A weak integral condition and its connections with existence and uniqueness of solutions for some abstract Cauchy problems in Banach spaces

In control theory, problems occur regarding the behavior of solutions of some abstract Cauchy problems like that 0.1 u (t)=-A(u(t))-f(t)b,t is an element of Ru(infinity)=limt -infinity u(t)=0 Here A generates a strongly continuous semigroup T={T(t)} acting on a complex Banach space X, f is a complex valued measurable function defined on R+verifying a certain integral condition (as in Theorem 4.1 below), b is an element of X is a randomly chosen vector and the limit is considered in the norm of X. We prove that the Cauchy Problem (0.1 ) has at least one solution (that is unique when X is a complex Hilbert space) provided the semigroup T is phi-weakly stable, that is, for every x is an element of X and x is an element of X of norms less than or equal to 1 the map. Concrete examples and even the expression of solutions are also provided in this paper. Here phi is a given N-function, X denotes the strong dual of X and denotes the duality map between X and X It is known (Storozhuk in Sib Math J 51:330-337, 2010) that the uniform spectral bound is negative whenever the semigroup T generated of A is phi-weakly stable for the above phi. We complete this result by proving that if the semigroup is phi-weakly stable then there exists a positive number nu such that s0(A)=-nu. An implicit expression of nu phi, is also given. The condition that phi is positive near to 0 is necessary in the proofs. A counterexample showing this is provided in the last section of the paper.