Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations. (English) Zbl 1408.43005

The paper investigates Weyl almost periodic and asymptotically Weyl almost periodic solutions of abstract Volterra integral equations and inclusions. The class of Weyl almost periodic functions is an extension of the class of the classical Stepanov almost periodic functions and it was defined by H. Weyl (1927).
The paper is organized as follows: Section 2 is devoted to the definitions of multivalued linear operators in Banach spaces, by repeating some things from the author’s previous work [“Perturbation results for abstract degenerate Volterra integro-differential equations”, J. Fract. Calc. Appl. 9, 137–152 (2018); http://fcag-egypt.com/Journals/JFCA/], as the \((a,k)\)-regularized C-resolvent families with subgenerator a multivalued linear operator \(\mathcal{A}\).
In Section 3, some facts from the theory of the Stepanov almost periodic and asymptotically almost periodic functions are presented. In Section 4, several new classes as the Weyl almost periodic and asymptotically Weyl almost periodic functions are introduced and their relationship is investigated.
In Section 5, the author gives information on the Weyl almost periodic properties of finite and infinite convolution products. For instance in Proposition 5.1 among others it is proved that if \((R(t))_{t\geq 0}\) is a strongly continuous family of linear operators from \(X\) to \(Y\) satisfying \(\|R\|_{L1}<+\infty\) and \(g:\mathbb{R}\to X\) is bounded and (equi)-Weyl almost periodic, then the function \(G(t):=\int_{-\infty}^tR(t-s)g(s)ds\), \(t\geq 0,\) is bounded and (equi)-Weyl almost periodic.
In Section 6, the author states (without proofs) some results concerning abstract Volterra integro-differential equations, while, in Section 7, some examples and applications to fractional inclusions with Weyl-Liapunov and Caputo type derivatives are presented.


43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
47D06 One-parameter semigroups and linear evolution equations
45N05 Abstract integral equations, integral equations in abstract spaces
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