On a Functional Equation Associated with ( a , k )-Regularized Resolvent Families

Abstract

Let $a\in L_{\text{l}\text{o}\text{c}}^1({ℝ}_{+})$ and $k\in C({ℝ}_{+})$ be given. In this paper, we study the functional equation $R(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s)$, for bounded operator valued functions $R(t)$ defined on the positive real line ${ℝ}_{+}$. We show that, under some natural assumptions on $a(·)$ and $k(·)$, every solution of the above mentioned functional equation gives rise to a commutative $(a,k)$-resolvent family $R(t)$ generated by $Ax={\lim \hspace{0.17em}}_{t\to 0^{+}}(R(t)x-k(t)x/(a*k)(t))$ defined on the domain $D(A):=\{x\in X:{\lim \hspace{0.17em}}_{t\to 0^{+}}(R(t)x-k(t)x/(a*k)(t))$ exists in $X\}$ and, conversely, that each $(a,k)$-resolvent family $R(t)$ satisfy the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems.