Stepanov-like almost automorphic functions and monotone evolution equations

Abstract

In this paper we are concerned with a (new) class of (Stepanov-like) almost automorphic ($S^p$-a.a.) functions with values in a Banach space $X$. This class contains the space $\mathrm{AA}\left(X\right)$ of all (Bochner) almost automorphic functions. We use the results obtained to prove the existence and uniqueness of a weak $S^p$-a.a. solution to the parabolic equation $u^{\prime }\left(t\right)+A\left(t\right)u=f\left(t\right)$ in a reflexive Banach space, assuming some appropriate conditions of monotonicity, coercitivity of the operators $A\left(t\right)$ and $S^{p^{\prime }}$-almost automorphy of the forced term $f\left(t\right)$. This result extends a known result in the case of almost periodicity. An application is also given.

Introduction

Let us consider in a Banach space $X$ with dual $X^{\ast }$, parabolic equations of the form

$$u^{\prime }\left(t\right)+A\left(t\right)u=f\left(t\right)$$

where the operator $A\left(t\right):X\to X^{\ast }$ is monotone and continuous for each $t\in \mathbb{R}$ and $f\in L_{\mathrm{loc}}^p(a,b;X^{\ast })$, $1\le p<\infty$, and is almost automorphic in a sense (Stepanov-like) defined below. We are interested in conditions under which Eq. (1) has a unique (weak) almost automorphic solution.

The concept of almost automorphic functions was introduced by S. Bochner [1], [2], [3], [4] in relation to some aspects of differential geometry. It turns out to be a generalization of almost periodicity. Some major contributions include works by W.A. Veech who studied almost automorphic functions on groups [20]. Almost automorphic functions with values in a Banach space were studied in [25], [22]. In [6] this study were conducted for functions in locally convex spaces. G.M. N’Guérékata introduced the concept of asymptotically almost automorphic functions with applications to evolution equations [12], [13], [16]. It is worth to quote S. Zaidman [22], [23], [24]. In recent years, there has been a considerable interest in the existence of almost automorphic solutions of various kinds of evolution equations (see for instance, [7], [14], [15], [17], [21], and references therein).

Actually, the problem of finding almost automorphic solutions to inhomogeneous differential equations goes back to S. Bochner and W.A. Veech, and has attracted many mathematicians. In [8], Russel A. Johnson has given an example of an almost periodic ODE of the form $u^{\prime }\left(t\right)+A\left(t\right)u\left(t\right)=B\left(t\right)$ which admits an almost automorphic solution which is not almost periodic. In fact he constructed a nice example of a two-dimensional almost periodic system whose projective flow has an almost automorphic minimal subset which is not almost periodic. He also proved that some equation in the hull of the above equation possesses an almost automorphic solution which is not almost periodic. Recently, in [11], the authors proved that every bounded solution of the above equation in a finite dimension $X$ is almost automorphic in Bochner’s sense if $A\left(t\right)$ is a $\tau$-periodic (unbounded) linear operator-valued function and $f$ is (Bochner) almost automorphic. In [17], the same problem was investigated in a general Banach space $X$ which does not contain a subspace isomorphic to $c_0$ and $A\left(t\right)$ generates a family of 1-periodic evolutionary process ${\left(U(t,s)\right)}_{t\ge s}$.

In the present work, we will deal with almost automorphic (in Stepanov-like sense) solutions in an infinite dimensional space, with $A\left(t\right)$ not necessarily periodic, but satisfying some abstract conditions. Recall that this concept of almost automorphy was first studied in [5]; it generalizes the Bochner almost automorphy in a natural way.

The work is organized as follows. We present in Section 2 the definitions and properties of almost automorphy of functions in both the Bochner and the Stepanov senses. In Section, we introduce the notion of weak almost automorphy in Stepanov’s sense. Our main result is studied in Section 4.