| In this Thesis, our first task consists of introducing a new class of functions called C(n)-pseudoalmost automorphic functions, which generalizes in a natural fashion the concept of C(n)-almost automorphy (respectively, C(n)-almost periodicity). Next, we present some of their properties, and then study their generalization called Stepanov-like C(n)-pseudo-almost automorphy.;Our second task consists of examining the existence of solutions to some systems of differential equations in C(n)-pseudo-almost automorphic spaces. Specifically, we first investigate the existence of solutions to the higher-order systems of differential equations of the form wN t+k=0N-1 aktw kt =ft,t∈R, .;where ak:R→R is almost automorphic for k = 0, 1, ..., N --1 and the function f:R→R is C(m)-pseudo-almost automorphic (respectively, Stepanov-like C(m)-pseudo almost automorphic).;Our third task consists of studying the existence of solutions to some general higher-order differential equations with operator cocients given by un t+k=0n-1 aktw kt =ft,t∈R, .;where A:DA⊂ X→X and Ak:D Ak⊂X→X k=1,2,...,n-1 are densely defined closed linear operators on a Banach space X , and the forcing term F:R→DA a+m,p is C(m)-pseudo-almost automorphic (respectively, Stepanov-like C(m)-pseudo almost automorphic), with DAa+m,p being a real interpolation space. In this Thesis, we are exclusively interested in the case when Ak=Aak (if it exists), with Aak being fractional powers of the operator A having exponents 0 ≤ alphak < 1 for k = 1,2, ..., n-1.;In order to illustrate the above-mentioned results, the existence of C(m+1)-pseudo almost automorphic solutions to some damped second-order abstract differential equations will be investigated. We will then make extensive use of these results to study the existence of C(m+1)-pseudo almost automorphic solutions to some (second order in time and fourth order in space) PDEs. |
