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On a functional equation associated with $$(a, k)$$-regularized resolvent families. (English) Zbl 1250.39012
Summary: Let $$a \in L^1_{\text{loc}}(\mathbb R_+)$$ and $$k \in C(\mathbb R_+)$$ 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 $$\mathbb R_+$$. 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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