Bochner's theorem and Stepanov almost periodic functions (Q931372)

The paper considers the functions \(f\in L^{loc}_p(R,X)\) with \(X\) a Banach space, together with the norm \[ S_l^p=\displaystyle{\sup_{t\in R}\left(\frac{1}{l}\,\int_t^{t+l}|f(\theta)|^pd\theta\right)^{1/p}} \] called the Stepanov norm. By introducing the associated notions of \(S_l^p\) continuity, uniform continuity, boundedness, the authors are thus led to Stepanov almost periodicity, defined in one of the several equivalent ways to the original Stepanov definition. Further a Bochner-type theorem is formulated and proved for the Stepanov almost periodicity. The paper goes on with the theory of the Stepanov almost periodic differential equations, ending with a Favard type theorem for Stepanov almost periodic systems.