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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.

39B52 Functional equations for functions with more general domains and/or ranges
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