c-Almost periodic type functions and applications


 In this paper, we introduce several various classes of c-almost periodic type functions and their Stepanov generalizations, where c ∈ ℂ and |c| = 1. We also consider the corresponding classes of c-almost periodic type functions depending on two variables and prove several related composition principles. Plenty of illustrative examples and applications are presented.


Introduction and preliminaries
The notion of almost periodicity was studied by H. Bohr around 1925 and later generalized by many others. The interested reader may consult the monographs by Besicovitch [6], Bezandry and Diagana [7], Corduneanu [11]- [12], Diagana [14], Fink [15], Guérékata [16], Kostić [22] and Zaidman [28] for the basic introduction to the theory of almost periodic functions. Almost periodic functions and almost automorphic functions play a significant role in the qualitative theory of differential equations, physics, mathematical biology, control theory and technical sciences.
The class of (ω, c)-periodic functions and various generalizations have recently been introduced and investigated by Alvarez, Gómez, Pinto [1] and Alvarez, Castillo, Pinto [2]- [3] (see also Pinto [27], where the notion of (ω, c)-periodicity has been analyzed for the first time). In [19]- [20], we have recently considered various generalizations of (ω, c)-periodic functions. Besides the notion depending on two parameters ω and c, it is meaningful to consider the notion depending only on the parameter c. The main aim of this paper is to introduce and analyze the classes of c-almost periodic functions, c-uniformly recurrent functions, semi-c-periodic functions and their Stepanov generalizations (see [10] for further information concerning semi-Bloch k-periodic functions (k ∈ R) and semi-anti-periodic functions; the class of uniformly recurrent functions has recently been analyzed in [23]), where c ∈ C and |c| = 1. We also introduce and investigate the corresponding classes of c-almost periodic type functions depending on two variables; several composition principles for c-almost periodic type functions are established in this direction. We provide some illustrative examples and applications to the abstract fractional semilinear integro-differential inclusions.
Before briefly explaining the organization of paper and notation used, the authors would like to thank Professor Toka Diagana for his invitation to submit this article to the special issue of Nonautonomous Dynamical Systems dedicated to the memory of Professor Constantin Corduneanu.
By (E, · ) we denote a complex Banach space; I denotes the interval R or [0, ∞). If X is also a complex Banach space, then L(E, X) stands for the space of all continuous linear mappings from E into X; L(E) ≡ L(E, E).
In Subsection 1.1 we collect the basic definitions and results about almost periodic type functions and their Stepanov generalizations. The notion of c-almost periodicity and the notion of c-uniform recurrence are introduced in Definition 2.1 and Definition 2.3, respectively (if c = 1, then we recover the usual notions of almost periodicity and uniform recurrence). After that, in Definition 2.4 and Proposition 2.5, we introduce the notion of semi-c-periodicity and prove some necessary and sufficient conditions for a continuous function f : I → E to be semi-c-periodic. Proposition 2.6 is important in our analysis because it states that there does not exist a c-uniformly recurrent function f : I → E if |c| = 1 (see also Proposition 2.7 and Proposition 2.8 for some expected results on the introduced classes of functions). Further on, in Proposition 2.9, we show that for any c-almost periodic function (c-uniformly recurrent function, semi-c-periodic function) f : I → E and for any positive integer l ∈ N the function f (·) is c l -almost periodic (c l -uniformly recurrent, semi-c l -periodic); after that, in Corollary 2.10, we clarify the basic consequences in case that arg(c) ∈ π · Q. In Proposition 2.11, we analyze the situation in which arg(c) / ∈ π · Q; if this is the case, then we prove that the c-almost periodicity of function f (·) implies the c ′ -almost periodicity of function f (·) for all c ′ ∈ S 1 , where S 1 := {z ∈ C ; |z| = 1}. Furthermore, if the function f (·) is bounded and c-uniformly recurrent, then f (·) is c ′ -uniformly recurrent for all c ′ ∈ S 1 (to sum up, any c-almost periodic function is always almost periodic and any bounded cuniformly recurrent function is always uniformly recurrent; the converse statement is false in general); see also Proposition 2.12.
The main structural properties of introduced classes of functions are stated in Theorem 2.13. After that, in Theorem 2.14, we clarify that a bounded continuous function f : I → E is semi-c-periodic if and only if there exists a sequence (f n ) of bounded continuous c-periodic functions which uniformly converges to f (·). If the function f (·) is real-valued, c-uniformly recurrent (semi-c-periodic) and f = 0, then c = ±1 and moreover, if f (t) ≥ 0 for all t ∈ I, then c = 1; see Proposition 2.17. Furthermore, if f : I → E is c-uniformly recurrent (semi-c-periodic) and f = 0, then f (·) cannot vanish at infinity; see Proposition 2.18.
An interesting extension of [26,Theorem 2.3] is proved in Theorem 2.24, which is one of the main results of the second section. The c-almost periodic extensions (semi-c-periodic extensions) of functions from the nonnegative real axis to the whole real axis are analyzed in Proposition 2.25. The notion of asymptotical c-almost periodicity (asymptotical c-uniform recurrence, asymptotical semi-c-periodicity) is introduced in Definition 2.26, while the notion of corresponding Stepanov classes is introduced in Definition 2.27.
Without going into full details, let us only say that the composition theorems for c-almost periodic type functions are analyzed in Subsection 2.1, while the invariance of c-almost type periodicity under the actions of convolution products is analyzed in Subsection 2.2 (the structural results in these subsections are given without proofs, which can be deduced similarly as in our previous research studies; it is also worth noting that, in Section 2, we present numerous illustrative examples and comments about the problems considered). The final section of paper is reserved for applications of our abstract theoretical results. By L p loc (I : E), C(I : E), C b (I : E) and C 0 (I : E) we denote the vector spaces consisting of all p-locally integrable functions f : I → E, all continuous functions f : I → E, all bounded continuous functions f : I → E and all continuous functions f : I → E satisfying that lim |t|→+∞ f (t) = 0, respectively (1 ≤ p < ∞). As is well known, C 0 (I : E) is a Banach space equipped with the sup-norm, denoted henceforth by · ∞ . If f : R → E, then we definef : We will use the following auxiliary result, whose proof follows from the argumentation used in the proof that every orbit under an irrational rotation is dense in S 1 (see e.g. the solution given by C. Blatter in [8]): is not rational. Then for each c ′ ∈ S 1 there exists a strictly increasing sequence (l k ) of positive integers such that sup k∈N (l k+1 − l k ) < ∞ and |c l k − c ′ | < ǫ.
1.1. Almost periodic type functions and generalizations. Given ǫ > 0, we call τ > 0 an ǫ-period for f (·) if The set constituted of all ǫ-periods for f (·) is denoted by ϑ(f, ǫ). It is said that f (·) is almost periodic if for each ǫ > 0 the set ϑ(f, ǫ) is relatively dense in [0, ∞), which means that there exists l > 0 such that any subinterval of [0, ∞) of length l meets ϑ(f, ǫ). The vector space consisting of all almost periodic functions is denoted by AP (I : E). This space contains the space consisting of all continuous periodic functions f : I → E.
Let f ∈ AP (I : E). Then the Bohr-Fourier coefficient exists for all r ∈ R; furthermore, if P r (f ) = 0 for all r ∈ R, then f (t) = 0 for all t ∈ R, and σ(f ) := {r ∈ R : P r (f ) = 0} is at most countable. The function f : I → E is said to be asymptotically almost periodic if and only if there exist an almost periodic function h : I → E and a function φ ∈ C 0 (I : E) such that f (t) = h(t) + φ(t) for all t ∈ I. This is equivalent to saying that, for every ǫ > 0, we can find numbers l > 0 and M > 0 such that every subinterval of I of length l contains, at least, one number τ such that For any almost periodic function f : I → E, the spectral synthesis states that where the closure is taken in the space C b (I : E). By AP Λ (I : E), where Λ is a non-empty subset of R, we denote the vector subspace of AP (I : E) consisting of all functions f ∈ AP (I : E) for which the inclusion σ(f ) ⊆ Λ holds. We have that AP Λ (I : E) is a closed subspace of AP (I : E) and therefore Banach space itself.
For the sequel, we need some preliminary results from the pioneering paper [5] by Bart is a linear surjective isometry and Ef (·) is a unique continuous almost periodic extension of a function f (·) from AP ([0, ∞) : E) to the whole real line.
Following Haraux and Souplet [17], we say that a continuous function f (·) is uniformly recurrent if and only if there exists a strictly increasing sequence (α n ) of positive real numbers such that lim n→+∞ α n = +∞ and It is well known that any almost periodic function is uniformly recurrent, while the converse statement is not true in general. For more details about uniformly recurrent functions, we refer the reader to [23].
Let 1 ≤ p < ∞. We continue by recalling that a function f ∈ L p loc (I : E) is said to be Stepanov p-bounded if and only if is almost periodic. Furthermore, we say that a function f ∈ L p S (I : E) is asymptotically Stepanov p-almost periodic if and only if there exist a Stepanov p-almost periodic function g ∈ L p S (I : E) and a function q ∈ L p S (I : E) such that f (t) = g(t) + q(t), t ∈ I andq ∈ C 0 (I : L p ([0, 1] : E)).
We also need the following definition from [23]. (ii) A function f ∈ L p loc (I : E) is said to be asymptotically Stepanov puniformly recurrent if and only if there exist a Stepanov p-uniformly recurrent function h(·) and a function q ∈ L p S (I : E) such that f (t) = h(t) + q(t), t ∈ I andq ∈ C 0 (I : L p ([0, 1] : E)).

c-Almost periodic type functions
With the exception of Proposition 2.6 and the paragraph preceding it, in this paper we will always assume that c ∈ C and |c| = 1. Let f : I → E be a continuous function and let a number ǫ > 0 be given. We call a number τ > 0 an (ǫ, c)-period for f (·) if f (t+τ )−cf (t) ≤ ǫ for all t ∈ I. By ϑ c (f, ǫ) we denote the set consisting of all (ǫ, c)-periods for f (·).
We are concerned with the following notion: It is said that f (·) is c-almost periodic if and only if for each ǫ > 0 the set ϑ c (f, ǫ) is relatively dense in [0, ∞). The space consisting of all c-almost periodic functions from the interval I into E will be denoted by AP c (I : E).
If c = −1, then we also say that the function f (·) is almost anti-periodic. The space of almost anti-periodic functions has recently been analyzed in [26].
In general case, it is very simple to prove that the following holds (see e.g., the proof of [6, Theorem 4 • , p. 2]): The following generalization of c-almost periodicity is meaningful, as well: Then a continuous function f : I → E is said to be c-uniformly recurrent if and only if there exists a strictly increasing sequence (α n ) of positive real numbers such that lim n→+∞ α n = +∞ and If c = −1, then we also say that the function f (·) is uniformly anti-recurrent. The space consisting of all c-uniformly recurrent functions from the interval I into E will be denoted by U R c (I : E).
We will also consider the following notion: The space of all semi-c-periodic functions will be denoted by SAP c (I : E).
Suppose that I = R, f ∈ C(R : E), p > 0 and m ∈ N. Then we have Therefore, we have the following: Furthermore, if I = R, then the above is also equivalent with It can be very simply shown that any semi-c-periodic function is bounded. Keeping in mind Proposition 2.5 and this observation, we may conclude that the notion introduced in Definition 2.4 is equivalent and extends the notion of semi-periodicity for case c = 1, introduced by Andres and Pennequin in [4], and the notion of semianti-periodicity for case c = −1, introduced by Chaouchi et al in [10].
The notion introduced in Definition 2.4 can be considered with general complex number c ∈ C \ {0}, but the situation is much more complicated in this case (cf. [24] for more details). The same holds with the notion introduced in Definition 2.1 and Definition 2.3, but then we have the following result: Proof. Without loss of generality, we may assume that I = [0, ∞). Suppose to the contrary that there exists t 0 ≥ 0 such that f (t 0 ) = 0. Inductively, (2.1) implies Then the contradiction is obvious because for each m ∈ N with m > n there exists k ∈ N such that t 0 + α m ∈ [kα n , (k + 1)α n ] and therefore On the other hand, we obtain inductively from (2.1) that which immediately yields a contradiction.
Using the same arguments as in the proof of [19, Lemma 3.4], we can clarify the following: Since for each numbers t, τ ∈ I and m ∈ N we have the following result simply follows: Further on, we have (x ∈ I, τ > 0, l ∈ N): The above estimate can be used to prove the following: Consider now the following condition: The next corollary of Proposition 2.9 follows immediately by plugging l = q : Therefore, if arg(c)/π ∈ Q, then the class of c-almost periodic functions (cuniformly recurrent functions, semi-c-periodic functions) is always contained in the class of almost periodic functions (uniformly recurrent functions, semi-periodic functions); in particular, we have that any almost anti-periodic function (uniformly anti-recurrent function, semi-anti-periodic function) is almost periodic (uniformly recurrent, semi-periodic). Now we will prove the following: Proposition 2.11. Let f : I → E be a continuous function, and let arg(c)/π / ∈ Q.
Proof. We will prove only (i). Clearly, it suffices to consider the case in which the function f (·) is not identical to zero. Let c ′ ∈ S 1 and ǫ > 0 be fixed; then the prescribed assumption implies that the set {c l : l ∈ N} is dense in S 1 and therefore there exists an increasing sequence (l k ) of positive integers such that lim k→+∞ c l k = c ′ . By Proposition 2.2, the function f (·) is bounded; let k ∈ N be such that |c l k − c ′ | < ǫ/(2 f ∞ ), and let τ > 0 be any (ǫ/2, c l k )-period for f (·).
Then we have for any x ∈ I. This simply completes the proof.
Proposition 2.12. Let f : I → E be a continuous function. Then we have the following: Proof. Let ǫ > 0 be fixed. To prove (i), it suffices to show that f (·) is c-almost periodic (see Proposition 2.9). Since arg(c)/π ∈ Q and (2.3) holds, then we have c 1+2lq = c for all l ∈ N. Then there exists p > 0 such that, for every m ∈ N and x ∈ I, ≤ ǫ so that the conclusion follows from the fact that the set {(1 + 2lq)p : l ∈ N} is relatively dense in [0, ∞). Assume now that arg(c)/π / ∈ Q. To prove (ii), it suffices to consider case f = 0. Observe first that Lemma 1.1 yields that there exists a strictly increasing sequence (l k ) of positive integers such that sup k∈N (l k+1 − l k ) < ∞ and |c l k − c ′ | < ǫ/ f ∞ for all k ∈ N. With this sequence and the number p > 0 chosen as above, we have: Since the set {pl k : k ∈ N} is relatively dense in [0, ∞), the proof is completed.
In connection with Proposition 2.12(ii), it is natural to ask whether the assumptions that the function f (·) is semi-c-periodic and arg(c)/π / ∈ Q imply that f (·) is semi-c ′ -almost periodic for all c ′ ∈ S 1 ?
With t = π, the above implies Now we will state and prove the following Proof. We will consider the class of c-uniformly recurrent functions, only, when we may assume without loss of generality that I = [0, ∞). Then f / ∈ C 0 ([0, ∞) : R); namely, if we suppose the contrary, then there exists a strictly increasing sequence (α n ) of positive real numbers such that lim n→+∞ α n = +∞ and (2.1) holds. In particular, for every fixed number t 0 ≥ 0 we have lim n→+∞ |f (t 0 +α n )−cf (t 0 )| = 0. This automatically yields f (t 0 ) = 0 and, since t 0 ≥ 0 was arbitrary, we get f = 0 identically, which is a contradiction. Therefore, there exist a strictly increasing sequence (t l ) l∈N of positive real numbers tending to plus infinity and a positive real number a ≥ lim sup t→+∞ |f (t)| > 0 such that |f (t l )| ≥ a/2 for all l ∈ N. Let ǫ > 0 be fixed. Then there exist two real numbers t 0 > 0 and n 0 ∈ N such that |f (t + α n ) − f (t)| ≤ ǫ for all t ≥ t 0 and n ≥ n 0 . If arg(c) = ϕ ∈ (−π, π], then we particularly get that for each t ≥ t 0 and n ≥ n 0 we have: Plugging in the second estimate t = t l for a sufficiently large l ∈ N we get that | sin ϕ| ≤ 2ǫ/a. Since ǫ > 0 was arbitrary, we get sin ϕ = 0 and c = ±1. Suppose, finally, that f (t) ≥ 0 for all t ≥ 0 and c = −1. Then we have f (t + α n ) + f (t) ≤ 2ǫ for all t ≥ t 0 and n ≥ n 0 . Plugging again t = t l for a sufficiently large l ∈ N we get that a ≤ ǫ for all ǫ > 0 and therefore a = 0, which is a contradiction.
By the proof of Proposition 2.17, we have: If the functions f 1 (·), · · ·, f n−1 (·) are already defined, set and the function is well defined, even and satisfies that 0 ≤ f (x) ≤ 1 for all x ∈ R (see Bohr, the first part of [9], pp. 113-115). Then we know that the function f : R → R, given by (2.6), is bounded, uniformly continuous and uniformly recurrent ( [23]). Therefore, f (·) is c-uniformly recurrent if and only if c = 1. (iii) The following function has been used by de Vries in [13, point 6., p. 208]. Let (p i ) i∈N be a strictly increasing sequence of natural numbers such that p i |p i+1 , i ∈ N and lim i→∞ p i /p i+1 = 0. Define the function and extend the function f i (·) periodically to the whole real axis; the obtained function, denoted by the same symbol f i (·), is of period 2p i (i ∈ N). Set Then the function f : R → R, given by (2.7), is bounded, uniformly continuous and uniformly recurrent ( [23]). Therefore, f (·) is c-uniformly recurrent if and only if c = 1. Any of the above three functions is not asymptotically (Stepanov) almost automorphic (see [23] for more details).
The function f (·) constructed in the following example is also not asymptotically (Stepanov) almost automorphic since it is not Stepanov bounded: Example 2.20. The function g : R → R, given by is unbounded, Lipschitz continuous and uniformly recurrent; furthermore, we have the existence of a positive integer k 0 ∈ N such that This can be proved in exactly the same way as in the proof of [17,Theorem 1.1]. Define now f (t) := sin t · g(t), t ∈ R. Then (2.9) easily implies Therefore, f (·) is uniformly anti-recurrent and Proposition 2.17 yields that the function f (·) is c-uniformly recurrent if and only if c = ±1. To prove that f (·) is Stepanov unbounded, obeserve that (2.8) implies the existence of a sequence (t k ) k∈N of positive real numbers such that g(t k ) ≥ (1/2)(ln k − 1) for all k ≥ k 0 . If we denote by L ≥ 1 the Lipschitzian constant of mapping g(·), then the above implies The existence of a constant M > 0 such that t+1 t | sin s| · g(s) ds < M for all t ∈ R would imply by (2.10) the existence of a sequence (a k ) of positive integers such that [2a k π + (π/2), 2a k π + (π/2)+ 1] ⊆ [t k , t k + 8π and therefore (take t = 2a k π + (π/2)) sin((π/2) + 1) · (1/2)(ln k − 1) − 8Lπ ≤ M, k ≥ k 0 , which is a contradiction.
In connection with Proposition 2.17 and Proposition 2.18, we would like to present an illustrative example with the complex-valued functions: Example 2.21. Let h : I → R, q : I → R and f (t) := h(t) + iq(t), t ∈ I. Suppose that f : I → C is c-uniformly recurrent, where c = e iϕ and sin ϕ = 0. Then h ∈ C 0 (I : R) or q ∈ C 0 (I : R) implies f ≡ 0. To show this, observe that the c-uniform recurrence of f (·) implies the existence of a strictly increasing sequence (α n ) of positive real numbers tending to plus infinity such that lim n→+∞ sup t∈I h(t + α n ) − cos ϕ · h(t) + sin ϕ · q(t) = 0, and lim n→+∞ sup t∈I q(t + α n ) − cos ϕ · q(t) − sin ϕ · h(t) = 0.
Since we have assumed that sin ϕ = 0, the assumption h ∈ C 0 (I : R) (q ∈ C 0 (I : R)) implies by the first (second) of the above equalities that q ∈ C 0 (I : R) (h ∈ C 0 (I : R)). Hence, f ∈ C 0 (I : C) and the claimed statement follows by Proposition 2.18. Proof. Assume first that f ∈ AP R\{0} (I : E). By (1.1), we have where the closure is taken in the space C b (I : E). Since σ(f ) ⊆ R \ {0} and the function t → e iµt , t ∈ I (µ ∈ R \ {0}) is c-almost periodic for all c ∈ S 1 , we have that span{e iµ· x : µ ∈ σ(f ), x ∈ R(f )} ⊆ AP c,0 (I : E). Hence, f ∈ AP c,0 (I : E). To complete the proof, it remains to consider case arg(c) ∈ π · Q and show that any function f ∈ AP c,0 (I : E) belongs to the space AP R\{0} (I : E). Furthermore, it suffices to consider case in which (2.3) holds with the number p even because otherwise we can apply Corollary 2.10(ii) and Proposition 2.11(i) to see that AP c,0 (I : E) ⊆ AN P 0 (I : E) and therefore AP c,0 (I : E) ⊆ AN P (I : E), so that the statement directly follows from [26,Theorem 2.3]. We will prove that lim t→∞ 1 t t 0 f (s) ds = 0; (2.11) clearly, by almost periodicity of f (·), the limit in (2.11) exists. Let ǫ > 0 be fixed, and let l > 0 satisfy that every interval of [0, ∞) of length l contains a point τ such that f (t+τ )−cf (t) ≤ ǫ, t ≥ 0. We have c q = 1 and therefore 1+c+···+c q−1 = 0; using this equality and decomposition (s ≥ 0, n ∈ N) we immediately get that there exists a finite constant A ≥ 1 such that, for every s ≥ 0 and n ∈ N, f (s + (n − 1)τ ) + f (s + (n − 2)τ ) + · · · + f (s) ≤ Aǫ⌈n/q⌉ + A f ∞ .
Dividing the both sides of the above inequality with nτ , we get that lim n→+∞ 1 nτ nτ 0 f (s) ds ≤ Aǫ/q.
Since ǫ > 0 was arbitrary, this immediately yields (2.11). Now we will state and prove the following result: Then Ef : R → E is a unique c-almost periodic extension (semi-cperiodic extension) of f (·) to the whole real axis.
Proof. The proof for the class of c-almost periodic functions is very similar to the proof of [26, Proposition 2.2] and therefore omitted. For the class of semi-c-periodic functions, the proof can be deduced as follows (see also [10,Proposition 4]). Due to Proposition 2.12, we have that the function f : [0, ∞) → E is almost periodic, so that the function Ef : R → E is a unique almost periodic extension of f (·) to the whole real axis. Therefore, it remains to be proved that Ef (·) is semi-c-periodic.
Let ǫ > 0 be fixed. Then there exists p > 0 such that for all m ∈ N and x ≥ 0 we have f (x + mp) − c m f (x) ≤ ǫ. For every fixed number m ∈ N, the function Ef (· + mp) − c m Ef (·) is almost periodic so that the supremum formula implies This completes the proof.
We continue by introducing the following notion: For the Stepanov classes, we will use the following notion:

2.1.
Composition principles for c-almost periodic type functions. In this subsection, we will clarify and prove several composition principles for c-almost periodic functions and c-uniformly recurrent functions; the composition theorems for semi-c-periodic functions will be investigated in [24]. Suppose that F : I × X → E is a continuous function and there exists a finite constant L > 0 such that Define F (t) := F (t, f (t)), t ∈ I. We need the following estimates (τ ≥ 0, c ∈ C\{0}, t ∈ I):  and lim n→+∞ sup t∈I F t + α n , cf (t) − cF (t, f (t)) = 0, (2.15) then the mapping F (t) := F (t, f (t)), t ∈ I is c-uniformly recurrent.
(ii) Suppose that f : I → X is c-almost periodic. If for each ǫ > 0 the set of all positive real numbers τ > 0 such that For the class of asymptotically c-almost periodic functions, the following result simply follows from the previous theorem and the argumentation used in the proof of [14,Theorem 3.49]: Theorem 2.29. Suppose that F : I × X → E and Q : I × X → E are continuous functions and there exists a finite constant L > 0 such that (2.12) holds as well as that (2.12) holds with the function F (·, ·) replaced therein with the function Q(·, ·).
(i) Suppose that g : I → E is a c-uniformly recurrent function, h ∈ C 0 (I : E) and f (x) = g(x) + h(x), x ∈ I. If there exists a strictly increasing sequence (α n ) of positive reals tending to plus infinity such that (2.14) and (2.15) hold with the function f (·) replaced therein with the function g(·), lim |t|→+∞ Q(t, y) = 0 uniformly for y ∈ R(f ), then the mapping H(t) := (F + Q)(t, f (t)), t ∈ I is asymptotically c-uniformly recurrent. (ii) Suppose that g : I → E is a c-almost periodic function, h ∈ C 0 (I : E) and f (x) = g(x) + h(x), x ∈ I. If for each ǫ > 0 the set of all positive real numbers τ > 0 such that (2.16) and (2.17) hold with the function f (·) replaced therein with the function g(·), lim |t|→+∞ Q(t, y) = 0 uniformly for y ∈ R(f ), then the mapping H(t) := (F +Q)(t, f (t)), t ∈ I is asymptotically c-almost periodic.
For the Stepanov classes, the following result slightly generalizes the well known result of Long and Ding [18,Theorem 2.2]. The proof can be deduced by using the argumentation used in the proofs of the above-mentioned theorem and [19,Theorem 3.18]: Theorem 2.30. Let p, q ∈ [1, ∞), r ∈ [1, ∞], 1/p = 1/q + 1/r and the following conditions hold: (i) Let F : I × X → E and let there exist a function L F ∈ L r S (I) such that F (t, x) − F (t, y) ≤ L F (t) x − y X , t ∈ I, x, y ∈ X. Then the function F (·, f (·)) is Stepanov (p, c)-uniformly recurrent. Furthermore, the assumption that F (·, 0) is Stepanov p-bounded implies that the function F (·, f (·)) is Stepanov p-bounded, as well.
The above results can be simply reformulated for the class of Stepanov (p, c)almost periodic functions. For the classes of asymptotically Stepanov (p, c)-uniformly recurrent (asymptotically Stepanov (p, c)-almost periodic) functions, we can simply extend the assertions of [22,Proposition 2.7.3,Proposition 2.7.4]. Details can be left to the interested readers.

2.2.
Invariance of c-almost type periodicity under the actions of convolution products. In this subsection, we investigate the invariance of c-almost periodicity, c-uniform recurrence and semi-c-periodicity under the actions of finite and infinite convolution products.
We start by stating the following slight generalizations of [26, Proposition 3.1, Proposition 3.2], which can be deduced by using almost the same arguments as in this paper (see also [ is well-defined and c-almost periodic (bounded c-uniformly recurrent/bounded and semi-c-periodic). Proposition 2.33. Suppose 1 ≤ p < ∞, 1/p + 1/q = 1 and (R(t)) t>0 ⊆ L(E, X) is a strongly continuous operator family satisfying that, for every s ≥ 0, we have is well-defined, bounded and asymptotically Stepanov (p, c)-almost periodic (asymptotically Stepanov (p, c)-uniformly recurrent/asymptotically Stepanov semi-(p, c)periodic).
In the following slight extension of [23, Proposition 3.2], we consider the case in which the forcing term f (·) is not Stepanov p-bounded, in general (the proof is essentially the same and therefore omitted; Proposition 2.33 can be reformulated in this context, as well): Proposition 2.34. Suppose that 1 ≤ p < ∞, 1/p + 1/q = 1, f : R → E is Stepanov (p, c)-almost periodic (Stepanov (p, c)-uniformly recurrent/Stepanov semi-(p, c)-periodic), there exists a continuous function P : and (R(t)) t>0 ⊆ L(E, X) is a strongly continuous operator family satisfying that If the functionf : R → L p ([0, 1] : E) is uniformly continuous, then the function F : R → X, given by (2.21), is well-defined and c-almost periodic (c-uniformly recurrent/semi-c-periodic).

Applications to the abstract Volterra integro-differential inclusions
In this section, we will present some illustrative applications of our abstract results in the analysis of the existence and uniqueness of c-almost periodic type solutions to the abstract (semilinear) Volterra integro-differential inclusions.
First of all, we would like to note that the results established in Subsection 2.2 can be applied at any place where the variation of parameters formula takes effect. Concerning semilinear problems, we can apply our results in the study of the existence and uniqueness of c-almost periodic solutions and c-uniformly recurrent solutions of the fractional semilinear Cauchy inclusion where D γ t,+ denotes the Riemann-Liouville fractional derivative of order γ ∈ (0, 1), F : R × X → E satisfies certain properties, and A is a closed multivalued linear operator satisfying condition [22, (P)]. To explain this in more detail, fix a strictly increasing sequence (α n ) of positive reals tending to plus infinity and define BU R (αn);c (R : Equipped with the metric d(·, ·) := ·− · ∞ , BU R (αn);c (R : E) becomes a complete metric space. Let (R γ (t)) t>0 be the operator family considered in [22] and [25]. Then we know that It is said that a continuous function u : R → E is a mild solution of (3.1) if and only if there exists an integer n ∈ N such that M n < 1, where Suppose that u ∈ BU R (αn);c (R : E). Then R(u) = B is a bounded set and the mapping t → F (t, u(t)), t ∈ R is bounded due to the prescribed assumption. Applying Theorem 2.31, we have that the function F (·, u(·)) is Stepanov (q, c)-uniformly recurrent. Define q ′ := q/(q − 1). By (3.2) and (3.3), we have R γ (·) ∈ L q ′ [0, 1] and ∞ k=0 R γ (·) L q ′ [k,k+1] < ∞. Applying Proposition 2.32, we get that the function t → t −∞ R γ (t − s)F (s, u(s)) ds, t ∈ R is bounded and c-uniformly recurrent, implying that Υu ∈ BU R (αn);c (R : E), as claimed. Furthermore, a simple calculation shows that Υ n u 1 − Υ n u 2 ∞ ≤ M n u 1 − u 2 ∞ , u 1 , u 2 ∈ BU R (αn);c (R : E), n ∈ N.
Since there exists an integer n ∈ N such that M n < 1, the well known extension of the Banach contraction principle shows that the mapping Υ(·) has a unique fixed point, finishing the proof of the theorem.
The examples and results presented by Zaidman [28,Examples 4,5,7,8; pp. 32-34] can be used to provide certain applications of our results, as well. For example, the unique regular solution of the heat equation u t (x, t) = u xx (x, t), x ∈ R, t ≥ 0, accompanied with the initial condition u(x, 0) = f (x), is given by f (s) ds, x ∈ R, t > 0; see [28,Example 4]. Let the number t 0 > 0 be fixed, and let the function f (·) be bounded c-uniformly recurrent (c-almost periodic, semi-c-periodic). Since e −· 2 /4t0 ∈ L 1 (R), we can use the fact that the space of bounded c-uniformly recurrent functions (c-almost periodic functions, semi-c-periodic functions) is convolution invariant in order to see that the solution x → u(x, t 0 ), x ∈ R is bounded and c-uniformly recurrent (c-almost periodic, semi-c-periodic). It is clear that the concepts of Weyl almost periodicity, Doss almost periodicity and Besicovitch-Doss almost periodicity, among many others, can be reconsidered and generalized following the approach based on the use of difference f (·+τ )−cf (·) in place of the usual one f (· + τ ) − f (·). We close the paper with the observation that the class of c-almost automorphic functions will be analyzed in our forthcoming paper [21].