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On a functional equation associated with $(a, k)$-regularized resolvent families. (English) Zbl 1250.39012
Summary: Let $a \in L^1_{\text{loc}}(\Bbb R_+)$ and $k \in C(\Bbb R_+)$ be given. We study the functional equation $R(s)(a \ast R)(t) - (a \ast R)(s)R(t) = k(s)(a \ast R)(t) - k(t)(a \ast R)(s)$, for bounded operator valued functions $R(t)$ defined on the positive real line $\Bbb R_+$. We show that, under some natural assumptions on $a(\cdot)$ and $k(\cdot)$, every solution of the above mentioned functional equation gives rise to a commutative $(a, k)$-resolvent family $R(t)$ generated by $Ax = \lim_{t \rightarrow 0^+}(R(t)x - k(t)x/(a \ast k)(t))$ defined on the domain $D(A) : = \{x \in X \lim_{t \rightarrow 0^+}(R(t)x - k(t)x/(a \ast k)(t))$ exists in $X\}$ and, conversely, that each $(a, k)$-resolvent family $R(t)$ satisfies 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.

##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges
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##### References:
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