Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions

Abstract

In this paper, we analyze the existence and uniqueness of generalized weighted pseudo-almost automorphic solutions of abstract Volterra integro-differential inclusions in Banach spaces. The main results are devoted to the study of various types of weighted pseudo-almost periodic (automorphic) properties of convolution products. We illustrate our abstract results with some examples and possible applications.2010 Mathematics Subject Classification. 44A35, 42A75, 47D06, 34G25, 35R11. Key words and phrases. Weighted pseudo-almost periodicity, weighted pseudo-almost automorphy, convolution products, abstract Volterra integro-differential inclusions, multivalued linear operators.The author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.where A(t) : D ⊆ X → X is a family of densely defined closed linear operators on a common domain D, independent of time t ∈ R, the history u t : (0, ∞] → X defined by u t (·) := u(t + ·) belongs to some abstract phase space B defined axiomatically, and f : R × B → X, g : R × B → X fulfill some conditions. Fractional calculus and fractional differential equations have received much attention recently due to its wide range of applications in various fields of science, such as mathematical physics, engineering, aerodynamics, biology, chemistry, economics etc (see e.g.[5], [24], [32]-[33] and [44]-[45]). In this paper, we essentially use only the Weyl-Liouville fractional derivatives (for more details, we refer the reader to the paper [42] by J. Mu, Y. Zhoa and L. Peng). The Weyl-Liouville fractional derivative 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 D γ t,+ u(t) := d dt t −∞ g 1−γ (t − s)u(s) ds, t ∈ R;

Almost automorphic and asymptotically almost automorphic type functions in Lebesgue spaces with variable exponents $L^{p(x)}$

“…In the following proposition, we state some invariance properties of generalized asymptotical almost automorphy in Lebesgue spaces with variable exponents L p(x) under the action of finite convolution products. This proposition generalizes [19,Proposition 6] provided that p > 1 in its formulation. Proposition 5.2.…”

“…Proof. The proof of theorem is very similar to that of above-mentioned proposition since the Hölder inequality holds in our framework (see Lemma 1.1(ii)) and any element of L p(x) ([0, 1] : X) is absolutely continuous with respect to the norm • L p(x) (see [10, Definition 1.12, Theorem 1.13]), which clearly implies that the translation mapping t → ǧ(• − t) ∈ L p(x) ([0, 1] : X), t ∈ R is continuous (we need this fact for proving the continuity of mapping F k (•) appearing in the proof of [19,Proposition 5], k ∈ N). The remaining part of proof can be given by copying the corresponding part of proof of above-mentioned proposition.…”

A note on (asymptotically) Weyl-almost periodic properties of convolution products

“…, t ≥ 0 can be shown following the lines of the proof of [13,Proposition 5] since the mapping F k (•) is continuous by the dominated convergence theorem and the series ∞ k=0 F k (t) converges uniformly in t ≥ 0 due to the Weierstrass criterion. To prove the continuity of mapping…”

“…Further on, let A(x; D) be a second order linear differential operator on Ω with coefficients continuous on Ω; see [4,Example 6.1] for more details. Based on the examination carried out in [13,Example 4], we can apply our main results in the study of existence and uniqueness of asymptotically (equi-)Weyl-p-almost periodic solutions of the following fractional damped Poisson-wave type equation in the space X := H −1 (Ω) or X := L p (Ω) :    D γ t m(x)D γ t u + 2ωm(x) − ∆ D γ t u + A(x; D) − ω∆ + ω 2 m(x) u(x, t) = f (x, t), t ≥ 0, x ∈ Ω ; u = D γ t = 0, (x, t) ∈ ∂Ω × [0, ∞), u(0, x) = u 0 (x), m(x) D γ t u(x, 0) + ωu 0 = m(x)u 1 (x), x ∈ Ω.…”

Weyl almost automorphic functions and applications

“…Furthermore, we introduce and analyze the multi-dimensional analogues of these concepts by using the definitions and results from the theory of Lebesgue spaces with variable exponents (the introduced classes of functions seem to be not considered elsewhere even in the constant coefficient case); in such a way, we continue our recent analysis of multi-dimensional almost periodicity and multi-dimensional almost automorphy carried out in our joint research studies [9]- [11] with A. Chávez, K. Khalil and M. Pinto. Several illustrative examples, open problems and applications to the abstract Volterra integro-differential equations are presented (for the Weyl and Besicovitch generalizations of the almost periodic functions and the almost automorphic functions, we refer the reader to [2,3,5,6,7,11,32,19,24,25,27] and [28,29,33,34,35,37,38,39]). It should be also noted that we present some new results about the (equi-)Weyl almost periodic functions here; for example, in [19], we have emphasized that some relations presented in [2,Table 2,p.…”