On weak-almost periodic mild solutions of some linear abstract differential equations (Q1422578)

The author considers the linear abstract differential equation \[ x'(t)= Ax(t)+ f(t), \quad t\in \mathbb R, \tag{1} \] in a uniformly convex Banach space under the hypotheses H1: \(A\) is a linear operator that generates a \(C_0\)-semigroup of bounded linear operators \(T(t)\) for each \(t\in \mathbb R^+\) and \(\sup_{t\in \mathbb R^+} \|T(t)\|< \infty\), H2: \(f\) is a nontrivial strongly continuous function. Let \(\Omega\) represent the set of all mild solutions of (1) which are bounded. It has been proved that under assumptions H1 and H2 and also assuming \(\Omega\) to be nonempty, equation (1) has a unique optimal mild solution. The result is accomplished by showing that \(\Omega\) is closed and convex. Under the additional assumptions that \(f\in L(\mathbb{R})\) and is almost-periodic and the adjoint operator \(T^*(t)\) is linear, it is proven that the optimal mild solution of equation (1) is weakly almost-periodic.