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

Abstract

In control theory, problems occur regarding the behavior of solutions of some abstract Cauchy problems like that 0.1$$\begin{aligned} \begin{array}{ll} u'(t) =&{} -A(u(t))-f(t)b,\quad t\in \mathbb {R} \\ u(\infty )=&{}\lim \nolimits _{t\rightarrow \infty }u(t)=0. \end{array} \end{aligned}$$Here A generates a strongly continuous semigroup $$\mathbf{T}=\{T(t)\}$$ acting on a complex Banach space X, f is a complex valued measurable function defined on $$\mathbb {R}_+$$ verifying a certain integral condition (as in Theorem 4.1 below), $$b\in X$$ 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 $$\mathbf{T}$$ is $$\Phi $$-weakly stable, that is, for every $$x\in X$$ and $$x'\in X'$$ of norms less than or equal to 1 the map $$t\mapsto \Phi (|\langle e^{tA}x, x'\rangle |$$ belongs to $$L^1(\mathbb {R}_+)$$. 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 $$\langle \cdot , \cdot \rangle $$ 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 $$s_0(A)$$ is negative whenever the semigroup $$\mathbf{T}$$ 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 $$s_0(A)\le -\nu $$. An implicit expression of $$\nu $$, in terms of $$\Phi $$ and $$\mathbf{T}$$, 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.