This monograph is devoted to the study of the almost periodic solutions, asymptotically almost periodic solutions, almost automorphic solutions and asymptotically almost automorphic solutions for abstract degenerate Volterra integro-differential equations and abstract (degenerate) fractional differential equations and inclusions in Banach spaces.
The book is organized into: Preface, Notation, Introduction, three chapters, Bibliography and Index.
In Chapter 1, the author presents some preliminary results that will be used in the next chapters, namely: fundamentals of the theory of operators and integration theory, properties of the multivalued linear operators in Banach spaces, the Laplace transform of functions with values in Banach spaces, operators of fractional differentiation, the Mittag-Leffler and Wright functions, and the definitions and properties of the degenerate \((a,k)\)-regularized \(C\)-resolvent families subgenerated by multivalued linear operators.
Chapter 2 contains 16 sections and it is focused on some abstract degenerate Volterra integro-differential equations and abstract fractional differential equations in Banach spaces, with a special attention on various classes of semilinear Cauchy problems and inclusions. There are investigated the almost periodic solutions, the asymptotically almost periodic solutions, the Stepanov almost periodic solutions, the aymptotically Stepanov almost periodic solutions, the Weyl almost periodic solutions, the asymptotically Weyl almost periodic solutions, the pseudo-almost periodic solutions, the Besicovitch-Doss almost periodic solutions, the generalized almost periodic solutions, the generalized asymptotically almost periodic solutions, and the vector-valued almost periodic ultradistributions.
Chapter 3 has 11 sections and it deals with some abstract degenerate Volterra integro-differential equations and inclusions, and abstract degenerate (multiterm) fractional differential inclusions in Banach spaces. There are studied here the generalized asymptotically almost periodic solutions, the generalized almost automorphic solutions, the generalized asymptotically almost automorphic solutions, the generalized weighted pseudo-almost periodic solutions, the generalized weighted pseudo-almost automorphic solutions, the Besicovitch almost automorphic solutions, the vector-valued almost automorphic distributions and vector-valued almost automorphic ultradistributions, the asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized solutions. In Section 3.10 the author gives several illustrative examples and applications of the established results.
We remark that in the last sections of Chapter 2 and Chapter 3, the author presents notes on some recent and interesting problems related to the subjects of the chapter. The extensive Bibliography has 422 titles of papers and books which are connected with the topics considered in this monograph. This text is a great resource for graduated students and researchers interested in the existence and properties of the almost periodic solutions and almost automorphic solutions to integro-differential equations.