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Stepanov-like pseudo almost periodic functions and their applications to differential equations. (English) Zbl 1286.44007
Summary: This paper introduces and examines a new class of functions called Stepanov-like pseudo almost periodic functions (or \(S^p\)-pseudo almost periodic functions), which generalizes in a natural fashion the classical notion of pseudo almost periodicity. We then make extensive use of these new functions to study the existence and uniqueness of a pseudo almost periodic solution to the semilinear equation \(u(t) = Au( t) + F( t, u( t))\), where \(A:D( A) \to X\) is the infinitesimal generator of an exponentially stable \(C_0\)- semigroup on a Banach space \(X\) and \(F : R \times X \to X\) is \(S^p\)-pseudo almost periodic for \(p > 1\) and jointly continuous.

MSC:
44A35 Convolution as an integral transform
42A85 Convolution, factorization for one variable harmonic analysis
42A75 Classical almost periodic functions, mean periodic functions
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