In this paper, we prove an existence theorem for the pseudo-non-local Cauchy problem $x^{\text{'}}\left(t\right)+Ax\left(t\right)=f(t,x\left(t\right),{\int }_{t₀}^{t}k(t,s,x\left(s\right))ds)$, x₀(t₀) = x₀ - g(x), where A is the infinitesimal generator of a C₀ semigroup of operator ${T\left(t\right)}_{t>0}$ on a Banach space. The functions f,g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.

Vengono dati nuovi teoremi di regolarità per le soluzioni dell'equazione $u^{\prime }(t)=\mathrm{\Lambda }u(t)+f(t)$ nel caso in cui $\mathrm{\Lambda }$ è il generatore infinitesimale di un semigruppo analitico in uno spazio di Banach $E$ e $f$ è una funzione continua.

Given a family of (W) contractions $T_1,...,T_N$ on a reflexive Banach space X we discuss unrestricted sequences $T_{r_n}\circ ...\circ T_{r_1}\left(x\right)$. We show that they converge weakly to a common fixed point, which depends only on x and not on the order of the operators $T_{r_n}$ if and only if the weak operator closed semigroups generated by $T_1,...,T_N$ are right amenable.

The existence, uniqueness and asymptotic stability of weak solutions of functional-differential abstract nonlocal Cauchy problems in a Banach space are studied. Methods of m-accretive operators and the Banach contraction theorem are applied.