WEYL – ALMOST PERIODIC AND ASYMPTOTICALLY WEYL – ALMOST PERIODIC PROPERTIES OF SOLUTIONS TO LINEAR

The main purpose of paper is to consider Weyl-almost periodic and asymptotically Weyl-almost periodic solutions of linear and semilinear abstract Volterra integrodifferential equations. We focus our attention to the investigations of Weyl-almost periodic and asymptotically Weyl-almost periodic properties of both, finite and infinite convolution product, working in the setting of complex Banach spaces. We introduce the class of asymptotically (equi)-Weyl-p-almost periodic functions depending on two parametres and prove a composition principle for the class of asymptotically equi-Weylp-almost periodic functions. Basically, our results are applicable in any situations where the variation of parameters formula takes a role. We provide several new contributions to abstract linear and semilinear Cauchy problems, including equations with the WeylLiouville fractional derivatives and the Caputo fractional derivatives. We provide some applications of our abstract theoretical results at the end of paper, considering primarily abstract degenerate differential equations, including the famous Poisson heat equation and its fractional analogues. DOI: 10.25587/SVFU.2018.98.14232


Introduction and Preliminaries
It is well known that the notion of an almost periodic function was introduced by H. Bohr around 1924-1926 and later generalized by V. V. Stepanov, H. Weyl, A. S. Besicovitch and many other mathematicians.In this paper, we primarily consider Weyl's and Kovanko's classes of generalized almost periodic functions, which were introduced in 1927 and 1944, respectively.The analysis of various classes of generalized almost periodic solutions of abstract Volterra integro-differential equations is still very attractive field of research of many mathematicians.For more details on the subject, we refer the reader to the monographs [1] by D. Cheban, [2] by T. Diagana, [3] by G. M. N'Guérékata and [4] by the author.mappings from X into Y ; L(X) ≡ L(X, X).Let I = R or I = [0, ∞).By C b (I : X) and BU C(I : X) we denote the vector spaces consisting of all bounded continuous functions from I into X and all bounded continuous functions from I into X vanishing at infinity, respectively.Endowed with the usual sup-norm, any of these spaces becomes one of Banach's.
In this paper, we use the Weyl-Liouville fractional derivatives D γ t,+ u(t) of order γ ∈ (0, 1) and the Caputo fractional derivatives of order α > 0. The Weyl-Liouville fractional derivatives D γ t,+ u(t) of order γ ∈ (0, 1) is defined for those continuous functions u : R → X such that t → t −∞ g 1−γ (t − s)u(s) ds, t ∈ R, is a well-defined continuously differentiable mapping, by For more details about this type of fractional derivatives, we refer the reader to the paper [22] by J. Mu, Y. Zhoa and L. Peng.If α > 0 and m = ⌈α⌉, the Caputo fractional derivative Fractional differential equations with Caputo derivatives have been investigated, among many other research papers and monographs, in [13,18,19].
1.1.Degenerate semigroups and degenerate fractional resolvent families generated by multivalued linear operators.Multivalued linear operators in Banach spaces have been analyzed by many authors.A multivalued linear operator (MLO, for short) : X → P (Y ) is any mapping satisfying the following conditions: (i) D( ) := {x ∈ X : x = ∅} is a linear subspace of X; If X = Y, then we say that is an MLO in X.It is well known that, if x, y ∈ D( ) and λ, η ∈ C with |λ| + |η| = 0, then λ x + η y = (λx + ηy).IF is an MLO, then 0 is a linear submanifold of Y and x = f + 0 for any x ∈ D( ) and The resolvent set of , ρ( ) for short, is defined as the union of those complex numbers λ ∈ C for which ) is called the resolvent of (λ ∈ ρ( )).For further information regarding multivalued linear operators, we refer the reader to the monographs [23] by R. Cross and [10] by A. Favini, A. Yagi.
The basic source of information about degenerate (a, k)-regularized C-resolvent families may be obtained by consulting the forthcoming monograph [13].Suppose that a closed MLO satisfies the condition [10, p. 47] introduced by A. Favini and A. Yagi: (P) There exist finite constants c, M > 0 and β ∈ (0, 1] such that If this condition holds and β > θ, then degenerate strongly continuous semigroup (T (t)) t>0 ⊆ L(X) generated by satisfies estimate for some finite constant M 0 > 0 [24].
Without specifying particular conditions on function f (•), which will be done later, we will use the following formal definition henceforth: Definition 1.1.Let be a closed MLO.(i) A continuous function u : R → X is said to be a mild solution of the abstract Cauchy inclusion of first order (ii) A continuous function u : R → X is said to be a mild solution of fractional relaxation inclusion (1.1) iff (iii) By a mild solution of (DFP) f,γ , we mean any function u ∈ C([0, ∞) : X) satisfying that Stepanov almost periodic functions and asymptotically Stepanov almost periodic functions.Assume that I = R or I = [0, ∞), and f : I → X is continuous.Given ǫ > 0, we call τ > 0 an ǫ-period for f (•) iff The set consisting of all ǫ-periods for f (•) is denoted by ϑ(f, ǫ).A function f (•) is said to be almost periodic, a.p. for short, iff for any ǫ > 0 the set ϑ(f, ǫ) is relatively dense in I, which means that there exists l > 0 such that any subinterval of I of length l meets ϑ(f, ǫ).By AP (I : X) we denote the space consisted of all almost periodic functions from the interval I into X.
The class of asymptotically almost periodic functions was introduced by M. Fréchet in 1941.A function f ∈ C b ([0, ∞) : X) is said to be asymptotically almost periodic iff for every ǫ > 0 we can find numbers l > 0 and M > 0 such that every subinterval of [0, ∞) of length l contains, at least, one number τ such that f (t + τ ) − f (t) ≤ ǫ for all t ≥ M. The space consisting of all asymptotically almost periodic functions from [0, ∞) into X is denoted by AAP ([0, ∞) : X).For any function f ∈ C([0, ∞) : X), we have the equivalence of the statements (a)-(c), where: (iii) The set H(f Let 1 ≤ p < ∞, let l > 0, and let f, g ∈ L p loc (I : X), where I = R or I = [0, ∞).The Stepanov 'metric' is defined by Then we know that, for every two numbers l 1 , l 2 > 0, there exist two positive real constants k 1 , k 2 > 0 independent of f, g, such that as well as that there exists respectively.The choice of length l > 0 is basically irrelevant for considerations of Stepanov class.In the sequel, for this class we will assume that l = 1.It is said that a function Endowed with the above norm, the space L p S (I : X) consisting of all S p -bounded functions is a Banach space; ap (I : X) for short, iff for each ǫ > 0 we can find two real numbers l > 0 and L > 0 such that any interval (ii) It is said that the function f (•) is Weyl-p-almost periodic, f ∈ W p ap (I : X) for short, iff for each ǫ > 0 we can find a real number L > 0 such that any interval In the set theoretical sense, we have the following inclusions: As it is well-known, any of these two inclusions can be strict (25).Let us recall that the space of scalar-valued functions W p ap (R : R) is introduced for the first time by A. S. Kovanko [26] in 1944.
It is well known that for any function f ∈ L p loc (I : X) its Stepanov boundedness is equivalent to its Weyl boundedness, i.e., For more details about Weyl-almost periodic functions, we refer the reader to [25,Section 4].
Asymptotically (equi)-Weyl-p-almost periodic functions have been recently introduced by the author [6].If q ∈ L p loc ([0, ∞) : X), then we define the function Replacing the limits in (1.3), we come to the class of equi-Weyl-p-vanishing functions.It is said that a function q By W p 0 ([0, ∞) : X) and e-W p 0 ([0, ∞) : X), we denote the vector spaces consisting of all Weyl-p-vanishing functions and equi-Weyl-p-vanishing functions, respectively.

Weyl almost periodic and asymptotically Weyl almost periodic solutions to abstract linear Volterra integro-differential equations
We start this section by stating the following proposition: Proposition 2.1.Suppose that 1 ≤ p < ∞, 1/p + 1/q = 1 and (R(t)) t>0 ⊆ L(X, Y ) is a strongly continuous operator family satisfying that (2.1) is well-defined, bounded continuous and (equi)-Weyl-p-almost periodic.
Proof.Without loss of generality, we may assume that X = Y.We will follow the proof of [24,Proposition 2.11] with appropriate changes.The integral is absolutely convergent due to the Hölder inequality, (2.1) and S p -boundedness of function g(•) : This also implies the boundedness of function G(•), while its continuity can be proved as in [27,Proposition 5].It remains to be shown that G(•) is (equi)-Weyl-p-almost periodic.The proof is completely same for both classes and we will show this only for (equi)-Weyl-p-almost periodic functions.So, let a number ǫ > 0 be given.By definition, there exist two finite numbers l > 0 and L > 0 such that any subinterval Arguing as in the proof of [24, Proposition 2.11], we get Using the monotone convergence theorem and this estimate, it readily follows that For any x ∈ R, we have Hence II ≤ M ǫ p and therefore This completes the proof.
Remark 2.3.Suppose that g : R → X is only Stepanov p-bounded.Then the same argumentation as above shows that the function G(•) given by (2.2) is bounded and continuous.[25].Taking again X = Y = C and R(t) = e −t for t > 0, the function G(•) will be given by G(t) = 0 for t ≤ 0 and G(t) = 1 − e −t for t ≥ 0. This function cannot be equi-Weyl-1-almost periodic, so that our result is, in a certain sense, optimal concerning the scale of generalized almost periodic functions.Let us remind ourselves that the Stepanov p-almost periodicity of function g(•) implies that the function G(•) is almost periodic [24].
Remark 2.7.In our previous research study [6], we have used a different condition on the function q(•) provided that p = 1.For any locally integrable function q ∈ L 1 loc (R : X) and for any strongly continuous operator family (R(t we have formally set In [6], we have used the condition lim t→∞ lim l→∞ J(t, l) = 0, resp., lim

Weyl almost periodic and asymptotically Weyl almost periodic solutions to abstract semilinear Volterra integro-differential equations
For the sake of brevity and better expostion, in this section we will use only one pivot space X = Y.For our study of semilinear problems, we need the following definition introduced recently in [5], with I = R : Definition 3.1.A function f : I × X → X is said to be equi-Weyl-almost periodic in t ∈ I uniformly with respect to compact subsets of X iff the function f (•, x) ∈ L p loc (I : X) for each fixed element x ∈ X and if for each ǫ > 0 there exists l > 0 such that for all compacts K of X we have that the set is relatively dense.We denote by e-W p ap,K (I × X : X) the vector space consisting of such functions.
A useful characterization of equi-Weyl-p-almost periodic functions established by L. I. Danilov [29] has been essentially employed by F. Bedouhene, Y. Ibaouene, O. Mellah and P. Raynaud de Fitte [5] for proving the following composition principle with I = R (see [5,Theorem 3]); the proof also works in the case that I = [0, ∞) : Lemma 3.2.Suppose that p, q, r ≥ 1, 1/r+1/q = 1/p and f (•, •) ∈ e-W p ap,K (I× X : X).If there exists a Stepanov r-bounded function L(•) such that M. Kostić then for each function x ∈ e-W q ap (I : X) we have f (•, x(•)) ∈ e-W p ap (I : X).The composition principles for Weyl-p-almost periodic functions will be considered somewhere else and, in this section, we will consider only the class of equi-Weyl-p-almost periodic functions (it is also worth mentioning an interesting result of S. Abbas [28], concerning composition principles for the class of Weyl pseudo almost automorphic functions).
We need to introduce the following definition: (i) We say that q(•, •) is Weyl-p-vanishing uniformly with respect to compact subsets of X iff for each compact set K of X we have: (ii) We say that q(•, •) is equi-Weyl-p-vanishing uniformly with respect to compact subsets of X iff for each compact set K of X we have: We denote by W p 0,K (I × X : X) and e-W p 0,K (I × X : X) the classes consisting of all Weyl-p-vanishing functions, uniformly with respect to compact subsets of X and equi-Weyl-p-vanishing functions, uniformly with respect to compact subsets of X, respectively.Now we are ready to state the following composition principle: and there exists a Stepanov r-bounded function L(•) such that (3.1) holds with the function f (•, •) replaced with the function g(•, •) therein.Suppose, further, that the following conditions hold: vanishing uniformly with respect to compact subsets of X, resp., equi-Weyl-p-vanishing uniformly with respect to compact subsets of X. Denote by W p 0,Q ([0, ∞) : X) the class W p 0 ([0, ∞) : X), resp.e-W p 0 ([0, ∞) : X), in the first, resp., the second case.
(iii) The function y : [0, ∞) → X is equi-Weyl-q-almost periodic and there exists a set Due to Lemma 3.2, it suffices to show that the function g( . This is evident for the second function since we have assumed (iii) and (3.2), resp., (3.3) holds.For the first function, the corresponding statement holds on account of the fact that p > q, the estimate g(t, x(t)) − g(t, y(t)) ≤ L(t) z(t) , t ≥ 0, and the following calculation involving the Hölder inequality: In the remaining part of this section, we will consider abstract semilinear Cauchy problems.First of all, we will state and prove the following slight generalization of the first part of [5, Theorem 5], stated only in the case that p ≥ 2 (see also [5,Remark 2], where the authors have imposed the constant value for the function L(•)): Theorem 3.5.Suppose that 1 ≤ p < ∞, 1/p + 1/q = 1 and (R(t)) t>0 ⊆ L(X) is a strongly continuous operator family satisfying (2.1).Suppose that the function t → f (t, v(t)), t ∈ R is locally p-integrable for any function  The afore-mentioned results can be also applied to abstract non-degenerate differential equations with almost sectorial operators [31].For example, we can apply Proposition 2.5 in the analysis of existence and uniqueness of asymptotically Weyl-p-almost periodic solutions of the following fractional equation in the Hölder space X = C α ( ) : D γ t u(t, x) = − |β|≤2m a β (t, x)D β u(t, x) − σu(t, x) + f (t, x), t ≥ 0, x ∈ , u(0, x) = u 0 (x), x ∈ .
For more details, the interested reader may consult [32] and [6].
We close the paper with the observation that the assertions of Proposition 2.1 and Proposition 2.5 seem to be new even for non-degenerate strongly continuous exponentially decaying semigroups.

g
(s + τ ) − g(s) p dt ds + II + III, with the clear meaning of I, II and III.For I and III, the length of segments [x − k, x − k + 1] and [x + l − k, x + l − k + 1] equals one while the length of segments [x, s + k] and [s − k + 1, x + l] does not exceed l.Keeping in mind (2.3), we get
v ∈ C b (R : X) and there exists a function L ∈ L p S (R) such that (3.1) holds with I = R.If M L S p < 1, then there exists a unique function u ∈ C b (R : X) such that u(t) := − s)f (s, u(s)) ds, t ∈ R.(3.4)   analysis of solutions to abstract degenerate differential equations and inclusions of first order.Typical examples are the Poisson heat equations∂ ∂t [m(x)v(t, x)] = ( − b)v(t, x) + f (t, x), t ∈ R, x ∈ ; v(t, x) = 0, (t, x) ∈ [0, ∞) × ∂ ,and∂ ∂t [m(x)v(t, x)] = ( − b)v(t, x) + f (t, x), t ≥ 0, x ∈ ; v(t, x) = 0, (t, x) ∈ [0, ∞) × ∂ , m(x)v(0, x) = u 0 (x), x ∈ ,in the space X := L p ( ), where is a bounded domain in R n , b > 0, m(x) ≥ 0 a.e.x ∈ , m ∈ L ∞ ( ) and 1 < p < ∞.The starting point in the whole analysis is the fact that the multivalued linear operator := AB −1 , where A := − b acts on X with the Dirichlet boundary conditions, and B is the multiplication operator by the function m(x), satisfies the condition (P) with β = 1/p and certain finite constants c, M > 0. Proposition 2.1 and Proposition 2.5 can be applied in the analysis of existence and uniqueness of (equi)-Weyl-p-almost periodic solutions and asymptotically (equi)-Weyl-p-almost periodic solutions of the fractional Poisson heat equations containing Weyl-Liouville of Caputo fractional derivatives, as well.