Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

Lemma 1. Let 0 < σ < (1/2)π. Then, for every $$ and $$,

where Z_s is defined by Z_s≔z^1/αe^2πis/α and the first summation is taken over all those integers s satisfying |arg(z) + 2πs| < α(π/2 + σ).

Definition 2 (see [35].)Assume that $$ or I = [0, ∞). Let 1 ≤ p < ∞ and $$.

• (i)

  It is said that the function f(·) is equi-Weyl-p-almost periodic, $$ for short, iff for each ϵ > 0 we can find two real numbers l > 0 and L > 0 such that any interval I^′⊆I of length L contains a point τ ∈ I^′ such that

  $$ (14)

• (ii)

  It is said that the function f(·) is Weyl-p-almost periodic, $$ for short, iff for each ϵ > 0 we can find a real number L > 0 such that any interval I^′⊆I of length L contains a point τ ∈ I^′ such that

  $$ (15)

Definition 3. We say that $$ is Weyl-p-vanishing iff

Proposition 4. Let (f_n) be a uniformly convergent sequence of functions from e  −  W^p([0, ∞) : X)∩C_b([0, ∞) : X), respectively, W^p([0, ∞) : X) ∩ C_b([0, ∞) : X), where 1 ≤ p < ∞. If f(·) is the corresponding limit function, then f ∈ e − W^p([0, ∞) : X)∩C_b([0, ∞) : X), respectively, f ∈ W^p([0, ∞) : X)∩C_b([0, ∞) : X).

Proof. We will prove the part (i) only for the equi-Weyl-p-almost periodic functions. It is clear that f ∈ C_b([0, ∞) : X). Let ϵ > 0 be given in advance. Then there exists an integer n₀(ϵ) such that for each n ≥ n₀(ϵ) we have that

By definition, we know that there exist two real numbers $$ and $$ such that any interval I^′⊆I of length $$ contains a point $$ such that Then, for the proof of equi-Weyl-p-almost periodicity of function f(·), we can choose the same $$ and $$, and the same $$ from any subinterval I^′⊆[0, ∞); speaking-matter-of-factly, we have for all t ≥ 0, so that a simple calculation involving (18) gives the existence of a finite constant c_p > 0 such that Then the final result simply follows from (19).

Lemma 5. Let Ω be a locally compact, separable metric space, and let μ be a locally finite Borel measure defined on Ω. Suppose that $$ is a closed MLO. Let f : Ω → X and g : Ω → Y be μ-integrable, and let $$, x ∈ Ω. Then $$ and $$

Lemma 6. Assume that $$ is a closed MLO and that f∈ (P1)-X, l∈ (P1)-Y and $$, $$ for $$ Then $$ for any t ≥ 0 which is a point of continuity of both functions f(t) and l(t).

Lemma 7. Let B, D ∈ L(X) and let $$ be an MLO. If BD = DB, $$, $$, and $$, then one has

Definition 10. A function $$ is called a (strong) solution of (1) iff $$ for 0 ≤ i ≤ n − 1, $$, and (1) holds.

Definition 11. Suppose that 0 < τ ≤ ∞, k ∈ C([0, τ)), C, C₁, C₂ ∈ L(X), and  C and C₂ are injective. A sequence $$ of strongly continuous operator families in L(X) is called a (local, if τ < ∞):

• (i)

  k-regularized C₁-existence propagation family for (1) iff the following holds:

  $$ (30)

•  

  for any i = 0, …, m_n − 1.

• (ii)

  k-regularized C₂-uniqueness propagation family for (1) iff the following holds:

  $$ (31)

•  

  provided $$ and $$.

• (iii)

  k-regularized C-resolvent propagation family for (1), in short k-regularized C-propagation family for (1), iff $$ is a k-regularized C-uniqueness propagation family for (1), and if for every t ∈ [0, τ), $$, and $$, one has R_i(t)A_j⊆A_jR_i(t), R_i(t)C = CR_i(t), and CA_j⊆A_jC.

Definition 12. Let f ∈ C([0, ∞) : X). Consider the following inhomogeneous Cauchy inclusion:

A function u ∈ C([0, ∞) : X) is said to be

• (i)

  a strong solution of (33) iff there exists a continuous function $$ such that $$ for all t ≥ 0 and

  $$ (34)

• (ii)

  a mild solution of (33) iff

  $$ (35)

Theorem 13. Suppose k(t) satisfies (P1), ω ≥ max(0, abs(k)), (R_i(t)) _t≥0 is strongly continuous, and the family {e^−ωtR_i(t) : t ≥ 0}⊆L(X) is bounded, provided 0 ≤ i ≤ m_n − 1. Let $$ be a closed MLO on X, let C₁, C₂ ∈ L(X), and let C₂ be injective. Set

• (i)

  Suppose A_j ∈ L(X), $$ Then $$ is a global k-regularized C₁-existence propagation family for (1) iff the following conditions hold:

  • (a)

    The inclusion

    $$ (37)

  •  

    holds provided x ∈ X, $$, m − 1 < i, and $$.

  • (b)

    The inclusion

    $$ (38)

  •  

    holds provided x ∈ X, $$, m − 1 ≥ i, and $$.

• (ii)

  Suppose A_j ∈ L(X), $$ Then $$ is a global k-regularized C₂-uniqueness propagation family for (1) iff, for every $$ with $$, $$, and $$, the following equality holds:

  $$ (39)

Theorem 14. Suppose k(t) satisfies (P1), ω ≥ max(0, abs(k)), (R_i(t)) _t≥0 is strongly continuous, and the family {e^−ωtR_i(t) : t ≥ 0}⊆L(X) is bounded, provided 0 ≤ i ≤ m_n − 1.

• (I)

  Let the following two conditions hold:

  • (i)

    CA_j⊆A_jC, $$, A_j ∈ L(X), $$, A_iA_j = A_jA_i, $$, and $$, $$.

  • (ii)

    There exists an index $$ satisfying exactly one of the following two conditions:

    • (a)

      m − 1 < i and the operator $$ is injective for every $$ with $$ and $$,

    • (b)

      m − 1 ≥ i, $$, and the operator $$ is injective for every $$ with $$ and $$

•  

  If $$ is a global k-regularized C-resolvent propagation family for (1) and (30) holds, then P_λ is injective for every $$ with $$ and $$ and equalities (37)-(38) are fulfilled.

• (II)

  Suppose that P_λ is injective for every $$ with $$ and $$ and equalities (37)-(38) are fulfilled. If condition (I)(i) holds, then $$ is a global k-regularized C-resolvent propagation family for (1).

Proof. Concerning assertion (I), we will only sketch the main details of the proof of the injectivity of operator P_λ for every $$ with $$ and $$ (we know that (37)-(38) hold on account of Theorem 13). Observe that we do not need the condition (I)(ii) for the proof of (II), where we only use an elementary argumentation as well as Lemmas 5–7 (the composition property (31) follows by applying the Laplace transform and Lemma 7, while the commutation of operator families R_i(·) with the operators C and A_j for 0 ≤ j ≤ n − 1 is much simpler to show). The consideration is quite similar in the case that the condition (II) holds and, because of that, we will consider only the first case. Let $$ with $$ and $$ be fixed, and let $$ for some x ∈ X. Using the fact that $$ is a global k-regularized C-uniqueness propagation family for (1), we can simply prove that

by performing the Laplace transform at the both sides of the composition property (31). By the injectivity of the operator $$ for λ = λ₀, we obtain that x = 0 and the claimed assertion follows.

Definition 15. (i) Let β ∈ (0, π], and let $$ be a k-regularized C-resolvent propagation family for (1). Then it is said that $$ is an analytic k-regularized C-resolvent propagation family of angle β, iff for each $$ there exists a function R_i : Σ_α → L(X) which satisfies that, for every x ∈ X, the mapping z ↦ R_i(z)x, z ∈ Σ_β is analytic and that

• (a)

  R_i(t) = R_i(t), t > 0, and

• (b)

  $$ for all γ ∈ (0, β) and x ∈ X.

(ii) Suppose that $$ is an analytic k-regularized C-resolvent propagation family of angle β. Then it is said that $$ is an exponentially bounded, analytic k-regularized C-resolvent propagation family of angle β, respectively, bounded k-regularized C-resolvent propagation family, iff for every γ ∈ (0, β), there exists ω_γ ≥ 0, respectively, ω_γ = 0, such that the family $$ is bounded for all $$. Since there is no risk for confusion, we will identify in the sequel R_i(·) and R_i(·).

Theorem 16. Assume k(t) satisfies (P1), ω ≥ max(0, abs(k)), β ∈ (0, π/2], and, for every $$, the function (k∗g_i)(t) can be analytically extended to a function $$ satisfying that, for every γ ∈ (0, β), the set {e^−ωzk_i(z) : z ∈ Σ_γ} is bounded.

Let the following three conditions hold:

• (i)

  CA_j⊆A_jC, $$, A_j ∈ L(X), $$, A_iA_j = A_jA_i, $$, and $$, $$.

• (ii)

  The operator P_λ is injective for all ω + Σ_β+π/2.

• (iii)

  Let q_i : ω + Σ_π/2+β → L(X)  (0 ≤ i ≤ m_n − 1) satisfy that, for every x ∈ X, the mapping λ ↦ q_i(λ)x, λ ∈ ω + Σ_π/2+β is analytic and that for each $$ there exists an operator D_i ∈ L(X) such that

  $$ (41)

•  

  provided m − 1 < i,

  $$ (42)

•  

  provided m − 1 ≥ i,

  $$ (43)

•  

  and, in the case $$,

  $$ (44)

Then there exists an exponentially bounded, analytic k-regularized C-resolvent propagation family $$ for (1), of angle β. Furthermore, (30) holds, the family {e^−ωzR_i(z) : z ∈ Σ_γ} is bounded for all $$ and γ ∈ (0, β), and R_i(z)A_j⊆A_jR_i(z), z ∈ Σ_β, and $$.

Remark 17. For the sequel, it will be very important to note that the notion introduced in Definition 11(iii) and Definition 15 can be introduced for any single operator family (R_i(t)) _t≥0 of the tuple $$ The assertions of Theorems 14 and 16 can be simply reformulated for (R_i(t)) _t≥0; for example, if the index $$ is given in advance, then in the formulation of Theorem 16 it suffices to assume that the function (k∗g_i)(·) can be analytically extended to a function $$ satisfying that, for every γ ∈ (0, β), the set {e^−ωzk_i(z) : z ∈ Σ_γ} is bounded, and that (i)-(ii) hold and (iii) holds only for this specified index i. It will be said that (R_i(t)) _t≥0 is an (exponentially bounded, analytic/analytic) k_i-regularized C-resolvent propagation family. All terminological agreements explained before will be accepted for k_i-regularized C-resolvent propagation families; the classes of k_i-regularized C₁-existence propagation families and k_i-regularized C₂-uniqueness propagation families are introduced similarly.

Theorem 18. Suppose that c_j ≥ 0 for 1 ≤ j ≤ n − 1, ζ^′ ≥ 0, $$ is a closed MLO, C ∈ L(X) is injective, $$, and the following condition holds:

• (H)

  There exist finite constants c < 0, M > 0, 0 < θ < π, and β ∈ (0,1] such that

  $$ (45)

Assume that the mapping $$, $$ is strongly continuous. Assume also that $$ satisfies m − 1 < i, and Set ζ≔ζ^′ if $$ is densely defined, ζ > ζ^′ otherwise, and k_i(·)≔g_ζ+1(·). Then there exists an exponentially bounded, analytic k_i-regularized C-propagation family (R_i(t)) _t≥0 for (1), of angle ν≔min(ν^′, π/2). Moreover, (30) holds and there exists a finite constant M^′ > 0 such that

Proof. Since we have assumed that the mapping $$, $$ is strongly continuous, and its restriction to c + Σ_π−θ is strongly analytic on this region; see [19, Proposition  1.2.6(iii)]. Taking into account (47) and c_j ≥ 0 for 1 ≤ j ≤ n − 1, we get

It is clear that (46) implies Using an elementary argumentation, (46), (49), and (50), we can simply verify that the conditions of Theorem 16 hold with ω > 0 sufficiently large, k(t) = g_ζ+1(t), t ≥ 0, and D_i = 0, in the case that the operator $$ is not densely defined. Hence, $$ is a subgenerator of an exponentially bounded, analytic ζ-times integrated C-propagation family (R_i(t)) _t≥0 for (1), of angle ν≔min(ν^′, π/2), as claimed. Estimate (48) remains to be proved. Fix the numbers t > 0 and 0 < γ < ν. By the proof of [2, Theorem 2.6.1] and Cauchy theorem, we have that, for every x ∈ X, where Γ is oriented counterclockwise and consists of Γ_±≔{re^i(π/2+γ) : r ≥ t⁻¹} and Γ₀≔{t⁻¹e^iθ:|θ | ≤ π/2 + γ}. Keeping in mind (49) and the estimate from the condition (H), it readily follows that for each λ ∈ Γ we have that $$, so that But, then estimate (48) follows from a simple integral computation that is very similar to that appearing in the proof of [2, Theorem 2.6.1].

Remark 19. (i) It is worth noting that the value of exponent β in (H) does not depend on the final estimate (48). In our proof, we only use the estimate $$, λ ∈ Γ.

(ii) As mentioned in the introductory part, the proof of important result of Cuesta [15, Theorem 2.1] for the classical fractional oscillation resolvent families generated by densely defined linear operators [11], satisfying the condition (H) with β = 1, C = I, and ω = c < 0, follows completely different lines. Furthermore, in our approach, the case in which π/2 < θ < π or α_n − α < 1 can occur, Theorem 18 is applicable in the qualitative analysis of fractional relaxation multiterm differential inclusions (but not in the analysis of generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions; see (53)).

Corollary 20. Let the requirements of Theorem 18 hold, let $$, and let k_i(·) = (g_ζ+1∗f)(·). Set u_x(t)≔(R_i∗f)(t)x, t ≥ 0, x ∈ X. Assume that

Then, for every x ∈ X, $$ is a unique mild solution of the abstract Cauchy inclusion Furthermore, u_x(·) is a strong solution of (54) provided that $$.

Proposition 21. Suppose that k(t) satisfies (P1) and (R_i(t)) _t≥0 is a strongly Laplace transformable k_i-regularized C-propagation family for (1).

• (i)

  For every $$, there exists a function f_λ(·) satisfying (P1)-L(X) and

  $$ (55)

•  

  provided that m − 1 < i, respectively,

  $$ (56)

•  

  provided that m − 1 ≥ i.

• (ii)

  Denote by D the set consisting of all eigenvectors x of operator $$ which corresponds to eigenvalues $$ of operator $$ for which the mapping

  $$ (57)

belongs to the space $$. Then the mapping t ↦ R_i(t)x, t ≥ 0, belongs to the space $$ for all x ∈ span(D); furthermore, the mapping t ↦ R_i(t)x, t ≥ 0, belongs to the space $$ for all $$ provided additionally that (R_i(t)) _t≥0 is bounded.

Proof. We will examine the case m − 1 < i only. The proof of (i) can be given following the lines of the proof of [9, Theorem 1.1.11], with appropriate changes briefly described as follows. Since the operator $$ is invertible in L(X) for all $$ with |z| sufficiently large, because the norm of bounded linear operator $$ for such values of z is strictly less than 1, we get that the term

is well-defined for all $$ with $$, for some ω₀ > max(0, abs(k)). Set and Then it is clear that and that there exists a finite constant c > 0 such that where $$ denotes the kth convolution power of H₀(·). The function $$, t > 0, is well-defined since there exists a finite constant c^′ > 0 such that and, due to Lemma 1, For the remaining part of proof of (i), it suffices to repeat literally the arguments from the proof of [9, Theorem 1.1.11]. For the proof of (ii), observe first that, if $$ for some $$, then performing the Laplace transform at the both sides of the composition property (31), as it has been done as in our previous examinations, immediately yields that for $$ suff. large, and therefore R_i(t)x = f_λ,x(t), t ≥ 0. As a consequence, we have that the mapping t ↦ R_i(t)x, t ≥ 0, belongs to the space $$ for all x ∈ span(D). The boundedness of (R_i(t)) _t≥0 implies the uniform convergence of R_i(t)x_n to R_i(t)x  (t ≥ 0) for any sequence (x_n) ∈ span(D) converging to some element $$; then the final result follows by combining the previously proved statement and the fact that the limit of a uniform convergent sequence of bounded continuous functions belonging to any space from $$ belongs to this space again (see Proposition 4 for the class of (equi-) Weyl-almost periodic functions).

Remark 22. If m − 1 < i and A_j = c_jI for some $$  (1 ≤ i ≤ n − 1), then a simple calculation shows that

for x ∈ X satisfying $$  $$. To the best knowledge of the authors, in the handbooks containing tables of Laplace transforms, the explicit forms of functions like f_λ,x(·) are not known, with the exception of some very special cases of the coefficients α_j, c_j (see, e.g., [12, Remark  3.3.10(vi)]).

Theorem 23. Assume that (R_i(t)) _t≥0 is an exponentially bounded k_i-regularized C-propagation family (R_i(t)) _t≥0 for (1). Let m − 1 < i, and let there exist a number ν > 1 such that ‖R_i(t)‖ = O(t^−ν), t ≥ 1. Assume, further, that the following conditions hold:

• (i)

  Let f ∈ C([0, ∞) : X) satisfy that there exists a function g ∈ AAP([0, ∞) : X) such that (C⁻¹f)(t) = (g_i∗k∗g)(t), t ≥ 0.

• (ii)

  Assume that g(t) = g_ap(t) + g₀(t), t ≥ 0, where g_ap ∈ AP([0, ∞) : X) and g₀ ∈ C₀([0, ∞) : X).

• (iii)

  Let $$ for all j ∈ D_i.

• (iv)

  Assume that g_k∗g ∈ L_∞([0, ∞) : X) for all $$ and c₀⊈X (i.e., X does not contain an isomorphic copy of c₀), or R(g_k∗g) is weakly relatively compact in X for all $$

• (v)

  For every $$, we have

  $$ (67)

Then there exists a unique exponentially bounded mild solution u(·) of the abstract Cauchy inclusion (33). Furthermore, u ∈ AAP([0, ∞) : X).

Proof. From our previous considerations of nondegenerate case, it is well known that any mild solution of the abstract Cauchy inclusion (33) has to satisfy the following equality:

See, for example, [12, Theorem 2.10.7]. Taking the Laplace transform, we get Since (C⁻¹f)(t) = (g_i∗k∗g)(t), t ≥ 0, we get By the proofs of Proposition 21 and [9, Theorem 1.1.11], the right-hand side of above equality is really the Laplace transform of a continuous exponentially bounded function u(·) given by With the help of Laplace transform and a simple calculation, it can be simply verified that the function u(·), whose Laplace transform is given by (69), is a mild solution of abstract fractional inclusion (33). The growth order of R_i(·) implies that the function t ↦ (R_i∗g)(t), t ≥ 0, is in AAP([0, ∞ : X). Since AAP([0, ∞ : X) is closed in C_b([0, ∞ : X) and $$ converges uniformly to $$ for t ≥ 0, the asymptotical almost periodicity of u(·) immediately follows if we prove that the function G_k(t) = (g_k∗g)(t), t ≥ 0, is asymptotically almost periodic for all $$; see (iii). But, this can be shown by making use of (iv)-(v) and applying successively [24, Theorem 2.2.2]; here, we only want to observe that the vanishing term of function G₁(t) is given by $$  (t ≥ 0), since $$ The uniqueness of exponentially bounded mild solutions of (33) can be proved as follows. Let u(·) be such a solution. Taking the Laplace transform and multiplying after that with $$, we get Hence, $$ and $$ for $$, $$ By the uniqueness theorem for the Laplace transform, u(·) must be uniquely determined. The proof of the theorem is thereby complete.

Remark 24. Concerning Theorem 23, the case in which m − 1 ≥ i is a little bit complicated: it seems that the assumption A_j = c_j for some complex numbers $$  (1 ≤ j ≤ n − 1) has to be imposed for establishing of any relevant result. Details can be left to the interested reader.

Example 1. In [16], the authors have considered the abstract two-term fractional differential equation

where c₁ > 0, A is a densely defined linear operator satisfying the condition (H) with β = 1, C = I and c < 0, 0 < α^′ ≤ β^′ ≤ 1, and f(t) is a given X-valued function; here, we have been forced to slightly change the notation used in [16]. For this, the notion of an $$-regularized family generated by A, which is a special case of the notion introduced in Definition 9 with $$, t ≥ 0, and $$, t ≥ 0, has been introduced; compare [16, Lemma 2.5]. Our main contributions will be given in the case that 0 < α^′ < β^′ ≤ 1, which is used in the formulations and proofs of [16, Theorems 4.3 and 5.3], the main results of afore-mentioned paper (although possible applications can be given in the study of two-term fractional Poisson heat equations on the space H⁻¹(Ω), where Ω is a bounded domain in $$ with smooth boundary [18], we will pay our attention to the case that $$ is a single-valued linear operator).

Let us define an exponentially bounded, analytic $$-regularized family $$ of angle ν ∈ (0, π/2], subgenerated by A, as the exponentially bounded, analytic (a, k)-regularized C-resolvent family of angle ν, subgenerated by the same operator, with a(t) and k(t) being defined as above. Then we have

and the assertion of [16, Theorem 3.1] holds with the initial values x, y ∈ X and the inhomogeneity f(t) replaced therein with the initial values C⁻¹x, C⁻¹y ∈ X and the inhomogeneity C⁻¹f(t), respectively, with the meaning clear.

Assume that the condition (H) holds with β = 1, c < 0, and θ = π/2 − γ^′π/2, where α^′ < γ^′ < β^′. Then we can argue as follows. Similarly as in the proof of Theorem 18, we have that the operator A is a subgenerator of an exponentially bounded, analytic $$-regularized family $$ of angle ν≔(π/2(γ^′ + 1))/(α^′ + 1) − π/2; compare also [12, Theorem 2.2.5]. Estimating the term $$ uniformly, as in the proof of Theorem 18, and using the integral computation given in the final part of this theorem and [2, Theorem 2.6.1], we get

here we would like to note that the arguments used in proof of [16, Theorem 4.1] are much more complicated than ours and that our estimate is better even in the case that C = I because then, with the notion introduced in [16] accepted, we only require that the operator A is of sectorial angle γ^′π/2, not of β^′π/2, as the stronger estimate in [16] requires. Using the estimate (75), we can repeat literally the proofs of [16, Theorems 4.3 and 5.3] in order to see that their validity hold with C-sectorial operators of smaller angle γ^′π/2, with the meaning clear. In the case that C ≠ I, we can make applications to generators of analytic C-regularized semigroups generated by nonelliptic differential operators in L^p-spaces [44].

In [17], Luong has investigated the following abstract two-term fractional differential equation with nonlocal conditions in a Banach space X:

where 0 < α^′ ≤ β^′ ≤ 1, the functions g(·) and h(·) satisfy certain conditions, and A is of sectorial angle β^′π/2. The improvements of main results of this paper, Theorem 13, to C-sectorial operators of angle γ^′π/2, with γ^′ being clarified above, can be proved straightforwardly (0 < α^′ < β^′ ≤ 1).

Example 2. (i) ([45]) Assume that η ∈ (0,1), $$, Ω is a bounded domain in $$ with boundary of class C^4q, and $$ Consider the operator $$ given by

with domain $$ In this place, $$, $$, and $$ Let $$ satisfy the following:

• (i)

  $$ for all $$ and |β| = 2q.

• (ii)

  $$ for all |β| ≤ 2q, and

• (iii)

  there exists a constant M > 0 such that

  $$ (78)

Then there exists a sufficiently large number σ > 0 such that the operator −A_σ ≡ −(A + σ) satisfies the condition (H) with β = 1 − η/2q, C = I, c < 0, and some θ ∈ (0, π/2). Let us remind ourselves that A is not densely defined and that the value of exponent β = 1 − η/2q is sharp. Applications of Corollary 20 are clear and here we would like to illustrate just one of them: c_j = 0 for j < n − 1, α_n−1 = α_n−, α = 0+, i = 0, and α_n(η/2q) < ζ < α_n − 1.

(ii) ([18]) Let X≔L^p(Ω), where Ω is a bounded domain in $$, b > 0, m(x) ≥ 0 a.e. x ∈ Ω, m ∈ L^∞(Ω), and 1 < p < ∞. Suppose that the operator A≔Δ − b acts on X with the Dirichlet boundary conditions and that B is the multiplication operator by the function m(x). By the analysis contained in [18, Example 3.6], the condition (H) is satisfied for the multivalued linear operator $$ with β = 1/p, C = I, and some numbers c < 0 and θ ∈ (0, π/2); here it is worth noting that the validity of additional condition [18, (3.42)] on the function m(x) enables us to get the better exponent β in (H), provided that p > 2. Applications in the study of the existence and uniqueness of asymptotically almost periodic and asymptotically almost automorphic solutions of multiterm fractional integrodifferential Poisson heat equation

where $$ for all t ≥ 0 and x ∈ Ω, are immediate (φ ∈ X).

Example 3. In [12, Subsection 3.3.2], we have analyzed hypercyclic and topologically mixing properties of solutions of abstract multiterm fractional Cauchy problem (1) with $$ being single-valued and A_j = c_jI for some $$  (1 ≤ i ≤ n − 1). Let it be the case. With the help of Proposition 21, we can reconsider a great number of examples given in the above-mentioned part of [12] and provide several interesting applications in the investigation of problem about the existence of a dense linear subspace X^′ of X such that the mapping t ↦ R_i(t)x, t ≥ 0, is asymptotically almost periodic for all x ∈ X^′; here, (R_i(t)) _t≥0 is a given k_i-regularized C-propagation family for (1) with a subgenerator A (cf. [7] for further information in this direction). For the sake of illustration, we will consider only the situation of [12, Example  3.3.12(ii)]; see also Ji and Weber [31]. Suppose that X is a symmetric space of noncompact type and rank one, p > 2, and the parabolic domain P_p and the positive real number c_p possess the same meaning as in [31]. Suppose, further, that $$, $$ is a nonconstant complex polynomial with a_k > 0, n = 2, 0 < a < 2, α₂ = 2a, α₁ = 0, α = a, c₁ > 0, i = 0, and $$ Then we already know that the operator $$ is the integral generator of an exponentially bounded, analytic resolvent propagation family ((R_θ,P,0(t)) _t≥0, …, (R_{θ,P,⌈2a⌉−1}(t)) _t≥0) of angle $$, and that (R_θ,P,0(t)) _t≥0 is topologically mixing provided the condition

compare [12] for the notion. Our new assumption will be, instead of (80), that there exist a number $$ and a sufficiently small number ϵ > 0 such that λ^a + c₁λ^−a ∈ −e^iθP(int(P_p)) and that λ^a ∉ Σ_aπ/2 for |λ − λ₀| ≤ ϵ. Let |λ − λ₀| ≤ ϵ. Since there exists an x ≠ 0 such that $$ due to our assumption $$, we can employ [12, Lemma 3.3.7] in order to see that see also [12, p. 418]. Due to our assumption λ^a ∉ Σ_aπ/2, the asymptotic expansion formulae (8)-(9), and the fact that the first term in the above expression can be continuously extended to the nonnegative real axis, we get the mapping t ↦ R_θ,P,0(t)x, t ≥ 0, is asymptotically almost periodic. Since the set $$ s.t. |λ − λ₀ | ≤ ϵ and $$ is total in X, Proposition 21 implies that there exists a dense linear subspace X^′ of X such that the mapping t ↦ R_θ,P,0(t)x, t ≥ 0, is asymptotically almost periodic for all x ∈ X^′.