {x ∈ X : lim t→0 + T (t)x − x t exists in X}.In 1966, Sova [35] introduces the concept of strongly continuous cosine operator functions, C(t), by means of D'Alembert's functional equation: which characterizes the well posedness of the abstract Cauchy problem of second order: where now Ax = 2 lim t→0 + C(t)x − x t 2 is defined on D(A) := {x ∈ X : lim t→0 + C(t)x − x t 2 exists in X}.Let A be a linear operator defined on a Banach space X.In [32] Prüss proved that the Volterra equation of scalar type: (1.1) u(t) = t 0 a(t − s)Au(s)ds + f (t), is well posed if and only if it admits a resolvent family, i.e. a strongly continuous family S(t) of bounded and linear operators which commutes with A and satisfies the so called resolvent equation [32,Definition 1.3]: Resolvent families of operators have been known for a long time.They have many applications in the study of abstract differential and integral equations.However, at present there are associated functional equations for resolvent families only in special cases of the scalar kernel a(t) : For example a(t) = 1 or a(t) = t, which corresponds to the well known cases of strongly continuous semigroups and cosine operator functions, respectively.Recently, Peng and Li in [30] have proposed an interesting functional equation for resolvent families in case a(t) = g α (t) := t α−1 Γ(α) which works for 0 < α < 1 (see also [31] for the scalar case).A functional equation for the kernel a(t) = g α (t) in case α > 0 has been proposed by Chen and Li in [6] (see [6,Definition 3.1 and Theorem 3.4]).
In this paper, we shall be concerned with a commutative one parameter family of strongly continuous operators R a,k (t), depending on two scalar kernels a(t) and k(t), satisfying R a,k (0) = k(0)I and the functional equation In case k(t) ≡ 1 and a(t) positive, one of our main results in this paper show that the functional equation (1.2) characterizes a resolvent family, and hence the well-posedness of the Volterra equation (1.1).Moreover, we have the representation exists in X}, which includes the case of semigroups, cosine operator functions and resolvent families for a(t) = g α (t), α > 0. Our discussion will not be restricted to resolvent families; The more general case of (a, k)-regularized resolvent families [20] is included in our results.Indeed, in such case we will see that in addition to (1.2) the condition is necessary to characterize an (a, k)-regularized resolvent family.A remarkable consequence is that the domain of A must be necessarily dense on the Banach space X.In particular, setting k(t) ≡ 1 we obtain a new (but equivalent) functional equation for strongly continuous semigroups (i.e. the case a(t) ≡ 1) and strongly continuous cosine operator functions (i.e.a(t) ≡ t), respectively.On the other hand, we prove that the condition lim that include e.g the theory of α-times integrated semigroups, is also necessary to characterize an (a, k)regularized resolvent family.However, the immediate denseness of D(A) is not automatically obtained in such case, in concordance with the theory of integrated semigroups [3].We remark that our results recover and extends Chen and Li ([6, Theorem 3.4] and [6, Theorem 3.12]), where the case k(t) = g β+1 (t), a(t) = g α (t), α > 0, β ≥ 0 was considered (see [6,Definition 3.7]).In particular, our formula extend the functional equations stated recently by Peng and Li [31], [30] (see Remark 3.11 below).
Let us fix some notations.From now on, we take X to be a complex Banach space with norm • .We denote by B(X) the Banach algebra of all bounded linear operators on X endowed with the operator norm, which again is denoted by • .The identity operator on X is denoted by I ∈ B(X), and R + denotes the interval [0, ∞).For a closed operator A, we denote by σ(A), σ p (A), σ r (A) and σ a (A) the spectrum, the point spectrum, the residual spectrum, and the approximate spectrum of A respectively.
, a = 0 be given.Assume that A is a linear operator with domain D(A).A strongly continuous family {R(t)} t≥0 ⊂ B(X) is called an (a, k)-regularized resolvent family on X having A as a generator if the following properties hold: We emphasize that the main properties of this theory admit very clear and simple proofs, and what is more interesting, it is easy to associate a suitable regularized resolvent family to a wide class of linear evolution equations, including e.g.fractional abstract differential equations.See [21].
Assume that a and k are both positive and one of them is non-decreasing.Let {R(t)} t≥0 be an (a, k)-regularized resolvent family with generator A such that (2.1) for some constant M > 0. Then we have (2.2) Here we denote (a * k)(t) := t 0 k(t − s)a(s)ds the finite convolution between a and k.The above representation of A in terms of R(t) was established in [24] (see also [26]).We note that there is a one-to-one correspondence between (a, k)-regularized resolvent families and their generators.Moreover, we can prove that an (a, k)-regularized resolvent family is uniformly continuous if and only if its generator is a bounded linear operator [20].
We say that {R(t)} t≥0 is of type (M, ω) if there exists constants M ≥ 0 and ω ∈ R such that R(t) ≤ M e ωt for all t ≥ 0.
The next result corresponds to the generation theorem for the theory.We assume that the Laplace transform for a(t) and k(t) exists for all λ > ω.
Theorem 2.2.( [20]) Let A be a closed linear densely defined operator on X.Then {R(t)} t≥0 is (a, k)regularized resolvent family of type (M, ω) if and only if the following conditions hold: (1) a(λ) = 0 and 1 a(λ) ∈ ρ(A) for all λ > ω (2) H(λ) := k(λ)(I − a(λ)A) −1 satisfies the estimates In the case where k(t) ≡ 1, Theorem 3.4 is well known.In fact, if a(t) ≡ 1, then it is just the Hille-Yosida theorem; if a(t) ≡ t, then it is the generation theorem for generators of cosine functions due to Sova and Fattorini; for arbitrary a(t), it is the generation theorem due essentially to Da Prato and Iannelli [32].In the case where k(t) = t n n! and a(t) ≡ 1, it is the generation theorem for n-times integrated semigroups [13]; if k(t) = t n n! and a(t) is arbitrary, it corresponds to the generation theorem for integrated solutions of Volterra equations due to Arendt and Kellermann [4].
Although the generation theorem characterizes all generators of (a, k)-regularized resolvent families, it is difficult to verify the estimate of all derivatives of the operator H(λ) in concrete applications.Thus, one tries to build up the operator from simpler ones using perturbation techniques.The following is the main result available until now.It corresponds to the extension of the Miyadera-Voigt perturbation theorem in the theory of C 0 -semigroups.In this theorem, the perturbation B is bounded only from the domain of the generator D(A), endowed with the graph norm x A := x + Ax .
Theorem 2.3.( [24]) Let A be a closed operator on X. Assume that A generates an (a, k)-regularized resolvent family {R(t)} t≥0 of type (M, ω) and suppose that for all x ∈ D(A).
The next result show, roughly speaking, the continuous dependence of an (a, k)-regularized resolvent family R(t) on its generator A. More precisely, the theorem below show that the convergence -in an appropriate sense -of a sequence of generators is equivalent to the convergence of the corresponding (a, k)-regularized resolvent families.
Theorem 2.4.( [25]) Let {k n } n≥0 ∈ L 1 loc (R + ) and {a n } n≥0 ∈ AC loc (R + ) be of type (M, ω), ω ≥ 0, such that a n (λ) = 0 for λ > ω and ∞ 0 e −ωs |a n (s)|ds < ∞.Let A n be closed and linear operators in X such that A 0 is densely defined.For each fixed n ∈ N 0 , assume that R n (t) is an (a n , k n )-regularized resolvent family generated by A n in X, and that there exists constants M > 0 and ω ∈ R, independent of n, such that R n (t) ≤ M e ωt , for all t ≥ 0. Suppose also a n (t) → a 0 (t) and k n (t) → k 0 (t) as n → ∞.Then the following statements are equivalent: (1) lim x for all x ∈ X, t ≥ 0. Moreover the convergence is uniform in t on every compact subset of R + .
Note that the above theorem is the extension of the Trotter-Kato theorem for the theory of C 0semigroups, which follows in case a(t) ≡ k(t) ≡ 1.
In our next result, of concern are ergodic type theorems.Here the contributions to the theory are contained in the references [27], [18].We below cite only a simple, but typical, result.
Theorem 2.5.Let A be the generator of an (a, k)-regularized resolvent family {R(t)} t≥0 such that f or all t ≥ 0.
Suppose that (i) a(t) is positive, and k(t) is nondecreasing and positive as well.
Then the following holds: (1) The mapping P x := lim t→∞ A t x is a bounded linear projection with Ran(P ) = Ker(A), Ker(P ) = Ran(A), and (2) For 0 < β ≤ 1 and x ∈ Ker(A) ⊕ Ran(A), we have In the next result, we are interested in the relation between the spectrum of A and the spectrum of each one of the operators R(t), t ≥ 0. We denote by s(t, λ) the unique solution of the convolution equation We also define From a purely formal point of view one would expect the relation σ(R(t)) = r(t, σ(A)).This, however, is not true in general.The following result corresponds to the inclusion theorem.
Recent results on the theory of (a, k)-regularized resolvent families include conditions under which the complex inversion formula for the Laplace transform holds for (a, k)-regularized regularized families [22] and Kallman-Rota type inequalities [23].However, following the analogy with the theories of C 0semigroups and cosine operator functions, many problems are still to be solved.

A functional equation
Let X be a complex Banach space and f ∈ L 1 loc (R + , X).The Laplace integral is defined by Also define abs(f ):=inf{Reλ : f (λ) exist }.Recall that a function f is called Laplace transformable if abs(f ) < ∞.Note that a locally Bochner integrable function f is Laplace transformable if and only if its antiderivative The following theorem is the main result in this section.Theorem 3.1.Let {R(t)} t≥0 ⊂ B(X) be an (a, k)-regularized resolvent family generated by a closed operator A such that ρ(A) = ∅.Then, for all t, s ≥ 0 we have R(t)R(s) = R(s)R(t), and the functional equation Proof.Note that by [20, Lemma 2.2] we have that for all x ∈ X, (a * R)(t)x ∈ D(A) and where we used the item (ii) in Definition 2.2 and equation (3.2).This show that (F E) holds for all x ∈ X and t, s ≥ 0.
Ax where we used Definition 2.2, the fact that A is closed and R(t)Ax = AR(t)x for all x ∈ D(A), t ≥ 0. It then follows that W (t) := F (t) − H(t) satisfies Note that by (ii) in the Definition 2.2, W (t)x ∈ D(A) for all x ∈ D(A) and hence by (iii) we obtain Now let λ ∈ ρ(A), y ∈ X and define x = (λ−A) −1 y.Then (λ−A)k * W (t)x = 0 implies that k * W (t)y = 0 for each y ∈ X.Therefore, by Titchmarsh's Theorem, we obtain that W (t)x = 0 for each x ∈ X which ends the proof.
Remark 3.2.Assume that R(t) is Laplace transformable.We note that an application of the double Laplace transform to (FE) gives the following identity Let S, T : R + → B(X) be strongly continuous functions satisfying ||S(t)|| ≤ M e ωt and ||T (t)|| ≤ M e ωt (t ≥ 0) for some ω ∈ R, M ≥ 0 ( for simplicity we may assume the same constants).For h ∈ R we shall denote S h the translation S h (u the convolution product between T and S. We will need the following result.Lemma 3.3.([14, Lemma 4.1]) Let S, T : [0, ∞) → B(X) be strongly continuous functions satisfying the assumptions above.For λ > µ > ω, the following identities are valid: Defining S(t) = S(−t) for t < 0 we have, and and defining S(t) := −S(−t) for t < 0 we obtain In what follows, we restate and analyze consequences of Theorem 3.1 in several particular cases.They are important because includes different theories of strongly continuous operators and, as consequence, involves the well posedness of wide classes of abstract evolution equations.
Example 3.4.(Semigroups) k(t) ≡ a(t) ≡ 1.In this case we have that R(t) corresponds to a C 0 -semigroup and the associated functional equation (FE) reads: Corollary 3.5.Assume that R(t) is Laplace transformable.Then equation (3.9) is equivalent to Abel's functional equation.
Proof.By (3.4) we have the identity: Hence, applying the double Laplace transform to the Abel's functional equation Hence, inversion of the double Laplace transform to the above identity gives (3.9).The converse is analogue.
Example 3.6.(Cosine operator families) k(t) ≡ 1, a(t) = t.In this case we have that R(t) corresponds to a cosine operator family [3, Section 3.14] and the associated functional equation (FE) reads: Corollary 3.7.Assume that R(t) is Laplace transformable.Then equation (3.10) is equivalent to the D'Alembert's functional equation.
Proof.By (3.6) we have the identity: valid whenever R(t) is extended as an even function to R, which is indeed the case of cosine operator families.
We apply the double Laplace transform to the D'Alembert's functional equation: and we obtain R(λ) − R(µ) which, after an algebraic manipulation, is equivalent to (compare with Remark 3.2 in case a(t) ≡ t and k(t) ≡ 1).The above identity is equivalent to: Hence, inversion of the double Laplace transform to the above identity gives (3.10).The converse is analogue.
Example 3.8.(Sine operator family) k(t) ≡ t, a(t) = t.In this case we have that R(t) corresponds to a Laplace transformable sine family [3, Section 3.15] and the associated functional equation (FE) reads: Remark 3.9.The functional equation (3.11) is equivalent to the following To show this, we apply the double Laplace transform to the equation (3.12) and then Lemma 3.3, to conclude that: µ 2 − λ 2 .Finally, using Remark 3.2 in case a(t) ≡ t and k(t) ≡ t we see from the last identity that (3.11) is equivalent to the equation (3.12).We notice that the functional equations (3.11) and (3.12) seems to be new for the theory of sine operator functions [3].
In this case we have that R(t) corresponds to a Laplace transformable (g α , g β+1 )-regularized resolvent family [6, Theorem 3.12] and the associated functional equation (FE) reads: Note that, taking a(t) = k(t) = g α (t), α > 0 (i.e.β + 1 = α) this example also includes the concept of α-resolvent families introduced in [2], which characterizes the well posedness of the fractional Cauchy problem (3.13) D α t u(t) = Au(t), t ≥ 0, where D α t denotes the Riemann-Liouville fractional derivative.On the other hand, the special case k(t) ≡ 1 and a(t) = g α (t), α > 0 gives the theory of solution operators introduced by Bazhlekova [5] for the fractional Cauchy problem (3.13)where now D α t denotes Caputo's fractional derivative instead of Riemann-Liouville fractional derivative.Note that the functional equation has now the following form: Observe the remarkable fact that in the scalar case, i.e.X = R and A = ρ ∈ R, we have that the Mittag-Leffler function E α (ρt α ) satisfies the functional equation (3.14), because it is the unique solution of (3.13) with the fractional derivative considered in the sense of Caputo.In other words, we have for α > 0. In particular, it shows that the functional equation (3.14) is a proper generalization of Abel's functional equation (corresponding to the case α = 1) and D'Alembert functional equation (corresponding to the case α = 2) since in case α = 1 we have E 1 (ρt) = e ρt and in case α = 2 we have E 2 (ρt 2 ) = cosh( √ ρt).
Remark 3.11.Suppose the family {R(t)} t≥0 of bounded linear operators of X is Laplace transformable.Then for 0 < α < 1 the functional equation (3.14) is equivalent to the functional equation proposed by Peng and Li in [31] and [30].Indeed, we apply the double Laplace transform to the equation (3.15) and use Lemma 3.3 to conclude that the following identity holds for 0 < α < 1: Then, algebraic manipulation shows that it is equivalent to the identity Finally, using Remark 3.2 in case a(t) = t α−1 Γ(α) and k(t) ≡ 1 we see from the last identity and inversion of the double Laplace transform that (3.15) is equivalent to the functional equation (3.14).
Example 3.12.(Laplace transformable k-times integrated semigroups).We take in this example k(t) = t k Γ(k+1) , k = 0, 1, ... and a(t) ≡ 1.We have that R(t) is an k-times integrated semigroup and the functional equation has the form Following the same type of arguments as in Corollaries 3.5 and 3.7 (see also the following example), we note that the above equation is equivalent to the following well known formula that originally define k-times integrated semigroups (see [ : Example 3.13.(K-convoluted semigroups) Let K be a complex-valued, locally integrable function on [0, ∞).We take in this example k(t) = t 0 K(σ)dσ and a(t) ≡ 1.We have that R(t) is an K-convoluted semigroup (see [16, Definition 2.1] and references therein) and the functional equation has the form which is equivalent with the standard definition: Indeed, if R is Laplace transformable then we can apply the double Laplace transform to the equation (3.17) and use Lemma 3.3 to conclude that: Then observing the formula in Remark 3.2 we note that (3.16) is equivalent to the functional equation (3.17).
Example 3.14.(K-convoluted cosine functions).Let K be a complex-valued, locally integrable function on [0, ∞).We take in this example k(t) = t 0 K(σ)dσ and a(t) = t.We have that R(t) is an K-convoluted semigroup and the functional equation has the form.
in contrast with the (equivalent) recently discovered expression [14]: we obtain the following functional equation, which seems to be the first, for the theory of integral equations of convolution type [32]: Example 3.16.(Integral resolvents).Taking a(t) = k(t) we obtain, to our knowledge, the first functional equation for the theory of integral resolvents.Of course, it includes the scalar case [10]: Example 3.17.(An special case).Taking k(t) = 1 and a(t) = t α−1 Γ(α) γe −βt , γ = 0, we obtain In this example, the kernel a(t) is important in viscoelasticity theory [32].

Sufficient conditions
Let k, a ∈ L 1 loc (R + ) be given and R : R + → B(X) be a strongly continuous family such that (4.1) holds for all s, t ≥ 0. In this section, we study in what extent the functional equation (FE) is sufficient to imply that R(t) is an (a, k) resolvent family.In passing, we are going to unify and clarify from a general perspective a basic property of the theories of semigroups and cosine operator families: Automatic denseness of the domain of the generator.Then R(t) is an (a, k)-regularized resolvent family with generator B.Moreover, B is closed and D(B) is dense in X.
Proof.Since R(0) = k(0)I, we have to prove conditions (ii) and (iii) from Definition 2.1.Fix x ∈ D(B) and t ≥ 0. For s ≥ 0 we have from the commutativity of the family R(t) that Since R(t) is bounded, by definition of B we have that lim exist and equals to R(t)Bx.Then (4.4) implies that R(t)x ∈ D(B) and BR(t)x = R(t)Bx for all t ≥ 0. It proves condition (ii) in Definition 2.1.In order to show condition (iii), let x ∈ X be given and note that by (4.3) we have that It follows from (4.1) and (4.5) that as s → 0 + for all x ∈ X.Then, for all x ∈ X and t ≥ 0, (a * R)(t)x ∈ D(B) and Now let x ∈ D(B).Note that (a * R)(t)x ∈ D(B) and hence, by (4.2) and the commutativity of R(t), we have It then follows from (4.7) and (4.8) that, for all x ∈ D(B) It shows (iii) in Definition 2.1 and hence that R(t) is an (a, k)-regularized resolvent family generated by B.
We now show closedness.Let (x n ) ∈ D(B) be a sequence such that x n → x and Ax n → y as n → ∞.It follows from the second equality in (4.9) that Then by (4.5) we have Finally, we show that D(B) is dense in X.Let x ∈ X be given.Then it was proved that a * R(t)x ∈ D(B) for all t ≥ 0. Hence, defining it follows from (4.5) that x n ∈ D(B) and lim n→∞ x n = x, proving the claim and the theorem.
Remark 4.2.We notice that the denseness of D(A) is always present under the conditions of the above theorem.In particular, we recover a well known result in the theories of semigroups and cosine operator functions and, more important, gives new results for other theories of strongly continuous functions of operators.An immediate application of this result is that the Laplacian operator with Dirichlet boundary conditions cannot be the generator on L p spaces for p = 2, of those classes of (a, k)-regularized resolvent families that have a(t) and k(t) satisfying (4.3).
Our next result, studies a complementary case of Theorem 4.1.Here we deal with integrated versions of the operator families where, as we known, the density of the domain is not present, in general.
Then for all |µ| ≤ δ, It follows from (4.11), (4.12) and the condition (4.10) that Finally, note that (a * R)(t)x ∈ D(B) and ).We left the details to the interested reader.

Application
In this section, we give one example to show how this class of regularized families defined by functional equations appears in new concrete problems, while other methods not apply directly.
Consider the following nonlinear third order differential equation (5.1) u (t) + αu (t) + c 2 Au(t) + bAu (t) = f (t, u(t), u (t), u (t)) with given initial conditions u(0), u (0), u (0), and where α, b, c are positive real numbers and f is a vector-valued function.Equation (1.1) has recently attracted the attention of a number of authors because their applications in different fields as, for example, high intensity ultrasound and vibrations of flexible materials.See [12], [9], [7] and references therein.In such cases, usually the operator A is the negative Laplacian and f (t, u, u t , u tt ) = (K(u t ) 2 + |∇u| 2 ) t for a suitable constant K > 0. For example, in high intensity ultrasound, u is the velocity potential of the acoustic phenomenon described on some bounded R 3domain.In the abstract case, A is a non-negative, self-adjoint operator (possibly with compact resolvent) defined on a Hilbert space H.
Mathematical understanding of the linearized equation (5.2) u (t) + αu (t) + c 2 Au(t) + bAu (t) = 0 is meant as a preliminary critical step for the subsequent analysis of the full nonlinear model (5.1).The usual operator-theoretic method to solve (5.2) is to rewrite it as a first order abstract Cauchy problem of the form U (t) = AU (t), t ≥ 0, U (0) = U 0 , on the Banach space X := X × X × X with the usual norm, and where

Theorem 4 . 3 .
Let k ∈ C(R + ) and a ∈ L 1 loc (R + ).Let R(t) ⊂ B(X) be a commutative strongly continuous family such that R(0) = k(0)I and satisfies (FE).Define D(B) and B as in the above theorem.Suppose the following condition(4.10)lim s→0+ (a * (a * k))(s) a * k(s) = 0.Then R(t) is an (a, k)-regularized resolvent family with generator B.Proof.Since R(0) = k(0), we have to prove conditions (ii)-(iii) in Definition 2.1.The proof of (ii) is the same as in Theorem 4.1.To show (iii), fix x ∈ D(B) and note that a

2 A(λ 2 +
−bA −αI   is defined on D(A) := D(A) × D(A) × X.It is known that if A is unbounded then, for b = 0, the matrix of operators A cannot be the generator of a C 0 -semigroup on X , or even in some subspaces of it (see[12,  Theorem 1.1]).It can be seen directly observing the entries of the resolvent operator(λI − A) −1 =   αλ + bA)H(λ) (λ + α)H(λ) H(λ) −c 2 AH(λ) (λ 2 + αλ)H(λ) λH(λ) −λc 2 AH(λ) −(λb + c 2 )H(λ) λ 2 H(λ)   [17,(a * R)(t)Bx, proving the theorem.Example 4.4.Suppose that a(t) and k(t) are positive kernels, then the condition (4.10) holds.In fact, denote c(t) = (a * k)(t).Then c(t) is positive non-decreasing and therefore Remark 4.5.In view of Theorems 3.1, 4.1 and 4.3, the Definition 2.1 is equivalent, under certain conditions on the kernels a(t) and k(t), to the functional equation (FE).This fact can be used to define (a, k)-regularized families in a local way, and avoid the use of Laplace transform in the development of the theory.We note that more general classes of families of bounded operators like, e.g.(a, k)-regularized C-regularized families can be understood using (FE) (see e.g.[17, Definition 1.2]