A Massera Type Criterion for Almost Automorphy of Nonautonomous Boundary Differential Equations∗

A classical result of Massera in his landmark paper [1] says that a necessary and sufficient condition for an ω-periodic linear scalar ordinary differential equation to have an ω-periodic solution is that it has a bounded solution on the positive half line. Since then, there has been an increasing interest in extending this classical result to various classes of functions (such as anti-periodic functions [2], quasi-periodic functions [3], almost periodic functions [4, 5, 6], almost automorphic functions [7, 8]) and also to various classes of dynamical systems (such as ordinary differential equations [1], functional differential equations [9, 10], quasi-linear partial differential equations [11], dynamic equations on time scale [12]). Recently, there has been an increasing interest in the almost automorphy of dynamical systems, which is first introduced by Bochner [13] and is more general than the almost periodicity and attracts more and more attention. One can see [14, 15] for a complete background on almost automorphic functions and see the important Memoirs [16] for almost automorphic dynamics. Many different kinds of criteria are established for the existence of almost automorphic solutions of various kinds of dynamical systems [14, 16, 17, 18, 19, 20, 21, 22, 23].